Noncommutative Geometry and Physics 2005: Proceedings of the International Sendai-Beijing Joint Workshop

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noncommutative geometry and physics 2005 Proceedings of the International Sendai–Beijing Joint Workshop

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edited by

Ursula Carow-Watamura Satoshi Watamura Tohoku University, Japan Yoshiaki Maeda Hitoshi Moriyoshi Keio University, Japan Zhangju Liu Peking University, China Ke Wu Capital Normal University, China

noncommutative geometry and physics 2005 Proceedings of the International Sendai–Beijing Joint Workshop Sendai, Japan 1 – 4 November 2005 Beijing, China 7 – 10 November 2005

World Scientiﬁc NEW JERSEY

•

LONDON

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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NONCOMMUTATIVE GEOMETRY AND PHYSICS 2005 Proceedings of the International Sendai-Beijing Joint Workshop Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-270-469-6 ISBN-10 981-270-469-8

Printed in Singapore.

ZhangJi - Noncommutative.pmd

1

6/11/2007, 6:27 PM

Preface The workshop “Noncommutative Geometry and Physics 2005” was organized by mathematicians and physicists from Keio Universty, Tohoku University and Beijing University in cooperation. The first part of this workshop was held at Tohoku University in Sendai on November 1-4, and the second part was held at Beijing University in Beijing on November 7-11. This workshop was the fifth in a series of joint workshops for mathematicians and physicists working in noncommutative geometry, deformation quantization and related topics, with the aim to stimulate discussions and the exchange of new ideas between both disciplines. Since the subject of noncommutative geometry has undergone rapid developments in the past few years, one of the important functions of our meetings is to elucidate these recent advances and the current status of research projects from mathematical point of view as well as from the physics’ side. In physical applications of noncommutative geometry, many key results have emerged on solutions of field theory on noncommutative spaces. Therefore, this was naturally one of the main subjects of the workshop. This volume includes disscussions of solitons such as monopoles and instantons in noncommutative spaces as well as in nonanticommutative superspaces. All contributions in this volume were submitted by conference speakers and participants, and were duly refereed. The articles contain presentations of new results which have not appeared yet in professional journals, or comprehensive reviews including an original part of the present developments in those topics. Effort was to provide comprehensive introductions to each subject such that the volume becomes accessible to researchers and graduate students interested in mathematical areas related to noncommutative geometry and its impact on modern theoretical physics. The workshop was held in the framework and with the support of the 21st century Center of Excellence (COE) program at Keio University, “Integrative Mathematical Sciences: Progress in Mathematics v

vi

Preface

motivated by Natural and Social Phenomena”, and it was also supported by a Grant-in-Aid for Scientific Research (No.13135202) of the Japanese Ministry of Education, Culture, Sports, Science and Technology at Tohoku University. We are also grateful to the Tohoku University, Japan, and to the Beijing University and the Academia Sinica, China, for providing the lecture rooms and their facilities which made a smooth performance of the meeting possible. The World Scientific Publishing company has been very helpful in the production of this volume, and we would like to thank Ms. Zhang Ji for her editorial guidance. In this place we wish to express our special thanks to all authors for their continuous effort in preparing these articles and the referees for their valuable comments and suggestions. Mathematical section: Zhangju Liu (Beijing Univ.) Yoshiaki Maeda (Keio Univ.) Hitoshi Moriyoshi (Keio Univ.) Physical section: Ursula Carow-Watamura (Tohoku Univ.) Satoshi Watamura (Tohoku Univ.) Ke Wu (CNU, Beijing)

Contents

I

Preface

v

DEFORMATIONS AND NONCOMMUTATIVITY

1

1 Expressions of Algebra Elements and Transcendental Noncommutative Calculus Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka 3 2 Quasi-Hamiltonian Quotients as Disjoint Unions of Symplectic Manifolds Florent Schaffhauser 31 3 Representations of Gauge Transformation Groups of Higher Abelian Gerbes Kiyonori Gomi 55 4 Algebroids Associated with Pre-Poisson Structures Kentaro Mikami, Tadayoshi Mizutani

71

5 Examples of Groupoid Naoya Miyazaki

97

6 The Cohomology of Transitive Lie Algebroids Z. Chen, Z.-J. Liu

109

7 Differential Equations and Schwarzian Derivatives Hajime Sato, Tetsuya Ozawa, Hiroshi Suzuki

129

8 Deformation of Batalin-Vilkovsky Structure Noriaki Ikeda

151

vii

viii

II

Contents

DEFORMED FIELD THEORY AND SOLUTIONS

173

9 Noncommutative Solitons Olaf Lechtenfeld

175

10 Non-anti-commutative Deformation of Complex Geometry Sergei V. Ketov

201

11 Seiberg-Witten Monopole and Young Diagrams Akifumi Sako

219

12 Instanton Counting, Two Dimensional Yang-Mills Theory and Topological Strings Kazutoshi Ohta 239 13 Instantons in Non(anti)commutative Gauge Theory via Deformed ADHM Construction Takeo Araki, Tatsuhiko Takashima, Satoshi Watamur 253 14 Noncommuative Deformation and Drinfel’d Twisted Symmetry Yoshishige Kobayashi 261 c (2) k and Twisted Conformal Field 15 Affine Lie Superalgebra gl(2|2) Theory Xiang-Mao Ding, Gui-Dong Wang, Shi-Kun Wang 273 16 A Solution of Yang-Mills Equation on BdS Spacetime Xin’an Ren, Shikun Wang

289

17 Solitonic Information Transmission in General Relativity Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

297

18 Difference Discrete Geometry on Lattice Ke Wu, Wei-Zhong Zhao, Han-Ying Guo

301

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Part I

DEFORMATIONS AND NONCOMMUTATIVITY

1

Expressions of algebra elements and transcendental noncommutative calculus Hideki Omori1 , Yoshiaki Maeda2 , Naoya Miyazaki3 , and Akira Yoshioka4 1

2

3

4

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama, 223-8825, Japan Department of Mathematics, Faculty of Economics, Keio University, Hiyoshi, Yokohama, 223-8521, Japan Department of Mathematics, Faculty of Science, Tokyo University of Science, Kagurazaka, Tokyo, 102-8601, Japan [email protected], [email protected], [email protected], [email protected] 2 Partially supported by Grant-in-Aid for Scientific Research (#18204006.), Ministry of Education, Science and Culture, Japan. 3 Partially supported by Grant-in-Aid for Scientific Research (#18540093.), Ministry of Education, Science and Culture, Japan. 4 Partially supported by Grant-in-Aid for Scientific Research (#17540096.), Ministry of Education, Science and Culture, Japan.

Abstract Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that 1 i~ uv in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set N+ 21 or −(N+ 21 ) . This may yield a more mathematical understanding of Dirac’s positron theory. A.M.S Classification (2000): Primary 53D55, 53D10; Secondary 46L65

1 Introduction Quantum theory is treated algebraically by Weyl algebras, derived from differential calculus via the correspondence principle. However, since the algebra is noncommutative, the so-called ordering problem appears. Orderings are treated in the physics literature of quantum mechanics (cf. [1]) as the rules of association from classical observables to quantum observables, which are supposed to be self-adjoint operators on a Hilbert space.

3

4

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Typical orderings are, the normal (standard) ordering, the anti-normal (antistandard) ordering, the Weyl ordering, and the Wick ordering in the case of complex variables. However, from the mathematical viewpoint, it is better to go back to the original understanding of Weyl, which says that orderings are procedures of realization of the Weyl algebra W~ . Since the Weyl algebra is the quotient algebra of the universal enveloping algebra of the Heisenberg Lie algebra, the Poincar´e-Birkhoff-Witt theorem shows that this algebra can be viewed as an algebra defined on a space of polynomials. As we show in §1, this indeed gives product formulas on the space of polynomials which produce algebras isomorphic to W~ . This gives the unique way of expressions of elements, and as a result one can treat transcendental elements such as exponential functions, which are necessary to solve differential equations (cf. §2.2). However, we encounter several anomalous phenomena, such as elements with two different inverses (cf. §4) and elements which must be treated as double valued (cf. [16],[17]). 1 uv should be In this note, we treat the phenomenon which shows that i~ 1 1 viewed as an indeterminate living in the set N+ 2 or −(N+ 2 ). We reach this interpretation in two different ways, by analytic continuation of inverses of 1 z+ i~ uv, and by defining star gamma functions using various ordered expressions. We emphasise in this paper, that our approach to show the discrete picture 1 uv is not to use operator representation at all, but for the element z + i~ to express it in various orderings instead, under a leading principle that a physical object should be free from the choice of orderings(the ordering free principle), just as classical, geometric objects are expressed independent of the choice of local coordinates. Since similar discrete pictures of elements is familiar in quantum observables, treated as a self-adjoint operator, our observation gives for their justification for the operator theoretic formalism of quantum theory. However, in this note we restrict our ordering to a particular subset to avoid the multi-valued expressions. In some cases, we should be more careful about the convergence of integrals and the continuity of the product, so the detailed computations and the proof of continuity of the products will appear elsewhere.

2 K-ordered expressions for algebra elements We introcuce a method to realize the Weyl algebra via a family of expressions. This leads to a transcendental calculus in the Weyl algebra. 2.1 Fundamental product formulas and intertwiners Let SC (n) and AC (n) be the spaces of complex symmetric matrices and skewsymmetric matrices respectively, and MC (n)=SC (n)⊕AC (n). For an arbitrary

2 K-ordered expressions for algebra elements

5

fixed n×n-complex matrix Λ∈MC (n), we define a product ∗Λ on the space of u] by the formula polynomials C[u i~

f ∗Λ g = f e 2 (

P ←− ij − − → ∂ ui Λ ∂ uj )

g=

X (i~)k k!2k

k

Λi1 j1· · ·Λik jk ∂ui1· · ·∂uik f ∂uj1· · ·∂ujk g.

(1) u], ∗Λ ) is an associative algebra. It is known and not hard to prove that (C[u u], ∗Λ ) is determined by the skew-symmetric (a) The algebraic structure of (C[u part of Λ (in fact, by its conjugacy class A → t GAG). u], ∗Λ ) is isomorphic to the (b) In particular, if Λ is a symmetric matrix, (C[u usual polynomial algebra. Set Λ=K+J, K∈SC (n), J∈AC (n). Changing K for a fixed J will be called a deformation of expression of elements, as the algebra remains in the same isomorphism class. Example of computations: ui ∗Λ uj =ui uj +

i~ ij Λ , 2

ui ∗Λ uj ∗Λ uk =ui uj uk +

i~ ij (Λ uk +Λik uj +Λjk ui ). 2

By computing the ∗Λ -product using the product formula (1), every element of the algebra has a unique expression as a standard polynomial. We view these expressions of algebra elements as analogous to the “local coordinate expression” of a function on a manifold. Thus, changing K corresponds to a local coordinate transformation on a manifold. In this context, we call the product formula (1) the K-ordered expression by ignoring the fixed skew part J. Following a familiar notion in quantum we call the K-ordered mechanics, 0 −Im 0 Im , the Weyl ordering, , expression for the particular K=0, −Im 0 Im 0 normal ordering, anti-normal ordering, respectively. The intertwiner between a K-ordered expression and a K 0 -ordered expression, which we view as a local coordinate transformation, is given in a concrete form : Proposition 2.1 For symmetric matrices K, K 0 ∈ SC (n), the intertwiner is given by K0

IK (f ) = exp

i~ X 4

i,j

(K

0

ij

K K0 −K ij )∂ui ∂uj f (= I0 (I0 )−1 (f )),

(2)

K0

u]; ∗K+J ) → (C[u u]; ∗K 0 +J ) between algebras. giving an isomorphism IK : (C[u u] : Namely, for any f, g ∈ C[u K0

K0

K0

IK (f ∗K+J g) = IK (f ) ∗K 0 +J IK (g).

(3)

6

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka K

u], the set {I0 (f ); K∈SC (n)} forms a family of eleThus, for every f ∈C[u ments which are mutually intertwined. We denote this family by f∗ viewing K as an algebraic object, and we use often the notation :f∗ :K+J =I0 (f ). 0 −Im u], ∗Λ ) is called the Weyl algebra, , (C[u In the case n=2m and J= Im 0 with isomorphism class denoted by W2m . In fact, if J is non-singular, then u], ∗Λ ) is isomorphic to the Weyl algebra. (C[u 1 t(z+s i~ uk )

2.2 The star exponential function e∗

Using the various ordered expression of elements of algebra, we can treat elementary transcendental functions. For an object H∗ , the ∗-exponential func∗ ∗ tion etH is defined as the family :etH ∗ ∗ :Λ of real analytic solutions in t of the evolution equations d ft =:H∗ :Λ ∗Λ ft , f0 =1. (4) dt For instance, for every z∈C, we have 1 z+s i~ uk

:e∗

s

1

:Λ =ez :e∗ i~

uk

:Λ =ez es

2

kk 1 4i~ K

1

es i~ uk .

(5)

When we fix the skew part J of Λ, we often abbreviate the notation to : :K , ∗K for : :K+J , ∗K+J respectively. Since the exponential law 1 (z+w)+(s+t) i~ uk

:e∗

1 z+s i~ uk

:K =:e∗

1 w+t i~ uk

:K ∗K :e∗

:K

holds for every K, it is better to write 1 (z+w)+(s+t) i~ uk

e∗ 1 z+s i~ uk

by viewing :e∗

1 z+s i~ uk

=e∗

1 w+t i~ uk

∗e∗

:K as the K-ordered expression of the (ordering free) ex-

z+s

1

u

ponential element e∗ i~ k . Under this convention, one may write for instance ij :ui ∗uj :K =ui uj + i~ 2 (K+J) . s

1

u

We remark that even for the simplest exponential function e∗ i~ k , formula (5) gives the following (cf. [12]). 1 P 2n i~ uk Proposition 2.2 If Im K kk <0, the K-ordered expression of n∈Z e∗ 1 P 2n u converges unifromly on every compact domain, and : n∈Z e∗ i~ k :K is pre1 i kk cisely the Jacobi theta function θ3 (− ~ uk , ~ K ). This shows that deformations of expressions of a fixed algebraic system are interesting in their own right (cf. [5]). However, it should be remarked 1 2 P∞ 2n i~ P−1 uk 2n 1 u u that , and − n=−∞ e∗ i~ k converge to inverses of 1−e∗i~ k n=0 e∗ respectively. This leads to a breakdown of associativity. Such phenomena occur very often in a transcendentally extended algebraic system.

3 Star exponential functions of quadratic forms

R

7

Moreover, if Im K kk <0, then the K-ordered expression of the integral t1 u e i~ k dt converges uniformly on every compact domain, and R ∗ Z Z 1 1 z i~ t i~ t1 u uk uk e∗ ∗ e∗ dt= e∗i~ k dt, ∀z∈C. (6) R

R

However, we have shown in [18] that

R

t1 u e i~ k dt R ∗

is double valued.

3 Star exponential functions of quadratic forms In this note we mainly deal with the Weyl algebra W2 over C. Putting u1 =u, u2 =v, we have the commutation relation where [u, v]=u∗v−v∗u. [u, v]=−i~, 0 −1 realizes W2 . The product formula (1) with Λ=K+J, J= 1 0 In what follows, we use the following notations: 1 uv= (u∗v+v∗u), 2

u∗v=v∗u−i~,

1 v∗u=uv+ i~. 2

(7)

0κ . The product ∗κ and the ordered expression : :κ stand for κ0 ∗K and : :K , respectively. Namely, ∗0 and ∗1 correspond to the Moyal product and the standard product. We also denote the intertwiner from the ∗κ -product 0 to the ∗κ0 product by Iκκ . Let Hol(C2 ) be the set of holomorphic functions f (u, v) on the complex 2-plane C2 endowed with the topology of uniform convergence on compact subsets. Hol(C2 ) is viewed as a Fr´echet space. The following fundamental lemma follows easily from the product formula (1). Let K =

Lemma 3.1 For every polynomial p(u, v), left multiplication p(u, v)∗ (resp. right multiplication ∗p(u, v) ) is a continuous linear mapping of Hol(C2 ) into itself. 1 t(z+ i~ uv)

3.1 The star exponential function e∗ 0

If ft = h(uv) in (4), then Iκκ (h(uv)) is also a function of uv. ¿From here on, 2 u we mainly concern with functions of uv alone. We set i~ uv=hu A, u i, where 01 κ0 u =(u, v) and A= . The intertwiner Iκ is given as follows: 10 0

2

Iκκ (get i~ uv )=g

1 1 1 2uv e i~ 1−t(κ0 −κ) 0 1−t(κ −κ)

(8)

Solving the evolution equation (4) for the exponential function, we see that t

1

e∗i~

2uv

is given by

8

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

1 1 e i~ 2uv tanh t cosh t in the Weyl ordering (cf. [16]), and by

t

1

2uv

:e∗i~

t

1

:e∗i~

:0 =

2uv

1

:I = et e i~ (e

2t

(9)

−1)uv

(10)

in the normal ordering (cf.[14]). t

2

Since :e∗i~

uv

1

1 i~ 2uv tanh t ), we see that :κ = I0κ ( cosh te

1 2 et −e−t exp 2uv . (11) t −t t −t (1−κ)e +(1+κ)e (1−κ)e +(1+κ)e i~ 0κ . The product ∗(κ,τ ) and the ordered expression : :(κ,τ ) Let K = κτ stand for ∗K and : :K , respectively. It is not hard to obtain the (κ, τ )-ordered expression: t

1

:e∗i~

2uv

:κ =

t

1

:e∗i~

2uv

:(κ,τ ) =

2 et −e−t 2 1 2 et −e−t 1 exp ( ) τ u + 2uv , ∆ ∆ i~ ∆ i~

(12)

where ∆=(et +e−t )−κ(et −e−t ). The general ordered expression is a little more complicated involving the squre root in the amplitude. t

1

i~ Note that (1−κ)et +(1+κ)e−t =0 if and only if e2t = κ+1 κ−1 . Hence, :e∗

2uv

:(κ,τ )

1 2uv t i~

κ+1 κ−1 +2πiZ.

:(±1,τ ) However, if κ=±1, then :e∗ has a singular point at 2t= log are entire functions with respect to t. In general we have the following: Lemma 3.2 If κ∈C−{κ≥1}∪{κ≤−1}, then t

2

uv

the (κ, τ )-ordered expression :e∗i~ :(κ,τ ) is real analytic and rapidly decreasing with respect to t∈R.

•

0

•

Formula (12) gives also the following: Proposition 3.1 Suppose κ6=0, z∈C. Then the (κ, τ )-ordered expression 1 uv):(κ,τ ) is holomorphic in (z, uv), and vanishes for z∈Z+ 12 . : sin∗ π(z+ i~ 1 πi i~ 2uv

Proof. By (12), :e∗ κ=0) of

±πi 1 uv e∗ i~

1 πi i~ uv

:e∗

:(κ,τ ) +1=0. Although the Weyl ordering (the case

diverges by (9), other ordered expressions exist, e.g. 1

:1 =ie− i~ 2uv ,

1 −πi i~ uv

:e∗

1

:1 = − ie− i~ 2uv

(in normal ordering).

Thus, we have 1 uv −πi i~

0=e∗

1 πi i~ 2uv

∗(e∗

1 πi i~ uv

+1)=e∗

1 −πi i~ uv

+e∗

The result follows from the the exponential law.

=2 cos∗ (π

1 uv). i~

3 Star exponential functions of quadratic forms

9

1 uv)∗f (uv) is defined on some domain containing Lemma 3.3 If sin∗ π(z+ i~ 1 z= 12 , then sin∗ π( 21 + i~ uv)∗f (uv)=0. 1 These observations lead to viewing 21 + i~ uv is an indeterminate in the 1 set of integers Z, that is, i~ v∗u behaves as if it were an indeterminate in Z. However, we have to keep in mind the following remark: z 1 uv Remark 1 There are two definitions of the product e∗ i~ ∗f (u, v). The first is to define as the real analytic solution of

d 1 ft = uv∗ft , dt i~

f0 =f (u, v),

if a real analytic solution exists. The second is to define z

1

e∗ i~

uv

z

1

∗f (u, v)= lim e∗ i~

uv

∗fn (u, v),

n→∞

if f (u, v)= lim fn (u, v), n

where fn are polynomials. These two definitions do not agree in general, since z 1 uv the multiplication e∗ i~ ∗ is not a continuous linear mapping of Hol(C2 ) into itself (cf.(16)).

3.2 Several estimates t

1

We have already known that :e∗i~

uv

:κ ∈Hol(C2 ) for every fixed t whenever t

1

uv

defined. By (11), we see also that if κ∈C−{κ≥1} ∪ {κ≤−1}, then :e∗i~ :κ is rapidly decreasing with respect to t. R ∞ t 1 uv In this section, we first show that −∞ e∗i~ dt∈Hol(C2 ) in the Weyl ordering. t

1

The Weyl ordering of e∗i~ :

Z

t

1

e∗i~ R

uv

uv

t

1

is :e∗i~

dt:0 =

Z

∞ −∞

uv

t

1

1 (tanh 2 ) i~ 2uv . Hence :0 = cosh t e 2

t 1 1 e(tanh 2 ) i~ 2uv dt. cosh 2t

By setting cos s= tanh 2t , −2 sin sds= sin2 sdt, the integral on the right hand side becomes into Z 0 Z π 1 1 2 e(cos s) i~ 2uv ds= e(cos s) i~ 2uv ds. −π

−π

By the Hansen-Bessel formula, we have r Z ∞ 2 π t 1 uv : e∗i~ dt :0 = J0 ( uv), 2 ~ −∞

(13)

10

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

where J0 is Bessel function of eigenvalue 0. 1 Since g(s) = e(cos s) i~ uv is a continuous curve in Hol(C2 ), its integral (13) on a compact domain belongs to Hol(C2 ). Applying the intertwiner I0κ for (13), we see that Z Z π 1 t 1 uv : e∗i~ dt:κ = :e(cos s) i~ 2uv :κ ds. −π

R

Since 1

:e(cos s) i~ 2uv :κ =

1 ecos s −e− cos s 2 exp 2uv , cos s cos s cos s − cos s (1−κ)e +(1+κ)e (1−κ)e +(1+κ)e i~

we have the following: Proposition 3.2 For every κ∈C−{κ≥1}∪{κ≤−1}, the κ-ordered expression R ∞ t 1 uv of the integral : −∞ e∗i~ dt:κ is contained in the space Hol(C2 ). Further1 R ∞ eiθ t i~ uv iθ d e dt=0 whenever defined. more, integration by parts gives dθ −∞ e∗ The ∗-delta function is defined by the following integral: Z Z ∞ 1 1 −it 1 uv uv t i~ dt= e∗ ~ dt=δ∗ ( uv). e∗ ~ R −∞

1+ cos s Note that cos s= tanh 2t implies t= log 1− cos s . Hence, we have 1+ cos s Lemma 3.4 If f (t) is a continuous function such that f (log 1− cos s ) is con1 R∞ t i~ uv 2 tinuous on [−π, 0], then −∞ f (t)e∗ dt is in Hol(C ) in the κ-ordered expression such that for every κ∈C−{κ≥1}∪{κ≤−1}. t

Applying Lemma 3.4 to the function f (t)=e−at (a>0) and e−e , we have R t t 1 uv t 1 uv : R e−at e∗i~ dt:κ and : R e−e e∗i~ dt:κ are elements of Hol(C2 ). We denote the second integral by Z t t 1 uv 1 e−e e∗i~ dt=Γ∗ ( uv) (cf. §6). i~ R R

Since v∗u=uv+ 21 i~, (12) shows the uniform convergence on each compact domain of the limits 1 τ 2 i~1 1−κ (2uv+ 1−κ u2 ) , e t→∞ 1−κ 2 1 τ 1 2 − i~ t 1 2u∗v 1+κ (2uv− 1+κ u ) , lim :e∗i~ :(κ,τ ) = e t→−∞ 1+κ

t

1

lim :e∗i~

t

1

lim :e∗i~

t→−∞

We call

2v∗u

2v∗u

:(κ,τ ) =

:(κ,τ ) =0,

t

1

lim :e∗i~

t→∞

2u∗v

:(κ,τ ) =0.

(14)

3 Star exponential functions of quadratic forms t

1

$00 = lim e∗i~

2u∗v

t→−∞

,

t

1

$00 = lim e∗i~

11

2u∗v

t→∞

vacuums. The exponential law gives $00 ∗$00 =$00 ,

$00 ∗$00 =$00 .

However, we easily see Theorem 3.1 The product $00 ∗$00 diverges in any ordered expression. The existence of the limit (14) gives also u∗v∗$00 = 0 = $00 ∗u∗v. But the “bumping identity” v∗f (u∗v)=f (v∗u)∗v gives the following: Lemma 3.5 v∗$00 =0=$00 ∗u. Proof. Using the continuity of v∗, we see that t

1

v∗ lim e∗i~

2u∗v

t

1

= lim v∗e∗i~

2u∗v

t→−∞

t→−∞

.

Hence, the bumping identity (proved by the uniqueness of the real analytic t

1

2v∗u

solution for linear differential equations) gives limt→−∞ e∗i~ ∗v=0 by using (14). However, we note that associativity is not easily ensured. The following is the simplest condition which ensures associativity for certain calculations: Proposition 3.3 For every polynomial and for every f ∈Hol(C2 ), the products p∗f and f ∗p are defined as elements of Hol(C2 ), and associativity (f ∗g)∗h=f ∗(g∗h) holds whenever two of f, g, h are polynomials. In general (f ∗g)∗h=f ∗(g∗h) does not hold even if g is a polynomial. Example 1 By Lemma 3.2, 1 −1 uv +∗ = i~

Z

0 −∞

1 i~ uv

has two different inverses

t 1 uv e∗i~ dt,

1 −1 uv −∗ =− i~

Z

∞

t

1

e∗i~

uv

dt

0

as elements of Hol(C2 ). Hence, we see the failure of associativity : 1 1 1 1 1 1 −1 −1 −1 ( uv)−1 ∗( uv) ∗( uv) = 6 ( uv) ∗ ( uv)∗( uv) +∗ −∗ +∗ −∗ , i~ i~ i~ i~ i~ i~ −1 1 1 uv)−1 and indeed ( i~ +∗ ∗( i~ uv)−∗ diverges in any ordered expression. In what follows, we use the notation

1 1 1 δ∗ ( uv)=( uv)−1 uv)−1 +∗ −( −∗ . ~ i~ i~

(15)

12

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

In spite of this general failure of associativity, we have another primitive criterion for associativity. We remark that if all terms are considered as formal power series in i~ in the product formula (1), then the product is always defined, and it is easy to show associativity, as it holds for polynomials (cf. [14] for details). Applying these remarks carefully, we give the following: Lemma 3.6 $00 ∗(up ∗$00 )=0, and ($00 ∗v p )∗$00 =0. Proof. By taking the formal power series expansion with respect to i~ for esu∗v , associativity holds, and the following computation is permitted by the ∗ bumping identity: (s+t)u∗v+i~ps

esu∗v ∗(up ∗e∗tu∗v )=(esu∗v ∗up )∗e∗tu∗v =up ∗e∗ ∗ ∗

.

The right hand side of the above equality is continuous in s, t. In particular, lim e∗su∗v ∗(up ∗e∗tu∗v )=esu∗v ∗ lim (up ∗e∗tu∗v ). ∗

t→a

t→a

Using the bumping identity, we have (s+t)u∗v+i~ps

esu∗v ∗(up ∗ lim e∗tu∗v )=esu∗v ∗ lim up ∗e∗tu∗v = lim up ∗e∗ ∗ ∗ t→−∞

t→−∞

t→−∞

(s+t)u∗v+i~ps

=up ∗ lim e∗ t→−∞

=up ei~ps ∗$00 .

It follows that s

1

$00 ∗(up ∗$00 )= lim e∗ i~ s→−∞

u∗v

t

1

∗( lim up ∗e∗i~ t→−∞

u∗v

)= lim up eps ∗$00 =0. s→−∞

Similarly, we also have ($00 ∗v p )∗$00 =0. Lemma 3.7 For every polynomial f (u, v)=

P

aij ui ∗v j ,

$00 ∗(f (u, v)∗$00 )=f (0, 0)$00 =($00 ∗f (u, v))∗$00 . Consequently, associativity holds for $00 ∗p(u, v)∗$00 for a polynomial p(u, v). A similar computation gives the following associativity ($00 ∗v q )∗(up ∗$00 )=δp,q p!(i~)p =$00 ∗(v q ∗up ∗$00 )=($00 ∗v q ∗up )∗$00 . Since $00 ∗v q ∗up ∗$00 =δp,q p!(i~)p $00 , we have the following: Proposition 3.4 √

1 up ∗$00 ∗v q p!q!(i~)p+q

is the (p, q)-matrix element.

4 Inverses and their analytic continuation z

1

As mentioned in Remark 1 in § 3.1, we have two definitions of e∗ i~ However both definitions give the formula z

1

e∗ i~ On the other hand, since t

1

e∗i~

lim

N →∞

Z

1

∗$00 =e− 2 z ∗$00 .

1 1 i~ uv∗δ∗ ( ~ uv)=0, uv

as the real analytic solution of t 1 uv e∗i~ ∗

uv

N

uv

13

∗f (u, v).

(16)

we must set

1 1 ∗δ∗ ( uv)=δ∗ ( uv) ~ ~

d 1 dt ft = i~ uv∗ft . s 1 uv e∗ i~ ds

= lim

However, computing

N →∞

−N

Z

N

1 uv (t+s) i~

e∗

ds

−N

gives the following: 1 (x+iy) ~ uv

e∗

1 1 iy 1 uv ∗δ∗ ( uv)=e∗ ~ ∗δ∗ ( uv). ~ ~

(17)

Hence (16) is holomorphic with respect to z, while (17) is only continuous, that is, there is no real analyticity with respect to z = x+iy.

4 Inverses and their analytic continuation Formula (5) and the exponential law give in particular 1 t(z+ i~ v)

:e∗

1

2

1

:(κ,τ ) =e 4i~ t τ et(z+ i~ v) . 1

2

It follows that if Im τ < 0, then e 4i~ t τ is rapidly decreasing in t and the integrals Z 0 Z ∞ 1 t(z+ i~ v) t(z+ 1 v) : e∗ dt:(κ,τ ) , −: e∗ i~ dt:(κ,τ ) (18) −∞

0

1 v, and are denoted converge. Both integrals are respectively inverses of z+ i~ −1 −1 1 1 by (z+ i~ v)+∗ , (z+ i~ v)−∗ , respectively, with the subscript (κ, τ ) ommitted.

Proposition 4.1 If Im τ < 0, then the (κ, τ )-ordered expression of the difference of the two inverses is given by Z ∞ 2 1 1 1 1 −1 :(z+ v)−1 −(z+ v) : = e 4i~ t τ et(z+ i~ v) dt. (κ,τ ) i~ +∗ i~ −∗ −∞ This difference is holomorphic in z.

14

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Similarly, by formula (9), we have the convergence of the two integrals :

Z

:−

0

t 1 uv etz e∗i~ dt:0

Z

=

−∞

Z

∞

t

1

etz e∗i~

uv

1

0

1 e 2 tz i~ 2uv tanh 21 t dt, 1 e cosh 2 t

−∞

dt:0 = −

0

Z

∞ 0

1 Re z > − , 2

(19)

1 . 2

(20)

1

1 1 e 2 tz i~ 2uv tanh 2 t dt, 1 e cosh 2 t

Re z <

1 Both (19) and (20) give inverses of z+ i~ uv. By a similar computation, there are two inverses for every (κ, τ ) such that κ∈C−{κ≥1}∪{κ≤−1}, which will −1 1 1 uv)−1 be denoted by (z+ i~ +∗ , (z+ i~ uv)−∗ . The following may be viewed as a Sato hyperfunction:

Proposition 4.2 If − 21 < Re z < 12 , then the difference of the two inverses is given by Z ∞ 1 1 t(z+ 1 uv) −1 (z+ uv)−1 (21) −(z+ uv) = e∗ i~ dt. +∗ −∗ i~ i~ −∞ Its (κ, τ )-ordered expression is holomorphic on this strip. One can see the right hand side more closely. For − 12 < Re z ≤ 0, the change of variables tanh 12 t= cos s from forms the right hand side of (21) into 2

Z

0

( −π

1+ cos s z (cos s) 1 2uv i~ ds. ) e 1− cos s

For 0 ≤ Re z< 12 and for − cos s= tanh 2t , 2 sin sds= sin2 sdt, the right hand side of (21) transforms into Z π 1+ cos s −z (cos s) 1 uv i~ 2 ( ) e ds. 1− cos s 0 Hence, Lemma 3.4 gives that

1 uv) t(z+ i~ dt e −∞ ∗

R∞

is an element of Hol(C2 ).

On the other hand, note that a change of variables gives Z 0 Z ∞ 1 1 uv) (z− 1 uv) −t(z− i~ dt=− e∗ i~ dt. ((−z)+ uv)−1 =− e ∗ −∗ i~ −∞ 0 Thus, we see that (z−

1 1 uv)−1 uv)−1 −∗ =−((−z)+ −∗ . i~ i~

(22)

This is holomorphic on the domain Re z> − 12 , which is also the holomorphic 1 domain for (z+ i~ uv)−1 −∗ . All of these results are easily proved for the Weyl ordering. However, if t 1 uv κ∈C−{κ≥1}∪{κ≤−1}, then :e∗i~ :κ is rapidly decreasing in t, and the same computation gives the following:

4 Inverses and their analytic continuation

15

1 uv)−1 Proposition 4.3 For every z with Re z>− 21 , the two inverses (z+ i~ +∗ 1 and (z− i~ uv)−1 −∗ are defined in the κ-ordered expression for κ∈C−{κ≥1} ∪ {κ≤−1}. −1 1 1 Note that (z+ i~ uv)−1 +∗ ∗(−z− i~ uv)−∗ diverges for any ordered expression. However, the standard resolvent formula gives the following:

Proposition 4.4 If z+w6=0, then 1 1 1 −1 (z+ uv)−1 uv) +(w− +∗ −∗ z+w i~ i~ 1 1 uv)∗(w− i~ uv). In particular, for every positive integer is an inverse of (z+ i~ n, and for every complex number z such that Re z> − (n+ 12 ), the κ-ordered −1 1 1 1 1 (1+ n1 (z+ i~ uv))−1 expression of 2n +∗ +(1− n (z+ i~ uv))−∗ gives an inverse of 1 1− n12 (z+ i~ uv)2∗ for κ∈C−{κ≥1}∪{κ≤−1}.

4.1 Analytic continuation of inverses 1 1 Recall that (z± i~ uv)−1 ±∗ are holomorphic on the domain Re z > − 2 . It is −1 −1 1 1 natural to expect that (z± i~ uv)±∗ =C(C(z± i~ uv))±∗ for any non-zero constant C. To confirm this, we set C=eiθ and consider the θ-derivative Z 0 1 uv) eiθ t(z± i~ dt. e∗ eiθ −∞

In the (κ, τ )-ordered expression, the phase part of the integrand is bounded in t and the amplitude is given by (1−κ)e

2eiθ tz , + (1+κ)e−eiθ t/2

eiθ t/2

κ6=1.

Hence, the integral converges whenever Re eiθ (z± 12 ) > 0, and by integration 1 by parts this convergence does not depend on θ. It follows that (z± i~ uv)−1 ±∗ 1 are holomorphic on the domain C−{t; −∞
1 1 uv)−1 uv)−1 +∗ ∗v=(z−1+ +∗ , i~ i~

−1 v+ ∗(z−

1 1 uv)−1 uv)−1 −∗ ∗v=(z+1− −∗ i~ i~

whenever one can use the inverse of v in a suitable ordered expression. In this section, analytic continuation will be produced via these sliding identities. In this note, we state the sliding identity by using, instead of v −1 , the left inverse v ◦ of v given below. First of all, we remark that formula (9) also gives

16

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

(u∗v)−1 −∗ = −

1 i~

Z

∞

t

1

e∗i~

u∗v

(v∗u)−1 +∗ =

dt,

0

1 i~

Z

0

t

1

e∗i~

v∗u

dt.

−∞

These gives left/right inverses of u, v v ◦ =u∗(v∗u)−1 +∗ ,

u• =v∗(u∗v)−1 −∗ ,

for it is easy to see that v∗v ◦ =1,

v ◦ ∗v=1−$00 ,

u∗u• =1,

u• ∗u=1−$00 .

The bumping identity gives v∗(z+

1 1 uv)∗v ◦ =z+1+ uv, i~ i~

v ◦ ∗(z+

1 1 uv)∗v=(1−$00 )∗(z−1+ uv). i~ i~

The successive use of the bumping identity gives the following useful formula: n (u∗(v∗u)−1 +∗ ) ∗$00 =

1 1 n ( u) ∗$00 . n! i~

(23)

Using v ◦ instead of v −1 , we can give the analytic continuation of inverses. However, we have to be careful about the continuity of the ∗-product. We compute 1 v ∗(z+ uv)−1 +∗ =u ∗ i~ ◦

=u ∗ =

Z

=

Z

0

Z

t( 1 uv+ 12 ) e∗ i~ dt

−∞ Z 0

Z

0

1 t( i~ uv+ 12 )

e∗ e

−∞

Z

0

1 s(z+ i~ uv)

e∗

ds

−∞

−∞ −∞ 0 Z 0 t 21 +sz

−∞ 0

∗

Z

−∞ 0

1

1 s(z+ i~ uv)

∗e∗

1 (t+s) i~ uv

u∗e∗

dtds

1 (t+s) i~ uv

et 2 +sz−(t+s) e∗

dtds

∗udtds.

−∞

Hence, we have the identity whenever both sides are defined: Z 0 Z 0 1 1 1 (t+s) i~ uv ◦ −1 e−t 2 +s(z−1) e∗ ∗(u∗v)dtds (v ∗(z+ uv)+∗ )∗v= i~ −∞ −∞ Z 0 Z 0 1 s(z−1+ i~ uv) t 1 u∗v e∗ ds = (u∗v)∗e∗i~ dt ∗ −∞

−∞

1 =(1−$00 )∗(z−1+ uv)−1 +∗ . i~ Remarking that $00 ∗(z−1+

1 1 −1 uv)−1 $00 , +∗ =(z− ) i~ 2

4 Inverses and their analytic continuation

17

1 uv)−1 whenever (z−1+ i~ +∗ is defined, we have

v ◦ ∗(z+

1 1 −1 1 −1 uv)−1 $00 = z−1+ uv ∗+ . +∗ ∗v+(z− ) i~ 2 i~

(24)

Since (z− 12 )−1 $00 is always defined, we see that (24) gives the formula for analytic continuation. Using this, we have the following (see [11] and [14] for more details): −1 1 1 uv)−1 Theorem 4.1 The inverses (z+ i~ +∗ , (z− i~ uv)−∗ extend to holomorphic 1 1 functions in z on C−{−(N+ 2 )}. In particular, (z 2 −( i~ uv)2 )−1 ±∗ extend to holomorphic functions of z on this domain. −1 1 1 uv)−1 The product (z+ i~ +∗ ∗(w+ i~ uv)+∗ is naturally defined, but the for1 uv is not mula in Theorem 4.1 looks strange at the first glance, because z+ i~ −1 1 1 1 zero at z=n+ 2 and (z+ i~ uv)+∗ is singular at z=n+ 2 , but

(z+

1 1 uv)∗(z+ uv)−1 +∗ =1 i~ i~

for z6∈−(N+ 12 ). Note that Z 0

1 1 1 Re z > − 12 t(z+ i~ uv) , dt= (z+ uv)∗e∗ 1−$00 z=− 21 i~ −∞ Z 0 1 1 1 Re z > − 21 t(z− i~ uv) dt= (z− uv)∗e∗ . 1−$00 z=− 21 i~ −∞

As suggested by these formulas, we extend the definition of the ∗-product as s 1 uv follows: For every polynomial p(u, v) or p(u, v)=e∗ i~ , Z 0 1 t(z± 1 uv) e∗ i~ dt. p(u, v)∗(z± uv)−1 = lim p(u, v) ∗ (25) +∗ N →∞ i~ −N Hence we have the formula 1 1 1 Re z > − 12 (z+ uv)∗(z+ uv)−1 = . +∗ 1−$00 z=− 21 i~ i~

(26)

Considering 1 1 1 1 ◦ n n n uv)∗(z+ uv)−1 uv)∗v n ∗(v ◦ )n ∗(z+ uv)−1 +∗ ∗v =(v ) ∗(z+ +∗ ∗v i~ i~ i~ i~ and using the formula (23), we have the following:

(v ◦ )n ∗(z+

Theorem 4.2 If we use definition (25) for the ∗-product, then 1 1 1 uv)∗(z+ uv)−1 = 1 1 +∗ 1− n! ( i~ u)n ∗$00 ∗v n i~ i~ 1 1 1 (z− uv)∗(z− uv)−1 = 1 1 −∗ 1− n! ( i~ v)n ∗$00 ∗un i~ i~ (z+

z6∈−(N+ 12 ) , z=−(n+ 21 )

(27)

z6∈−(N+ 12 ) . z=−(n+ 12 )

(28)

18

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Although z=−(n+ 12 ) are all removable singularities for (27) and (28) as a function of z, it is better to retain these singular points. These formulas give in particular for every fixed positive integer m 1 1 1 1 1 z6∈−(N+m+ 12 ) −1 (1+ (z+ uv))∗(1+ (z+ uv))+∗ = 1 1 k k 1− k! ( i~ u) ∗$00 ∗v z=−(k+m+ 12 ) m i~ m i~ (29) for arbitrary k ∈ N. We state the following identity for later use: 1 1 1 $00 ∗v n ∗(−n− + uv)=$00 ∗( u∗v)∗v n =0. 2 i~ i~

(30)

5 An infinite product formula Recall the classical formula sin πx=πx ∞ Y

k=1

(1−

1 x2 )= k 2 2i

Z

x2 k=1 (1− k2 ).

Q∞

χ[−π,π] (t)eitx dt= lim

n→∞

Z Y n

Rewrite this as follows:

(1+

k=1

1 2 ∂ )δ(t)eitx dt, k2 t

where χ[−π,π] (t) is the characteristic function of the interval [−π, π]. It follows that χ[−π,π] (t)=2i lim

n→∞

n Y

k=1

(1+

1 2 ∂ )δ(t) k2 t

in the space of distributions. it 1 uv κ+1 |6=1 , so that :e∗ i~ :κ is not singular on t ∈ R, we For κ uch that | κ−1 compute as follows: Z Z 1 uv) ±it 1 uv) it(z± i~ :κ dt= χ[−π,π] (t)eitz :e∗ i~ :κ dt. χ[−π,π] (t):e∗ Fixing a cut-off function ψ(t) of compact support such that ψ=1 on [−π, π], we see that Z Z Y n 1 t(z± 1 uv) ±it 1 uv χ[−π,π] (t):e∗ i~ :κ dt=2i lim (1+ 2 ∂t2 )δ(t)ψ(t)etz :e∗ i~ :κ dt. n→∞ k k=1

Integration by parts gives Z n n Y Y 1 1 ±it 1 uv t(z± 1 uv) lim δ(t) (1+ 2 ∂t2 )ψ(t)etz :e∗ i~ :κ dt= lim :(1+ 2 ∂t2 )e∗ i~ :κ . n→∞ n→∞ k k k=1

k=1

Hence we have in the κ-ordered expression that Z n Y 1 1 1 uv) it(z± i~ dt=2i lim (1− 2 (z± uv)2 )∗ . χ[−π,π] (t)e∗ n→∞ k i~ k=1

5 An infinite product formula

19

Noting that sin∗ π(z±

1 1 uv)=π(z± uv)∗ i~ i~

Z

1 uv) it(z± i~

χ[−π,π] (t)e∗

dt ∈ Hol(C2 ),

we have n

sin∗ π(z±

Y 1 1 1 1 uv)=π(z± uv)∗ lim ∗ (1− 2 (z± uv)2 )∗ n→∞ i~ i~ k i~

(31)

k=1

in Hol(C2 ). In particular, we have κ+1 Proposition 5.1 In the κ-ordered expression with | κ−1 |6=1, we have n

sin∗ π(z+

Y 1 1 1 1 uv)=π(z+ uv)∗ lim ∗(1− 2 (z+ uv)2 ). n→∞ i~ i~ k i~ k=1

This is identically zero on the set z∈Z+ 12 . The formula in Proposition 5.1 may be rewritten as n

sin∗ π(z+

Y 1 1 1 1 1 1 (z+ i~ uv) uv)=π(z+ uv)∗ lim ∗(1− (z+ uv))∗e∗k n→∞ i~ i~ k i~ k=1 n Y

∗

k=1

1 1 1 − 1 (z+ i~ uv) ∗(1+ (z+ uv))∗e∗ k . k i~

In §6, we will define a star gamma function via the two different inverses mentioned previously and give an infinite product formula for the star gamma function. −1 1 1 1 uv)+∗ and with 1+ m (z+ i~ uv) 5.1 The product with (z+ i~

−1 +∗

1 1 uv)−1 First we consider the product (z+ i~ ±∗ ∗ sin∗ π(z+ i~ uv) in two different ways. One way is by defining:

(z+

1 1 uv)−1 uv) ±∗ ∗ sin∗ π(z+ i~ i~ n (32) Y 1 1 1 1 2 = lim (z+ uv)−1 ∗ (z+ uv)∗ ∗(1− (z+ uv) ) . ±∗ n→∞ i~ i~ k2 i~ k=1

Qn

1 uv)∗ Since (z+ i~ (27) and (30) give

k=1

1 ∗(1− k12 (z+ i~ uv)2 ) is a polynomial, Proposition 3.3,

∞

Y 1 1 1 1 uv)= ∗(1− 2 (z± uv)2 ). (z+ uv)−1 ±∗ ∗ sin∗ π(z± i~ i~ k i~ k=1

(33)

20

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

The second way is by defining 1 1 (z+ uv)−1 uv)= lim ±∗ ∗ sin∗ π(z+ N →∞ i~ i~

Z

0

1 t(z+ i~ uv)

e∗

∗ sin∗ π(z+

−N

1 uv). (34) i~

This may be written as the complex integral 1 2i

Z

0+πi

t(z+ 1 uv) e∗ i~ dt−

−∞+πi

1 2i

Z

0−πi

1 t(z+ i~ uv)

e∗

dt.

−∞−πi

Rπ 1 Adding − 21 −π eit(z+ i~ uv) dt to this expression gives the clockwise contour integral along the boundary of the domain D={z∈C; Re z<0, −π

−D

D

z

1

Proof. :e∗ i~

uv

:κ =

2 z z (1−κ)e 2 +(1+κ)e− 2

1

uv

Lemma 5.1 :e∗ i~ :κ has at most one singular point in the domain D ∪ (−D). If Re κ>0, then there is no singular point in D. z

exp z 2

z

e 2 −e− 2 2 uv. z z (1−κ)e 2 +(1+κ)e− 2 i~ − z2

Thus, the sin-

κ+1 gular points are given by (1−κ)e +(1+κ)e =0. This gives ez = κ−1 . If κ+1 κ6= ± 1, then z= log κ−1 +2πni. Thus, the domain D ∪ (−D) contains at most one singular point. κ+1 κ+1 If Re κ>0, then | κ−1 | > 1 and the singular point z= log κ−1 +2πni has a positive real part.

Proposition 5.2 Suppose Re κ>0 and κ∈C−{κ≥1}∪{κ≤−1}. Then in the κ-ordered expression, we have Z 0 Z 1 1 1 π it(z+ i~ 1 t(z+ i~ uv) uv) lim e∗ ∗ sin∗ π(z+ uv)= dt. e∗ N →∞ −N i~ 2 −π According to (31) the right hand side gives the same result as (33), that is, Q∞ 1 1 2 ∗(1− 1 k2 (z+ i~ uv) ).

Proposition 5.3 Suppose Re κ>0 and κ∈C−{κ≥1}∪{κ≤−1}. Then in the 1 1 κ-ordered expression, the product sin∗ π(z+ i~ uv)∗(z+ i~ uv)−1 +∗ is an entire −1 1 function of z. Namely, all singularities of (z+ i~ uv)+∗ at −(N+ 12 ) are cancelled out in formulas (29) and (30). By a proof similar to that of Proposition 5.3, we obtain Proposition 5.4 Suppose Re κ>0, and κ∈C−{κ≥1}∪{κ≤−1}. Then in the κ-ordered expression,

6 Star gamma functions

sin∗ π(z−

21

1 1 uv)∗(z− uv)−1 −∗ i~ i~

is a well defined entire function of z. 1 1 In particular, sin∗ π(z+ i~ uv)∗(z 2 −( i~ uv)2 )−1 ±∗ is a holomorphic function of z in C. 1 1 1 (z+ i~ uv))−1 Consider next the product (1+ m +∗ ∗ sin∗ π(z+ i~ uv). Since

(1+

1 1 1 (z+ uv))−1 uv)−1 +∗ =m(m+z+ +∗ , m i~ i~

1 1 and sin∗ π(z+m+ i~ uv)=(−1)m sin∗ π(z+ i~ uv) by the exponential law, the product formula is essentially the same as above. Hence we see the following:

Proposition 5.5 Suppose Re κ>0, and κ∈C−{κ≥1}∪{κ≤−1}. Then in the 1 1 1 uv)∗(1+ m (z+ i~ uv))−1 κ-ordered expression, the product sin∗ π(z+ i~ +∗ is an entire function of z with no removable singularity. Remark 2 Suppose Re κ<0, and κ∈C−{κ≥1}∪{κ≤−1}. Then the residue t(z+

1

uv)

κ+1 of e∗ i~ at the singular point t= log κ−1 +2πni in D gives the difference between the twosides of the equality in Proposition 5.3. This observation shows that continuity does not hold for κ-ordered expressions. 1 R uv) t(z+ i~ 1 dt e Lemma 5.1 and formula (11) show that the integral 2πi ∂D ∗ gives the residue at the singular point in D. This residue will be computed in the subsection 7.2.

6 Star gamma functions We first recall the ordinary gamma function and beta function: Z ∞ Z 1 −t z−1 e t dt, B(x, y)= tx−1 (1−t)y−1 dt. Γ(z)= 0

0

s

Substituting t = e gives Z ∞ s Γ(z) = e−e esz ds, −∞

B(x, y)=

Z

0

esx (1−es )y−1 ds.

−∞

The star gamma function and the star beta function may be defined by replacing x with z ± uv i~ : Z ∞ τ τ (z± uv ) uv i~ dτ, )= e−e e∗ Γ∗ (z ± i~ −∞ (35) Z 0 uv τ (z± uv i~ ) e∗ B∗ (z ± , y) = (1−eτ )y−1 dτ. ~i −∞

22

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

These Weyl orderings are uv ):0 = :Γ∗ (z ± i~ : B∗ (z ±

Z

Z

uv , y) :0 = i~

∞ −∞

0 −∞

τ

e−e +zτ ± 1 uv tanh 1 τ 2 dτ, e i~ cosh 12 τ

(1−eτ )y−1 eτ z ± 1 uv tanh 1 τ 2 dτ. e i~ cosh 12 τ

The κ-ordered expressions are obtained by applying the intertwiner I0κ for κ∈C−{κ≥1}∪{κ≤−1}. uv ) :κ = lim : Γ∗ (z ± N,N 0 →∞ ~i

Z

N0 −N

τ

e−e +zτ κ ± 1 uv tanh 1 τ 2 )dτ. I (e ~i cosh 12 τ 0

(36)

The right hand side converges on a dense open domain of κ. Proposition 6.1 For every uv ∈ C, and for every z ∈ C such that Re z > − 12 , the right hand side of (36) converges and is holomorphic with respect to z . However, Γ∗ (− 21 ± uv ~i ) is singular. Throughout this section, ordered expressions are always restricted to κ∈C−{κ≥1}∪{κ≤−1}. 6.1 Analytic continuation of Γ∗ (z ±

uv ) ~i

As with the usual gamma function, integration by parts gives the identity Γ∗ (z+1 ±

uv uv uv )=(z ± )∗Γ∗ (z ± ). ~i ~i ~i

(37)

Using

uv uv −1 uv )=(z ± ) ∗Γ∗ (z+1 ± ), ~i ~i ±∗ ~i and careful treating continuity inssues, we have Γ∗ (z ±

Proposition 6.2 Γ∗ (z ± C−{−(N+ 21 )}. τ (z± uv )

uv ~i )

extends to a holomorphic function on z ∈

~i ∗$00 =(z ± 21 )−1 $00 , we see the following remarkable feature Since e∗ of these star functions: Z N τ τ (z± uv ) uv ~i Γ∗ (z ± e−e e∗ )∗$00 ≡ lim dτ ∗$00 = Γ(z ± 21 )$00 , N →∞ −N ~i Z N uv τ (z± uv ~i ) (1−eτ )y−1 dτ ∗$00 = B(z ± 21 , y)$00 . , y)∗$00 ≡ lim e∗ B∗ (z ± N →∞ −N ~i (38)

6 Star gamma functions

23

6.2 An infinite product formula We see in the same notation as above Z 0 uv −1 uv τ (z± uv ~i ) e∗ dτ = z+ , B∗ (z± , 1) = ~i i~ ∗± −∞

1 Re z > − . 2

(39)

We now compute uv )Γ(y) = Γ∗ (z ± ~i

ZZ

τ σ τ (z± uv ~i ) σy −(e +e )

e∗

e e

dτ dσ.

R2

We change variables by setting eσ = et (1−es ),

τ = t+s,

where − ∞ < t < ∞, −∞ < s < 0.

Since eτ +eσ = et , this gives a diffeomorphism of R×R− onto R2 . The Jacobian 1 is given by dτ dσ = 1−e s dtds. Hence we have the fundamental relation between the gamma function and the beta function Z ∞Z 0 t uv t(y+z± uv s(z± uv ~i ) −e ~i ) )Γ(y) = e∗ e ∗e∗ (1−es )y−1 dtds Γ∗ (z ± ~i −∞ −∞ (40) uv uv =Γ∗ (y+z ± )∗B∗ (z ± , y). ~i ~i Integration by parts gives (z ±

uv uv uv )∗B∗ (z ± , y+1) = yB∗ (1+z ± , y+1). ~i ~i ~i

To prove this, note that uv d τ (z± uv τ (z± uv ~i ) ~i ) = (z ± , e∗ )∗e∗ dτ ~i lim e−e

τ →±∞

τ

uv +zτ ±τ ~i e∗

τ d −eτ e = −eτ e−e , dτ

1 = 0 for Re z > − . 2

uv uv Since B∗ (z ± uv ~i , y+1) = B∗ (z ± ~i , y)−B(1+z ± ~i , y), we have the functional equation y+z ± uv uv uv ~i , y) = ∗B∗ (z ± , y+1). (41) B∗ (z ± ~i y ~i

Iterate (41) to obtain B∗ (z ±

(y+z ± uv , y) = ~i

uv ~i )∗(y+1+z

y(y+1)

±

uv ~i )

∗B∗ (z ±

uv , y+2). ~i

Using the notation (a)n = a(a+1) · · · (a+n−1),

{A}∗n = A∗(A+1)∗ · · · ∗(A+n−1),

24

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

we have B∗ (z ±

{y+z ± uv uv uv ~i }∗n , y) = ∗B∗ (z ± , y+n). ~i (y)n ~i

(42)

Similarly, integration by parts gives the formula Γ∗ (1+z ±

uv uv uv ) = (z ± )∗Γ∗ (z ± ), ~i ~i ~i

1 for Re z > − . 2

(43)

Iterate (43) to obtain uv uv uv . ) = Γ∗ (z ± )∗ z ± ~i ~i ~i ∗n Qn uv −1 Lemma 6.1 B∗ (z ± uv k=0 ∗(k+z ± ~i )±∗ . ~i , n+1) = n!

(44)

Γ∗ (n+1+z ±

Proof. The right hand side of the above equality will be denoted by

n! . (±) {z± uv ~i }∗n+1

The case n = 0 is given by (39). Suppose the formula holds for n. For the case n+1, we see that uv B∗ (z ± , n+2) ~i Z 0 τ (z± uv ~i ) = (1−eτ )(1−eτ )n dτ = e∗ −∞

It follows that B∗ (z ±

n! {z ±

uv (±) ~i }∗n+1

−

n! {1+z ±

uv (±) ~i }∗n+1

.

uv (n+1)! , n+2) = . (±) ~i {z ± uv } ∗n+2 ~i

In this subsection, we give an infinite product formula for the ∗-gamma function. By Lemma 6.1, we see that Z 0 1 n! τ (z± uv ~i ) , Re z > − . e∗ (1 − eτ )n dτ = uv (±) 2 −∞ {z ± ~i }∗n+1 Replacing eτ by

1 τ0 ne ,

namely setting τ = τ 0 − log n in the left hand side, and (log n)(z± uv ~i )

multiplying both side by e∗ Z

log n

τ 0 (z± uv ~i )

e∗ −∞

, we have

n! 1 0 (log n)(z± uv ~i ) (1− eτ )n dτ 0 = . ∗e∗ (±) uv n {z ± ~i }∗n+1

(45)

Lemma 6.2 The Weyl ordering of the left hand side of (45) converges when R ∞ τ 0 (z± uv τ0 ~i ) −e n→∞ to −∞ e∗ e dτ 0 in Hol(C2 ).

6 Star gamma functions 0

25

τ0

Proof. Obviously, limn→∞ (1− n1 eτ )n =e−e uniformly on each compact subset as a function of τ 0 . In the Weyl ordering, it is easy to show that Z log n 0 Z ∞ 0 τ0 τ0 τ (z± uv τ (z± uv 0 ~i ) −e ~i ) −e e∗ e∗ lim e dτ = e dτ 0 n→∞

−∞

−∞

2

in Hol(C ). Thus it is enough to show that Z log n 0 τ0 1 0 τ (z± uv ~i ) e∗ lim (e−e −(1− eτ )n )dτ 0 =0 n→∞ −∞ n in Hol(C2 ). This is easy in the Weyl ordering. Applying the intertwiner gives the desired result. The right hand side of (45) equals for Re z > − 21 1 (log n−(1+ 12 +···+ n ))(z± uv ~i )

e∗

∗(z ±

n z± uv ~i z ± uv uv −1 Y ~i −1 k ∗e 1+ )∗± ∗ . ∗ ±∗ ~i k k=1

(log n−(1+ 1 +···+ 1 ))(z± uv )

2 n ~i The left hand side of (45) converges, and limn→∞ e∗ = −γ(z± uv ~i ) obviously, where γ is Euler’s constant. By the continuity of the ∗e∗ s uv multiplication e∗ ~i ∗, we have the convergence in Hol(C2 ) of

n Y uv −1 k1 (z± uv 1 ~i ) . ) ±∗ ∗e∗ lim ∗ 1+ (z ± n→∞ k ~i k=1

Hence we have the convergence in Hol(C2 ) of the infinite product formula Γ∗ (z+

∞ Y 1 −1 k1 (z+ i~ 1 uv 1 uv −γ(z+ uv uv) ~i ) 1+ ) = e∗ ∗ ∗ (z+ uv) ∗e ∗(z+ )−1 ∗ +∗ ~i ~i ∗+ k i~ k=1 (46) −

1

(z+ uv )

1 m ~i (z+ uv to both side of (46) and Fix m∈N. Multiplying (1+ m ~i )e∗ using the abbreviated notation Y Y 1 −1 k1 (a+ i~ 1 1 uv uv) (a+ uv) ∗e 1+ (z=a) = (z+ )−1 ∗ ∗ +∗ ~i +∗ k i~ k6=m

k6=m

we have uv 1 uv − 1 (z+ uv ~i ) (1+ (z+ ))∗e∗ m ∗Γ∗ (z+ ) m i~ ~i Q 1 (z=z) k6=m z6∈ − (N+m+ 12 ) (47) = Q 1 1 1 n n z= − (n+m+ 2 ) k6=m (z=−n−m− 2 )∗ 1− n! ( i~ u) ∗$00 ∗v

−1 1 1 where n∈N. As opposited to the case that (1+ m (z+ uv i~ ))+∗ ∗ sin∗ π(z+ i~ uv) is entire function (cf. Proposition 5.5), there are removable singularities with respect to z.

26

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Multiplying (47), we have

− k1 (z+ uv ~i ) 1+ k1 (z+ uv to both sides of (46) and using ~i ) e∗

Q∞

k=1

N Y − 1 (z+ i~1 uv) 1 1 1 ∗Γ∗ (z+ uv) ∗ 1+ (z+ uv) ∗e∗ k N →∞ k i~ i~ k=1 z6∈ − (N+ 21 ) Pn 1 11 k = k 1− k=0 k! ( i~ u) ∗$00 ∗v , z= − (n+ 12 ),

lim

in Hol(C2 ), where n∈N.

1 uv) 7 Products with sin∗ π(z+ i~ 1 1 In this section we show that sin∗ π(z+ i~ uv)∗Γ∗ (z+ i~ uv) is well defined as an entire function of z. By recalling Euler’s reflection formula, this product may 1 be understood as Γ (1−(z+ . We define the product by the integral 1 uv)) ∗

2i sin∗ π(z+

i~

1 1 uv)∗Γ∗ (z+ uv) i~ i~ Z T0 1 1 τ τ (z+ uv ) uv) −πi(z+ i~ uv) πi(z+ i~ i~ −e∗ )∗e−e e∗ dτ = lim (e∗ 0 T,T →∞ −T Z ∞ τ (τ +πi)(z+ uv (τ −πi)(z+ uv i~ ) i~ ) = e−e (e∗ −e∗ )dτ. −∞

(48) The κ-ordered expression of (48) is given as follows: :(48):κ = By using e−e

Z

τ −πi

∞+πi

e−e

τ −πi

τ (z+ uv i~ )

e∗

dτ −

−∞+πi

=e−e

τ +πi

(

Z

Z

∞−πi

e−e

τ +πi

τ (z+ uv i~ )

e∗

dτ.

−∞−πi

, this is given by the integral

∞+πi

− −∞+πi

Z

∞−πi

τ

τ (z+ uv i~ )

)ee e∗

dτ.

−∞−πi

Note this is not a contour integral, but is defined for κ∈C−{κ≥1}∪{κ≤−1}. The following is our main result: 1 1 Theorem 7.1 sin∗ π(z+ i~ uv)∗Γ∗ (z+ i~ uv) is defined as an entire function 1 of z, vanishing at z∈N+ 2 in any κ-ordered expression such that Re κ<0, and κ∈C−{κ≥1}∪{κ≤−1}.

After careful arguments about associativity, (48) can be expressed as an infinite product

1 7 Products with sin∗ π(z+ i~ uv)

27

∞

sin∗ π(z+

Y 1 1 1 1 uv (z+ uv ~i ) ∗ 1− (z+ ) ∗ ∗e∗k . uv)∗Γ∗ (z+ uv)= i~ i~ k ~i

(49)

k=1

Recalling the reflection formula, we define 1 1 1 1 (1−(z+ uv))= sin∗ π(z+ uv)∗Γ∗ (z+ uv). Γ∗ i~ i~ i~ By this we see that

1 1 (1−(z+ uv)) 1 = 0. Γ∗ i~ z= 2

1 This supports the interpretation that ( 12 + i~ uv)) is an indeterminate living in the set of positive integers N={1, 2, 3, · · · }. 1 1 (z+ i~ uv) −n

1 1 We can form the product Γ1 ∗ (1−(z+ i~ uv))∗(1− n1 (z+ i~ uv))−1 −∗ ∗e∗ At first glance, this looks like

Y

k6=n

1 1 1 1 (z+ i~ uv) ∗(1− (z+ uv))∗e∗n k i~

and hence as an entire function with respect to z. 1 1 However, note that (1− n1 (z+ i~ uv))−1 −∗ is singular at n−z ∈ −N− 2 , i.e. 1 z∈k+ 2 for k ≥ −n, and the same calculation as in (47) shows that sin∗ π(z+

1 1 1 1 uv)∗ Γ∗ (z+ uv)∗(1− (z+ uv))−1 −∗ i~ i~ n i~

is not defined as an entire function, since some matrix elements appear in the formula as removable singularities. 7.1 Additional support for the discrete interpretation 1 We give another formula to support the discrete interpretation for ~i uv. Recall Hankel’s formula Z 1 1 et t−s dt, (cf. [19] p. 244) = Γ(s) 2πi C

where C is taken to be a line from −∞ to −δ, then a circle of radius δ in the positive direction, and finally a line from −δ to −∞. 1 R u∗v 1 1 1 Setting s= 2 − i~ uv=− i~ u∗v, we want to prove C et t∗~i dt = 0 as additional support for the discrete interpretation. f

By setting t = eτ +πi , it is easy to see that the Weyl ordering of this integral is equal to

.

28

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

:

Z

0

(

1

et t∗~i

uv)

dt:0 =:

−∞

=

Z

Z

∞

ee

−∞ ∞

ee

τ +πi

τ +πi

−∞

1 (τ +πi)(1+ ~i uv)

e∗

dτ :0

1 eτ +πi e ~i uv tanh(τ +πi) dτ. cosh(τ +πi)

Hence the integral :

Z

0

1

et t∗~i

uv

−∞

dt:0 =

Z

∞

e−e

−∞

τ

eτ 1 e ~i uv tanh(τ ) dτ cosh(τ )

exists and our integral vanishes for the axis part of C. Thus, setting t = eτ eiθ , we consider for a fixed real τ << 0 Z 2π 1 τ iθ 1 uv uv) (τ +iθ)( ~i :0 dθ. :ee e +(τ +iθ) e∗ : A ∗ ( ) :0 = ~i 2π 0 This can be written as Z 2π τ iθ 1 eτ +iθ 1 ee e e ~i uv tanh(τ +iθ) dθ. 2π 0 cosh(τ +iθ) We easily see the following 2 R 2π eτ eiθ +(τ +iθ) (τ +iθ)( ~i uv) 1 Lemma 7.1 limτ →−∞ 2π dθ = 0. e∗ 0 e Lemma 7.1 suggests that we write Γ1∗ (z+ u∗v ~i ) z=0 = 0, although this is not rigorous. 1 t(z+ i~ uv)

7.2 The residue of e∗

We first use the Weyl ordering. The general κ-ordered expression is obtained via the intertwiner. Lemma 7.2 Let Ck be a small circle of radius π4 with the center at ζ = 1 R ζ(z+ i~ 2uv) 1 iπ(k+ 12 ). Then the contour integral 2πi :0 dζ gives the residue Ck :e∗ 1

of et t∗~i

uv

1 2uv). and this is an entire function of X = (z, i~

1 The continuity of the multiplication (z+ i~ 2uv)∗ requires that this function must satisfy the equation Z 1 ζ(z+ 1 2uv) :e∗ i~ (z+ 2uv)∗0 :0 dζ = 0, (50) i~ Ck

since (50) equals

R

1 d ζ(z+ i~ 2uv) dζ. Ck dζ e∗

For simplicity, we set

1 7 Products with sin∗ π(z+ i~ uv)

w=

1 2uv, ~

Φk (z, w) =

1 2πi

Z

1 2uv) ζ(z+ i~

:e∗

29

:0 dζ.

Ck

Equation (50) is (iz+w)∗0 Φk (z, w) = 0. Hence by the Moyal product formula, Φk (z, w) must satisfy the equation (iz+w)f (x)+f (w)0 +wf (w)00 = 0,

(51)

independent of k. It is not difficult to see that equation (51) has the unique holomorphic solution f with initial condition f (0) = 1. For f (w) = eaw g(bw), (51) can be rewritten as b2 wg 00 (bw)+(2abw+b)g 0 (bw)+((a2 +1)w+a+iz)g(bw) = 0. Thus g(w) must satisfy the equation wg 00 (w)+(1+

a2 +1 a+iz 2a w)g 0 (w)+( 2 w+ )g(w) = 0. b b b

(52)

Setting a = − 12 b = ±i, we have a Laguerre equation 1 wg 00 (w)+(1−w)g 0 (w)+ (∓z−1)g(w) = 0, 2

(53)

where solution is known to be an entire function of exponential growth with respect to w. Equation (53) gives two expressions for the solutions of (51) using the (0) Laguerre functions Lν (2iw): (0)

Ψz (w) = e−iw L 1 (z−1) (2iw), 2

where L(0) ν (w) =

(0)

Ψz (w) = eiw L− 1 (z+1) (−2iw), 2

∞ X (−ν)n n w , (n!)2 n=0

ν=

1 (∓z−1). 2

Here we use the notation (a)n = a(a+1) · · · (a+n−1),

(a)0 = 1.

By this observation, we see that Φk (z, x) = ck Ψz (x), but the constant ck is not fixed by this method. To fix the constant we remark that Ψz (x) is also analytic in the variable z. The constant ck is fixed by investigating the case z = 0. t 1 uv The residue of e∗i~ is obtained in the Weyl ordering by the contour integral Z ∞

1 uv (t−πi) i~

(e∗

1 (t+πi) i~ uv

−e∗

)dt.

−∞

1 uv (t−πi) i~

Since e∗

1 (t+πi) i~ uv

=−e∗

, this is given by (13). Comparing the following 1 t(z+ i~ uv)

lemma, we know the residue of e∗

.

30

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Lemma 7.3 The residue of

1 1 ( i~ tanh ζ)2uv cosh ζ e

at ζ = iπ(k+ 21 ) is

√ 2 (−1)k (−i) 2πJ0 ( uv), ~ where J0 is the Bessel function with the eigenvalue 0.

References 1. G. S. Agawal, E. Wolf, Calculus for functions of noncommuting operators and general phase-space method of functions, Physcal Review D, 2 (1970), 2161-2186. 2. G. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia Math, Appl. 71, Cambridge, 2000. 3. A. Connes, Noncommutative geometry. Academic Press, 1994. 4. I. M. Gel’fand, G. E. Shilov, Generalized Functions, 2 Academic Press, 1968. 5. M. Gerstenhaber, A. Giaquinto, Deformations associated to rigid algebras. preprint, 2005. 6. V. Guillemin, S. Sternberg, Geometric Asymptotics. A.M.S. Mathematical Surveys, 14, 1977. 7. N. Hitchin, Lectures on special Lagrangian submanifolds, Winter School on Mirror Symmetry, Vector bundles and Lagrangian Submanifolds, AMS/IP Stud. Adv. Math. 23 2001, 151-182. 8. M. Kontsevitch Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003), 157-216. 9. H. Omori, One must break symmetry in order to keep associativity, Banach Center Publ. 55 (2002), 153-163. 10. H. Omori, Noncommutative world, and its geometrical picture, A.M.S translation of Sugaku expositions, 2000. 11. H. Omori, Physics in mathematics, (in Japanese) Tokyo Univ. Publ., 2004. 12. H. Omori, Toward geometric quantum theory, Progr. Math. 252 (2006), 213251. 13. H. Omori, T. Kobayashi, Singular star-exponential functions, SUT J. Math. 37 (2001), 137-152. 14. H. Omori, Y. Maeda, Quantum theoretic calculus, (in Japanese) Springer-Verlag, Tokyo, 2004. 15. H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, Deformation quantization of Fr´echet-Poisson algebras, –Convergence of the Moyal product–, in Conf´erence Mosh´e Flato 1999, Quantizations, Deformations, and Symmetries, Vol II, Math. Phys. Stud. 22, (2000), 233-246. 16. H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, Strange phenomena related to ordering problems in quantizations, J. Lie Theory , 13, ( 2003), 481-510. 17. H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, Star exponential functions as two-valued elements, in The breadth of symplectic and Poisson Geometry, Progr. Math. 232 (2004), 483-492. 18. H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, Geometric objects in an approach to quantum geometry, Progr. Math. 252 (2006), 303-324. 19. E. T. Whittaker, G. N. Watson, A course of modern analysis. Cambridge Press 1940.

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds Florent Schaffhauser Department of Mathematics, Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Keywords:Momentum maps, quasi-hamiltonian spaces, moduli spaces Abstract The main result of this paper is Theorem 2.12 which says that the quotient µ−1 ({1})/U associated to a quasi-hamiltonian space (M, ω, µ : M → U ) has a symplectic structure even when 1 is not a regular value of the momentum map µ. Namely, it is a disjoint union of symplectic manifolds of possibly different dimensions, which generalizes the result of Alekseev, Malkin and Meinrenken in [AMM98]. We illustrate this theorem with the example of representation spaces of surface groups. As an intermediary step, we give a new class of examples of quasihamiltonian spaces: the isotropy submanifold MK whose points are the points of M with isotropy group K ⊂ U . The notion of quasi-hamiltonian space was introduced by Alekseev, Malkin and Meinrenken in their paper [AMM98]. The main motivation for it was the existence, under some regularity assumptions, of a symplectic structure on the associated quasi-hamiltonian quotient. Throughout their paper, the analogy with usual hamiltonian spaces is often used as a guiding principle, replacing Lie-algebra-valued momentum maps with Lie-group-valued momentum maps. In the hamiltonian setting, when the usual regularity assumptions on the group action or the momentum map are dropped, Lerman and Sjamaar showed in [LS91] that the quotient associated to a hamiltonian space carries a stratified symplectic structure. In particular, this quotient space is a disjoint union of symplectic manifolds.. In this paper, we prove an analogous result for quasi-hamiltonian quotients. More precisely, we show that for any quasi-hamiltonian space (M, ω, µ : M → U ), the associated quotient M//U := µ−1 ({1})/U is a disjoint union of symplectic manifolds (Theorem 2.12): G µ−1 ({1})/U = (µ−1 ({1}) ∩ MKj )/LKj . j∈J

Here Kj denotes a closed subgroup of U and MKj denotes the isotropy submanifold of type Kj : MKj = {x ∈ M | Ux = Kj }. Finally, LKj is the quotient 31

32

Florent Schaffhauser

group LKj = N (Kj )/Kj , where N (Kj ) is the normalizer of Kj in U . As an intermediary step in our study, we show that MKj is a quasi-hamiltonian space when endowed with the (free) action of LKj . Acknowledgements. This paper was written during my post-doctoral stay at Keio University, which was made possible thanks to the support of the Japanese Society for Promotion of Science (JSPS). I would like to thank the referee for comments and suggestions to improve the paper.

1 Quasi-hamiltonian spaces 1.1 Definition Throughout this paper, we shall designate by U a compact connected Lie group whose Lie algebra u = Lie(U ) = T1 U is equipped with an Ad-invariant positive definite product denoted by (. | .). We denote by χ (half) the Cartan 3-form of U , that is, the left-invariant 3-form on U defined on u = T1 U by: χ1 (X, Y, Z) :=

1 1 (X | [Y, Z]) = ([X, Y ] | Z). 2 2

Recall that, since (. | .) is Ad-invariant, χ is also right-invariant and that it is a closed form. Further, let us denote by θ L and θR the respectively left-invariant and right-invariant Maurer-Cartan 1-forms on U : they take value in u and are the identity on u, meaning that for any u ∈ U and any ξ ∈ Tu U , θuL (ξ) = u−1 .ξ

and θuR (ξ) = ξ.u−1

(where we denote by a point . the effect of translations on tangent vectors). Finally, we denote by M a manifold on which the group U acts, and by X # the fundamental vector field on M defined, for any X ∈ u, by the action of U in the following way: d |t=0 exp(tX).x Xx# := dt for any x ∈ M . We then recall the definition of a quasi-hamiltonian space, which was first introduced in [AMM98]. Definition 1.1 (Quasi-hamiltonian space, [AMM98]) Let (M, ω) be a manifold endowed with a 2-form ω and an action of the Lie group (U, (. | .)) leaving the 2-form ω invariant. Let µ : M → U be a U -equivariant map (for the conjugacy action of U on itself ). Then (M, ω, µ : M → U ) is said to be a quasi-hamiltonian space with respect to the action of U if the map µ : M → U satisfies the following three conditions:

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

33

(i) dω = −µ∗ χ (ii) for all x ∈ M , ker ωx = {Xx# : X ∈ u | (Ad µ(x) + Id).X = 0} (iii) for all X ∈ u, ιX # ω = 21 µ∗ (θL + θR | X) where (θL + θR | X) is the real-valued 1-form defined on U for any X ∈ u by (θL + θR | X)u (ξ) := (θuL (ξ) + θuR (ξ) | X) (where u ∈ U and ξ ∈ Tu U ). In analogy with the usual hamiltonian case, the map µ is called the momentum map. 1.2 Examples In this subsection, we recall the fundamental examples of quasi-hamiltonian spaces. We will use them in section 3 to illustrate Theorem 2.12. Proposition 1.2 ([AMM98]) Let C ⊂ U be a conjugacy class of a Lie group (U, (. | .)). The tangent space to C at u ∈ C is Tu C = {X.u − u.X : X ∈ u}. The 2-form ω on C given at u ∈ C by ωu (X.u − u.X, Y.u − u.Y ) =

1 (Ad u.X | Y ) − (Ad u.Y | X) 2

is well-defined and makes C a quasi-hamiltonian space for the conjugacy action with momentum map the inclusion µ : C ,→ U . Such a 2-form is actually unique. The following theorem explains how to construct a new quasi-hamiltonian U -space out of two existing quasi-hamiltonian U -spaces. Theorem 1.3 (Fusion product of quasi-hamiltonian spaces, [AMM98]) Let (M1 , ω1 , µ1 ) and (M2 , ω2 , µ2 ) be two quasi-hamiltonian U spaces. Endow M1 × M2 with the diagonal action of U . Then the 2-form 1 ω := (ω1 ⊕ ω2 ) + (µ∗1 θL ∧ µ∗2 θR ) 2 makes M1 × M2 a quasi-hamiltonian space with momentum map: µ1 · µ2 : M1 × M2 −→ U (x1 , x2 ) 7−→ µ1 (x1 )µ2 (x2 ) Corollary 1.4 The product C1 × · · · × Cl of l conjugacy classes of U is a quasi-hamiltonian space for the diagonal action of U , with momentum map the product µ(u1 , . . . , ul ) = u1 . . . ul . Proposition 1.5 ([AMM98]) The manifold D(U ) := U × U equipped with the diagonal conjugacy action of U , the U -invariant 2-form ω=

1 ∗ L 1 1 (α θ ∧ β ∗ θR ) + (α∗ θR ∧ β ∗ θL ) + (α · β)∗ θL ∧ (α−1 · β −1 )∗ θR 2 2 2

and the equivariant momentum map

34

Florent Schaffhauser

µ : D(U ) = U × U −→ U (a, b) 7−→ aba−1 b−1 (where α and β are the projections respectively on the first and second factors of D(U )) is a quasi-hamiltonian U -space, called the internally fused double of U . Corollary 1.6 The product manifold Mg,l := (U × U ) × · · · × (U × U ) × C1 × · · · × Cl {z } | g times

equipped with the diagonal U -action and the momentum map µg,l : (U × U ) × · · · × (U × U ) × C1 × · · · × Cl −→ U (a1 , b1 , . . . , ag , bg , u1 , . . . , ul ) 7−→ [a1 , b1 ]. . .[ag , bg ]u1 . . .ul is a quasi-hamiltonian space. This space plays a very important role in the description of symplectic structures on representation spaces of fundamental groups of Riemann surfaces (see [AMM98] and section 3 below). 1.3 Properties of quasi-hamiltonian spaces We now give the properties of quasi-hamiltonian spaces that we shall need when considering the reduction theory of quasi-hamiltonian spaces. The results in the Proposition below are quasi-hamiltonian analogues of classical lemmas entering the reduction theory for usual hamiltonian spaces. Proposition 1.7 ([AMM98]) Let (M, ω, µ : M → U ) be a quasi-hamiltonian U -space and let x ∈ M . Then: (i) The map Λx : ker(Ad µ(x) + Id) −→ ker ωx X 7−→ Xx# =

d dt |t=0

exp(tX).x

is an isomorphism. (ii) ker Tx µ ∩ ker ωx = {0} (iii) The left translation U −→ U −1 u 7−→ µ(x) u induces an isomorphism Im Tx µ ' u⊥ x where ux = {X ∈ u | Xx# = 0} is the Lie algebra of the stabilizer Ux of x and u⊥ x denotes its orthogonal with respect to (. | .). Equivalently, ∗ R ⊥ Im (µ∗ θL )x = u⊥ x (and likewise, Im (µ θ )x = ux ).

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

35

(iv) (ker Tx µ)⊥ω = {Xx# : X ∈ u}, where (ker Tx µ)⊥ω ⊂ Tx M denotes the subspace of Tx M orthogonal to ker Tx µ with respect to ωx . We end this subsection with a result that we will need in subsection 2.2. This theorem relates quasi-hamiltonian spaces to usual hamiltonian spaces and we quote it from [AMM98] (see remark 3.3, see also [HJS06]). Theorem 1.8 (Linearization of quasi-hamiltonian spaces, [AMM98]) Let (M0 , ω0 , µ0 : M0 → U ) be a quasi-hamiltonian U -space. Suppose there exists an Ad-stable open subset D ⊂ u such that exp |D : D → exp(D) is a diffeomorphism onto a open subset of U containing µ0 (M0 ). Denote by exp−1 : exp(D) → D the inverse of exp |D . Then, there exists a symplectic 2-form ω f0 on M0 such that (M0 , ω f0 , µ f0 := exp−1 ◦µ0 : M0 → u) is a hamiltonian U -space in the usual sense, for the same U -action. Furthermore, one has: µ−1 f0 −1 ({0}) 0 ({1U }) = µ and therefore f0 µ−1 0 ({1U })/U = µ

−1

({0})/U

2 Reduction theory of quasi-hamiltonian spaces In this section. we show that for any quasi-hamiltonian space (M, ω, µ : M → U ), the associated quotient M red := µ−1 ({1})/U is a disjoint union of symplectic manifolds. We begin by reviewing the usual hamiltonian case and the reduction theorem of Alekseev, Malkin and Meinrenken for quasi-hamiltonian spaces (Theorem 2.2). We then begin our study of the stratified case and prove the main result of this paper (Theorem 2.12). We also prove that isotropy submanifolds are always quasi-hamiltonian spaces (Theorem 2.5). 2.1 Symplectic reduction in the usual hamiltonian setting In this subsection, we recall how to obtain a symplectic manifold from a usual hamiltonian space by a reduction procedure, that is to say, by taking the quotient of a fiber µ−1 ({u}) of the momentum map by the action of the stabilizer group Uu , which preserves the fiber µ−1 ({u}) since µ is equivariant. This reduction procedure is usually called the Marsden-Meyer-Weinstein procedure. Let us first recall how to obtain differential forms on an orbit space N/G where N is a manifold acted on by a Lie group G. We will assume that G is compact and that it acts freely on N so that N/G is a manifold and the submersion p : N → N/G is a locally trivial principal fibration with structural group G. Let [x] denote the G-orbit of x ∈ N . Since p is surjective, one has T[x] (N/G) = Im Tx p ' Tx N/ ker Tx p. And ker Tx p consists exactly of the vectors tangent to N at x which are actually tangent to the G-orbit of x in N . Those are exactly the values at x of fundamental vector fields:

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ker Tx p = Tx (G.x) = {Xx# : X ∈ g = Lie(G)}. Let then α be a differential form on N (say, a 2-form). Under what conditions does α define a 2-form α on N/G verifying p∗ α = α ? This last condition amounts to saying that α[x] ([v], [w]) = αx (v, w) for all x ∈ N and all v, w ∈ Tx N . One then checks that the left-hand side term of this equation is welldefined by this relation if and only if the 2-form α is G-invariant. Further, since Xx# is sent to 0 in T[x] (N/G) by the map Tx p, the relation p∗ α = α implies that ιX # α = 0 for all X ∈ g. These two conditions turn out to be enough: Lemma 2.1 Let p : N → B = N/G be a locally trivial principal fibration with structural group G and let α be a differential form on N . If α satisfies g ∗ α = α for all g ∈ G

(G−invariance)

and ιX # α = 0 for all X ∈ g = Lie(G) then there exists a unique differential form α on B satisfying p∗ α = α. In such a case, the differential form α on N is said to be basic. Observe that if G is compact and connected (so that the exponential map is surjective), the condition g ∗ α = α for all g ∈ G may be replaced by LX # α = 0 for all X ∈ g (which is always implied by the G-invariance). Further, observe that if α is basic then dα is also basic (the first condition is obvious and the second follows from the Cartan homotopy formula: ιX # (dα) = LX # α − d(ιX # α)). We can now use this result to construct differential forms on orbit spaces associated to level manifolds of the momentum map. Let (M, ω) be a symplectic manifold endowed with a hamiltonian action of a compact connected Lie group U with momentum map µ : M → u∗ , and take N := µ−1 ({ζ}) where ζ ∈ u∗ . Because of the equivariance of µ, the stabilizer G := Uζ of ζ for the co-adjoint action of U on u∗ acts on N = µ−1 ({ζ}). Assuming that Uζ (which is compact) acts freely on µ−1 ({ζ}), one has that ζ is a regular value of µ (see the proof of Theorem 2.2 for similar reasoning) and we then have a principal fibre bundle p : µ−1 ({ζ}) → µ−1 ({ζ})/Uζ and the following diagram: µ−1 ({ζ})

i

/M

p

µ−1 ({ζ})/Uζ where i : µ−1 ({ζ}) ,→ M is the inclusion map. The 2-form ω on M induces a 2-form i∗ ω on µ−1 ({ζ}), which turns out to be basic (again, see the proof of Theorem 2.2 for similar reasoning). Therefore, by Lemma 2.1,

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

37

there exists a unique 2-form ω red on µ−1 ({ζ})/Uζ such that p∗ ω red = i∗ ω. Since ω is closed, so is ω red . And one may then notice that a vector red v ∈ Tx N = ker Tx µ is sent by Tx p to a vector in ker ω[x] if and only if v ⊥ω ⊥ω # is contained in (Tx N ) = (ker Tx µ) = {Xx : X ∈ u} as well. But then v = Xx# ∈ ker Tx µ ∩ (ker Tx µ)⊥ω , so that by the equivariance of µ, one has, denoting by X † the fundamental vector field on u∗ associated to X by the co† adjoint action of U : Xζ† = Xµ(x) = Tx µ.Xx# = 0, so that X ∈ uζ = Lie(Uζ ). red We have thus proved that Tx p.v ∈ ker ω[x] if and only if v ∈ {Xx# : X ∈ uζ }. Consequently, for such a v, one has Tx p.v = 0, so that ω red is non-degenerate and µ−1 ({ζ})/Uζ is a symplectic manifold. When ζ = 0 ∈ u∗ , Uζ = U and one usually denotes µ−1 ({0})/U by M//U . This manifold is called the symplectic quotient of M by U . Observe that in this case µ−1 ({0}) is a co-isotropic submanifold of M , since, if µ(x) = 0, then for all X ∈ u, Tx µ.Xx# = X0† = 0, so that (ker Tx µ)⊥ω = Tx (U.x) ⊂ ker Tx µ. And the 2-form ω red is then symplectic because the leaves of the null-foliation of ω|N (that is, the foliation corresponding to the distribution x 7→ ker(ω|N )x = (Tx N )⊥ω = (ker Tx µ)⊥ω ) are precisely the U -orbits. In [LS91], the authors study the case where regularity assumptions (such as assuming the action of U on µ−1 ({0}) to be free, or the weaker assumption that 0 is a regular value of µ) are dropped. More precisely, Lerman and Sjamaar showed that when the above regularity assumptions are dropped, the reduced space M//U is a union of symplectic manifolds which are the strata of a stratified space. Their proof relies on the Guillemin-Marle-Sternberg normal form for the momentum map. See subsection 2.3 for further comments. 2.2 The smooth case Let us now come back to the quasi-hamiltonian setting. In [AMM98], Alekseev, Malkin and Meinrenken showed how to construct new quasi-hamiltonian spaces from a given quasi-hamiltonian U -space (M, ω, µ : M → U ) by a reduction procedure, assuming that U is a product group U = U1 × U2 (so that µ has two components µ = (µ1 , µ2 )). Their result says that the reduced space µ−1 1 ({u})/(U1 )u is a quasi-hamiltonian U2 -space. As a special case, when U2 = {1}, they obtain a symplectic manifold. Since this is the case we are interested in, we will state their result in this way and give a proof that is valid in this particular situation. We refer to [AMM98] for the general case. It is quite remarkable that one can obtain symplectic manifolds from quasihamiltonian spaces by a reduction procedure. As a matter of fact, this is one of the nicest features of the notion of quasi-hamiltonian spaces: it enables one to obtain symplectic structures on quotient spaces (typically, moduli spaces) using simple finite dimensional objects as a total space. The most important example in that respect is the moduli space of flat connections on a Riemann surface Σ, first obtained (in the case of a compact surface) by Atiyah and

38

Florent Schaffhauser

Bott in [AB83] by symplectic reduction of an infinite-dimensional symplectic manifold. We refer to [AMM98] and to section 3 below to see how one can recover these symplectic structures using quasi-hamiltonian spaces. Let us now state and prove the result we are interested in. Theorem 2.2 (Symplectic reduction of quasi-hamiltonian spaces, the smooth case, [AMM98]) Let (M, ω, µ : M → U ) be a quasihamiltonian U -space. Assume that U acts freely on µ−1 ({1}). Then 1 is a regular value of µ. Further, let i : µ−1 ({1}) ,→ M be the inclusion of the level manifold µ−1 ({1}) in M and let p : µ−1 ({1}) → µ−1 ({1})/U be the projection on the orbit space. Then there exists a unique 2-form ω red on the reduced manifold M red := µ−1 ({1})/U such that p∗ ω red = i∗ ω on µ−1 ({1}) and this 2-form ω red is symplectic. We call this case the smooth case because in this case the quotient is a smooth manifold. We see from the statement of the theorem that this case arises when the action of U on µ−1 ({1}) is a free action. Proof. Take x ∈ µ−1 ({1}). Then, by Proposition 1.7, one has Im Tx µ = u⊥ x. Since the action of U on µ−1 ({1}) is free, one has ux = 0 and therefore Im Tx µ = u. Consequently, 1 ∈ U is a regular value of µ and µ−1 ({1}) is a submanifold of M . The end of the proof consists in showing that i∗ ω is basic with respect to the principal fibration p and then verifying that the unique 2-form ω red on µ−1 ({1})/U such that p∗ ω red = i∗ ω is indeed symplectic. Let us first show that i∗ ω is basic: u∗ (i∗ ω) = i∗ ω for all u ∈ U and ιX # i∗ ω = 0 for all X ∈ u The first condition is obvious since ω is U -invariant. Consider now X ∈ u. Then: ιX # (i∗ ω) = i∗ (ιX # ω) 1 = i∗ µ∗ (θL + θR | X) 2 1 = i∗ ◦ µ∗ (θL + θR | X) 2 1 = (µ ◦ i)∗ (θL + θR | X) 2 =0 since µ◦i is constant on µ−1 ({1}) and therefore T (µ◦i) = 0, hence (µ◦i)∗ = 0. Then there exists, by Lemma 2.1, a unique 2-form ω red on µ−1 ({1})/U such that p∗ ω red = i∗ ω. Let us now prove that ω red is a symplectic form. First:

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

39

p∗ (dω red ) = d(p∗ ω red ) = d(i∗ ω) = i∗ (dω) = i∗ (−µ∗ χ) = −(µ ◦ i)∗ χ | {z } =0

=0 so that dω red = 0. Second, take [x] ∈ µ−1 ({1})/U , where x ∈ µ−1 ({1}), red and [v] ∈ ker ω[x] , where v ∈ Tx µ−1 ({1}) = ker Tx µ. Then, for all w ∈ −1 Tx µ ({1}) = ker Tx µ, one has: red (i∗ ω)x (v, w) = (p∗ ω red )x (v, w) = ω[x] ([v], [w]) = 0 red since [v] ∈ ker ω[x] . Hence:

v ∈ ker(i∗ ω)x = {s ∈ ker Tx µ | ∀ w ∈ ker Tx µ, ωx (s, w) = 0} = ker Tx µ ∩ (ker Tx µ)⊥ω ⊂ Tx M But, by Proposition 1.7, (ker Tx µ)⊥ω = {Xx# : X ∈ u}, so v = Xx# for some X ∈ u. Hence: [v] = Tx p.v = Tx p.Xx# = 0 so that ω red is non-degenerate. 2.3 The stratified case What happens if we now drop the regularity assumptions of Theorem 2.2? First one may observe that if instead of assuming the action of U on µ−1 ({1}) to be free one assumes that 1 is a regular value of µ, then one still has ux = (Im Tx µ)⊥ = {0} so that the stabilizer Ux of any x ∈ µ−1 ({1}) is a discrete, hence finite (since U is compact), subgroup of U . Consequently, µ−1 ({1})/U is a symplectic orbifold (this is the point of view adopted in [AMM98]). Following the techniques used in [LS91] for usual hamiltonian spaces, we will show that if we do not assume that U acts freely on µ−1 ({1}) nor that 1 is a regular value of µ : M → U then the orbit space µ−1 ({1})/U is a disjoint union, over 0 red ) : subgroups K ⊂ U , of symplectic manifolds (NK G 0 red µ−1 ({1})/U = (NK ) K⊂U 0 red each (NK ) being obtained by applying Theorem 2.2 to a quasi-hamiltonian 0 0 0 , ωK , µc space (NK K : NK → LK ). Actually, the study conducted in [LS91] is far more precise and ensures that the reduced space M red := µ−1 ({1})/U is a stratified space M red = ∪K⊂U SK (in particular, there is a notion of smooth function on M red , and the set C ∞ (M red ) of smooth functions is an algebra over the field R), with strata (SK )K⊂U , such that:

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Florent Schaffhauser

- each stratum SK is a symplectic manifold (in particular C ∞ (SK ) is a Poisson algebra). - C ∞ (M red ) is a Poisson algebra. - the restriction maps C ∞ (M red ) → C ∞ (SK ) are Poisson maps. A stratified space satisfying these additional three conditions is called a stratified symplectic space. In [LS91], to show that M red is always a stratified symplectic space, Lerman and Sjamaar actually obtain this space as a disjoint union of symplectic manifolds in two differents ways. The first one enhances the stratified structure of M red (the stratification being induced by the partition of M according to orbit types for the action of U ), and relies on the Guillemin-Marle-Sternberg normal form for the momentum map. It also shows that each stratum carries a symplectic structure. The second description of M red as a disjoint union of symplectic manifolds then aims at relating this reduction to the regular Marsden-Meyer-Weinstein procedure: the symplectic structure on each stratum is obtained by symplectic reduction from a submanifold of M endowed with a free action of a compact Lie group. We also refer to [OR04] for a detailed account on the stratified symplectic structure of symplectic quotients in usual hamiltonian geometry. Here, we shall not be dealing with the notion of stratified space and we will content ourselves with a description of µ−1 ({1})/U as a disjoint union of symplectic manifolds obtained by reduction from a quasi-hamiltonian space 0 NK ⊂ M . We will nonetheless call the case at hand the stratified case. Isotropy submanifolds We start with a quasi-hamiltonian space (M, ω, µ : M → U ) and use the partition of M given by what we may call the isotropy type: G M= MK K⊂U

where K ⊂ U is a closed subgroup of U and MK is the set of points of M whose stabilizer is exactly K: MK = {x ∈ M | Ux = K}. Observe that if one wants K to be the stabilizer of some x ∈ M , one has to assume that K is closed, since a stabilizer always is. If MK is non-empty, it is a submanifold of M (see Proposition [GS84], p.203), called the manifold of symmetry K in [LS91]. As for us, we will follow [OR04] and call MK the isotropy submanifold of type K. The tangent space at some point x ∈ MK consists of all vectors in Tx M which are fixed by K: Tx MK = {v ∈ Tx M | for all k ∈ K, k.v = v}

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

41

where k ∈ K acts on Tx M as the tangent map of the diffeomorphism y ∈ M 7→ k.y which sends x to itself by definition. The action of U does not preserve MK but MK is globally stable under the action of elements n ∈ N (K) ⊂ U , where N (K) denotes the normalizer of K in U : N (K) := {u ∈ U | for all k ∈ K, uku−1 ∈ K}. It is actually the largest subgroup of U leaving MK invariant, since the stabilizer of u.x for some x ∈ MK and some u ∈ U is still Ux if and only if uUx u−1 = Ux , that is, uKu−1 = K. Observe that we have: Lie N (K) ⊂ {X ∈ u | for all Y ∈ k, [X, Y ] ∈ k}. That is, the Lie algebra of the normalizer of K in U is included in the normalizer of n(k) of the Lie algebra k := Lie(K) in u = Lie(U ). The subgroup K is normal in N (K) and acts trivially on MK by definition of the isotropy submanifold of type K, so that MK inherits an action of the quotient group N (K)/K. It actually follows from the definition of MK that this induced action is free: if n ∈ N (K) stabilizes some x in MK , then n ∈ K and so is the identity in N (K)/K. We now wish to show that MK is a quasihamiltonian space with respect to this action. We need to find a momentum map µK : MK → N (K)/K and a 2-form ωK satisfying the axioms of definition 1.1. The natural candidates are µK := µ|MK and ωK := ω|MK , but the problem is that µK does not take its values in N (K)/K. We will now show that µ(MK ) ⊂ N (K) and that we can therefore consider the composed map µc K := pK ◦ µK : MK → N (K)/K, where pK is the projection map pK : N (K) → N (K)/K. Denote then by LK the group LK := N (K)/K. As K is closed in U , so is N (K), and since U is compact, N (K) is compact. Therefore LK = N (K)/K is a compact Lie group. We will then show that (MK , ω|MK , µc K ) is a quasi-hamiltonian space. Moreover, we will show that −1 1 ∈ LK is a regular value of µc c ({1}), so K and that LK acts freely on µ K −1 red ({1})/LK is a symthat, by Theorem 2.2, the reduced space MK := µc K plectic manifold. To do so, we start by studying µ(MK ). This whole analysis adapts the ideas of [LS91] to the quasi-hamiltonian setting. Let us denote ωK := ω|MK and µK := µ|MK . First, for all X ∈ k, we have: ιX # ωK = 21 µ∗K (θL + θR | X)

(1)

(where θL and θR denote as usual the Maurer-Cartan 1-forms of U , so that the above relationship simply follows from the fact that (M, ω, µ : M → U ) is a quasi-hamiltonian space). Second, since K acts trivially on MK , we have, for all x ∈ MK and all k ∈ K: µK (x) = µK (k.x) = kµK (x)k −1

42

Florent Schaffhauser

so that µ(x) belongs to the centralizer of K in U : C(K) := {u ∈ U | for all k ∈ K, uku−1 = k} Since C(K) ⊂ N (K), we have: µ(MK ) ⊂ C(K) ⊂ N (K). We can therefore consider the map µc K := pK ◦ µK : MK → LK = N (K)/K, where pK : N (K) → N (K)/K. Furthermore, we may identify the Lie algebra of LK to Lie(N (K))/k. Under this identification, the Maurer-Cartan L R 1-forms θL and θL of LK are obtained by restricting those of U to N (K) K K (which gives Lie(N (K))-valued 1-forms) and composing by the projection Lie(N (K)) → Lie(N (K))/k. It is then immediate from relation (1), that for all X ∈ Lie(LK ), one has: ιX # ωK =

1 ∗ L R µc K (θLK + θLK | X) 2

(2)

Likewise, the Cartan 3-form χLK of LK is obtained by restricting that of U to N (K) and composing the Lie(N (K))-valued 3-form thus obtained by the projection Lie(N (K)) → Lie(N (K))/k. Then, it follows from the fact that dω = −µ∗ χ that we have: ∗

dωK = −µ∗K χ|N (K) = −µc K χLK

(3)

Thus, we have almost proved that (MK , ωK , µc K ) is a quasi-hamiltonian LK space. In order to compute ker(ωK )x for all x ∈ MK , we observe the following two facts, the first of which is classical in symplectic geometry and the second of which is a quasi-hamiltonian analogue: Lemma 2.3 Let (V, ω) be a symplectic vector space and let K be a compact group acting linearly on V preserving ω. Then the subspace VK := {v ∈ V | for all k ∈ K, k.v = v} of K-fixed vectors in V is a symplectic subspace of V . Proof. Since K is compact, there exists a K-invariant positive definite scalar product on V , that we shall denote by (. | .). Since ω is non-degenerate, there exists, for any v ∈ V , a unique vector Av ∈ V satisfying (v | w) = ω(Av, w) for all w ∈ V , and the map A : V → V thus defined is an automorphism of V . Moreover, it satisfies A(VK ) ⊂ VK . Indeed, if v ∈ VK , then for all k ∈ K, one has, for all w ∈ V :

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

43

ω(k.Av, w) = ω(Av, k −1 .w) = (v | k −1 .w) = (k.v | w) = ω(A(k.v), w) = ω(Av, w) and therefore k.Av = Av for all k ∈ K (incidentally, if one forgets the last equality, which used the fact that k.v = v, this also proves that Ak = kA for all k ∈ K), hence Av ∈ VK . If now v ∈ VK satisfies ω(v, w) = 0 for all w ∈ VK , then in particular for w = Av, one obtains ω(v, Av) = 0, that is, (v | v) = 0, hence v = 0, since (. | .) is positive definite. Lemma 2.4 Let (V, ω) be a vector space endowed with a possibly degenerate antisymmetric bilinear form and let K be a compact group acting linearly on V preserving w. Then the 2-form wK := ω|VK defined on the subspace VK := {v ∈ V | for all k ∈ K, k.v = v} of K-fixed vectors of V has kernel: ker ωK = ker ω ∩ VK Proof. If ω is non-degenerate then this is simply Lemma 2.3. Assume now that ker ω 6= {0}. Observe that ker ωK = VK⊥ω ∩ VK ⊃ ker ω ∩ VK . We now consider the reduced vector space V red := V / ker ω. The 2-form ω induces a 2-form ω red on V red , which is non-degenerate by construction. The map VK ,→ V → V / ker ω induces an inclusion VK /(ker ω ∩ VK ) ,→ V / ker ω. Further, the action of K on V induces an action k.[v] := [k.v] on V red : this action is well-defined because K preserves ω and therefore if r ∈ ker ω then k.r ∈ ker ω. The subspace (V red )K of K-fixed vectors for this action can be identified with VK /(ker ω ∩ VK ). Indeed, if [v] ∈ V red satisfies, for all k ∈ K, [k.v] = [v], then set: Z w :=

(k.v)dλ(k) k∈K

where λ is the Haar measure on the compact Lie group K (such that λ(K) = 1). Then for all k 0 ∈ K: Z k 0 .w = k 0 . (k.v)dλ(k) k∈K Z = (k 0 k.v)dλ(k) k∈K Z = (h.v)dλ(h) h∈K

=w

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Florent Schaffhauser

since the Haar measure on K is invariant by translation. Thus w ∈ VK and we have: hZ i [w] = (k.v)dλ(k) Z k∈K = [k.v]dλ(k) k∈K |{z} =[v]

Z = [v] ×

dλ(k) k∈K

= [v]. Thus [v] ∈ VK /(ker ω ∩VK ) ⊂ V red , which proves that (V red )K ⊂ VK /(ker ω ∩ VK ), and therefore: (V red )K = VK /(ker ω ∩ VK ) (the converse inclusion being obvious). Consequently, since V red is a symplectic space, Lemma 2.3 applies and we obtain: ker ω red |(V red )K = {0}. Now ωK = ω|VK induces a 2-form (ωK )red on VK /(ker ω ∩ VK ) = (V red )K , whose kernel is, by definition: ker(ωK )red = ker ωK /(ker ω ∩ VK ). But, again by definition, (ωK )red = ω red |(V red )K , so that ker(ωK )red = {0}, hence ker ωK = ker ω ∩ VK , which proves the lemma. We then obtain a new class of examples of quasi-hamiltonian spaces: Theorem 2.5 For each closed subgroup K ⊂ U , the compact Lie group L K := N (K)/K acts freely on the isotropy submanifold MK = {x ∈ M | Ux = K}. In addition to that, µ(MK ) ⊂ N (K) and (MK , ωK := ω|MK , µc K := pK ◦ µ|MK ), where pK is the projection map pK : N (K) → N (K)/K = LK , is a quasi-hamiltonian space. Proof. Observe first that µc K is LK equivariant because µ is U -equivariant and pK : N (K) → N (K)/K is a group morphism. Second, recall that we have obtained the relations (2) and (3), so that, to prove that (MK , ωK , µc K : MK → LK ) is a quasi-hamiltonian LK -space, the only thing left to do is compute ker(ωK )x ⊂ Tx MK . Since Tx MK = {x ∈ Tx M | ∀k ∈ K, k.v = v}, Lemma 2.4 applies and one has: ker(ωK )x = ker ωx ∩ Tx MK = {Xx# : X ∈ u | Ad µ(x).X = −X} ∩ Tx MK .

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

45

But a vector of Tx M of the form Xx# lies in Tx MK ⊂ Tx M if and only of X ∈ Lie(N (K)) ⊂ u. Further, we have seen that for all x ∈ MK , µ(X) = µK (x) ∈ N (K). Therefore: ker(ωK )x = {Xx# : X ∈ Lie(N (K)) | Ad µK (x).X = −X}. Since K acts trivially on MK and on N (K)/K, this last statement is equivalent to: ker(ωK )x = {Xx# : X ∈ Lie(N (K))/k | Ad µc K (x).X = −X} which completes the proof. And we then observe that: Corollary 2.6 1 ∈ LK is a regular value of µc K and the reduced space −1 red MK := µc ({1})/L is a symplectic manifold. K K red Proof. Since the action of LK on MK is free, the fact that MK := µc K is a symplectic manifold follows from Theorem 2.2.

−1

({1})/LK

Structure of quasi-hamiltonian quotients We will now use the above analysis to show that, without any regularity assumptions on the action of U on M or on the momentum map µ : M → U , the orbit space M red := µ−1 ({1})/U is a disjoint union of symplectic manifolds. First, in analogy with [LS91], we observe: Lemma 2.7 Denote by (Kj )j∈J a system of representatives of conjugacy classes of closed subgroups of U (every closed subgroup K ⊂ U is conjugate to exactly one of the pairwise non-conjugate Kj ). Denote by MKj the isotropy submanifold of type Kj in the quasi-hamiltonian space (M, ω, µ : M → U ): MKj = {x ∈ M | Ux = Kj }. Then, the orbit space µ−1 ({1U })/U is the disjoint union: G µ−1 ({1U })/U = (µ−1 ({1U }) ∩ U.MKj )/U. j∈J −1

Proof. Take a U -orbit U.x in µ ({1U }). The stabilizer Ux of x is conjugate to one of the (Kj ), that is: Ux = uKj u−1 for some u ∈ U . Therefore, the stabilizer of y := u−1 .x ∈ µ−1 ({1U }) is exactly Kj , and we then have U.y = U.x with y ∈ MKj . Therefore, we have shown: [ µ−1 ({1U })/U = (µ−1 ({1U }) ∩ U.MKj )/U. j∈J

The above union is disjoint because if U.x is a U -orbit in µ−1 ({1U }) ∩ U.MKj , the stabilizer of x is conjugate to Kj and therefore not conjugate to any Kj 0 for j 0 6= j.

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We will now study each one of the sets (µ−1 ({1U })∩U.MKj )/U separately. We will show, in analogy with the result of Lerman and Sjamaar in [LS91], that each one of these sets is a smooth manifold that carries a symplectic structure, and that this symplectic structure may be obtained by reduction from a quasi-hamiltonian space endowed with a free action of a compact Lie group (that is, by applying Theorem 2.2). In [OR04], this procedure is called Sjamaar’s principle. The way this principle is developped in [OR04] is way more general than what we do here: they consider the quotients µ−1 ({ξ})/Uξ for an arbitrary ξ ∈ u∗ , which also makes the situation slightly more complicated (notably to find an equivariant momentum map for the isotropy submanifolds MK ). Here, we we begin by observing the following fact: Lemma 2.8 Let K ⊂ U be a closed subgroup of U and denote by MK the isotropy submanifold of type K in the quasi-hamiltonian space (M, ω, µ : M → U ): MK = {x ∈ M | Ux = K}. Denote by N (K) the normalizer of K in U and by LK the quotient group LK = N (K)/K. Then, the map: fK : (µ−1 ({1U }) ∩ MK )/LK −→ (µ−1 ({1U }) ∩ U.MK )/U LK .x 7−→ U.x sending the LK -orbit of a point x ∈ (µ−1 ({1U }) ∩ MK ) to its U -orbit in (µ−1 ({1U }) ∩ U.MK ) is well-defined and is a bijection: '

(µ−1 ({1U }) ∩ MK )/LK −→ (µ−1 ({1U }) ∩ U.MK )/U Consequently, we deduce from Lemma 2.7 that: G µ−1 ({1U })/U = (µ−1 ({1U }) ∩ MKj )/LKj . j∈J

Proof. The map fK is well-defined because if x, y ∈ µ−1 ({1U }) ∩ MK lie in a same LK -orbit then they lie in a same U -orbit in (µ−1 ({1U }) ∩ U.MK ). The map fK is onto because a U -orbit in (µ−1 ({1U }) ∩ U.MK ) is of the form U.x for some x ∈ (µ−1 ({1U }) ∩ MK ), and fK then sends the LK -orbit of such an x in (µ−1 ({1U }) ∩ MK ) to the U -orbit U.x in (µ−1 ({1U }) ∩ U.MK ). The map fK is one-to-one because if x, y ∈ (µ−1 ({1U })∩MK ) lie in a same U orbit in (µ−1 ({1U }) ∩ U.MK ), say y = u.x for some u ∈ U , then the stabilizer of y in (µ−1 ({1U }) ∩ U.MK ) is Uy = uUx u−1 . But since x, y ∈ MK we have Ux = Uy = K, hence u ∈ N (K) and LK .y = LK .x. The rest of the Proposition follows from Lemma 2.7. We will now prove that each of the sets (µ−1 ({1U })∩U.MK )/U = (µ−1 ({1U })∩ MK )/LK is a smooth, symplectic manifold. To do so, we will show that each

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

47

0 of these sets is the quasi-hamiltonian quotient NK //LK associated to a quasi0 0 0 hamiltonian space of the form (NK , ωK , µc : N K K → LK ) (see Theorem 2.5 and Corollary 2.6). More precisely, we have to show that 0

−1 (µ−1 ({1U }) ∩ MK )/LK = (µc ({1LK })/LK K ) 0

where µc K is the momentum map of a free action of LK on a quasi-hamiltonian 0 0 0 space (NK , ωK , µc K : NK → LK ). This last step is not entirely immediate. In fact, experience from the usual hamiltonian case dealt with by Lerman and Sjamaar in [LS91] shows that in that setting too, one has ∗ to replace (MK , ωK , µc K : MK → Lie(LK ) ) by another hamiltonian LK 0 0 0 ∗ 0 space (MK , ωK , µc K : MK → Lie(LK ) ), that space MK being the union of connected components of MK which have a non-empty intersection with 0 µ−1 ({0}). The point is that this space MK is in a way big enough to study −1 0 the quotient (µ ({0}) ∩ MK )/LK because by definition of MK one has −1 −1 0 (µ ({0}) ∩ MK )/LK = (µ ({0}) ∩ MK )/LK . And then one can prove 0 0 −1 0 red that (µ−1 ({0}) ∩ MK )/LK = µc ({0})/LK = (MK ) (whereas it is K −1 −1 not true that (µ ({0}) ∩ MK )/LK = µc ({0})/L ), thereby proving that K K 0 red (µ−1 ({0}) ∩ MK )/LK = (MK ) is a symplectic manifold. Trying an exactly analogous approach in the quasi-hamiltonian setting does not work: the union of connected components of MK containing points of µ−1 ({1U }) is still too big, and one has to introduce another quasi-hamiltonian LK -space, which we will denote by NK (see Lemma 2.10). This is what we do next (see al so remark 1). We begin with the following lemma: Lemma 2.9 Let B ⊂ u be an Ad-stable open ball centered at 0 ∈ u such that the exponential map exp |B : B → exp(B) is a diffeomorphism onto an open subset of U containing 1U . Denote by N ⊂ M the U -stable open subset of M defined by N := µ−1 (exp(B)). Then (N, ω|N , µ|N : N → U ) is a quasi-hamiltonian U -space, and one has: (µ|N )−1 ({1U })/U = µ−1 ({1U })/U. Proof. Any U -stable open subset of a quasi-hamiltonian space is a quasihamiltonian space when endowed with the restriction of the 2-form and the restriction of the momentum map. In the above case, one has (µ|N )−1 ({1U }) = µ−1 ({1U }) by construction of N = µ−1 (exp(B)). We can then compare the isotropy submanifolds of M and of N : Lemma 2.10 Let (N, ω|N , µ|N : N → U ) be the quasi-hamiltonian U -space introduced in Lemma 2.9. Let K ⊂ U be a closed subgroup of U and denote by MK = {x ∈ M | Ux = K} and NK = {x ∈ N | Ux = K} the isotropy submanifolds of type K of M and N respectively. Then one has: µ−1 ({1U }) ∩ MK = µ−1 ({1U }) ∩ NK .

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Proof. The equality µ−1 ({1U })∩MK = µ−1 ({1U })∩NK follows from the fact that µ−1 ({1U }) ⊂ N by construction of N = µ−1 (exp(B)). We will now show that the orbit space (µ−1 ({1U })∩NK )/LK has a symplectic structure. To do this, we apply Theorem 1.8 to the quasi-hamiltonian space M0 = N = µ−1 (exp(B)) constructed in Lemma 2.9 to obtain the following result: Lemma 2.11 Let (N = µ−1 (exp(B)), ω|N , µ|N : N → U ) be the quasihamiltonian U -space introduced in Lemma 2.9. Let K ⊂ U be a closed subgroup of U and let N (K) be its normalizer in U . Denote by LK the quotient group LK := N (K)/K and by pK the projection pK : N (K) → LK = N (K)/K. Let NK = {x ∈ N | Ux = K} be the istotropy submanifold of type K in N . Recall from Theorem 2.5 that µ(NK ) ⊂ N (K) and that (NK , ω|NK , µc K = pK ◦µ|NK : NK → LK ) is a quasi0 hamitonian LK -space. Denote by NK the union of connected components of 0 NK which have a non-empty intersection with µ−1 ({1U }), and by µc K the 0 0 0 0 0 0 ,µ restriction of µc c K to NK . Then: NK is LK -stable and (NK , ω|NK K : NK → LK ) is a quasi-hamiltonian LK -space. Furthermore, one has: 0

0 −1 µ−1 ({1U }) ∩ NK = µ−1 ({1U }) ∩ NK ({1LK }) = (µc K )

and consequently: 0 (µ−1 ({1U }) ∩ NK )/LK = (µ−1 ({1U }) ∩ NK )/LK 0

−1 0 red = (µc ({1LK })/LK = (NK ) K ) 0 is LK -stable and is a quasi-hamiltonian LK Proof. We first show that NK 0 0 , a point space. If x ∈ NK and n ∈ N (K) then there exists, by definition of NK −1 x0 ∈ µ ({1U }) ∩ NK which is connected to x by a path (xt ) in NK . Then (n.xt ) is a path from (n.x0 ) to (n.x) in NK . Since µ(n.x0 ) = nµ(x0 )n−1 = 1U and (n.x) lies in the same connected component of NK as (n.x0 ), we 0 0 0 0 , ω|NK0 , µc . The fact that (NK have (n.x) ∈ NK K : NK → LK ) is a quasi0 hamiltonian space then follows from the fact that NK is an LK -stable open subset of the quasi-hamiltonian space (NK , ω|NK , µc K : NK → LK ). 0 −1 0 = (µc ({1LK }). Let us now prove that µ−1 ({1U })∩NK = µ−1 ({1U })∩NK K ) 0 0 . Furthermore, , one has µ−1 ({1U })∩NK = µ−1 ({1U })∩NK By definition of NK 0 0 −1 0 0 c ({1LK }) since µc ⊂ (µc it is obvious that µ−1 ({1U })∩NK K |N K K = pK ◦ µ K ) and pK : N (K) → N (K)/K is a group morphism. Let us now prove the converse inclusion. We begin by observing that since the exponential map is invertible on B ⊂ u and N = exp(B), Theorem 1.8 applies: the map µ e := exp−1 ◦µ|N : N → u is a momentum map in the usual sense for the action of U on N and µ−1 ({1U }) = µ e−1 ({0}). In particular, one has, for all x ∈ 0 ⊥ ⊥ 0 0 NK , Im Tx µ e = ux = k and, since 0 ∈ µ e(NK ) by definition of NK , this 0 −1 0 0 implies µ e(NK ) ⊂ k⊥ . Take now x ∈ (µc ) ({1 }) ⊂ N . This means that K LK K

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

49

0 0 µ(x) ∈ (K ∩ µ(NK )) ⊂ exp(B), hence µ e(x) = exp−1 ◦µ(x) ∈ k ∩ µ e(NK ) ⊂ mathf rakk ∩ k⊥ = {0}. Consequently, µ e(x) = 0 and therefore µ(x) = 1U . 0 −1 0 Hence (µc ({1LK }) ⊂ µ−1 ({1U }) ∩ NK , which completes the proof. K )

Remark 1. Lemma 2.11 is crucial in our proof of forthcoming Theorem 2.12. Although our argument is similar to the one in [LS91], where the usual hamil0 tonian case is treated, extra difficulties arise to show that µ−1 ({1U }) ∩ NK = 0 −1 (µc ({1LK }). In particular, we were unable to obtain such a statement K ) 0 0 involving MK or MK instead of NK and NK . In the end this is not a problem −1 0 because we proved that µ ({1U })∩MK = µ−1 ({1U })∩NK = µ−1 ({1U })∩NK 0 (see Lemma 2.10). The point of introducing NK (and then later NK ) is to be able to linearize the quasi-hamiltonian space that we are dealing with without changing the associated quotient. This idea was suggested to us by the reading of [HJS06], where a description of quasi-hamiltonian quotients as disjoint unions of symplectic manifolds is also obtained. The main difference between Theorem 2.12 and Theorem 2.9 in [HJS06] is that in our case the symplectic structure on each component of the union is obtained by reduction from 0 0 0 a quasi-hamiltonian space (NK , ωK , µc K : NK → LK ) endowed with a free action of the compact Lie group LK . The linearization theorem enables us to reduce the case at hand to the usual hamiltonian case and mimic the argument in [LS91] (Theorem 3.5). It would be interesting to know if this detour can be avoided. Theorem 2.12 (Symplectic reduction of quasi-hamiltonian spaces, the stratified case) Let (M, ω, µ : M → U ) be a quasi-hamiltonian U -space. For any closed subgroup K ⊂ U , denote by MK the isotropy manifold of type K in M : MK = {x ∈ M | Ux = K}. Denote by N (K) the normalizer of K in U and by LK the quotient group LK := N (K)/K. Then the orbit space (µ−1 ({1U }) ∩ MK )/LK is a symplectic smooth manifold. Denote by (Kj )j∈J a system of representatives of closed subgroups U . Then the orbit space M red := µ−1 ({1U })/U is the disjoint union of the following symplectic manifolds: G µ−1 ({1U })/U = (µ−1 ({1U }) ∩ MKj )/LKj . j∈J

Proof. By Lemmas 2.10 and 2.11, we have: (µ−1 ({1U }) ∩ MK )/LK = (µ−1 ({1U }) ∩ NK )/LK 0 0 red = (µ−1 ({1U }) ∩ NK )/LK = (NK )

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where the compact group LK acts freely on the quasi-hamiltonian space 0 0 0 −1 (NK , ωK , µc ({1U }) ∩ K : NK → LK ), so that Theorem 2.2 shows that (µ 0 red MK )/LK = (NK ) is a symplectic manifold. By Lemmas 2.7 and 2.8, we then have: G G µ−1 ({1U })/U = (µ−1 ({1U }) ∩ U.MKj )/U = (µ−1 ({1U }) ∩ MKj )/LKj . j∈J

j∈J

Observe that to prove that the set (µ−1 ({1U })∩MK )/LK is a smooth symplec0 0 0 tic manifold, we found a quasi-hamiltonian LK -space (NK , ωK , µc K : NK → 0 red −1 LK ) on which LK acts freely such that (NK ) = (µ ({1U }) ∩ MK )/LK and then applied quasi-hamiltonian reduction in the smooth case (Theorem 0 −1 0 2.2) to NK . One key step in this proof is to show that (µc ({1LK })/LK = K ) −1 0 (µ ({1U }) ∩ NK )/LK and it was to obtain this equality that we used the linearization Theorem 1.8. We then showed that for any quasi-hamiltonian space (M, ω, µ : M → U ) the reduced space M red := µ−1 ({1})/U is a disjoint union of symplectic manifolds. We denote this reduced space by M//U , as in the usual hamiltonian case: Definition 2.13 (Quasi-hamiltonian quotient) The reduced space G M//U := µ−1 ({1U })/U = (µ−1 ({1U }) ∩ MKj )/LKj j∈J

associated, by means of Theorems 2.2 and 2.12, to a given quasi-hamiltonian space (M, ω, µ : M → U ) is called the quasi-hamiltonian quotient associated to M . Remark 2. Observe that when the action of U on M is free, then the only subgroup K ⊂ U such that the isotropy submanifold MK is non-empty is K = {1}, so that the results of Theorems 2.2 and 2.12 do coincide in this case. As we shall see in section 3, representation spaces of surface groups naturally arise as quasi-hamiltonian quotients. Since in this case it is known that representation spaces are stratified symplectic spaces in the sense of [LS91] (see for instance [Hue95]), it should be possible to obtain this stratified symplectic structure in the quasi-hamiltonian framework. Following [LS91], the first step to do so should be a normal form for momentum maps on quasi-hamiltonian spaces.

3 Application to representation spaces of surface groups In this section, we wish to briefly explain, following [AMM98], how the notion of quasi-hamiltonian space provides a proof of the fact that, for any Lie group

Quasi-hamiltonian quotients as disjoint unions of symplectic manifolds

51

(U, (. | .)) endowed with an Ad-invariant non-degenerate product and any collection C = (Cj )1≤j≤l of l conjugacy classes of U , there exists a symplectic structure on the representation spaces HomC (πg,l , U ) U (see 3.1 below for a precise definition of these spaces). This will serve as an example to illustrate Theorem 2.12. Here, πg,l = π1 (Σg,l ) denotes the fundamental group of the surface Σg,l := Σg \{s1 , . . . , sl }, Σg being a compact Riemann surface of genus g ≥ 0, l being an integer l ≥ 1 and s1 , . . . , sl being l pairwise distinct points of Σg . When l = 0, we set C := ∅ and Σg,0 := Σg . Everything we will say is valid for any g ≥ 0 and any l ≥ 0 but we will not always distinguish between the cases l = 0 and l ≥ 1, to lighten the presentation. Recall that the fundamental group of the surface Σg,l = Σg \{s1 , . . . , sl } has the following finite presentation: πg,l =< α1 , . . . , αg , β1 , . . . , βg , γ1 , . . . , γl |

g Y i=1

[αi , βi ]

l Y

γj = 1 >

j=1

each γj being the homotopy class of a loop around the puncture sj . In particular, if l ≥ 1, it is a free group on (2g + l − 1) generators. As a consequence of this presentation, we see that, having chosen a set of generators of πg,l , giving a representation of πg,l in the group U (that is, a group morphism from πg,l to U ) amounts to giving (2g + l) elements (ai , bi , uj )1≤i≤g,1≤j≤l of U satisfying: g Y i=1

[ai , bi ]

l Y

uj = 1.

j=1

Two representations (ai , bi , uj )i,j and (a0i , b0i , u0j )i,j are then called equivalent if there exists an element u ∈ U such that a0i = uai u−1 , b0i = ubi u−1 , u0j = uuj u−1 for all i, j. The original approach to describing symplectic structures on spaces of representations shows that, in order to obtain symplectic structures, one has to prescribe the conjugacy class of each uj , 1 ≤ j ≤ l. Otherwise, one may obtain Poisson structures, but we shall not enter these considerations and refer to [Hue01] and [AKSM02] instead. We are then led to studying the space HomC (πg,l , U ) of representations of πg,l in U with prescribed conjugacy classes for the (uj )1≤j≤l : Definition 3.1 We define the space HomC (πg,l , U ) to be the following set of group morphisms: HomC (πg,l , U ) = {ρ : πg,l → U | ρ(γj ) ∈ Cj for all j ∈ {1, . . . , l}}. Observe that this space may very well be empty, depending on the choice of the conjugacy classes (Cj )1≤j≤l . As a matter of fact, when g = 0, conditions on the

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Florent Schaffhauser

(Cj ) for this set to be non-empty are quite difficult to obtain (see for instance [AW98] for the case U = SU (n)). However, when g ≥ 1 and U is semi-simple, the above set is always non-empty, as shown in [Ho04]. As earlier, giving such a morphism ρ ∈ HomC (πg,l , U ) amounts to giving appropriate elements of U : HomC (πg,l , U ) ' { (a1 , . . . , ag , b1 , . . . , bg , u1 , . . . ul ) ∈ U × · · · × U × C1 × · · · × C l | | {z } 2g times

g Y

[ai , bi ]

i=1

l Y

uj = 1}.

j=1

In particular, two representations (ai , bi , uj )i,j and (a0i , b0i , u0j )i,j are equivalent if and only if they are in a same orbit of the diagonal action of U on U × · · · × U × C1 × · · · × Cl . The representation space RepC (πg,l , U ) is then defined to be the quotient space for this action: RepC (πg,l , U ) := HomC (πg,l , U ) U. Following for instance [Hue95], the idea to obtain a symplectic structure on the representation space, or moduli space, RepC (πg,l , U ) is then to see this quotient as a symplectic quotient, meaning that one wishes to identify HomC (πg,l , U ) with the fibre of a momentum map defined on an extended moduli space (the expression comes from [Jef94, Hue95]). The notion of quasihamiltonian space then arises naturally from the choice of U × · · · × U × C1 × · · · × C l | {z } 2g times

as an extended moduli space, and of the map µg,l (a1 , . . . , ag , b1 , . . . , bg , u1 , . . . , ul ) = [a1 , b1 ]. . .[ag , bg ]u1 . . .ul as U -valued momentum map, so that: RepC (πg,l , U ) = µ−1 g,l ({1})/U. Actually, because of the occurrence of the commutators [ai , bi ], it is more appropriate to re-arrange the arguments of the map µg,l in the following way: µg,l (a1 , b1 , . . . , ag , bg , , u1 , . . . , ul ) = [a1 , b1 ]. . .[ag , bg ]u1 . . .ul = 1 and to write the extended moduli space: (U × U ) · · · × (U × U ) × C1 × · · · × Cl . | {z } g times

In the case where g = 0, one simply has: µ0,l : C1 × · · · × Cl −→ U (u1 , . . . , ul ) 7−→ u1 . . .ul

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When g = 1 and l = 0, one has: µ1,0 : U × U −→ U (a, b) 7−→ aba−1 b−1 These two particular cases correspond to the examples we recalled in Propositions 1.4 and 1.5, and motivate the notion of quasi-hamiltonian space. Thus, in general, the extended moduli space is the following quasi-hamiltonian space: Mg,l := D(U ) × · · · × D(U ) × C1 × · · · × Cl . | {z } g times

(where D(U ) is the internally fused double of U of Proposition 1.5) equipped with the diagonal U -action and the momentum map µg,l : D(U ) × · · · × D(U ) × C1 × · · · × Cl −→ U (a1 , b1 , . . . , ag , bg , u1 , . . . , ul ) 7−→ [a1 , b1 ]. . .[ag , bg ]u1 . . .ul The representation space RepC (πg,l , U ) is then the associated quasi-hamiltonian quotient (see definition 2.13): RepC (πg,l , U ) = Mg,l //U = (D(U ) × · · · × D(U ) × C1 × · · · × Cl )//U. | {z } g times

In particular, in the case of an l-punctured sphere (g = 0), we have: HomC π1 (S 2 \{s1 , . . . , sl }), U U = (C1 × · · · × Cl )//U. We also spell out the case of torus: Hom π1 (T2 ), U

U = D(U )//U

(there are no conjugacy classes necessary here, as the surface T2 is closed) and of the punctured torus: HomC π1 (T2 \{s} , U ) U = (D(U ) × C)//U. We then know from Theorems 2.2 and 2.12 that these representation spaces RepC (πg,l , U ) = Mg,l //U carry a symplectic structure, obtained by reduction from the quasi-hamiltonian space Mg,l . More precisely, the representation spaces RepC (πg,l , U ) are disjoint unions of symplectic manifolds. Observe that one essential ingredient to obtain this symplectic structure was the fact that πg,l admits a finite presentation with a single relation, which was used as a momentum relation.

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References [AB83]

M.F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A, 308(1505):523–615, 1983. [AKSM02] A. Alekseev, Y. Kosmann-Schwarzbach, and E. Meinrenken. QuasiPoisson manifolds. Canad. J. Math., 54(1):3–29, 2002. [AMM98] A. Alekseev, A. Malkin, and E. Meinrenken. Lie group valued moment maps. J. of Differential Geom., 48(3):445–495, 1998. [AW98] S. Agnihotri and C. Woodward. Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett., 5(6):817–836, 1998. [GS84] V. Guillemin and S. Sternberg. Symplectic techniques in physics. Cambridge University Press, 1984. [HJS06] J. Hurtubise, L. Jeffrey, and R. Sjamaar. Group-valued implosion and parabolic structures. Amer. J. Math., 128(1):167–214, 2006. [Ho04] N-K. Ho. The real locus of an involution map on the moduli space of flat connections on a Riemann surface. Int. Math. Res. Not., 61:3263–3285, 2004. [Hue95] J. Huebschmann. Symplectic and Poisson structures of certain moduli spaces. I. Duke Math. J., 80(3):737–756, 1995. [Hue01] J. Huebschmann. On the variation of the Poisson structures of certain moduli spaces. Math. Ann., 319(2):267–310, 2001. [Jef94] L. Jeffrey. Extended moduli spaces of flat connections on Riemann surfaces. Math. Ann., 298(4):667–692, 1994. [LS91] E. Lerman and R. Sjamaar. Stratified symplectic spaces and reduction. Ann. of Math., 134(2):375–422, 1991. [OR04] J.P. Ortega and T. Ratiu. Momentum maps and Hamiltonian reduction. Number 222 in Progress in Mathematics. Birkh¨ auser, 2004.

Representations of gauge transformation groups of higher abelian gerbes Kiyonori Gomi Graduate school of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-Ku, Tokyo, 153-8914 Japan. e-mail: [email protected] Summary. Smooth Deligne cohomology groups can be thought of as gauge transformation groups of “higher abelian gerbes”. For a compact oriented Riemannian manifold of dimension 4k + 1, the smooth Deligne cohomology has certain projective unitary representations. We account for a meaning of these representations in a context of a field theory.

1 Introduction The gauge transformation group of a principal U (1)-bundle over a smooth manifold M is naturally isomorphic to the group C ∞ (M, U (1)) of U (1)-valued smooth functions on M , and is identified with the free loop group LU (1) in the case of M = S 1 . The smooth Deligne cohomology groups ([2, 5, 7]) of M generalize C ∞ (M, U (1)), so that we can think of them as generalizations of LU (1). We notice that the idea of regarding smooth Deligne cohomology groups as generalizations of loop groups is natural from a viewpoint of “higher gerbes”. The notion of gerbes was introduced by Giraud [11]. One can think of gerbes as fiber bundles whose “fibers” are categories. Similarly, one can think of 2-gerbes, due to Breen [1], as fiber bundles whose “fibers” are so called 2categories. At present, formulating general “higher gerbes” rigorously may remain as an issue to be studied. However, a cohomological argument allows us to grasp an outline of “higher abelian gerbes”. By the help of the argument, we can deduce that smooth Deligne cohomology groups realize “gauge transformation groups of higher abelian gerbes”, so that they provide generalizations of ordinary gauge transformation groups as well as loop groups. (A more detailed discussion of this deduction is given in [12].) As is well known, loop groups of compact Lie groups have certain projective representations with nice properties, positive energy representations [19]. As a generalization of positive energy representations of LU (1), the no-

55

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tion of admissible representations of smooth Deligne cohomology groups was introduced and classified in [13]: Theorem 1.1 ([13]) Let k be a non-negative integer, M a compact oriented (4k +1)-dimensional Riemannian manifold (without boundary), and G(M ) the smooth Deligne cohomology group H 2k+1 (M, Z(2k + 1)∞ D ) of M . Then admissible representations of G(M ) (of level 1) have the following properties: (a) An admissible representation is equivalent to a finite direct sum of irreducible admissible representations. (b) The number of the equivalence classes of irreducible admissible representations is 2b r, where b = b2k (M ) = b2k+1 (M ) is the Betti number, and r is the number of elements in the set {t ∈ H 2k+1 (M, Z)| 2t = 0}. In the case of k = 0 and M = S 1 , admissible representations of G(S 1 ) = H (S 1 , Z(1)∞ D ) = LU (1) give rise to positive energy representations of LU (1) of level 2, and vice verse. We can readily see that Theorem 1.1 recovers the classification of positive energy representations of LU (1) of level 2, ([19]). 1

Originally, Theorem 1.1 stemmed from a work generalizing the WessZumino-Witten models. In a formulation of the models, loop groups and their representations play some basic roles ([9, 10] for example). Hence a natural strategy for a generalization is to generalize representation theory of loop groups. As is mentioned, smooth Deligne cohomology groups provide generalizations of the loop group LU (1). Thus, representation theory of smooth Deligne cohomology groups arises as a key subject. The aim of this article is to give a field theory in dimension 4k + 2 in the context according to the motivation above. The theory leads us to study representations of smooth Deligne cohomology groups. We notice that the field theory is a generalization of theory of free scalar fields in two dimensions, rather than the Wess-Zumino-Witten models. So, our theory may better be regarded as a toy model of further non-abelian generalizations. The organization of this article is as follows. In Section 2, we recall the definition of smooth Deligne cohomology and its properties. In Section 3, we regard a smooth Deligne cohomology class as a field, and introduce a field theory in dimension 4k + 2 giving an action functional. Since our field theory is quite similar to the Wess-Zumino-Witten models, we study it along the lines in [9, 10], which motivates us to consider representations of smooth Deligne cohomology groups. Then, in Section 4, we deals with representations of smooth Deligne cohomology groups. A relationship with Henningson’s work is also explained here. At the end of this section, we give an analogy of the space of conformal blocks in the Wess-Zumino-Witten models, and state a result of a computation without proof.

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2 Smooth Deligne cohomology We give here the definition of smooth Deligne cohomology groups, and summarize some basic properties. For the details, we refer the reader to [2, 7, 5]. We also introduce a complexified version of smooth Deligne cohomology groups for later convenience. 2.1 Definition and property Definition 2.1 Let q be a non-negative integer, and X a smooth manifold. We define the smooth Deligne cohomology group H p (X, Z(q)∞ D ) of X to be the qth hypercohomology of the complex of sheaves: d

d

d

0 1 q−1 −→ 0 −→ · · · , Z(q)∞ D : Z −→ A −→ A −→ · · · −→ A

where Z is the constant sheaf located at degree 0 in the complex, and A p is the sheaf of germs of R-valued p-forms on X. We remark that the smooth Deligne cohomology group H q (X, Z(q)∞ D ) is b q−1 (X, R/Z) of differential characters naturally isomorphic to the group H invented by Cheeger and Simons [3]. (See [2, 7] for this fact.) The smooth Deligne cohomology H p (X, Z(0)∞ D ) is isomorphic to the ordinary cohomology H p (X, Z) for p ≥ 0. Lemma 2.2 Let q be a positive integer. ∼ p−1 (X, R/Z). (a) If 0 ≤ p < q, then H p (X, Z(q)∞ D)=H (b) If p = q, then H q (X, Z(q)∞ ) fits into the following exact sequences: D δ

q 0 → H q−1 (X, R/Z) → H q (X, Z(q)∞ D ) → A (X)Z → 0, ι

χ

q 0 → Aq−1 (X)/Aq−1 (X)Z → H q (X, Z(q)∞ D ) → H (X, Z) → 0,

where Aq (X) is the group of q-forms on X, and Aq (X)Z ⊂ Aq (X) is the subgroup of closed integral q-forms. ∼ p (c) If q < p, then H p (X, Z(q)∞ D ) = H (X, Z). By the exact sequence in (b), the smooth Deligne cohomology group H q (X, Z(q)∞ D ) turns out to be infinitely generated in general. For example, in the case of q = 1, we have the natural isomorphism 1 ∼ ∞ H 1 (X, Z(1)∞ D ) = C (X, R/Z). For f : X → R/Z, the image δ(f ) ∈ A (X)Z is expressed as δ(f ) = df by means of the exterior differential. The image χ(f ) ∈ H 1 (X, Z) is expressed as χ(f ) = f ∗ 1, where 1 ∈ H 1 (R/Z, Z) ∼ = Z is the generator. In the case of q = 2, the smooth Deligne cohomology H 2 (X, Z(2)∞ D ) is isomorphic to the group of isomorphism classes of principal U (1)-bundle with connection over X. Under this identification, the homomorphism δ assigns to

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−1 F (A), where a principal U (1)-bundle with connection (P, A) the 2-form 2πi F (A) is the curvature of A. The homomorphism χ assigns the Chern class c1 (P ) to P . A similar story exits in the case of q = 3, and we come to theory of (abelian) gerbes ([2, 17, 16]). For treatment in the case of general q, we refer the reader to [6, 8, 18].

For later convenience, we introduce a complexified version of the smooth Deligne cohomology H p (X, Z(q)∞ D ). Definition 2.3 For a non-negative integer q and a smooth manifold X, we denote by H p (X, Z(q)∞ D,C ) the pth hypercohomology of the complex of sheaves: d

d

d

q−1 0 1 Z(q)∞ −→ 0 −→ · · · , D,C : Z −→ AC −→ AC −→ · · · −→ AC

where AqC is the sheaf of germs of C-valued differential q-forms. ∼ ∞ We can easily see the natural isomorphism H 1 (X, Z(1)∞ D,C ) = C (X, C/Z). Similarly, there also exists a natural isomorphism between H 2 (X, Z(2)∞ D,C ) and the group of isomorphism classes of principal C∗ -bundles with connection over X. The cohomology group H p (X, Z(q)∞ D,C ) has properties similar to that in Lemma 2.2. In particular, we have the following exact sequences: δ

q 0 → H q−1 (X, C/Z) → H q (X, Z(q)∞ D,C ) → A (X, C)Z → 0, ι

χ

q 0 → Aq−1 (X, C)/Aq−1 (X, C)Z → H q (X, Z(q)∞ D,C ) → H (X, Z) → 0,

where Aq (X, C) is the group of C-valued q-forms on X, and Aq (X, C)Z the subgroup of closed integral q-forms. 2.2 Cup product and integration As ordinary cohomology groups have the operations of cup product and integration, there are corresponding operations for smooth Deligne cohomology. The cup product for smooth Deligne cohomology ([2, 7]) is a natural homomorphism: p+q q ∞ (X, Z(p + q)∞ ∪ : H p (X, Z(p)∞ D ). D ) ⊗Z H (X, Z(q)D ) −→ H

For α ∈ Ap−1 (X) and β ∈ Aq−1 (X), the monomorphism ι gives cohomology q ∞ classes ι(α) ∈ H p (X, Z(p)∞ D ) and ι(β) ∈ H (X, Z(q)D ). Then we can express their cup product as ι(α) ∪ ι(β) = ι(α ∧ β). In general, the cup product is graded commutative and associative: fp ∪ fq = (−1)pq fq fp ,

(fp ∪ fq ) ∪ fr = fp ∪ (fq ∪ fr )

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for fi ∈ H i (X, Z(i)∞ D ), (i = p, q, r). In addition, we have: δ(fp ∪ fq ) = fp ∧ fq ,

χ(fp ∪ fq ) = fp ∪ fq .

For the integration for smooth Deligne cohomology to be defined, we assume that the manifold X is compact, oriented and d-dimensional (see [10, 12, 14], for example). Under the assumption, the integration is given as the following natural homomorphism with its values in R/Z, rather than R: Z : H d+1 (X, Z(d + 1)∞ D ) −→ R/Z. X

Lemma 2.4 ([12]) Let X be a compact oriented smooth d-dimensional manifold without boundary. (a) Let p and q be non-negative integers such that p + q = d. For α ∈ Ap (X)/Ap (X)Z and f ∈ H q (X, Z(q)∞ D ), we have Z Z α ∧ δ(f ) mod Z. ι(α) ∪ f = X

R

R

X

In particular, we have X ι(α) = X α mod Z for α ∈ Ad (X)/Ad (X)Z . (b) Let W be a compact oriented smooth (d+1)-dimensional manifold with its boundary ∂W = X. For f ∈ H d+1 (W, Z(d + 1)∞ D ), we have Z Z δ(f ) mod Z. f |X = W

X q

For the hypercohomology H (X, Z(q)∞ D,C ), there are corresponding notions of cup product and integration. For example, the integration is a natural homomorphism: Z : H d+1 (X, Z(d + 1)∞ D,C ) −→ C/Z, X

where X is assumed to be a compact oriented smooth d-dimensional manifold without boundary. The cup product and the integration for H q (X, Z(q)∞ D,C ) ), which we omit here. have properties similar to those for H q (X, Z(q)∞ D

3 A field theory We introduce a field theory of smooth Deligne cohomology classes on (4k +2)dimensional manifolds. We study some nature of the theory along the lines in the study of the Wess-Zumino-Witten models [9, 10], which will motivate us to consider representations of smooth Deligne cohomology groups. Some of the statements in this section can be found in [12, 13], where the viewpoint of field theory is rather suppressed. In the following, we take and fix a non-negative integer k. We put G(X) = 2k+1 H 2k+1 (X, Z(2k + 1)∞ (X, Z(2k + 1)∞ D ) and G(X)C = H D,C ) for a smooth manifold X.

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3.1 Action functional Let W be a compact oriented (4k + 2)-dimensional Riemannian manifold (possibly with boundary). We consider smooth Deligne cohomology classes f ∈ G(W )C as fields. The homomorphism δ : G(W )C → A2k+1 (W, C)Z plays the role of differential. Now we introduce the action functional of our theory, fixing an integer `: Definition 3.1 We define the function EW : G(W )C → C by Z EW (f ) = `π δ(f ) ∧ ∗δ(f ), W

where ∗ : A2k+1 (W, C) → A2k+1 (W, C) is the Hodge star operator. The action of the Hodge star operator ∗ on A2k+1 (W, C) obeys ∗∗ = −1. If we introduce the homomorphisms δ ± : G(W )C → A2k+1 (W, C) by δ + (f ) =

δ(f ) + i ∗ δ(f ) , 2

δ − (f ) =

δ(f ) − i ∗ δ(f ) , 2

then we can write the function EW as Z EW (f ) = 2`πi δ + (f ) ∧ δ − (f ). W

Note that, on the subspace A2k+1 (W ) ⊂ A2k+1 (W, C), we cannot perform the eigenspace decomposition with respect to ∗. This is the main reason that we use the complexified version G(M )C of G(M ). For instance, we consider the case of k = 0 and W is a compact oriented 2-dimensional Riemannian manifold. In this case, we have the identifications G(W ) = C ∞ (W, R/Z) and G(W )C = C ∞ (W, C/Z). Since the homomorphism δ : G(W )C → A1 (W, C)Z is givenRby the exterior differential: δ(f ) = df , we can express EW as EW (f ) = `π W df ∧ ∗df . Thus, upon the restriction to C ∞ (W, R/Z) = G(W ) ⊂ G(W )C , we recover the standard functional of scalar fields in two dimensions. Notice that the Riemannian metric on W makes it into a Riemann surface. ¯ Hence we also have It is easy to see the identifications δ + = ∂ and δ − = ∂. R ¯ the expression EW (f ) = 2`πi W ∂f ∧ ∂f . In the above case (k = 0), the theory with its functional EW directly yields a σ-model whose target space is R/Z or C/Z. However, in this article, the case of k > 0 is not formulated as a σ-model. It may be better to think of our theory as a certain gauge theory in which higher-order differential forms provide “gauge fields”.

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3.2 Equation of motion To derive the equation of motion from the functional, we begin with some comments on Lie group structures on G(W ) and G(W )C : we can make G(W ) into an infinite dimensional Lie group whose Lie algebra g(W ) is the vector space g(W ) = A2k (W )/d(A2k−1 (W )) with the trivial Lie bracket. The exponential map exp : g(W ) → G(W ) is given by the following composition of homomorphisms: ι

A2k (W )/d(A2k−1 (W )) → A2k (W )/A2k (W )Z → G(W ). In a similar way, we can also make G(W )C into a complex Lie group. As the notation indicates, G(W )C gives rise to a complexification of G(W ). Since the tangent space of G(W )C is seen, we compute the equation of motion for the action functional EW to obtain: Lemma 3.2 A Deligne cohomology class f ∈ G(W )C such that δ(f )|∂W = 0 is a critical point of EW if and only if f satisfies: d∗ δ(f ) = 0, where d∗ = − ∗ d∗ is the formal adjoint of d : A2k+1 (W, C) → A2k+1 (W, C). d E (f + tα) for α ∈ g(W )C . Because δ(α) = dα, Proof. We compute dt t=0 W Stokes’ theorem leads to: Z Z d α ∧ d ∗ δ(f ), α ∧ ∗δ(f ) − 2`π EW (f + tα) = 2`π dt t=0

∂W

W

which implies the lemma. t u We note that f ∈ G(W )C always satisfies the equation dδ(f ) = 0. Thus, in the case where W has no boundary, f is a solution to the equation of motion if and only if δ(f ) is a harmonic form. So Lemma 2.2 allows us to identify the space of solutions with H 2k+1 (W, Z) × (H 2k (W, R)/H 2k (W, Z)). We also note that f ∈ G(W )C such that ∗δ(f ) = iδ(f ) or ∗δ(f ) = −iδ(f ) also gives rise to a solution to the equation of motion. This motivates us to introduce the following subgroups in G(W )C : Definition 3.3 We define the chiral subgroup G(W )+ C and the anti-chiral subgroup G(W )− to be the following subgroups in G(W ) C: C ∓ G(W )± C = Ker δ = {f ∈ G(W )C | δ(f ) ∓ i ∗ δ(f ) = 0}.

In the case of k = 0 and W is a Riemann surface, G(W )+ C is isomorphic to the group of holomorphic functions f : W → C/Z, and G(W )− C the group of anti-holomorphic functions f : W → C/Z. A 2k-form α ∈ A2k (W, C) satisfying the “self-dual” condition i ∗ δ(f ) = δ(f ) is called a chiral 2k-form (see [15, 20], for example). By means of the homomorphism ι in Lemma 2.2 (b), a chiral 2k-form induces an element in + G(W )+ C . This is the reason that G(W )C is named the chiral subgroup.

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3.3 Analogy of the Polyakov-Wiegmann formula The action functionals in the Wess-Zumino-Witten models obey the so-called Polyakov-Wiegmann formula. There is a similar formula for the functional EW . We define ΓW : G(W )C × G(W )C → C by Z ΓW (f, g) = 4`πi δ − (f ) ∧ δ + (g). W

Lemma 3.4 Suppose that ∂W = ∅. For f, g ∈ G(W )C we have: eEW (f ) · eEW (g) = (exp ΓW (f, g)) eEW (f +g) . Proof. A straight computation gives: Z EW (f + g) − EW (f ) − EW (g) + ΓW (f, g) = 2`πi

δ(f ) ∧ δ(g). W

Recall Rthat δ(f ), δ(g) ∈ A2k+1 (W, C)Z . Since ` is taken to be an integer, we have ` W δ(f ) ∧ δ(g) ≡ 0 in C/Z under the present assumption on W . t u The point R in the above proof is that W has no boundary: if W has a boundary, then W δ(f ) ∧ δ(g) is not necessarily an integer. To take into account contributions of the boundary, we introduce a complex line bundle. Definition 3.5 Let M be a compact oriented (4k + 1)-dimensional smooth manifold (without boundary). (a) We define the line bundle LM over G(M )C by LM = G(M )C × C. (b) We define the product structure LM × LM → LM by (f, z) · (g, w) = (f + g, zw exp 2`πiSM,C (f, g)), where SM,C : G(M )C × G(M )C → C/Z is defined to be Z SM,C (f, g) = f ∪g M

by using the cup product and the integration for smooth Deligne cohomology. Lemma 3.6 Suppose that ∂W 6= ∅. For f ∈ G(W )C we define an element eEW (f ) ∈ L∂W to be eEW (f ) = (f |W , exp EW (f )). Then we have: eEW (f ) · eEW (g) = (exp ΓW (f, g)) eEW (f +g) . Proof. By means of Lemma 2.4 (b), we have Z Z S∂W,C (f |∂W , g|∂W ) = (f ∪ g)|∂W = ∂W

Z δ(f ∪ g) = W

δ(f ) ∧ δ(g). W

Now this lemma follows from the formula in the proof of Lemma 3.4. t u

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3.4 Central extension The product structure on LM induces a group multiplication on the complement LM \{0} of the image of the zero section. We denote the group by ˜ )C . Obviously, G(M ˜ )C is a central extension of G(M )C : G(M ˜ )C −→ G(M )C −→ 1. 1 −→ C∗ −→ G(M ˜ ) of G(M ): By restriction, we also obtain the central extension G(M ˜ ) −→ G(M ) −→ 1. 1 −→ U (1) −→ G(M ˜ ) = G(M ) × U (1) as We can express the group multiplication in G(M (f, u) · (g, v) = (f + g, uv exp 2`πiSM (f, g)), R where SM (f, g) ∈ R/Z is defied to be SM (f, g) = M f ∪ g by using again the cup product and the integration for smooth Deligne cohomology. Proposition 3.7 Let M be a compact oriented smooth (4k + 1)-dimensional ˜ ) is non-trivial as a central extension. manifold. If ` 6= 0, then G(M In [12], the proof in the case of ` = 1 is given. We can easily generalize the ˜ )C is also non-trivial, proof to the case of ` 6= 0. The central extension G(M ˜ since it is a complexification of G(M ). As an example, we consider the case of k = 0 and M = S 1 . In this ˜ 1) ∼ b (1)/Z2 , where case, G(S 1 ) ∼ = LU (1) as mentioned, and we have G(S = LU b LU (1)/Z2 is the universal central extension of LU (1), ([19]). 3.5 Toward the quantum theory To approach the quantum theory, we appeal to a method by using path integrals formally. In the case of ∂W = ∅, we may describe the partition function of our theory as: Z eEW (F ) DF,

ZW = F ∈G(M )C

where DF is a formal invariant measure on G(M )C . In the case of ∂W 6= ∅, the probability amplitude eEW (F ) is formulated as an element in L∂W . Hence the formal path integral gives the section ZW ∈ Γ (L∂W ) by Z ZW (f ) = eEW (F ) DF. F ∈G(W )C ,F |∂W =f

On a formal level, we can reconstruct the partition function for a (4k + 2)dimensional manifold W without boundary from the above sections by cutting W along a submanifold M of dimension 4k + 1. This suggests that the space

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of sections Γ (LM ) contains the quantum Hilbert space of our theory, as is in the Wess-Zumino-Witten models [9, 10]. By construction, the central extension G(M )C acts on the line bundle LM ˜ )C covering the action of G(M )C on itself. Though G(M )C is abelian, G(M ˜ )C is not. Hence the space of sections Γ (LM ) gives rise to a two sided G(M module. Compared to the Wess-Zumino-Witten models, one may expect that the quantum Hilbert space in Γ (LM ) is of the form “⊕λL ,λR HλL ⊗ HλR ”, ˜ )C , where HλL and HλR are certain left and right irreducible modules of G(M ˜ respectively. This motivates us to study representations of G(M )C , which is the subject of the next section.

4 Representations of smooth Deligne cohomology In this section, we deal with representations of smooth Deligne cohomology groups, culling out some results from [13]. After the statement of the classification, we explain a relationship between the representations and the quantum Hilbert space of a chiral 2-form due to Henningson [15]. We also consider an analogy of the space of conformal blocks in the Wess-Zumino-Witten model. 4.1 Representations of smooth Deligne cohomology First of all, we make a general remark: for a compact oriented (4k + 1)˜ ) is constructed by using dimensional manifold M , the central extension G(M 2`πiS(·,·) the group 2-cocycle e : G(M ) × G(M ) → U (1). Hence a representation ˜ ) such that the center U (1) acts as the scalar multiplication (˜ ρ, H) of G(M corresponds bijectively to a projective representation (ρ, H) of G(M ) with its ˜ )C . We cocycle e2`πiSM . There is a similar correspondence in the case of G(M use the correspondences freely in the following. For the smooth Deligne cohomology group G(M ), admissible representations of level ` are certain projective unitary representations on Hilbert spaces with their cocycle e2`πiSM . They are characterized by the representations of the subgroup A2k (M )/A2k (M )Z ⊂ G(M ) obtained by restriction. Generalizing straightly the proof of Theorem 1.1 given in [13], we can obtain the following classification of admissible representations: Theorem 4.1 Let M be a compact oriented (4k+1)-dimensional Riemannian manifold. For a positive integer `, admissible representations of G(M ) of level ` have the following properties: (a) An admissible representation is equivalent to a finite direct sum of irreducible admissible representations. (b) The number of the equivalence classes of irreducible admissible representations is (2`)b r, where b = b2k (M ) = b2k+1 (M ) is the Betti number, and r is the number of elements in the set {t ∈ H 2k+1 (M, Z)| 2` · t = 0}.

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If H 2k+1 (M, Z) is torsion free, then the number of the equivalence classes of irreducible admissible representations of level ` is (2`)b . The outline of constructing these irreducible representations is as follows: 1. We let G 0 (M ) be the subgroup A2k (M )/A2k (M )Z in G(M ). Using the Riemannian metric on M , we decompose the subgroup G 0 (M ) as follows: G 0 (M ) = H2k (M )/H2k (M )Z × d∗ (A2k+1 (M )), where H2k (M ) is the group of harmonic 2k-forms on M , H2k (M )Z = H2k (M ) ∩ A2k (M )Z the subgroup of harmonic 2k-forms with integral periods, and d∗ : A2k+1 (M ) → A2k (M ) the formal adjoint of d given by d∗ = − ∗ d∗. Using the Riemannian metric again, we define an inner product ( , ) on d∗ (A2k+1 (M )) and a compatible complex structure J on the completion V of d∗ (A2k+1 (M )) such that: (ν, Jν 0 ) = `SM (ν, ν 0 ),

ν, ν 0 ∈ d∗ (A2k+1 (M )) ⊂ V.

2. We construct the projective representation (ρ, H) of d∗ (A2k+1 (M )). The representation is realized as the representation of the Heisenberg group associated to the symplectic form (·, J·) : V × V → R. The representation space H is a completion of the symmetric algebra S(W ), where W is the eigenspace in V ⊗ C of J with its eigenvalue i. 3. Let X (M ) denote the set of homomorphisms λ : H2k (M )/H2k (M )Z → R/Z. For λ ∈ X (M ), we construct the projective representation (ρλ , Hλ ) of G 0 (M ). The representation space is Hλ = H. The action of (η, ν) ∈ (H2k (M )/H2k (M )Z ) × d∗ (A2k+1 (M )) is ρλ (η, ν) = e2πiλ(η) ρ(ν). 4. We construct the projective representation (ρλ , Hλ ) as the representation induced from the representation (ρλ , Hλ ) of the subgroup G 0 (M ) ⊂ G(M ). The projective representations (ρλ , Hλ ) and (ρλ0 , Hλ0 ) are equivalent if and only if there is ξ ∈ H 2k+1 (M, Z) such that λ0 = λ + 2`s(ξ). Here the homomorphism s : H 2k+1 (M, Z) → X (M ) is defined by Z s(ξ)(η) = η ∧ ξR mod Z, M

where ξR is a de Rham representative of the real image of ξ. If H 2k+1 (M, Z) is torsion free, then s is an isomorphism by the Poincar´e duality. Thus, among the representations (ρλ , Hλ ), we have (2`)b inequivalent representations. For example, we again consider the case of k = 0 and M = S 1 . Then we have the isomorphism G(S 1 ) ∼ = LU (1), and an admissible representation of G(S 1 ) of level ` gives rise to a positive energy representation of LU (1) of level 2`, and vice verse. This is because the construction of irreducible admissible representations outlined above coincides with that of irreducible positive energy representations given in [19]. The number of the equivalence classes of irreducible positive energy representations of LU (1) of level 2` is 2`, which is consistent with Theorem 4.1.

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4.2 Relationship to Henningson’s work The irreducible admissible representations in the case of k = 1 have certain relationship to the quantum Hilbert space of a chiral 2-form studied in a work of Henningson [15]. To explain the relationship, some notations in [15] will be used here without change. Let M be a compact oriented 5-dimensional Riemannian manifold such that H 2 (M, Z) and H 3 (M, Z) are torsion free. Then, by means of Theorem 4.1, the number of the equivalence classes of irreducible admissible representations of G(M ) of level ` = 1 is 2b . These irreducible representations are parameterized by the set Coker{2s : H 3 (M, Z) → X (M )} ∼ = (Z/2Z)b . In [15], Henningson studied the quantum Hilbert space V+ of a chiral 2form on R × M . The Hilbert space V+ can be expressed as a Hilbert space 0 0 tensor product: V+ = V 0 ⊗ V+ . The Hilbert space V+ admits a further decom0 0 position: V+ = ⊕a+ ∈H 3 (M,Z2 ) Va+ . Accordingly, we have the following decomposition into different version of the chiral theory: M V 0 ⊗ Va0+ . V+ = a+ ∈H 3 (M,Z2 )

Under the present assumption on M , the homomorphism s : H 3 (M, Z) → X (M ) induces the natural isomorphism: H 3 (M, Z2 ) ∼ = Coker{2s : H 3 (M, Z) → X (M )}. Hence we can naturally identify the parameterization space of the Hilbert spaces V 0 ⊗ Va0+ with that of the equivalence classes of irreducible admissible representations. If we take fixed lifting a+ ∈ H 3 (M, Z) of elements a+ ∈ H 3 (M, Z2 ), then (ρs(a+ ) , Hs(a+ ) ) represents the equivalence class of irreducible admissible representations corresponding to V 0 ⊗ Va0+ . In addition, we can find a natural isomorphism between V 0 ⊗ Va0+ and Hs(a+ ) . On the one hand, the construction in [15] implies that the Hilbert space V 0 is the completion of the symmetric algebra S(W ), so that V 0 = H. The Hilbert space Va0+ is spanned by |k+ , a+ i, (k+ ∈ H 3 (M, Z)). Thus, we have the following expression by using a Hilbert space direct sum: M d V 0 ⊗ Va0+ = H ⊗ C|k+ , a+ i. 3 k+ ∈H (M,Z)

On the other hand, it is shown in [13] that, for each λ ∈ X (M ), the representation Hλ |G 0 (M ) of G 0 (M ) obtained by the restriction of Hλ is expressed as the following Hilbert space direct sum: M d Hλ+2s(ξ) . Hλ |G 0 (M ) = 3 ξ∈H (M,Z)

Now the natural isomorphism V 0 ⊗ Va0+ → Hs(a+ ) follows from the obvious identification H ⊗ C|k+ , a+ i ∼ = Hs(a+ +2k+ ) .

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4.3 An analogy of the space of conformal blocks Projective representations of G(M )C , rather than G(M ), concern the field theory in Section 3. It is known [19] that positive energy representations of LU (1) extend to that of LC∗ in a certain way. The following proposition generalizes the fact. Proposition 4.2 Let M be a compact oriented (4k + 1)-dimensional Riemannian manifold, and (ρ, H) an admissible representation of G(M ) of level `. Then there is an invariant dense subspace E ⊂ H, and (ρ, E) extends to a projective representation of G(M )C . We can prove the proposition above by a straight generalization of the proof in the case of ` = 1 described in [13]. By means of the representations of G(M )C , we consider below an analogy of the space of conformal blocks in higher dimensions. Before the consideration, we notice that Lemma 3.6 and Theorem 4.1 lead to: (i) For a compact oriented (4k+2)-dimensional Riemannian manifold W with boundary, the following map gives rise to a homomorphism: ˜ r+ : G(W )+ C → G(∂W )C ,

f 7→ (f |∂W , exp EW (f )).

(ii) For a compact oriented (4k + 1)-dimensional Riemannian manifold M , we can parameterize the equivalence classes of irreducible admissible representations of G(M ) of level ` by a finite set Λ` (M ). Now, for W and λ ∈ Λ` (M ), we define CB(W, λ) to be the vector space consisting of continuous linear maps ψ : Eλ → C such that ψ(˜ ρλ (r+ (f ))v) = + ψ(v) for all v ∈ Eλ and f ∈ G(M )C : +

CB(W, λ) = Hom(Eλ , C)G(W )C . As the simplest example, we let W = D 4k+2 be the standard (4k + 2)dimensional disk whose boundary is S 4k+1 . In a direct way, we can compute CB(W, λ) to obtain a finite dimensional vector space: in the case of k = 0, the parameterization set Λ` (S 1 ) is identified with Z/`Z, and we obtain: C, (λ = 0) CB(D2 , λ) ∼ = {0}. (λ 6= 0) In the case of k > 0, the parameterization set Λ` (S 4k+1 ) = {0} consists of a single element, and we obtain: CB(D4k+2 , 0) ∼ = C. At present, the only example available is the above one. Computations of other examples as well as a proof that CB(W, λ) is a finite dimensional vector space still remain as problems.

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Kiyonori Gomi

Acknowledgement. Thanks are due to organizers and audiences during International Workshop on Noncommutative Geometry and Physics 2005, November 1–4, 2005, at Tohoku University. The author’s research was supported by Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.

References 1. L. Breen, On the classification of 2-gerbes and 2-stacks. Asterisque No. 225 (1994), 160 pp. 2. J-L. Brylinski, Loop spaces, Characteristic Classes and Geometric Quantization. Birkh¨ auser Boston, Inc., Boston, MA, 1993. 3. J. Cheeger and J. Simons, Differential characters and geometric invariants. Lecture Notes in Math. 1167(1985), Springer Verlag, 50-80. 4. A. L. Carey, M. K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories. J. Geom. Phys. 21 (1997), no. 2, 183–197. 5. P. Deligne and D. S. Freed, Classical field theory. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 137–225, Amer. Math. Soc., Providence, RI, 1999. 6. J. L. Dupont and F. W. Kamber, Gerbes, simplicial forms and invariants for families of foliated bundles. Comm. Math. Phys. 253 (2005), no. 2, 253–282. 7. H. Esnault and E. Viehweg, Deligne-Be˘ılinson cohomology. Be˘ılinson’s conjectures on special values of L-functions, 43–91, Perspect. Math., 4, Academic Press, Boston, MA, 1988. 8. P. Gajer, Geometry of Deligne cohomology. Invent. Math. 127 (1997), no. 1, 155–207. 9. K. Gaw¸edzki, Conformal field theory: a case study. Conformal field theory (Istanbul, 1998), 55 pp., Front. Phys., 102, Adv. Book Program, Perseus Publ., Cambridge, MA, 2000. 10. K. Gaw¸edzki, Topological actions in two-dimensional quantum field theories. Nonperturbative quantum field theory (Carg`ese, 1987), 101–141, NATO Adv. Sci. Inst. Ser. B Phys., 185, Plenum, New York, 1988. 11. J. Giraud, Cohomologie non-ab´elienne. Grundl. 179, Springer Verlag (1971). 12. K. Gomi, Central extensions of gauge transformation groups of higher abelian gerbes. J. Geom. Phys. to appear. hep-th/0504075. 13. K. Gomi, Projective unitary representations of smooth Deligne cohomology groups. math.RT/0510187. 14. K. Gomi and Y. Terashima, Higher-dimensional parallel transports. Math. Res. Lett. 8 (2001), no. 1-2, 25–33. 15. M. Henningson, The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions. J. High Energy Phys. 2002, no. 3, No. 21, 15 pp. 16. N. Hitchin, Lectures on special Lagrangian submanifolds. Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 151–182, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001. 17. M. K. Murray, Bundle gerbes. J. London Math. Soc. (2) 54 (1996), no.2, 403416. 18. R. Picken, A cohomological description of abelian bundles and gerbes. Twenty years of Bialowieza: a mathematical anthology, 217–228, World Sci. Monogr. Ser. Math., 8, World Sci. Publ., Hackensack, NJ, 2005.

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19. A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. 20. E. Witten, Five-brane effective action in M -theory. J. Geom. Phys. 22 (1997), no. 2, 103–133.

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Algebroids associated with pre-Poisson structures Kentaro Mikami1 and Tadayoshi Mizutani2 1

2

Dep of Computer engineering, Akita University, Japan [email protected] Dep of Mathematics, Saitama University, Japan [email protected]

1 Introduction There are several ways to generalize Poisson structures. A Jacobi structure (or a local Lie algebra structure), in which we do not require the Leibniz identity for the bracket, and a Nambu-Poisson structure, where the brackets are not binary but n-ary operations satisfying a generalized Leibniz rule called fundamental identity, are well-known examples. Also, a Dirac structure is a natural generalization of a Poisson structure. As another direction of studying Poisson geometry, we would like to do some trial or attempt to generalize the concepts, ideas, or theories of Poisson geometry into some area where the Poisson condition is not fulfilled. In the first half of this note, we show briefly our trials in this context, namely in almost Poisson geometry. As we will see in short, a Poisson structure gives a Lie algebroid. It is natural to handle a Leibniz algebroid as generalization of a Lie algebroid. Thus, it is meaningful to study the fundamental properties of Leibniz algebra or super Leibniz algebra. In the second half of this note, after we recall some properties of Leibniz modules, we define super Leibniz algebras and super Leibniz modules keeping the exterior algebra bundle of the tangent bundle with Schouten bracket as a prototype of a super Lie algebra (and so a super Leibniz algebra). We will show that an abelian extension is controlled by the second super cohomology group. The notion of super Leibniz bundles is clear, but unfortunately we do not have the proper notion of anchor, so far. In near future, we hope we could find concrete examples of super Leibniz bundles tightly connected to the properties of Poisson geometry, and could understand what the anchor should be.

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2 Review of elements of Poisson geometry Definition 1. By a Poisson structure or a Poisson bracket on a manifold M , we mean a binary operation on the function space C ∞ (M ) of M , C ∞ (M ) × C ∞ (M ) 3 (f, g) 7→ {f, g} ∈ C ∞ (M ) satisfying 1. R-bilinearity 2. skew-symmetry 3. Jacobi identity 4. Leibniz rule

{λf + µg, h} = λ{f, h} + µ{g, h} (λ, µ ∈ R) {g, f } = −{f, g} {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 {f, gh} = {f, g}h + g{f, h}.

We call a manifold with a specified Poisson structure as a Poisson manifold. Definition 2. On a Poisson manifold, the Hamiltonian vector field Xf of f ∈ C ∞ (M ) is given by hXf , dgi := {f, g} Proposition 1. X{f,g} = [Xf , Xg ] holds for f, g ∈ C ∞ (M ). This comes from Jacobi identity of the Poisson bracket. Example 1. Every manifold has the trivial Poisson structure {f, g} := 0. Example 2. For a given symplectic structure ω, the Poisson bracket is defined by {f, g} := hXf , dgi where ω [ (Xf ) := −df or Xf := −ω ] (df ) . It is well-known that the cotangent bundle T∗ (Q) of a manifold Q has a canonical symplectic structure. The Poisson bracket satisfies {τ ∗ f, τ ∗ g} = 0,

{X, τ ∗ g} = τ ∗ hX, dgi,

{X, Y } = [X, Y ]

where τ : T∗ (Q) → Q is the bundle projection, and f, g ∈ C ∞ (Q), X, Y ∈ Γ (T(M )) and considered as linear functions along the fibres of T∗ (M ), and [X, Y ] is the usual Lie bracket of X and Y . Example 3. Let g be a Lie algebra of finite dimension. Consider the dual space g∗ as the underlying manifold. Since an element of g is a linear function on g∗ , g is a subspace of C ∞ (g∗ ). forF, H ∈ C ∞ (g∗ ), andµ ∈ g∗ , the Poisson bracket is defined by δF δH , ], µi {F, H}(µ) := h[ δµ δµ δF d δF where (ν) := F (µ + tν)|t=0 (ν ∈ g∗ ) and ∈ g∗ ∗ ∼ = g. In fact, δµ X dt δµ {zj , zk } = cijk zi holds where (zi ) is a basis of g whose structure constants i

are (cijk ). This bracket is called Lie-Poisson bracket on the dual space of a Lie algebra.

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Definition 3. For a given Poisson bracket {·, ·}, there exists a unique π ∈ Γ (Λ2 T(M )) satisfying π(df, dg) = hπ, df ∧ dgi = {f, g} . π is called the Poisson 2-vector field (or the Poisson tensor field) of the Poisson bracket. Proposition 2. Conversely, for a given 2-vector field π, define a new bracket by {f, g} := π(df, dg) = hπ, df ∧ dgi . This bracket is Poisson, namely satisfies the Jacobi identity if and only if the Schouten bracket [π, π]S vanishes. We recall here the definition and the related properties of the Schouten bracket . Now on, we abbreviate Γ (Λ• T(M )) to Λ• T(M ), and so on. Definition 4. The Schouten bracket [·, ·]S is the homogeneous bi-derivadimM X tion on Λ• (T(M )) of degree −1 •=0

Λp T(M ) × Λq T(M ) 3 (P, Q) 7→ [P, Q]S ∈ Λp+q−1 (T(M )) uniquely defined by the following five conditions. Property (6) is called super Jacobi identity. 1. [f, g]S = 0 f, g ∈ Λ0 (T(M )) = C ∞ (M ) 2. [X, f ]S = hX, df i = Xf X ∈ Λ1 (T(M )), f ∈ Λ0 (T(M )) 3. [X, Y ]S = [X, Y ]Lie bracket X, Y ∈ Λ1 (T(M )) 4. [P, Q]S = −(−1)(p−1)(q−1) [Q, P ]S 5. [P, Q ∧ R]S = [P, Q]S ∧ R + (−1)(p−1)q Q ∧ [P, R]S 6. (−1)(p−1)(r−1) [[P, Q]S , R]S + (−1)(q−1)(p−1) [[Q, R]S , P ]S + (−1)(r−1)(q−1) [[R, P ]S , Q]S = 0, where the small letter p means the ordinary degree of the capital letter P , i.e., P ∈ Λp (T(M )). Remark 1. The Schouten bracket on the decomposable elements is given by [X1 ∧ · · · Xp ,Y1 ∧ · · · Yq ]S =

p X q X i

ci · · · Y1 ∧ · · · c (−1)i+j [Xi , Yj ] ∧ X1 ∧ · · · X Yj · · · ∧ Y q

j

for Xi , Yj ∈ Γ (T(M )) (p, q ≥ 1). We often abbreviate [·, ·]S to [·, ·]. The sign convention of the Schouten bracket here is different from that in Vaisman’s book [9].

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Given a 2-vector field π on M , we have a bundle map from T∗ (M ) to T(M ) defined by β 7→ π(β, ·) = ιβ π = β π . This map is denoted by π ] , π ˜ , or often π itself if there is no danger of confusion. The Hamilton vector field Xf is then written as π ] (df ).

3 Algebroids related with Poisson structures 3.1 Lie algebroids We start this section by a famous result by B. Fuchssteiner [2]. Theorem 1 ([2]). Given a Poisson 2-vector field π on M , define a bracket by {α, β}π := Lπ] (α) β − Lπ] (β) α − d(π(α, β)) for each α, β ∈ Γ (T∗ (M )). Then {·, ·}π yields a Lie algebra structure on Γ (T∗ (M )) and the following equality holds: {α, f β}π = hπ ] α, df iβ + f {α, β}π . Replacing T(M ) by a general vector bundle we obtain the notion of Lie algebroids. Definition 5. A vector bundle L over M is a Lie algebroid if and only if (a) Γ (L) is endowed with a Lie algebra bracket [·, ·] over R, i.e., [·, ·] is skewsymmetric R-bilinear and satisfies Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (b) there exists a bundle map (called anchor ) a : L → T(M ) which induces a : Γ (L) → Γ (T(M )) is a(f x) = f a(x) (c) and satisfying [x, f y] = ha(x), df iy + f [x, y] where x, y, z ∈ Γ (L), f ∈ C ∞ (M ). Example 4. (1) T∗ (M ) with the bracket {·, ·}π defined from the Poisson 2vector field π is a Lie algebroid whose anchor is π ] . (2) T(M ) is a Lie algebroid with the identity map as the anchor. Assume that a 2-vector field π is not necessarily Poisson. Then we look at the space {α ∈ Γ (T∗ (M )) | α

[π, π]S = 0} =: ker[π, π]S

and ask the questions. Is ker[π, π]S closed with respect to {·, ·}π ? Does {·, ·}π satisfy Jacobi identity?

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We would like to say this trial is one of pre-Poisson attempts. The answers for the both are yes and we get a theorem. Theorem 2 (Mikami-Mizutani [7]). If rank ker[π, π]S is constant, ker[π, π]S is a Lie algebroid having bracket defined by {α, β}π := Lπ] (α) β − Lπ] (β) α − d(π(α, β)) and the anchor α 7→ π ] (α). 3.2 Dirac structures As stated already, a 2-vector field π is Poisson if and only if [π, π]S = 0. On the other hand, T. Courant and A. Weinstein studied Poisson condition from more geometrical point of view. They handle the bundle homomorphism π ] : T∗ (M ) → T(M ). They claim that “Poisson condition is equivalent to some property of the graph of π ] , (Dirac structure)”, and generalize their discussion from a graph to a relation. On T(M ) ⊕ T∗ (M ), Courant([1]) defined h(Y1 , β1 ), (Y2 , β2 )i+ := iY1 β2 + iY2 β1 (fibre wise) 1 [[(Y1 , β1 ), (Y2 , β2 )]] := [Y1 , Y2 ] , LY1 β2 − LY2 β1 − d (iY1 β2 − iY2 β1 ) 2 T(e1 , e2 , e3 ) := h[[e1 , e2 ]], e3 i+ where (Yj , βj ) = ej ∈ Γ (T(M ) ⊕ T∗ (M )) (j = 1, 2, 3). Remark 2. In general, T is not tensor field, and only skew-symmetric in the first two arguments. Definition 6. A sub-bundle L ⊂ T(M ) ⊕ T∗ (M ) is an almost Dirac structure if L is maximally isotropic with respect to the pairing h·, ·i+ , i.e., L is a sub-bundle of rank dimM , and the restriction of h·, ·i+ to L × L is identically zero. Proposition 3. If L is an almost Dirac structure, T|L (e1 , e2 , e3 ) = S hβ1 , [Y2 , Y3 ]i + LY1 hβ2 , Y3 i

123

where ej = (Yj , βj ) ∈ Γ (L), (j = 1, 2, 3). Especially, T|L is tensorial, and skew-symmetric in 3 arguments. Example 5. Let π be an arbitrary 2-vector field on M . The graph of π ] , L = {(π ] (β), β) | β ∈ T∗ (M )} is an almost Dirac structure. T|L ((π ] (β1 ), β1 ), (π ] (β2 ), β2 ), (π ] (β3 ), β3 )) =

1 [π, π]S (β1 , β2 , β3 ) . 2

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Example 6. Let ω be a 2-form on M . The graph of ω [ , L = {(X, iX ω) | X ∈ T(M )} is almost Dirac. T|L ((X1 , iX1 ω), (X2 , iX2 ω), (X3 , iX3 ω)) = (dω)(X1 , X2 , X3 ) . The Courant bracket [[·, ·]] is skew-symmetric, but does not satisfy Jacobi identity. In fact, let (J1 , J2 ) be the components of Jacobiator, i.e., (J1 , J2 ) := S [[[[(Y1 , β1 ), (Y2 , β2 )]], (Y3 , β3 )]] . 123

Then, J1 = 0 holds, but J2 is complicated. Proposition 4. J2 is given explicitly, and the restriction of J2 to an almost Dirac structure L is given by J2 |L (· · · ) =

1 d T|L (· · · ) 2

Definition 7. An almost Dirac structure L is a Dirac structure if Γ (L) is closed by bracket [[·, ·]], i.e., it satisfies T|L ≡ 0. In the case of an almost Dirac structure defined by a 2-vector field, a Dirac structure gives a Poisson structure of the base manifold. In the case of a 2-form, a pre-symplectic structure. When L is almost Dirac, we consider the following “sub-bundle” ker(T|L ) : = {e ∈ L | T|L (e1 , e2 , e) = 0, e1 , e2 ∈ L} . Again we would like to say this is one of pre-Poisson trials. Theorem 3 (Mikami-Mizutani[8]). Let L be an almost Dirac structure and ker(T|L ) be of constant rank. Then ker(T|L ) is a Lie algebroid with the bracket [[·, ·]], and the anchor ρker(T|L ) , which is the restriction of the first projection ρ : T(M ) ⊕ T∗ (M ) → T(M ) to ker(T|L ). Application of Theorem 3 to Theorem 2 (Mikami-Mizutani[8]) We know that the graph of a general 2-vector field π is almost Dirac. Since 1 TL (·, (π ] (βj ), βj ), ·) = [π, π]S (·, βj , ·), we have ker TL := {(π ] (γ), γ) | γ ∈ 2 ker[π, π]S }. The bracket is computed as follows [[(π ] (α), α), (π ] (β), β)]] = ([π ] (α), π ] (β)], {α, β}π ) = (π ] ({α, β}π ), {α, β}π ) . If we pick up the first components of the elements of ker TL , this computation gives another proof of Theorem 2 of Mikami-Mizutani.

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4 Leibniz algebra and Leibniz algebroid Definition 8. A vector bundle A over M is a Leibniz algebroid if (a) Γ (A) is endowed with a Leibniz algebra structure over R, i.e., there is a binary operation [·, ·] : Γ (A) × Γ (A) → Γ (A) which is non skew in general and R-bilinear, and satisfying [[x, y], z] = [x, [y, z]] − [y, [x, z]] (Leibniz identity) where x, y, z ∈ Γ (A). (b) there exists a bundle map (called anchor ) a : A → T(M ) (c) the following compatibility condition holds: [x, f y] =ha(x), df iy + f [x, y] where x, y, z ∈ Γ (A), ha(x), df i = La(x) f for f ∈ C ∞ (M ). Remark 3. (1) The condition [[x, y], z] = [x, [y, z]] − [y, [x, z]] is equivalent to [x, [y, z]] = [[x, y], z] + [y, [x, z]] and this means [x, ·] satisfies Leibniz identity or left-derivation rule. (2) If it is skew-symmetric, Jacobi identity is equivalent to Leibniz identity as we see [[x, y], z] = −[[y, z], x] − [[z, x], y] = [x, [y, z]] − [y, [x, z]] (using skew-symmetry) (3) The difference between Lie algebroids and Leibniz algebroids is just relaxing the skew-symmetric property. But, we can see some hidden “almost” skew-symmetric property for Leibniz algebroid. The reason is: [[x, y], z] = [x, [y, z]] − [y, [x, z]] (Leibniz rule) this shows RHS is skew-symmetric in x and y, thus 0 = [[x, y] + [y, x], z]

(x, y, z ∈ Γ (A)).

(4) It follows a([x, y]) = [a(x), a(y)] (x, y ∈ Γ (A)) by the following computations. Since [y, f z] = (La(y) f )z + f [y, z], we have [x, [y, f z]] = [x, La(y) f z + f [y, z]] LHS =[[x, y], f z] + [y, [x, f z]] =(La([x,y])f )z + f [[x, y], z] + [y, (La(x) f )z + f [x, z]] =(La[x,y] f )z + f [[x, y], z] + (La(y) La(x) f )z+ + (La(x) f )[y, z] + (La(y) f )[x, z] + f [y, [x, z]] RHS =(La(x) La(y) f )z + (La(y) f )[x, z] + (La(x) f )[y, z] + f [x, [y, z]] Thus, we have La([x,y])f − La(x) La(y) f + La(y) La(x) f z = 0. This means we have the result.

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Example 7. Let M be a manifold. A := T(M ) ⊕ T∗ (M ). For a pair of sections of A, define a bracket by [(X, α), (Y, β)] := ([X, Y ], LX β − ιY dα) and the anchor a(X, α) := X. Then this is a Leibniz algebroid, but not a Lie algebroid. 4.1 Abelian extension of Leibniz algebra In this subsection, we shall study an algebraic property of a Leibniz algebra and related bi-module (a Leibniz bi-module), namely the second Leibniz cohomology group with coefficient in the Leibniz module. Definition 9 (Leibniz module). Let (g, [·, ·]) be a Leibniz algebra. A module A is called a g-bi module if A is a module where g acts from both, left and right, and it holds the following three conditions. (a · g) · h = a · [g, h] − g · (a · h) (g · a) · h = g · (a · h) − a · [g, h] [g, h] · a = g · (h · a) − h · (g · a) where g, h ∈ g and a ∈ A, and we denoted the left action of g on A by g · a, and the right action of g on A by a · g (g ∈ g, a ∈ A). When a Leibniz g-module A is given, we can construct cochain complex and cohomology groups which are called Leibniz cohomology groups (cf. J.-L. Lodays’s works, for example [4] or [5]). Definition 10. For each non-negative integer k, k-th cochain complex consists of k-multilinear maps from g × · · · × g to A and the coboundary operator | {z } k−times

is given by (δψ)(g1 , . . . , gk+1 ) =

k X

(−1)i−1 gi · ψ(. . . gbi . . .) + (−1)k+1 ψ(g1 , . . . , gk ) · gk+1

i=1

+

X

j

(−1)i ψ(. . . gbi . . . [gi , gj ] . . .) .

i<j

We denote by H k (g, A) the k-th cohomology group. We take a bi-linear map ψ : g × g → A and a linear exact sequence Π

0 −→ A −→ g0 −→ g −→ 0

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and a linear section s of Π, i.e., Π ◦ s = idg which gives a linear isomorphism g0 ∼ = s(g) ⊕ A. Define a multiplication on g0 by [[s(g1 ) + a1 , s(g2 ) + a2 ]] := s[g1 , g2 ] + ψ(g1 , g2 ) + g1 · a2 + a1 · g2 for g1 , g2 ∈ g and a1 , a2 ∈ A, where “·” is the Leibniz action of g to A. We let g 0 i := s(gi ) + ai ∈ g0 (i=1,2,3). We have [[[[g 0 1 , g 0 2 ]], g 0 3 ]] =[[s[g1 , g2 ] + ψ(g1 , g2 ) + g1 · a2 + a1 · g2 , s(g3 ) + a3 ]] =s[[g1 , g2 ], g3 ] + ψ([g1 , g2 ], g3 ) + [g1 , g2 ] · a3 + (ψ(g1 , g2 ) + g1 · a2 + a1 · g2 ) · g3 =s[[g1 , g2 ], g3 ] + ψ([g1 , g2 ], g3 ) + [g1 , g2 ] · a3 + ψ(g1 , g2 ) · g3 + (g1 · a2 ) · g3 + (a1 · g2 ) · g3 =s[[g1 , g2 ], g3 ] + ψ([g1 , g2 ], g3 ) + ψ(g1 , g2 ) · g3 + [g1 , g2 ] · a3 + (g1 · a2 ) · g3 + (a1 · g2 ) · g3 and [[g 0 1 , [[g 0 2 , g 0 3 ]]]] =[[s(g1 ) + a1 , s[g2 , g3 ] + ψ(g2 , g3 ) + g2 · a3 + a2 · g3 ]] =s[g1 , [g2 , g3 ]] + ψ(g1 , [g2 , g3 ]) + g1 · (ψ(g2 , g3 ) + g2 · a3 + a2 · g3 ) + a1 · [g2 , g3 ] =s[g1 , [g2 , g3 ]] + ψ(g1 , [g2 , g3 ]) + g1 · ψ(g2 , g3 ) + g1 · (g2 · a3 ) + g1 · (a2 · g3 ) + a1 · [g2 , g3 ] and so [[g 0 2 , [[g 0 1 , g 0 3 ]]]] =s[g2 , [g1 , g3 ]] + ψ(g2 , [g1 , g3 ]) + g2 · ψ(g1 , g3 ) + g2 · (g1 · a3 ) + g2 · (a1 · g3 ) + a2 · [g1 , g3 ] , [[·, ·]] satisfies Leibniz property, i.e., [[[[g 0 1 , g 0 2 ]], g 0 3 ]] = [[g 0 1 , [[g 0 2 , g 0 3 ]]]] − [[g 0 2 , [[g 0 1 , g 0 3 ]]]] if and only if [·, ·] satisfies Leibniz property and ψ([g1 , g2 ], g3 ) + ψ(g1 , g2 ) · g3 + [g1 , g2 ] · a3 + (g1 · a2 ) · g3 + (a1 · g2 ) · g3 =ψ(g1 , [g2 , g3 ]) + g1 · ψ(g2 , g3 ) + g1 · (g2 · a3 ) + g1 · (a2 · g3 ) + a1 · [g2 , g3 ] − ψ(g2 , [g1 , g3 ]) − g2 · ψ(g1 , g3 ) − g2 · (g1 · a3 ) − g2 · (a1 · g3 ) − a2 · [g1 , g3 ]

(1)

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for each g1 , g2 , g3 ∈ g and for each a1 , a2 , a3 ∈ A. The equation (1) is equivalent to the following 4 conditions: (a1 · g2 ) · g3 =a1 · [g2 , g3 ] − g2 · (a1 · g3 ) (g1 · a2 ) · g3 =g1 · (a2 · g3 ) − a2 · [g1 , g3 ] [g1 , g2 ] · a3 =g1 · (g2 · a3 ) − g2 · (g1 · a3 ) and ψ([g1 , g2 ], g3 ) + ψ(g1 , g2 ) · g3 =ψ(g1 , [g2 , g3 ]) + g1 · ψ(g2 , g3 ) − ψ(g2 , [g1 , g3 ]) − g2 · ψ(g1 , g3 ) .

(2)

The first three equations are just the property of the Leibniz action of the Leibniz algebra g on A, and the equation (2) is equivalent to ψ being a 2cocycle of Leibniz coboundary operator. Thus, we have Proposition 5. For a given Leibniz bracket [·, ·] on g and a Leibniz action of mikamiLieg on A, the new bracket [[·, ·]] on mikamiLieg 0 becomes a Leibniz bracket if and only if ψ is a 2-cocycle. From the definition, we see that [[A, A]] = {0} and Π is a Leibniz algebra homomorphism. We call such an exact sequence as an abelian Leibniz extension of g with the kernel A. Also, from the definition of the multiplication [[·, ·]], we see that [[s(g1 ), s(g2 )]] − s[g1 , g2 ] = ψ(g1 , g2 )

(∀g1 , g2 ∈ g).

Theorem 4 ([5]). For a given Leibniz algebra g and its bi-module A, the class of Leibniz abelian extensions up to Leibniz isomorphisms one-to-one correspond to the second Leibniz cohomology group H 2 (g, A). Example 8. Let g0 = Γ (T(M ) ⊕ T∗ (M )) be the Leibniz algebra with usual Courant bracket. The natural exact sequence of bundles T∗ (M ) → T(M ) ⊕ T∗ (M ) → T(M ) gives an exact sequence of Leibniz algebras 0 → A → g0 → g → 0

(3)

where g = Γ (T(M )) and A = Γ (T∗ (M )). The left action of g on A is given by X·α = LX α and the right action by α·X = −ιX dα. The cocycle corresponding to (3) is the zero cocycle. Indeed, taking ψ ≡ 0, we have [[(X, α), (Y, β)]] = ([X, Y ], ψ(X, Y ) + X · β + α · Y ) = ([X, Y ], LX β − ιY dα).

Algebroids associated with pre-Poisson structures

81

Example 9. We put g0 = Γ (Λq+1 (T(M )) and let F be a transversely oriented regular foliation of codimension q on M . Choose a transverse volume form ω of F which is a locally decomposable q-form that is, locally, ω = ω1 ∧ · · · ∧ ωq and F is defined by ω1 = · · · = ωq = 0. The bracket on g0 is defined by [P, Q]ω = LP (ω) Q + (−1)q+1 P (dω)Q. Then g0 is a Leibniz algebra and the ‘anchor’ ρ : P 7→ P (ω) is a Leibniz algebra homomorphism whose image is the space of vector fields tangent to the foliation. Take g = Imageρ. Thus, we have an exact sequence which is a central extension in the sense that [P, Q] = 0 for any Q ∈ g0 if P ∈ ker ρ, 0 → ker ρ → g0 → g → 0. An element in ker ρ is a q + 1-vector field contained in the ideal generated by Γ (Λ2 g). The corresponding 2-cocycle can be described as follows. Choose a q-vector field W which satisfies hω, W i ≡ 1 and take a 1-form γ satisfying dω = γ ∧ ω. The integrability assures that we have such a 1-form γ. The 2-cocycle on g corresponding to the above extension, which takes values in ker ρ is defined by ψ(X, Y ) = LX W + γ(X)W. Example 10. As in the previous example, we assume that ω1 and ω2 are transverse volume forms of (regular) foliations on M , of codimension q1 and q2 , respectively. Also,we assume ω1 and ω2 are transverse to each other and ω1 ∧ω2 defines a foliation of codimension q1 + q2 . We have the following bundle map which is the contraction by ω1 Iω

Λq1 +q2 +1 T(M ) →1 Λq2 +1 T(M ). This induces a Leibniz algebra homomorphism I˜ω

Γ (Λq1 +q2 +1 T(M )) →1 Γ (Λq2 +1 T(M )). It can be seen that ker I˜ω1 is consisting of q1 + q2 + 1-vector fields on M which are in the ideal (in the exterior algebra) generated by Γ (Λq2 +2 T(M )). Taking g0 = Γ (Λq1 +q2 +1 T(M )) and g = ImageI˜ω1 , we have an abelian extension of Leibniz algebras 0 → ker I˜ω1 → g0 → g → 0. The corresponding 2-cocycle is given by the following formula ψ(X, Y ) = LX(ω2 ) W1 ∧ Y + γ1 ((X(ω2 ))W1 ∧ Y

X, Y ∈ g,

where W1 is a q1 -vector field satisfying hω1 , W1 i ≡ 1 and γ1 is a 1-form satisfying dω1 = γ1 ∧ ω1 . Likewise as in the previous example, we can verify that ψ is a well-defined 2-cocycle having the values in ker I˜ω1 .

82

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4.2 Super Leibniz algebra and its cochain complex We can rewrite super Jacobi identity of the Schouten bracket in the following form: [[P, Q]S , R]S = [P, [Q, R]S ]S − (−1)(p−1)(q−1) [Q, [P, R]S ]S for P ∈ Γ (Λp T(M )), Q ∈ Γ (Λq T(M )), and R ∈ Γ (Λr T(M )). Taking the above formula as a model, we generalize the notion of Leibniz algebras to that of graded (or super) Leibniz algebras. P Definition 11. A graded vector space g := j gj is a graded Leibniz algebra if there is a R-bilinear multiplication (we denote it by [·, ·] again) which satisfies the next two conditions. deg([x, y]) = deg(x) + deg(y) [[x, y], z] = [x, [y, z]] − (−1)deg(x) deg(y) [y, [x, z]] where deg(x) means the grade of x, i.e., x ∈ gdeg(x) . Example 11. If we consider the exterior algebra bundle of T(M ) and define deg(P ) := p − 1 (the reduced degree of P ) for P ∈ Γ (Λp T(M )), then we obtain a typical example of graded Leibniz algebras. We also construct super Leibniz algebras from Leibniz algebras just following the definition of Schouten bracket as follows. Example 12. Let g be a Leibniz algebra with the multiplication [·, ·]. Condim Xg 0 ∧j g, and define the degrees by deg(x) := sider the exterior algebra g := j=1

0 for each x ∈ g and deg(P ) := p − 1 for P ∈ ∧p g. In particular, deg(P ∧ Q) = deg(P ) + deg(Q) + 1 for homogeneous elements P and Q. It is convenient to consider the degree of ‘∧’ to be +1. Note that Q ∧ P = (−1)(deg(Q)+1)(deg(P )+1) P ∧ Q. A new multiplication on g0 is defined by [[x1 ∧ · · · xp , y1 ∧ · · · yq ]] :=

p X q X i

(−1)i+j [xi , yj ] ∧ x1 ∧ · · · xbi · · · y1 ∧ · · · ybj · · · ∧ yq

j

for xi , yj ∈ g (p, q ≥ 1). We extend it R-bi-linearly and we see that this satisfies Leibniz property. Indeed as we expected, we have the next three properties: [[x, y]] :=[x, y]

(x, y ∈ g)

[[P, Q ∧ R]] :=[P, Q] ∧ R + (−1)deg(P )(deg(Q)+1) Q ∧ [[P, R]] [[P ∧ Q, R]] :=P ∧ [Q, R] + (−1)(1+deg(Q) deg(R) [[P, R]] ∧ Q

Algebroids associated with pre-Poisson structures

83

As a complete analogue of Leibniz modules, we can define graded Leibniz modules. Definition 12. Let g be a P graded Leibniz algebra with the multiplication [·, ·]. A graded module A = j∈Z Aj is called (g-)graded Leibniz module if g operates on A on both sides, R-bilinearly (we use the same notations x · a or a · x as before) satisfying the following four conditions for the homogeneous elements. deg(x · a) = deg(a · x) = deg(x) + deg(a) [x, y] · a = x · (y · a) − (−1)deg(x) deg(y) y · (x · a) (x · a) · y = x · (a · y) − (−1)deg(x) deg(a) a · [x, y] (a · x) · y = a · [x, y] − (−1)deg(a) deg(x) x · (a · y) Definition 13. Let g be a graded Leibniz algebra and A a graded g-Leibniz bi-module. For each non-negative integer k, a k-multilinear map ψ from g ⊗ · · · ⊗ g to A is called homogeneous of degree p if it satisfies | {z } k−times

ψ(gd1 ⊗ · · · ⊗ gdk ) ⊂ Ap+d1 +···+dk

(∀dj ∈ Z,

j = 1 . . . k)

for a integer p. p is called the degree of ψ and denoted by deg(ψ). Now we define a R-multilinear operator by (δψ)(x1 , . . . , xk+1 ) :=(−1)k+1 ψ(x1 , . . . , xk ) · xk+1 +

k X

(−1)1+i+deg(xi )(deg ψ+

i=1

+

X

(−1)i+deg(xi )

i<j

P

i<`<j

P

`
deg(x` )

deg(x` ))

xi · ψ(. . . xbi . . .)

ψ(. . . xbi . . . [xi , xj ] . . .) .

For homogeneous ψ, δψ is also homogeneous and it turns out deg(δψ) = deg(ψ). Theorem 5. We have δ◦δ = 0. Thus, we have a cohomology theory for graded Leibniz modules. Proof. First, we consider 0-cochain ψ, i.e., ψ ∈ A, then (δ 2 ψ)(x1 , x2 ) =(δψ)(x1 ) · x2 + (−1)deg(x1 ) deg(ψ) x1 · δψ(x2 ) − δψ([x1 , x2 ]) = (−ψ · x1 ) · x2 + (−1)deg(x1 ) deg(ψ) x1 · (−ψ · x2 ) + ψ · [x1 , x2 ] =0 . Next, we fix a positive integer k, and consider a k-multilinear homogeneous map ψ from g ⊗ · · · ⊗ g to A. We use the notation | {z } k−times

S(i) := deg(xi )(deg(ψ) +

X `
deg(x` ))

84

Kentaro Mikami and Tadayoshi Mizutani

X

T (i, j) := deg(xi )

deg(x` ) ,

T (i, i + 1) := 0

i<`<j

U (i, j) :=U (j, i) := deg(xi ) deg(xj ) and so, we have T (i, j) =S(j) − S(i) − U (i, i) . We rewrite the definition and decompose δψ into 3 parts: (δψ)(x1 , . . . , xk+1 ) =(−1)k+1 ψ(x1 , . . . , xk ) · xk+1 + (−1)S(k+1) xk+1 ψ(x1 , . . . , xk ) +

k+1 X

(−1)1+i+S(i) xi · ψ(. . . xbi . . .)

i=1

+

X

(−1)i+T (i,j) ψ(. . . xbi . . . [xi , xj ] . . .)

i<j

=:A(ψ | x1 , . . . , xk+1 ) + B(ψ | x1 , . . . , xk+1 ) + C(ψ | x1 , . . . , xk+1 ) where A(ψ | x1 , . . . , xk+1 ) := (−1)k+1 (ψ(x1 , . . . , xk ) · xk+1 +(−1)S(k+1) xk+1 · ψ(x1 , . . . , xk ) B(ψ | x1 , . . . , xk+1 ) :=

k+1 X

(−1)1+i+S(i) xi · ψ(. . . xbi . . .)

i=1

C(ψ | x1 , . . . , xk+1 ) :=

X

(−1)i+T (i,j) ψ(. . . xbi . . . [xi , xj ] . . .) .

i<j

Remark 4. Since the super Leibniz module has “almost skew-symmetric” property, namely it holds that (a · x) · y = a · (x · y) − (−1)deg(a) deg(x) x · (a · y) = −(−1)deg(a) deg(x) (x · a) · y we see that A(ψ | x1 , . . . , xk+1 ) · y = 0 . Using these notations, we have (δ 2 ψ)(x1 , . . . , xk+2 ) =A(δψ | x1 , . . . , xk+2 ) + B(δψ | x1 , . . . , xk+2 ) + C(δψ | x1 , . . . , xk+2 ) X =A(δψ | x1 , . . . , xk+2 ) + (−1)1+λ+S(λ) xλ · (δψ)(. . . x cλ . . .) λ

Algebroids associated with pre-Poisson structures

X

+

85

(−1)λ+T (λ,µ) (δψ)(. . . x cλ . . . [xλ , xµ ] . . .)

λ<µ

=A(δψ | x1 , . . . , xk+2 ) X + (−1)1+λ+S(λ) xλ · (A(ψ | . . . x cλ . . .) + B(ψ | . . . x cλ . . .)

(4)

λ

X

+

+ C(ψ | . . . x cλ . . .)) (−1)λ+T (λ,µ) (A(ψ | . . . x cλ . . . [xλ , xµ ] . . .)

λ<µ

+ B(ψ | . . . x cλ . . . [xλ , xµ ] . . .) + C(ψ | . . . x cλ . . . [xλ , xµ ] . . .)) In order to show the total sum becomes zero, we separate those terms by three types. Type (0) is of terms which φ is free of the bracket [·, ·] in its variables. Type (I) is of terms which ψ has just one bracket [·, ·] in its variables. Type (II) is of terms which ψ has either two brackets [·, ·] and [·, ·], or double brackets like [[·, ·], ·] or [·, [·, ·]] in its variables. It turns out that the last term in (4) is Type (II) and so Type(II) X = (−1)λ+T (λ,µ)+i+T (i,j) ψ(. . . xbi . . . [xi , xj ] . . . x cλ . . . [xλ , xµ ] . . .)[3] i<j<λ<µ

+

(5)

X

(−1)

λ+T (λ,µ)+i+T (i,j)+U (i,λ)

ψ(. . . xbi . . . x cλ . . . [xi , xj ] . . . [xλ , xµ ] . . .)[3]

i<λ<j<µ

+

X

(6) (−1)λ+T (λ,µ)+i+T (i,µ)+U (i,λ) ψ(. . . xbi . . . x cλ . . . [xi , [xλ , xµ ]] . . .)[3]

i<λ<j=µ

+

X

(−1)λ+T (λ,µ)+i+T (i,j) ψ(. . . xbi . . . x cλ . . . [xλ , xµ ] . . . [xi , xj ] . . .)[3]

i<λ<µ<j

+

X

(7) (−1)

λ+T (λ,µ)+i−1+T (i,j)

ψ(. . . x cλ . . . xbi . . . [xi , xj ] . . . [xλ , xµ ] . . .)[3]

λ
+

X

(8) (−1)λ+T (λ,µ)+i−1+T (i,j) ψ(. . . x cλ . . . xbi . . . [xi , [xλ , xµ ]] . . .)[3]

λ
+

X

(−1)λ+T (λ,µ)+i−1+T (i,j)+U (i,λ) ψ(. . . x cλ . . . xbi . . . [xλ , xµ ] . . . [xi , xj ] . . .)

λ
+

X

(9) (−1)

λ
λ+T (λ,µ)+i−1+T (i,j)+

P

i<`<j

U (λ,`)

ψ(. . . x cλ . . . x cµ . . . [[xλ , xµ ], xj ] . . .)

86

Kentaro Mikami and Tadayoshi Mizutani

X

+

(−1)λ+T (λ,µ)+i−1+T (i,j) ψ(. . . x cλ . . . [xλ , xµ ] . . . xbi . . . [xi , xj ] . . .)

λ<µ
(10) (5) and (10), or (6) and (9), or (7) and (8) cancel out each other, and we can proceed the calculations X

=

(−1)λ+T (λ,µ)+i+T (i,µ)+U (i,λ) ψ(. . . xbi . . . x cλ . . . [xi , [xλ , xµ ]] . . .)

i<λ<j=µ

X

+

(−1)λ+T (λ,µ)+i−1+T (i,j) ψ(. . . x cλ . . . xbi . . . [xi , [xλ , xµ ]] . . .)

λ
P X cλ . . . x cµ . . . [[xλ , xµ ], xj ] . . .) + (−1)λ+T (λ,µ)+i−1+T (i,j)+ i<`<j U (λ,`) ψ(. . . x

λ
Using super Leibniz condition for the last term, we have Type(II) X = (−1)λ+T (λ,µ)+i+T (i,µ)+U (i,λ) ψ(. . . xbi . . . x cλ . . . [xi , [xλ , xµ ]] . . .) i<λ<j=µ

+

X

(−1)λ+T (λ,µ)+i−1+T (i,j) ψ(. . . x cλ . . . xbi . . . [xi , [xλ , xµ ]] . . .)

λ
+

X

(−1)λ+T (λ,µ)+i−1+T (i,j)+

λ
=

X

P

i<`<j

U (λ,`)

ψ(. . . x cλ . . . x cµ . . . [xλ , [xµ , xj ]] + (−1)1+U (λ,µ) [xµ , [xλ , xj ]] . . .) (−1)q+T (q,r)+p+T (p,r)+U (p,q) ψ(. . . x cp . . . x cq . . . [xp , [xq , xr ]] . . .)

p
+

X

(−1)p+T (p,r)+q−1+T (q,r) ψ(. . . x cp . . . x cq . . . [xq , [xp , xr ]] . . .)

p
+

X

(−1)p+T (p,q)+q−1+T (q,r)+

p
+

X

=

p
+

X p
q<`
U (p,`)

ψ(. . . x cp . . . x cq . . . [xp , [xq , xr ]] . . .) (−1)p+T (p,q)+q−1+T (q,r)+

p
X

P

P

q<`
U (p,`)

ψ(. . . x cp . . . x cq . . . (−1)1+U (p,q) [xq , [xp , xr ]] . . .) (−1)p+q+T (q,r) ψ(. . . x cp . . . x cq . . . [xp , [xq , xr ]] . . .)

(−1)T (p,r)+U (p,q) − (−1)T (p,q)+

P

q<`
U (p,`)

(−1)p+q−1+T (q,r) ψ(. . . x cp . . . x cq . . . [xq , [xp , xr ]] . . .)

Algebroids associated with pre-Poisson structures

(−1)T (p,r) − (−1)T (p,q)+

P

q<`
U (p,`)+U (p,q)

87

=0 from the elementary property of T (i, j). Since A(δψ|x1 , . . . , xk+2 ) =(−1)k+2 (δψ)(x1 , . . . , xk+1 ) · xk+2 + (−1)k+2+S(k+2) xk+2 · (δψ)(x1 , . . . , xk+1 ) we recall Remark 4 and reduce (δ 2 ψ)(x1 , . . . , xk+2 ) as (δ 2 ψ)(x1 , . . . , xk+2 ) =(−1)

k+2

+(−1)

k+2+S(k+2)

+

X

(11)

(B(ψ | x1 , . . . , xk+1 ) + C(ψ | x1 , . . . , xk+1 )) · xk+2 xk+2 · (A(ψ | x1 , . . . , xk+1 ) + B(ψ | x1 , . . . , xk+1 )

+C(ψ | x1 , . . . , xk+1 )) (−1)1+λ+S(λ) xλ · (A(ψ | . . . x cλ . . .) + B(ψ | . . . x cλ . . .) + C(ψ | . . . x cλ . . .))

λ

+

X

(−1)λ+T (λ,µ) (A(ψ | . . . x cλ . . . [xλ , xµ ] . . .) + B(ψ | . . . x cλ . . . [xλ , xµ ] . . .))

λ<µ

and we can gather the all terms of Type (I). Type(I) =(−1)k+2 C(ψ | x1 , . . . , xk+1 ) · xk+2 + (−1)k+2+S(k+2) xk+2 · C(ψ | x1 , . . . , xk+1 ) X + (−1)1+λ+S(λ) xλ · C(ψ | . . . x cλ . . .) λ

+

X

(−1)λ+T (λ,µ) A(ψ | . . . x cλ . . . [xλ , xµ ] . . .)

λ<µ
+

X

(−1)λ+T (λ,µ) B 0 (ψ | . . . x cλ . . . [xλ , xµ ] . . .)

λ<µ

(here B 0 means the partial sum of B except the term [xλ , xµ ] · ψ) =(−1)k+2 C(ψ | x1 , . . . , xk+1 ) · xk+2 + (−1)k+2+S(k+2) xk+2 · C(ψ | x1 , . . . , xk+1 ) X + (−1)1+λ+S(λ) xλ · C(ψ | . . . x cλ . . .) λ

+

X

(−1)λ+T (λ,µ)+k+1 ψ(. . . x cλ . . . [xλ , xµ ] . . .) · xk+2

λ<µ
+

X

λ<µ
(−1)λ+T (λ,µ)+k+1+S(k+2) xk+2 · ψ(. . . x cλ . . . [xλ , xµ ] . . .)

88

Kentaro Mikami and Tadayoshi Mizutani

+

X

(−1)λ+T (λ,µ) B 0 (ψ | . . . x cλ . . . [xλ , xµ ] . . .)

λ<µ

=

X λ

+

(−1)1+λ+S(λ) xλ · C(ψ | . . . x cλ . . .)

X

(−1)λ+T (λ,µ) B 0 (ψ | . . . x cλ . . . [xλ , xµ ] . . .)

λ<µ

=

X

(−1)1+λ+S(λ) xλ · (−1)i+T (i,j) ψ(. . . xbi . . . [xi , xj ] . . . x cλ . . .)

i<j<λ

+

X

(−1)1+λ+S(λ) xλ · (−1)i+T (i,j)+U (i,λ) ψ(. . . xbi . . . x cλ . . . [xi , xj ] . . .)

i<λ<j

+

X

(−1)1+λ+S(λ) xλ · (−1)i−1+T (i,j) ψ(. . . x cλ . . . xbi . . . [xi , xj ] . . .)

λ
+

X

(−1)λ+T (λ,µ)+1+i+S(i) xi · ψ(. . . xbi . . . x cλ . . . [xλ , xµ ] . . .)

i<λ<µ

+

X

(−1)λ+T (λ,µ)+1+i−1+S(i)+U (λ,i) xi · ψ(. . . x cλ . . . xbi . . . [xλ , xµ ] . . .)

λ
+

X

(−1)λ+T (λ,µ)+1+i−1+S(i) xi · ψ(. . . x cλ . . . [xλ , xµ ] . . . xbi . . .)

λ<µ
=0 as we see that the first and the last, the second and the fifth, and the third and the fourth cancel out each other. Finally, we will try to show Type(0), which is the sum of terms without bracket [·, ·] inside ψ, is identically to 0. ¿From the reduced equation (11), we see that Type(0) =(−1)k+2 B(ψ | x1 , . . . , xk+1 ) · xk+2 +(−1)k+2+S(k+2) xk+2 · (A(ψ | x1 , . . . , xk+1 ) + B(ψ | x1 , . . . , xk+1 )) X + (−1)1+λ+S(λ) xλ · (A(ψ | . . . x cλ . . .) + B(ψ | . . . x cλ . . .)) λ

+

X

(−1)λ+T (λ,k+2) A(ψ | . . . x cλ . . . [xλ , xk+2 ])

λ
+

X

(−1)λ+T (λ,µ)+1+µ−1+(deg(xλ )+deg(xµ ))(deg(ψ)+

λ<µ

P

`<µ

deg(x` )−deg(xλ ))

× [xλ , xµ ] · ψ(. . . x cλ . . . x cµ . . .) k+2

=(−1) B(ψ | x1 , . . . , xk+1 ) · xk+2 X (−1)1+λ+S(λ) xλ · (A(ψ | . . . x cλ . . .) + B(ψ | . . . x cλ . . .)) + λ
Algebroids associated with pre-Poisson structures

X

+

89

(−1)λ+T (λ,k+2) A(ψ | . . . x cλ . . . [xλ , xk+2 ])

λ
+

X

(−1)λ+µ+S(λ)+S(µ)−U (λ,µ) [xλ , xµ ] · ψ(. . . x cλ . . . x cµ . . .)

λ<µ

We pick up the terms where xk+2 is outside of ψ(), and denote it by Type(0)out , also pick up the terms where xk+2 is inside of ψ(), and denote it by Type(0)in . Then Type(0)in X (−1)1+λ+S(λ) xλ · (−1)1+i+S(i) xi · ψ(. . . xbi . . . x cλ . . .) = i<λ
X

+

(−1)1+λ+S(λ) xλ · (−1)1+i−1+S(i)−U (λ,i) xi · ψ(. . . x cλ . . . xbi . . .)

λ
X

+

(−1)λ+µ+S(λ)+S(µ)−U (λ,µ) [xλ , xµ ] · ψ(. . . x cλ . . . x cµ . . .)

λ<µ
X

=

(−1)i+λ+S(i)+S(λ) xλ · (xi · ψ(. . . xbi . . . x cλ . . .))

i<λ
X

+

(−1)1+λ+i+S(λ)+S(i)−U (λ,i) xλ · (xi · ψ(. . . x cλ . . . xbi . . .))

λ
X

+

(−1)λ+µ+S(λ)+S(µ)−U (λ,µ) [xλ , xµ ] · ψ(. . . x cλ . . . x cµ . . .)

λ<µ
=0 using the super Leibniz property [xλ , xµ ] · ψ() = xλ · (xµ · ψ()) − (−1)U (λ,µ) xµ · (xλ · ψ()). Concerning to Type(0)out , we have Type(0)out =(−1)k+2 B(ψ | x1 , . . . , xk+1 ) · xk+2 X + (−1)1+λ+S(λ) xλ · (A(ψ | . . . x cλ . . .)) λ
X cλ . . . x[ + (−1)1+λ+S(λ) xλ · ((−1)1+k+2−1+S(k+2)−U (k+2,λ) xk+2 · ψ(. . . x k+2 )) λ
+

X

(−1)λ+T (λ,k+2) A(ψ | . . . x cλ . . . [xλ , xk+2 ])

λ
+

X

(−1)λ+k+2+S(λ)+S(k+2)−U (λ,k+2) [xλ , xk+2 ] · ψ(. . . x cλ . . . x[ k+2 )

λ
=(−1)k+2 B(ψ | x1 , . . . , xk+1 ) · xk+2 X + (−1)1+λ+S(λ) xλ · (A(ψ | . . . x cλ . . .)) λ
90

+

Kentaro Mikami and Tadayoshi Mizutani

X

(−1)λ+T (λ,k+2) A(ψ | . . . x cλ . . . [xλ , xk+2 ])

λ
+

X

(−1)λ+k+2+S(λ)+S(k+2)+1 xk+2 · (xλ · ψ(. . . x cλ . . . x[ k+2 ))

λ
=(−1) +

k+2

X

X

! (−1)

1+i+S(i)

xi · ψ(. . . xbi . . .)

· xk+2

i
(−1)1+λ+S(λ)+k+1 xλ · (ψ(. . . x cλ . . . x[ k+2 ) · xk+2 )

λ
+

X

(−1)1+λ+S(λ)+k+1+S(k+2)+U (λ,k+2) xλ · (xk+2 · ψ(. . . x cλ . . . x[ k+2 ))

λ
+

X

(−1)λ+T (λ,k+2)+k+1 ψ(. . . x cλ . . . x[ k+2 ) · [xλ , xk+2 ]

λ
+

X

(−1)λ+k+1+S(λ)+S(k+2)−U (k+2,λ) [xλ , xk+2 ] · ψ(. . . x cλ . . . x[ k+2 )

λ
+

X

(−1)λ+k+2+S(λ)+S(k+2)+1 xk+2 · (xλ · ψ(. . . x cλ . . . x[ k+2 ))

λ
We see that the third, the fifth, and the last terms cancel out because of super Leibniz property, and we continue the calculations: ! X 1+i+S(i) k+2 (−1) xi · ψ(. . . xbi . . .) · xk+2 Type(0)out =(−1) +

X

i
(−1)1+λ+S(λ)+k+1 xλ · (ψ(. . . x cλ . . . x[ k+2 ) · xk+2 )

λ
+

X

(−1)λ+T (λ,k+2)+k+1 ψ(. . . x cλ . . . x[ k+2 ) · [xλ , xk+2 ]

λ
=(−1)k+2+1+k+1+S(k+1) xk+1 · ψ(. . . x[ k+1 ) · xk+2 X +(−1)k+2 ( (−1)1+i+S(i) xi · ψ(. . . xbi . . . xk+1 )) · xk+2 i
+(−1) xk+1 · (ψ(x1 . . . xk ) · xk+2 ) X 1+i+S(i)+k+1 + (−1) xi · (ψ(. . . xbi . . . xk+1 ) · xk+2 ) i
+(−1)k+1+k+1 ψ(x1 . . . xk ) · [xk+1 , xk+2 ] X + (−1)i+T (i,k+2)+k+1 ψ(. . . xbi . . . xk+1 ) · [xi , xk+2 ] i
Algebroids associated with pre-Poisson structures

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=(−1)S(k+1) (xk+1 · ψ(x1 . . . xk )) · xk+2 +(−1)1+S(k+1) xk+1 · (ψ(x1 . . . xk ) · xk+2 ) +(−1)0 ψ(x1 . . . xk ) · [xk+1 , xk+2 ] =0 . Example 13. Let g be a super Leibniz algebra and take g itself as bi-module A. Choose and fix a homogeneous element π. We define a linear map fπ : g → A by fπ (x) = [π, x]. Since deg(fπ (x)) = deg(π) + deg(x) we see that deg(fπ ) = deg(π). (δfπ )(x, y) =fπ (x) · y + (−1)deg(fπ ) deg(x) x · fπ (y) − fπ ([x, y]) =[[π, x], y] + (−1)deg(π) deg(x) [x, [π, y]] − [π, [x, y]] =0 in fact, fπ is exact as (δπ)(x) = − [π, x] = −fπ (x) . Example 14. As a super Leibniz algebra g and its bimodule, we consider the exterior graded algebra of T(M ) with the Schouten bracket. Then, the map fπ is known as the coboundary operator of the Poisson cohomology when π is a Poisson 2-tensor field. Again choose and fix a homogeneous element π, and consider a linear map h : g → A by h(x) = π ∧ x. We see that deg(h) = deg(π) + 1, and (δh)(x, y) =h(x) · y + (−1)deg(h) deg(x) x · h(y) − h([x, y]) =[π ∧ x, y] + (−1)(deg(π)+1) deg(x) [x, π ∧ y] − π ∧ [x, y] =0 where we used the super skew-symmetric property and also the rule of wedge product and the Schouten bracket. h is exact if there is some element z satisfying −[z, x] = π ∧ x for all x. This looks like unimodular condition, but the meaning of this equation is not clear so far. Assume that we are given a super Leibniz algebra g, a super Leibniz bimodule A, and a super vector space g0 . We take a homogeneous bi-linear map ψ : g ⊗ g → A with degree 0, and a linear exact sequence Π

0 −→ A −→ g0 −→ g −→ 0 and a linear section s of Π, where Π and s are maps of homogeneous degree 0. Likewise of ordinary Leibniz modules, we define a multiplication on g0 by [[s(x1 ) + a1 , s(x2 ) + a2 ]] := s[x1 , x2 ] + ψ(x1 , x2 ) + x1 · a2 + a1 · x2

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for x1 , x2 ∈ g and a1 , a2 ∈ A, where “·” is the super Leibniz action of g on A. For x0 i := s(xi ) + ai ∈ g0 (i=1,2,3), we have [[[[x0 1 , x0 2 ]], x0 3 ]] =s[[x1 , x2 ], x3 ] + ψ([x1 , x2 ], x3 ) + ψ(x1 , x2 ) · x3 + [x1 , x2 ] · a3 + (x1 · a2 ) · x3 + (a1 · x2 ) · x3 and [[x0 1 , [[x0 2 , x0 3 ]]]] =s[x1 , [x2 , x3 ]] + ψ(x1 , [x2 , x3 ]) + x1 · ψ(x2 , x3 ) + x1 · (x2 · a3 ) + x1 · (a2 · x3 ) + a1 · [x2 , x3 ] and so [[x0 2 , [[x0 1 , x0 3 ]]]] =s[x2 , [x1 , x3 ]] + ψ(x2 , [x1 , x3 ]) + x2 · ψ(x1 , x3 ) + x2 · (x1 · a3 ) + x2 · (a1 · x3 ) + a2 · [x1 , x3 ] . [[·, ·]] satisfies super Leibniz property, i.e., 0

0

[[[[x0 1 , x0 2 ]], x0 3 ]] = [[x0 1 , [[x0 2 , x0 3 ]]]] − (−1)deg(x 1 ) deg(x 2 ) [[x0 2 , [[x0 1 , x0 3 ]]]] for homogeneous elements x0 i (i.e., xi or ai must become 0) if and only if (a1 · x2 ) · x3 =a1 · [x2 , x3 ] − (−1)deg(a1 ) deg(x2 ) x2 · (a1 · x3 ) (x1 · a2 ) · x3 =x1 · (a2 · x3 ) − (−1)

deg(x1 ) deg(a2 )

if a2 = a3 = 0

a2 · [x1 , x3 ] if a1 = a3 = 0

[x1 , x2 ] · a3 =x1 · (x2 · a3 ) − (−1)deg(x1 ) deg(x2 ) x2 · (x1 · a3 ) if a1 = a2 = 0 if we put a1 = a2 = a3 = 0 and split the result into Ims-part and ker Π = Apart, we get s[[x1 , x2 ], x3 ] = s[x1 , [x2 , x3 ]] − (−1)deg(x1 ) deg(x2 ) s[x2 , [x1 , x3 ]]

(12)

and ψ([x1 , x2 ], x3 ) + ψ(x1 , x2 ) · x3 = ψ(x1 , [x2 , x3 ]) + x1 · ψ(x2 , x3 ) − (−1)deg(x1 ) deg(x2 ) ψ(x2 , [x1 , x3 ]) − (−1)deg(x1 ) deg(x2 ) x2 · ψ(x1 , x3 ) (13) The first 3 equations are just the property of the super Leibniz action of the super Leibniz algebra g to A, and the equation (12) is just saying that [·, ·] is a super Leibniz multiplication, and the equation (13) is equivalent to ψ being a 0-homogeneous 2-cocycle of super Leibniz coboundary operator. Thus, we see that Proposition 6. For a given super Leibniz bracket [·, ·] and a super Leibniz action, the new bracket [[·, ·]] becomes a super Leibniz multiplication if and only if ψ is a 0-homogeneous 2-cocycle in the super Leibniz module sense.

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From the definition, we see that [[A, A]] = {0} and Π is a super Leibniz algebra homomorphism. We call such an exact sequence as an abelian super Leibniz extension of g with the kernel A. Also, from the definition of the multiplication [[·, ·]], we see that [[s(x1 ), s(x2 )]] − s[x1 , x2 ] = ψ(x1 , x2 )

(∀x1 , x2 ∈ g).

We can follow Theorem 4, and get the same result. Theorem 6. For a given super Leibniz algebra g and its super bi-module A, the class of super Leibniz abelian extensions up to super Leibniz 0homogeneous isomorphisms are in one-to-one correspondence to the second super Leibniz homogeneous cohomology group H02 (g, A), whose elements are classes of 2-cocycles which are homogeneous and with degree 0. Proof. In this proof, we assume every (multi-)linear maps are homogeneous of degree 0. Given an abelian super Leibniz extension g0 , which is of a super Leibniz algebra g with the kernel A, Π

0 −→ A −→ g0 −→ g −→ 0 , and take a homogeneous 0 linear section s of Π, i.e., a linear map s : g → g0 with Π ◦ s = idg and deg(s(x)) = deg(x) for each homogeneous element x ∈ g. Then for x ∈ g and a ∈ A, [[s(x), a]] ∈ A and [[a, s(x)]] ∈ A, and their values are independent of choice of section s. We see easily that A is a super Leibniz g-module by the left action x · a := [[s(x), a]] and the right action a · x := [[a, s(x)]]. For each x0 i ∈ g0 , put xi := Π(x0 i ) and ai := x0 i − s(xi ) (i = 1, 2, 3). Then [[x0 1 , x0 2 ]] = [[s(x1 ) + a1 , s(x2 ) + a2 ]] = [[s(x1 ), s(x2 )]] + [[s(x1 ), a2 ]] + [[a1 , s(x2 )]] + [[a1 , a2 ]] = [[s(x1 ), s(x2 )]] + [[s(x1 ), a2 ]] + [[a1 , s(x2 )]] . Using the section s, we define ψs (x1 , x2 ) := [[s(x1 ), s(x2 )]] − s[x1 , x2 ]

(x1 , x2 ∈ g) .

ψs is bi-linear homogeneous of degree 0 and valued in A. Now the bracket [[·, ·]] on g0 can be written as [[x0 1 , x0 2 ]] = s[x1 , x2 ] + ψs (x1 , x2 ) + x1 · a2 + a1 · x2 . We can prove directly that [[·, ·]] being a super Leibniz bracket leads to ψs being a 0-homogeneous 2-cocycle and vice versa. If we take another section s0 : g → g0 with Π ◦ s0 = idg , then t := s0 − s is A-valued and we see that ψs0 − ψs = δt ,

i.e., [ψs0 ] = [ψs ] ∈ H02 (g, A) .

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Take two abelian extensions of a super Leibniz algebra g, Π

0 −→ A −→ g0 −→ g −→ 0 and ˜ Π

0 −→ A −→ ˜ g0 −→ g −→ 0 and sections s and s˜ respectively. We will show here that those two extensions are isomorphic in super Leibniz algebraic sense if and only if [ψs ] = [ψs˜] ∈ H02 (g, A). Assume that [ψs ] = [ψs˜] ∈ H02 (g, A). Then we have some t : g → A satisfying ψs˜ − ψs = δt. We now define a linear map f : g0 → ˜ g0 by f|A = idA

and f (s(x)) := s˜(x) − t(x)

(∀x ∈ g) .

Similarly, define a linear map f˜ : g˜0 → g0 by f˜|A = idA Then we have

and

f˜(˜ s(x)) := s(x) + t(x)

f ◦ f˜ = idg˜0

and

(∀x ∈ g) .

f˜ ◦ f = idg0

and so f (and f˜) is bijective. ˜ is super Leibniz algebraic isomorphism as follows. We see that f (and f) ˜[[f (s(x) + a), f (s(y) + b)˜]] =˜[[˜ s(x) − t(x) + a), s˜(y) − t(y) + b)˜]] =˜[[˜ s(x), s˜(y)˜]] + x · (b − t(y)) + (a − t(x)) · b =˜ s[x, y] + ψs˜(x, y) + x · (b − t(y)) + (a − t(x)) · b =˜ s[x, y] + ψs (x, y) + (δt)(x, y) + x · (b − t(y)) + (a − t(x)) · b =˜ s[x, y] + ψs (x, y) − t[x, y] + x · b + a · b and f ([[s(x) + a, s(y) + b]]) =f ([[s(x), s(y)]] + x · b + a · y) =f (s([x, y]) + ψs (x, y) + x · b + a · y) =˜ s[x, y] − t[x, y] + ψs (x, y) + x · b + a · y . Now assume that given two extensions are isomorphic, namely there is a super ˜ with the properties F|A = idA and g0 , Π) Leibniz isomorphism F : (g0 , Π) → (˜ ˜ and so ˜ ◦ F = Π. Define s˜0 := F ◦ s : g → ˜ g0 . Then s˜0 is a section of Π Π [ψs˜0 ] = [ψs˜]. Since

Algebroids associated with pre-Poisson structures

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ψs˜0 (x, y) = ˜[[F s(x), F s(y)˜]] − F s[x, y] = F [[s(x), s(y)]] − F s[x, y] = F (ψs (x, y) + s[x, y]) − F s[x, y] = F (ψs (x, y)) = ψs (x, y) , we thus have [ψs˜0 ] = [ψs˜] = [ψs ]. Example 15. We deal with a concrete super Leibniz algebra and whose the second cohomology group. Starting from (ax + b) Lie algebra, we make use of a central extension of it, and get a Leibniz algebra of dimension 3 whose table of multiplication is given by z1 z2 z1 κz3 z2 − z 3 z2 −z2 + z3 0 z3 0 0

z3 0 0 0

where κ 6= 0 is a constant, and the table means [z1 , z2 ] is given by z2 − z3 , and so on. We give them 0 degree. Keeping zi ∧ zi+1 in mind, we prepare linearly independent elements U1 , U2 , and U3 , and give them degree 1. Also, we prepare a non-zero element V of degree 2. We fix the multiplication [zi , Uj ] and [zi , V ] by U1 z 1 U1 − U 2 + U 3 z2 −U2 z3 0

U2 U2 0 0

U3 0 U2 0

V V 0 0

We collect all the equations coming from requiring super Leibniz identities, and solve the linear system, and find a solution as following. z1 z2 z3 U1 z1 κz3 z2 − z 3 0 U 1 − U 2 + U 3 z2 −z2 + z3 0 0 −U2 z3 0 0 0 0 U1 −U1 + αU2 − U3 U2 /κ U2 /κ −2βV U2 0 0 0 0 U3 0 0 0 βV V −V 0 0 0

U2 U2 0 0 0 0 0 0

U3 0 U2 0 βV 0 0 0

V V 0 0 0 0 0 0

where α, β are arbitrary constants, and κ 6= 0. Assuming κ = 1 for simplicity, we manipulate the second super cohomology group of the super Leibniz algebra and get the two cases. If β = 0 then the space of exact 2-cochains is 14 dimensional, and the space of closed 2-cochains is 25 dimensional, and so the second cohomology group is 11 dimensional. On the other hand, if β 6= 0, then the space of exact 2-cochains is 15 dimensional, and the space of closed 2-cochains is 22 dimensional, and so the second cohomology group is 7 dimensional.

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References 1. T. Courant. Dirac manifolds. Trans. Amer. Math. Soc., 319:63l–661, 1990. 2. B. Fuchssteiner. The Lie algebra structure of degenerate Hamiltonian and biHamiltonian systems. Progr. Theoret. Phys., 68(4):1082–1104, 1982. 3. Y. Kosmann-Schwarzbach, Quasi, twisted, and all that...in Poisson geometry and Lie algebroid theory, The breadth of symplectic and Poisson geometry, Progr. Math.,232, Birkh¨ auser Boston, 363–389, 2005. 4. J. L. Loday. Cyclic Homology, volume 301 of Grund. Math. Wissen. SpringerVerlag, 1992. xviii+454p. ISBN:3-540-53339-7. 5. J. L. Loday. Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann., 296:139–158, 1993. 6. K. Mikami and T. Mizutani. Lie algebroids associated with 2-vector fields through deformed Schouten bracket. preprint, 2003. 7. K. Mikami and T. Mizutani. Integrability of plane fields defined by 2-vector fields. International Journal of Mathematics, 16(1):197–212, 2005. 8. K. Mikami and T. Mizutani. Lie algebroid associated with an almost Dirac structure. Travaux math´ematiques, 16:255–264, 2005. 9. I. Vaisman. Lectures on the geometry of Poisson manifolds. Birkh¨ auser, Basel, 1994.

Examples of Groupoid Naoya MIYAZAKI Department of Mathematics, Hiyoshi Campus, Keio University, Yokohama, 223-8521, JAPAN, [email protected]

Keywords: stack, groupoid, torsor, spin gerbe, index theorem. Mathematics Subject Classification (2000): Primary 53C27; Secondary 58H05 Abstract In the present paper, we introduce spin groupoids associated with the degree two Stiefel-Whitney classes, and we also introduce spinor torsors equipped with certain connections compatible with an action of spin groupoid. Using the connection, we propose a Dirac-like operator acting on the space of smooth sections of the spinor torsor. We can show that its Dirac-Laplacian admits a heat kernel in a formal sense.

1 Introduction The notion of stack (sheaf of category), groupoid, torsor are regarded as bridges joining physics to mathematics. Other than particle physics, they appear in several fields such as microlocal analysis, extended notion of quantum groups, Courant algebroids and deformation quantization, etc. In this paper we would like to review examples of groupoids and torsors, and to introduce spin groupoids and spinor torsors endowed with suitable connections. It is known that, through the notion of characteristic class, the degree two cohomology classes in H 2 (M, Z) of a manifold M have geometric realizations, that is, hermitian line bundles (or principal circle bundles). For example, the hermitian line bundle is constructed in the following way: Let ω be an integral symplectic structure, then we can take a family of local functions (resp. local 1-forms) {fαβ }αβ (resp. {θα }α ) defined on open sets Uα ∩ Uβ (resp. Uα ), satisfying d(θα ) = (δω)α , d(fαβ ) = (δθ)αβ , cαβγ = (δf )αβγ , where U = {Uα } is a good covering of a symplectic manifold (M, ω), d is the ˇ deRham exterior differential operator, δ is the Cech coboundary operator, and ˇ {cαβγ } is a Cech 2-cocycle corresponding to ω. Setting hαβ = exp[2πifαβ ], 1 we see θα − θβ = 2πi d log hαβ . This equation ensures the exsistence of a line bundle defined by 97

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Naoya MIYAZAKI

L=

a

hαβ

(Uα × C)/ ∼ ,

∇ξ (φα 1α ) = (ξφα + 2πiθα (ξ)φα )1α ,

which gives the desired bundle equipped with a connection whose curvature equals ω. It is also known that the line bundle plays an important role in the theory of Souriau-Kostant quantization [29] and then the line bundle is referred to as a prequantum line bundle. On the other hand, the degree two cohomology classes in certain cohomology groups such as H 2 (M, S 1 ), H 2 (M, R) and H 2 (M, R[[ν]]) admit various geometric interpretations which appear in several fields such as the ChernSimons theory, Toeplitz operator algebras and deformation quantization. The background informations on these studies can be traced back to the works by Giraud [10]. The end of 1960’, he initiated the theory of gerbes. Shortly after his works the notion of gerbes is deeply and widely investigated by Brylinski. Especially, in [6], he studied connective structures and curvings in order to analyse gerbes in detail. In recent years, notable progress has been made in the theory of gerbes as the passage from mathematical physics to twisted Ktheory [1], [3], [5], [6], [7], [8], [9], [11], [12], [14], [17] [18], [19], [20], [22], [23], [24] and [25] etc. In this paper, we introduce a groupoid and a torsor with a certain connection (these are called a spin groupoid and a spinor torsor with a spinor connection. cf. Theorem 1). We are also concerned with a Dirac-like operator naturally defined by the spinor connection, and study the fundamental properties of its square operator (Dirac-Laplacian operator) (cf. Theorem 2).

2 Groupoids In the present section, we recall the precise definitions of Lie groupoids and torsors. To illustrate what are groupoids, we begin with a basic example. Example 1. (Transformation groupoids) Let M be a manifold and ast // M ” gives a sume that a Lie group G acts on M . Then “ M × G s typical example of Lie groupoid in the sense of Definition 1 below, where s(m, g) = m, (resp. t(m, g) = mg) is referred to as a source map (resp. a target map), and a partial product is defined by (m, g) • (n, h) = (m, gh) when mg = n (see the picture below). mg=n (m,g)

(n,h)

Examples of Groupoid

99

Extracting the essential properties used in Example 1, we arrive on the definition of Lie groupoid as follows: Definition 1. Let Γ and Γ0 be smooth manifolds. Then the diagram t / / Γ0 ” is called a Lie (smooth) groupoid over a manifold M if the “Γ s following conditions are satisfied (1) ∃ • (partial product) : Γ ×t,Γ0 ,s Γ := {(γ1 , γ2 )|t(γ1 ) = s(γ2 )} → Γ s.t. γ1 • (γ2 • γ3 ) = (γ1 • γ2 ) • γ3 , (2) ∃ ε (unit) : Γ0 → Γ s.t. ε(s(γ)) • γ = γ, γ • ε(t(γ)) = γ, (3) ∃ i (inverse) : Γ → Γ s.t. i(γ) • γ = ε(t(γ)), γ • i(γ) = ε(s(γ)), (4) Γ and Γ0 are smooth manifolds, and all structure maps are smooth maps, (5) the source and target maps are surjective submersions. The triple (•, ε, i) is called a groupoid operation. A pair of smooth maps Φ = (φ1 , φ0 ) from a smooth groupoid into another one is called a groupoid homomorphism if the following diagram Φ:

φ1

Γ˜ s ↓↓ t φ0 ×φ0 ˜ Γ0 × Γ0 −→ Γ0 × Γ˜0 Γ s ↓↓ t

−→

is compatible with respect to all smooth maps. Our aregments in the present paper are based on the following example. π

Example 2. (Fiber product) Let X → M be a fibration. Then Γ := X ×π X

t s

// Γ := X 0

(1)

is a Lie groupoid, where s and t are the projection with respect to the first and second variables. Especially, `it is important to consider the case of M = ∪λ Uλ ˇ (Cech covering) and X := λ Uλ . Then we have the following diagram: a s,t s(x; i, j) = (x; i) ∈ Ui ⊂ Γ0 . (2) Ui ∩ Uj 3 (x; i, j) =⇒ Γ = t(x; i, j) = (x; j) ∈ Uj ij

Remark that this groupoid is useful to introduce a geometric realization of a degree two cohomology class in H 2 (M, S 1 ), and the groupoid above is referred ˇ to as a Cech groupoid. Example 3.(Kronecker foliation) Let T 2 be a torus and Fθ a foliation defined as a flow corresponding to a vector field Xθ = ∂/∂x + θ∂/∂y (θ:irrational). Set Γ = {γ : an oriented segment along a leaf} = (R × R × S 1 )/ ∼, where (x, y, t) ∼ (x + 1, y + 1, t + θ), s(γ) = s(x, y, t) = “the initial point of γ” = (x, t), t(γ) = t(x, y, t) = “the end point of γ” = (y, t).

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We identify T 2 = (R × S 1 )/ ∼, where (x, t) ∼ (x + 1, t + θ), and then each leaf is defined by the image of Fθ . Thus, we have an example of groupoid t / / Γ := T 2 called the Kronecker holonomy groupoid1 . Γ s

0

Example 4. (Z2 -gerbes) We give here a typical example of S 1 -central exπ tension [24]. Let SO(M ) → M be the orthonormal-frame bundle (a principal SO(n)-bundle). We consider the following fiber product t

SO(M ) ×π SO(M )

s

// SO(M ) .

Thanks to Example 2, it gives a Lie groupoid. By the central extension below, $

1 → Z2 → Spin(n) → SO(n) → 1,

(3)

^ ) of the Lie groupoid SO(M ) ×π SO(M ) we can define a Z2 -extension SO(M in the following way: Set ^) SO(M := (p1 , g, p2 ) (p1 , p2 )∈SO(M )×π SO(M ), g ∈ Spin(n), p1 $(g) = p2 (4) then we obtain a Z2 -central extension (a Lie groupoid morphism) ^) [ SO(M

t s

// SO(M ) ] −→ [ SO(M ) × SO(M ) π

t s

// SO(M ) ].

(5)

Next we recall the definition of torsors. t

Definition 2. Suppose that Γ

s

// Γ0 is a Lie groupoid. Then a manifold

t

// Γ acts on a manifold P ) if there exist a P is called a Γ -space (or Γ 0 s smooth map J : P → Γ0 referred to as a momentum map and a smooth map called an action : Γ ×t,γ0 ,J P → P such that (1) (γ1 • γ2 ) x = γ1 (γ2 x), (2) ε(J(x))x = x. In addition, if the action is effective2 and proper then P is referred to as a Γ -torsor. Remark. In a similar way, we can give the definition of bitorsor. We also recall the precise definition of Γ -equivariant morphism.

1 2

Remark that this groupoid is not a Hausdorff space This might be replaced by free

Examples of Groupoid

Definition 3. Let P and Q be [ Γ J

t s

101

// Γ ] -spaces with momentum maps 0

JQ

P P → Γ0 and Q → Γ0 . A smooth map Ψ = (ψ1 , identity) from (P, Γ0 ) into (Q, Γ0 ) is called a Γ -equivariant morphism if it satisfies

JP (x) = tP (γ) ⇒ “JQ ◦ ψ1 (x) = tQ (x) and ψ1 ( • x) = γ ψ1 (x)”. Under the notations above, we can recall the notion of quasi-Morita equivalence. t / t / / Γ0 ] are Lie groupoids. / Γ0 ] and [ Γ˜ Definition 4. Assume that [ Γ s s Then Γ and Γ˜ are said to be quasi-Morita equivalent (denoted by Γ ∼q−Morita Γ˜ ) if they satisfy one of the following equivalent conditions: (i) ∃ Γˆ :a Lie groupoid and ∃ Γˆ → Γ, Γˆ → Γ˜ : groupoid morphisms, (ii) ∃ (Γ, Γ˜ )-bitorsor P , s.t. (γ x) γ˜ = γ (x γ˜ ).

As for the relation above it is easy to see that Proposition 1. The relation ∼q−Morita above is an equivalence relation.

3 Spin groupoids and spinor torsors In the present section we introduce spin groupoids and spinor torsors. As well known (cf. [6] and [9]) that for a short exact sequence (central extension) $

1→A→B→C →0

(6)

of sheaves of groups on a space M , there exists an exact sequence of pointed sets 3 , δ

0 1 → H 0 (M, A) → H 0 (M, B) → H 0 (M, C) →

δ1

H 1 (M, A) → H 1 (M, B) → H 1 (M, C) → H 2 (M, A).

(7)

ˇ Here δ1 : H 1 (M, C) → H 2 (M, A) is defined as follows. Let cij be a Cech 1-cocycle of the good covering {Ui } with values in C such that cij = $(bij ) ˇ for bij ∈ Γ (Uij , B). Then aijk = b−1 ik bij bjk gives a Cech 2-cocycle {aijk } with values in A. δ1 then maps the cohomology class of {cij } to that of {aijk }. ¿From the above point of view, we wish to relate an explicit groupoid to the degree two sheaf cohomology H 2 (M, A) when Eq. (6) = [1 → Z2 → Spin(n) → SO(n) → 1]. We begin with basics of Clifford algebras. 3

Note that since A is a sheaf of abelian group H 2 (M, A) is defined. It is also known that, under the same assumption above, if B is soft then δ1 : H 1 (M, C) → H 2 (M, A) is a bijection.

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Definition 5. Let (V ∗ , g) be an n = 2l dimensional4 Euclidean space. Clifford algebra C`(V ∗ ) is an algebra formally generated by an orthonomal basis e1 , · · · , en with respect to g over C satisfying eα eβ + eβ eα = −2δ αβ .

(8)

We can also define the spinor space S and Clifford multiplication c by the standard manner: Set D √ 1 n E W = wj = {e2α−1 − −1e2α } : 1 ≤ α ≤ , (9) 2 2 C S = ΛW, S± = Λ± W, (10) and for any element s ∈ S, set c(w)s = 21/2 w ∧ s (w ∈ W ), c(w)s ¯ = −2

1/2

ι(w)s ¯ = −2

1/2

(11)

¯ ). g(w, ¯ s) (w ¯∈W

(12)

We can construct an extended Lie groupoid and a torsor in the following way. t / / Γ0 a Theorem 1. Let U be a good covering of a manifold M and Γ s ˇ Cech groupoid. Set ` −1 ^ ` (a) “SP IN ”(M ):= π (Ui ∩ Uj ) → Γ0 , where π −1 ^ (Ui ∩ Uj ) := (Ui ∩ Uj ) × Spin(n), (b) s(x; i, j; g1 ) := (x, i), t(x; i, j; g2 ) := (x, j) (source and target maps), (c) (x; i, j; g1 ) • (x; j, k; g2 ) := (x; i, k; Ckij Adφ˜−1 (g1 ) · g2 ) (partial product) jk

(d) ε(x; i, j) := (x; i, j; 1) (unit), −1 (e) i(x; i, j; g) := (x; j, i; Adφij ) (inverse), where φ˜ := (φ˜ij ) is a family ˜ g of Spin(n)-valued functions obtained by lifting of the given family of SO(n)valued transition functions (φij ) corresponding to the orthonormal frame bundle SO(M ) and Cijk := φ˜ij φ˜jk φ˜ki is the cocycle which corresponds to the s // Γ with the opStiefel-Whitney class. Then the diagram “SP IN ”(M ) t

0

erations •, ε, i is a Lie groupoid. Define S :=

a

−1 (U ) := π^ i

a

Ui × S → Γ 0 ,

(13)

which are endowed with the Clifford multiplication, and a spin groupoid action, in fact, using the trivialization above, the action5 is defined by (x; i, j; g) (x; j; s) := (x; i; φ˜ij gs). 4 5

This condition is not essential. The moment map J = id.

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Proof. By direct computations, we can verify that the spaces and operations (a)-(e) defined as above satisfy the conditions (1)-(5) in Definition 1. Definition 6. “SP IN ”(M ) (resp. S) is called a spin groupoid (resp. a spinor torsor). Next we introduce a spinor torsor connection. Computing the push2:1 forward of $ : Spin(2n) → SO(2n), we have $∗−1 : so(2n) → spin(2n) : eα ∧ eβ 7→

1 α β 1 [e , e ] = eα · eβ , 4 2

and then we consider the connection form: X 1 X −1 LC β β LC $∗ (ωi,αβ ) ⊗ [eα ωi = $∗−1 (ωiLC ) = $∗−1 ( ωi,αβ ⊗ eα i , ei ], i ∧ ei ) = 4 α<β

α<β

where ω LC is so(2n)-valued Levi-Civita connection 1-form on Γ0 = urally induced by the Levi-Civita connection on the base manifold.

`

Ui nat-

Definition 7. Set a family of covariant derivatives ∇S := {∇i } acting on the space of all smooth sections of S, where ωi is the connection form, ρ denotes the spinor representation, X X X i ∇i (xi , wiA siA ) := (xi , wiA (dsiA + ρ(ωi )B A sB )), A

A

B

and wiA stands for a local frame defined by w a1 ∧· · ·∧wap (A = (a1 , · · · , ap ), a1 < · · · < ap ) on the trivialization Ui × S. ∇S = {∇i } is referred to as a spinor torsor connection. Proposition 2. The connection ∇S = {∇i } is compatible with the “SP IN ”(M )action in the sense below; X (xij , g) ∇j (xj , wjB sjB ) i

= ∇ (xij , g) (xj ,

B

X

X j wjB sjB ) + Adφ˜ij Adg (ωj ) − ωj · (xij , g) ( wjB SB ) .

B

B

Thanks to the formula above, it is easy to see that Proposition 3. There is a spinor torsor connection ∇S on S, and exp curv(∇S ) is a closed form on Γ0 which descends to give a form on the manifold M . Thus it defines a cohomology class of H 2∗ (M, R). Under the notation above, we introduce the Dirac operator acting on S in the standard way.

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Definition 8. D /S =

X

c(eα )∇Seα .

(14)

We remark that it is easy to verify that Proposition 4. D /SS is anti-commutative with the Clifford multiplication and compatible with “SP IN ”(M )-action in the certain sense. With a slight modification, we can adapt the Bochner-Weizenb¨ och-Lichnerowicz principle in such a way that we can apply it to our Dirac operator. Then we have the following. r

Proposition 5. (D /S )2 = ∇∗ ∇ + Γ40 , where rΓ0 is a scalar curvature on Γ0 = ` Ui . Hence, the operator D /S is elliptic. Therefore the standard argument gives Proposition 6. There exists the heat kernel {kt (xi , xi )}i acting on S with respect to (D /S )2 in a formal sense 6 , where xi is a normal coordinate centered at the point p on Γ0 . Proof. In fact, solving the Hamilton-Jacobi and transport equations we obtain the heat kernel in a formal sense. Here we recall the following (cf. [4] and [26]). 1 X LC g R (∂β , ∂γ )∂α , ∂δ xδ eβ eγ 8 p β,γ,δ X + O(|x|2 )eβ eγ + O(|x|),

∇∂α (x) = ∂α −

(15)

β,γ

where RLC denotes the curvature of the Levi-Civita connection, and O(r) denotes Landau’s symbol. We see that the trace along the fiber of this form descends to give a form on the manifold M . Moreover we also have that Theorem 2. There exists an element k γ,t ∈ C ∞ (M ; C`(V ∗ ))[[t]] such that π ∗ (k Chirality, t (π(p))) = trγ kt (p, p) where γ =

√ n2 1 2 −1 e e · · · en is the chirality operator and trγ is the super trace.

Proof. With a slight modification, we can adapt the method used in the proof of Atiyah-Singer index theorem based on the heat kernel method in such a

6

Strictly speaking, the heat kernel kt (xi , xi ) should be regarded as an (S ⊗ S∗ ) ⊗ “density bundle”-valued section.

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way that we can apply it to the present case. Combining (15) with Getzler’s conformal rescaling method7 gives the index form, kt (xi , xi ) = (4πt)

−n 2

det 1/2

1 D tR tR E tR/2 × exp − xi coth xi , sinh(tR/2) 4t 2 2

where the matrix R is defined by R = [(Rpi ∂αi , ∂βi )]αβ ∈ M (n, R) ⊗ ∧V ∗ . Furthmore according to Proposition 3, trγ kt descends to give a form on the base manifold.

4 Analytic properties In this section, we are concerned with the wave front set (singular spectrum), (cf. [15], [16], [27] and [28]). If u ∈ E 0 (Rn ) then the singular support of u, denoted by sing supp u, is the set of all x ∈ Rn such that x has no open neighborhood on which the restriction of u is C ∞ . Define L(u) as the closed cone consisting of all η ∈ Rn − {0} such that η has no conic neightborhood C in which |ˆ u(ξ)| ≤ CN (1 + |ξ|)−N , N = 1, 2, . . . , ξ ∈ C. Then we have Lemma 1. If φ ∈ Cc∞ (Rn ) and u ∈ E 0 (Rn ), then L(φu) ⊂ L(u). Let M be a smooth manifold. The wave front set of u consists of certain (x, ξ) ∈ T ∗ M − M with x ∈ sing supp u. More precisely, n o \ W F (u) = (x, ξ) ∈ T ∗ M − M : ξ ∈ L(φu) . φ∈Cc∞ (M ), φ(x)6=0

We next recall the criterion of the wave front set of the kernel distribution ¯ x, ξ) ∈ S 0 (cf. [15], [16] and [27] u(¯ x, x) ∈ D0 (M × M ). For any symbol a ˜(¯ x, ξ, 0 for the definition of the space S of all symbols of order zero), put ¯ x, ξ) ∈ T ∗ (M × M ) − (M × M ) : lim ¯ x, τ ξ)| = 0}. γ(˜ a) = {(¯ x, ξ, a(¯ x, τ ξ, τ →0 |˜ (16) Then we obtain Proposition 7. Let W F (u) be the wave front set for the distribution u(¯ x, x). Then 7

Conformal rescaling means multiplication u (resp.u−1 ) to covariant (resp. con−1 1 travariant) time direction, and u 2 (resp. u 2 ) to covariant (resp. contravariant) space direction. As mentioned in the footnote of Proposition 6, since the heat kernel kt (xi , xi ) should be regarded as an (S ⊗ S∗ ) ⊗ “density bundle” -valued section, in order to define conformal rescaling, we took account of not only t, x and dx, but also “density bundle”. See [4] for details.

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W F (u) =

\

γ(˜ a),

(17)

P (˜ a)u∈C ∞ ,˜ a∈S 0

where P (˜ a) is a pseudo-differential operator with a symbol a ˜, that is, Z √ ¯ ¯ ¯ x, ξ)e ~−1 {(¯x,x)−(¯y,y)}(ξ,ξ) u(¯ y , y)dyd¯ ydξdξ. P (˜ a)u(¯ x, x) = a ˜(¯ x, ξ, The criterion (17) of the wave front set may be interpreted as follows: a ˜ in (16) plays the role of annihilating the directions of covectors which cause the singular support of u, and the wave front set is the minimal subset of such directions. Under the definitions above, we have the following: Theorem 3. The wave front set of the fundamental solution for the Diraclike operator D /S descends to give a conic subset CC of the cotangent bundle T ∗M . Proof. It is obvious that the wave front set of the fundamental solution deˇ termines a conic subset of the cotangent bundle T ∗ Γ0 where Γ0 is the Cech groupoid. Combining Propositions 2, 7 and Lemma 1, we can show this theorem. Definition 9. We call the conic subset CC the wave front set of the fundamental solution for D /S .

5 Concluding remarks To the end of this paper we give several remarks. Since “SP IN ”(M ) corresponds to the degree two Stiefel-Whitney class in H 2 (M, Z2 ), it is easy to imagine that the spin groupoid and the spin bundle ^ ) are quasi-Morita equivalent as Lie groupoids. In fact we can gerbe SO(M prove that ^ ). Proposition 8. “SP IN ”(M ) ∼q−M orita SO(M ^) However, we do not know whether the groupoids “SP IN ”(M ) and SO(M are Morita equivalent in genuine sense. Here the Morita equivalence is defined in the following way: t / t / / Γ0 ] and [ Γ˜ / Γ0 ] are Lie groupoids. Definition 10. Assume that [ Γ s s Then Γ and Γ˜ are said to be Morita equivalent (denoted by Γ ∼Morita Γ˜ ) if they are quasi-Morita equivalent and satisfy the following condition: Let P be a (Γ, Γ˜ )-bitorsor with momentum maps J : P → Γ0 and J˜ : P → Γ0 . Then J˜ J the Γ (resp. Γ˜ ) action on P → Γ0 (resp. P → Γ0 ) by left (resp. right) is principal.

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^ ), we can also see that If we can prove that “SP IN ”(M ) ∼M orita SO(M 8 de Rham type cohomologies of them coincide (K-theories of them coincide). Moreover, thanks to the compatibility of Clifford multiplication and the spin groupoid action, the Dirac operator defined in the previous section is elliptic without tensoring an auxiliary torsor and quotient, opposite to the ^ ) module. Dirac operator defined in [24] on a Z2 -gerbe SO(M We also remark that as to the central extension 1 → S 1 → Spinc (n) → SO(n) → 0, we can relate the Lie groupoids “SP IN c”(M ) and spinor torsors with spinc connections to the degree two cohomology classes in δ1 (H 1 (M, SO(n))) ⊂ H 2 (M, S 1 ). We believe that these topics are closely related to the fractional analytic index theorem (cf. [19]) and local line bundles (cf. [20]). We shall be concerned with them in the forthcoming paper. Acknowledgements. The author thanks Professor Hitoshi Moriyoshi, Doctors Yasushi Homma, Shingo Kamimura and Kiyonori Gomi for their helpful discussions. This research is partially supported by Grant-in-Aid for Scientific Rresearch #17540096, #18540093, Ministry of Education, Culture, Sports, Science and Technology, Japan and Keio Gijuku Academic Funds.

References 1. Aristide, T.:An Atiyah-Singer theorem for gerbes. math.DG/0302050v1 2. Atiyah, M. and Singer, I. :The index of elliptic operators I. Ann. Math. 87, 484-530 (1963) 3. Behrend, K. and Xu, P. : Differentiable stacks and Gerbes. math.DG/0605694v1 8

For a Lie groupoid Γ , set Γ [p] := Γ ×Γ0 · · · ×Γ0 Γ (p-times), πi := the deletion of i-the element,

∂ :=

p X

(−1)i πi : Γ [p] → Γ [p−1] .

i=0

Then

C p,q (Γ ) := Ω q (Γ [p]), ∂ ∗ , ddR · · · (#)

forms a double complex. We define

? HdR (Γ, R) := the cohomology of total complex (#).

K-theory for a Lie groupoid Γ is defined as K-theory for the groupoid Γ -algebra C ∗ (Γ ). In our case, because our Lie groupoid corresponds to a torsion class, K0 (C ∗ (“SP IN ”(M ))) coincides with the set of isomorphism classes of bundle ^ ) modules. See [3], [5], [8], [18], [24] and [25] gerbe SO(M

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4. Berline, N., Getzler, E. and Vergne, M.: Heat Kernels and Dirac operators. Grundl. 298 Springer-Verlag, (1996) 5. Bouwknegt, P., Carey, A., Mathai, V., Murray, M.K. and Stevenson, D. :Twisted K-theory and K-theory of bundle gerbes. Comm. Math. Phys. 228 no. 1, 17–45, (2002) 6. Brylinski,J-L.: Loop spaces, Characteristic classes and Geometric quantization. Progress in Mathematics 107, Birkh¨ auser, (1992) 7. Connes, A.: Noncommutative Geometry. Academic Press, (1994) 8. Cuntz, J. Tsygan, B. and Skandalis, G. :Cyclic Homology in Non-Commutative Geometry. EMS. 121, Springer-Verlag, (2004) 9. Dixmir, J. and Douady, A.: Champs continus d’espaces hilbertiens et de C ∗ alg`ebre, Bull. Soc. Math. Fr. 91, 227-284 (1963) 10. Giraud, J.: Cohomoloie non-ab´elienne. Grundl. 179 Springer-Verlag, (1971) 11. Gomi, K. :Connections and curvings on lifting bundle gerbes. J. London Math. Soc. 67, no.2, 510-529 (2003) 12. Gomi, K. :Reduction of strongly equivariant bundle gerbes with connection and curving. math.DG/0406144v2 ˆ 13. Grothendieck, A. et al: Revˆetments Etales et Groupe Fondamental. SGA1, Lecture Notes in Mathematics, 224, Springer-Verlag, (1971) 14. Hitchin, N.: Lectures on Special Lagrangian Submanifolds. math.DG/9907034 15. H¨ ormander, L.: The analysis of linear partial differential operators I. SpringerVerlag, (1983) 16. Kumano-go, H.: Pseudodifferential Operators. MIT, (1982) 17. Mackenzie, K. :General Theory of Lie Groupoids and Lie Algebroids , London Mathematical Society, Lecture Note Series, 213, Cambridge University Press, (2005) 18. Mathai, V., Melrose, R. and Singer, I. : The index of projective families of elliptic operators.math.DG/0206002 19. Mathai, V., Melrose, R. and Singer, I. : Fractional analytic index.math.DG/0406329 20. Melrose, R.:Star products and local line bundles, Annales de l’Institut Fourier, 54, 1581-1600 (2004) 21. Miyazaki, N.: On vectorial gerbes and Poincar´e-Cartan classes. Noncommutative geometry and physics, 291-300, World Scientific, (2005) 22. Moriyoshi, H. :Operator algebras and the index theorem on foliated manifolds. Proceedings of Foliations: Geometry and Dynamics, 127-155, World Scientific, (2002) 23. Murray, M.: Bundle gerbes. J. London Math. Soc. (2) 54 no.2 , 403-416 (1996) 24. Murray M. and Singer M.: Gerbes, Clifford modules and the index theorem. Ann. Global Anal. Geom. 26, 355-367 (2004) 25. Natsume, T. and Moriyoshi, H.: Operator algebras and Geometry. Mathematical Society of Japan, Memoire, 2 (2001), (in Japanese) 26. Sakai, T.:Riemannian geometry. Shokabo, (1992), (in Japanese) 27. Shubin, M. A.: Pseudodifferential operators and spectral theory. SpringerVerlag, (2001) 28. Sogge, C. D.: Fourier integrals in classical analysis. Cambridge, (1993) 29. Woodhouse, N.: Geometric quantization. Clarendon Press, Oxford, (1980)

The Cohomology of Transitive Lie Algebroids Z. Chen and Z.-J. Liu Department of Mathematics and LMAM Peking University, Beijing 100871, China [email protected], [email protected]

Abstract For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F , we study the localization map Υ 1 : H 1 (A, F ) → H 1 (Lx , Fx ), where Lx is the adjoint algebra at x ∈ M . The main result 1 in this paper is that: KerΥx1 = Ker(p1∗ ) = HdeR (M, F0 ). 1∗ 1 Here p is the lift of H (A, F ) to its counterpart over the p 1 f → universal covering space M M and HdeR (M, F0 ) is the 0 F0 = H (L, F )-coefficient deRham cohomology. We apply these results to study associated vector bundles with respect to some principal fiber bundle and the structure of transitive Lie bialgebroids. MSC: Primary 17B65. Secondary 18B40, 58H05. Research partially supported by NSF (19925105) of China, the Research Project of “Nonlinear Science” and China Postdoctoral Science Foundation (20060400017).

Keyword:Lie algebroid, cohomology, Lie bialgebroid

1 Introduction The theory of Lie algebroids is one of important fields in modern differential geometry, which gives an unified way to study Lie algebras and the tangent bundle of a manifold. Its global version is Lie groupoid. See [13] for a detailed introduction to the theory of Lie algebroids and Lie groupoids as well as [18] by Weinstein for their applications in Poisson geometry. Moreover, in [3] Connes also points out that groupoids play an essential role in non-commutative geometry. The purpose of this paper is to study the cohomology of Lie algebroids, which is a basic topic in this field and has wide applications in physics and 109

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other fields of mathematics (e.g., see [5], [6] and [7]), as well as its applications for associated vector bundles with respect to some principal fiber bundle and the structure of transitive Lie bialgebroids. The notion of a Lie bialgebroid was introduced by Mackenzie and Xu in [15] as a natural generalization of that of a Lie bialgebra, as well as the infinitesimal version of Poisson groupoids introduced by Weinstein [17]. It has been shown that much of the theory of Poisson groups and Lie bialgebras can be similarly carried out in this general context. It is therefore a basic task to study the structure of Lie bialgebroids. In particular, it is very interesting to figure out what special features a Lie bialgebroid, in which the Lie algebroid structure is transitive, would have. The paper is organized as follows. In Section 2, first we give a brief introduction to basic notion and then, for a transitive Lie algebroid A and its representation on a vector bundle F , we define a morphism of cohomology groups, called the localization map, and prove that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra under some topological condition. In Section 3 we study the kernel of the localization map mentioned above. To do it, the connection and parallelism are used(see [1] for more details). As an equivalent statement of the main theorem, we describe some conditions to make a Lie algebroid 1-cocycle to be coboudary. In section 4, we study some properties of the cohomology of associated vector bundles with respect to some principal fiber bundle. In Section 5, we recall some results on the structure of transitive Lie bialgebroids in ref. [1] as an another application of our localization theory.

2 Preliminaries Throughout the paper we suppose that any smooth manifold M under consideration is connected. 2.1 Lie algebroids and its representations First we introduce some basic concept used below: Definition 2.1 A Lie algebroid A with base space M , is a (real) vector bundle over M , together with a bundle map ρ : A → T M , called the anchor, and there is a (real) Lie algebra structure [ · , · ]A on Γ (A) satisfying the following conditions: 1. The induced map ρ : Γ (A) → X (M ) is a Lie algebra morphism; 2. for all f ∈ C ∞ (M ), A, B ∈ Γ (A), the Leibnitz law holds. That is, [A, f B]A = f [A, B]A + (ρ(A)f )B. In this paper, by (A, [ · , · ]A , ρ) we denote a Lie algebroid A with anchor map ρ : A → T M and Lie bracket [ · , · ]A on Γ (A). For a transitive Lie

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algebroid (A, [ · , · ]A , ρ) over M (i.e., the anchor ρ is surjective), the Atiyah sequence is as follows: ρ

i

0 → L −→ A −→ T M → 0,

(1)

where L = Ker(ρ) is called the adjoint bundle of A. In fact, L is a Lie algebra bundle. The fiber type can be taken as the Lie algebra g = Lp at any point p ∈ M . By [ · , · ]L , we denote the fiber-wise bracket on L (see ref. [13]). Let F be a vector bundle over M . By CDO(F ) we denote the bundle of covariant differential operators of F . (see III in ref. [13] and refs. [14, 10] for more details. In the last text, the authors use D(F ) instead of CDO(F )). Each element D ∈ CDO(F )x is an operator Dx : Γ (F ) → Fx , corresponding to a 4

unique X = σ(Dx ) ∈ Tx M , such that Dx (f µ) = X(f )µ(x) + f (x)Dx µ,

∀f ∈ C ∞ (M ), µ ∈ Γ (F ).

(2)

Actually, (CDO(F ), [ · , · ]CDO(F ) , σ) is a transitive Lie algebroid over M , with the the bracket of commutator [ · , · ]CDO(F ) . That is, for two D1 , D2 ∈ Γ (CDO(F )), 4

[D1 , D2 ]CDO(F ) = D1 ◦ D2 − D2 ◦ D1 , is also an element of Γ (CDO(F )). Moreover, the corresponding vector field σ[D1 , D2 ]CDO(F ) = [σ(D1 ), σ(D2 )]. Obviously, the adjoint bundle of this Lie algebroid is End(F )=Ker(σ). Definition 2.2 A representation of a Lie algebroid (A, [ · , · ]A , ρ) on a vector bundle F → M is a vector bundle map L : A → CDO(F ), which is also a Lie algebroid morphism. That is, 1) ∀A ∈ Ax , x ∈ M , ρ(A) = σ(L(A)); 2) ∀A, B ∈ Γ (A), L[A, B]A = [L(A), L(B)] = L(A) ◦ L(B) − L(B) ◦ L(A). We also denote L(A) by LA . For example, when A is transitive, one can define the adjoint representation of A on L: for each A ∈ Ax , define adA : Γ (L) → Lx ,

µ 7→ [A, µ]A (x),

∀µ ∈ Γ (L).

(3)

Here, we need to extend A to be a local section A˜ of A near x, and then the ˜ µ]A (x). Note that, it does not depend value of [A, µ]A (x) is defined to be [A, on the choice of the extension of A near x. Usually adA is written as [A, · ]A . As usual, a representation L of A on F defines the Chevalley complex (C k (A, F ), D) and cohomology groups H k (A, F ) = Ker(D)/Im(D) (see ref. [4]). In detail, we call a series of C ∞ (M )-modules 4 C k (A, F ) = bundle maps Ω : ∧k A → F , (k > 1),

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and C 0 (A, F ) = Γ (F ) the cochain space of L. The coboundary operators (also called the differentials) D = Dk : C k (A, F ) → C k+1 (A, F ) are defined in the traditional way and satisfy D2 = 0. For this complex (C k (A, V ), D), the corresponding cohomology groups are H k (A, F ) = Ker(Dk )/Im(Dk−1 ).

(k > 0)

We adopt the convention that H 0 (A, F ) = Ker(D0 ). In particular, a closed 0-cochain is a smooth section of Γ (F ), say ν, satisfying LA ν = 0, ∀A ∈ A. The group H 0 (A, F ) is the collection of all such closed 0-cochains. A 1-cochain Ω ∈ C 1 (A, F ) is a bundle map from A to F . It is called closed, or a 1-cocycle, denoted by Ω ∈ D1 (A, F ), if Ω[A, B]A = LA (Ω(B)) − LB (Ω(A)),

∀A, B ∈ Γ (A).

(4)

Especially, we call Ω a coboundary, denoted by Ω ∈ B 1 (A, F ), if Ω = Dµ, i.e., Ω(A) = LA (µ), ∀A ∈ Γ (A) for some µ ∈ Γ (F ). In what follows, cochains are simply called chains. Two 1-cocycles are called homologic, if their substraction is a coboundary. The group H 1 (A, F ) are the quotient group of all 1-cocycles in sense of homological equivalence: 4 H 1 (A, F ) = D1 (A, F )/B 1 (A, F ). We usually write the equivalence class of a 1-cocycle Ω by [Ω]. If the Lie algebroid degenerates to a Lie algebra, then the above construction of cohomology groups returns to that of the Lie algebras. 2.2 The localization of 1-cohomology of transitive Lie algebroids We choose an arbitrary point x ∈ M . For the Lie algebra (Lx , [ · , · ]Lx ), L induces a representation L of Lx on Fx . In fact, for any u ∈ Lx , we define 4 Lu (µ) = Lu (µ), here µ ∈ Fx , µ ∈ Γ (F ) is a locally smooth extension of µ. This is well defined, because Lu (f µ) = f (x)Lu (µ), ∀f ∈ C ∞ (M ). Since L is a Lie algebra morphism, so is L: Lx → End(Fx ) and hence L is indeed a representation. Consider the Chevalley complex C • (Lx , Fx ), where Fx is regarded as an Lx -module via the representation L. We denote the set of closed chains in C k (Lx , Fx ) by Dk (Lx , Fx ), and the corresponding Chevalley cohomology groups by H k (Lx , Fx ). We need the following theory of localizations and some results in ref. [2]. Proposition 2.3 Let L : A → CDO(F ) be a representation of a Lie algebroid (A, [ · , · ]A , ρ) on F → M . For any x ∈ M , there is a natural morphism Υxk : H k (A, F ) → H k (Lx , Fx ) defined by

The Cohomology of Transitive Lie Algebroids 4

Υxk ([Ω]) = [Ω],

113

∀Ω ∈ C k (A, F ).

Here Ω is the limitation of Ω on ∧k Lx ⊂ ∧k Ax . We call the group morphism Υxk defined above the localization of the homology group H k (A, F ) at x. For this Υ , we have the following claims. Theorem 2.4 Given a representation of a transitive Lie algebroid (A, [ · , · ]A , ρ) on F → M , L : A → CDO(F ), then we have 1) For any x, y ∈ M , there exits an isomorphism J: H 1 (Ly , Fy ) → H 1 (Lx , Fx ), such that the following diagram commutes H 1 (Ly , Fy ) 5 k k kk k k k kk Υy1

H 1 (A, F )

(5)

J

SSSS SSSS S) Υx1 H 1 (Lx , Fx ) 2) If M is simply connected, or H 0 (Lx , Fx ) = 0, then the localization Υx1 : H 1 (A, F ) → H 1 (Lx , Fx ) is an injection. In general, the isomorphism J in 1) depends on the choice of a path from x to y in M . It is shown in ref. [2] that, although the isomorphism J: H 1 (Ly , Fy ) ∼ = H 1 (Lx , Fx ) is not naturally defined, the two subgroups ImΥy1 and ImΥx1 are naturally isomorphic, under the condition of 2) of this theorem. In this paper, we are going to prove in Corollary 3.11 that even without this condition, the conclusion also holds.

3 Kernel of the localization map Υ 1 An interesting problem is that, if M is not simply connected, and H 0 (Lx , Fx ) 6= 0, then what is the kernel of Υx1 ? In this section, we give the answer to this problem in two different ways. Please see the following two equations (8) and (15). f → M be a covering map. The for any vector bundle Lemma 3.1 Let p : M F → M , we have CDO(p! F ) ∼ = p! CDO(F ). f, x = p(z), D0 ∈ Proof. For each pair (z, D0 ) ∈ p! CDO(F )z , where z ∈ M ! CDO(F )x , we define φ(z, D0 ) ∈ CDO(p F )z as follows: given an arbitrary λ ∈ Γ (p! F ), we find a decomposition near z: X f), µi ∈ Γ (F ), λ= fi µi , where fi ∈ C ∞ (M i

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and we then set 4

φ(z, D0 )λ =

X

(Z(fi )µi (x) + fi (z)D0 (µi )),

i

f is the unique tangent vector satisfying p∗ (Z) = σ(D0 ). It is where Z ∈ Tz M easy to prove that this definition does not depend on the choice of decompositions of λ. In this way, we obtain φ : p! CDO(F ) → CDO(p! F ), which is obviously an injection. f, x = p(z), and D ∈ CDO(p! F )z , σ(D) = Z ∈ Conversely, for any z ∈ M f Tz M, D induces D0 ∈ CDO(F )x , such that σ(D0 ) = p∗ (Z), φ(z, D0 ) = D. In fact, each section µ ∈ Γ (F ) can be naturally regarded as µ ∈ Γ (p! F ). Let D(µ) = ν ∈ (p! F )z = Fx and we define D0 (µ) = ν. Therefore, for the preceding λ ∈ Γ (p! F ), it is not hard to see X D(λ) = (Z(fi )µi (x) + fi (z)D(µi )) = φ(z, D0 )(λ). i

This shows that φ is also surjective. So φ is indeed an isomorphism from p! CDO(F ) to CDO(p! F ) . Lemma 3.2 Suppose that a Lie algebroid (A, [ · , · ]A , ρ) has a representation f → M be a covering map. Then on F → M , L : A → CDO(F ). Let p : M ! f, and it has the pull back bundle Ae = p A is also a Lie algebroid over M e : Ae → CDO(Fe), such that the an induced representation on Fe = p! F , L following diagram commutes. e L Ae −−−−→   p! y L

A −−−−→

CDO(F˜ ) ∼ = p! CDO(F )   p! y

(6)

CDO(F ).

f, x = p(z), A ∈ Ax , one has In other words, for z ∈ M e A) = (x, L(A)). L(x, e Moreover, if A is transitive, then so is A. We omit the proofs of this lemma and the following two theorems. Theorem 3.3 With the same assumptions as in Lemma 3.2, we have a group morphism e Fe ). pk∗ : H k (A, F ) → H k (A, Theorem 3.4 With the same assumptions as in Lemma 3.2, we have a commute diagram

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k∗

p e Fe ) H k (A, F ) −→ H k (A, Υxk ↓ ↓ Υexek H k (Lx , Fx ) = H k (Lxe, Fxe).

(7)

e Fe ) at some x f with p(e Here Υexek is the localization of H k (A, e∈M x) = x. And k k one naturally regards H (Lx , Fx ) = H (Lxe, Fxe).

f → M is a uniCorollary 3.5 If A is a transitive Lie algebroid and p: M versal covering, then Ker(Υx1 ) = Ker(p1∗ ). (8) f being Proof. By Theorem 3.4, Υx1 = Υexe1 ◦ p1∗ . By 2) of Theorem 2.4 and M 1 e simply connected, we know Υxe is an injection. So we get (8). The conclusion of Equation (8) of course describes Ker(Υx1 ), but it has no relationship with the group H 0 (Lx , Fx ). We now give another description of Ker(Υx1 ). For the Lie algebroid (A, [ · , · ]A , ρ) and its representation on F → M , L : A → CDO(F ), and any x ∈ M , we consider a sub vector space F0x = {ν ∈ Fx |Lu (ν) = 0, ∀u ∈ Lx } = H 0 (Lx , Fx ). Then by Theorem 2.4, when A is transitive, F0 = H 0 (L, F ) ⊂ F is a sub vector bundle. Since for each u ∈ Γ (L), A ∈ Γ (A), ν ∈ Γ (F0 ), we have Lu (LA ν) = L[u,A]A ν − LA (Lu ν) = 0, and hence LA ν ∈ Γ (F0 ). So we have an induced representation of A on F0 , also denoted by L. ¯ : Meanwhile, L induces a representation of T M on F0 , denoted by L T M → CDO(F0 ). In fact, for each X ∈ Tx M , and an arbitrary A ∈ Ax , ρ(A) = X, we set 4 ¯X ν = L LA ν, ∀ν ∈ Γ (F0 ). ¯ is well Obviously this definition does not depend on the choice of A, and L defined. A representation of the tangent bundle T M on F0 is also referred as ¯ the reduced (flat) connection of F0 a flat connection of F0 . We will call L coming from the represention L. Now, elements of C k (T M, F0 ) = Hom(∧k (T M ), F0 ) ( with C 0 (T M, F0 ) = Γ (F0 )) are also called the F0 -coefficient k-forms. With the usual exterior differential operator d : C k (T M, F0 ) → C k+1 (T M, F0 ), C • (T M, F0 ) is known as the F0 -coefficient de Rham complex. Especially, D 1 (T M, F0 ) is the kernel of d : C 1 (T M, F0 ) → C 2 (T M, F0 ) and 1 HdeR (M, F0 ) = D1 (T M, F0 )/d(Γ (F0 ))

is the first F0 -coefficient de Rham cohomology of M . We are going to prove that this group is just the kernel of Υx1 (Theorem 3.9).

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Lemma 3.6 There is a one-one correspondence between D1 (A, F0 )0 = Ω ∈ D1 (A, F0 )|Ω(v) = 0, ∀v ∈ L ¯ For the inversion map, we denote and D1 (T M, F0 ), denoted by Ω 7→ Ω. θ ∈ D1 (T M, F0 ) 7→ ρ∗ θ ∈ D1 (A, F0 )0 . 1 More over, this map induces an injection of HdeR (M, F0 ) into H 1 (A, F ): [θ] 7→ ∗ [ρ θ].

¯ to be a map sending X ∈ Tx M Proof. Given any Ω ∈ D 1 (A, F0 )0 , we define Ω 4 ¯ to Ω(X) = Ω(A), where A ∈ Ax , ρ(A) = X. This is of course well defined. It is also easy to check ¯ X Ω(Y ¯ )−L ¯ Y Ω(X) ¯ ¯ L − Ω([X, Y ]) = 0,

∀X, Y ∈ X (M ).

(9)

¯ ∈ D1 (T M, F0 ). I.e., Ω On the other hand, given any θ ∈ D 1 (T M, F0 ) satisfying Equation (9), we can define ρ∗ θ(A) = θ(ρ(A)), ∀A ∈ A. ρ∗ θ naturally satisfies ρ∗ θ|L = 0, and it is also a cocycle. It is just by the definitions to see that the map 1 ρ∗ (·) : HdeR (M, F0 ) → H 1 (A, F0 );

[θ] 7→ [ρ∗ θ],

is an injection of cohomology groups. But as we shall see in the following exact Sequence (12) that H 1 (A, F0 ) is embedded into H 1 (A, F ). So we conclude that 1 HdeR (M, F0 ) can be embedded into H 1 (A, F ) via [ρ∗ (·)]. Let F = F/F0 be the quotient bundle. Now one obtains an exact sequence i

j

0 → F0 −→ F −→ F → 0.

(10)

The algebroid A has an induced representation on F , in an obvious sense and also denoted by L: 4 LA [µ] = [LA µ], ∀µ ∈ Γ (F ). Now, we get an exact sequence of complexes i

j

0 → C k (A, F0 ) −→ C k (A, F ) −→ C k (A, F ) → 0.

(11)

Here i, j are both cochain maps. So we have the following long exact sequence (the Mayer-Vietoris) of cohomology groups i

j∗0

∗0 H 0 (A, F ) −→ H 0 (A, F )(= 0) 0 −→ H 0 (A, F0 ) −→

i

j∗1

∗1 −→ H 1 (A, F0 ) −→ H 1 (A, F ) −→ H 1 (A, F ) −→ · · · .

(12)

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Similarly, at any x ∈ M the Lie algebra Lx has the trivial representation on F0x and an induces representation on F x . So we have another exact sequence of complexes j

i

0 → C k (Lx , F0x ) −→ C k (Lx , Fx ) −→ C k (Lx , F x ) → 0. And it induces a long exact sequence, j

i

∗0 ∗0 H 0 (Lx , Fx ) −→ H 0 (Lx , F x )(= 0) 0 −→ H 0 (Lx , F0x ) −→

j

i

∗1 ∗1 −→ H 1 (Lx , F0x ) −→ H 1 (Lx , Fx ) −→ H 1 (Lx , F x ) −→ · · · .

(13)

Moreover, there is a series of vertical arrows Υ k between (12) and (13), namely the localization maps with respect to the preceding representations, such that the diagram commute. Here we pick out a part of the diagram as follows i

j∗1

i∗1

j ∗1

∗1 H 1 (A, F ) −→ H 1 (A, F ) −→ · · · 0 −→ H 1 (A, F0 ) −→ 1 1 ↓ Υ0x ↓ Υx1 ↓ Υx

(14)

0 −→ H 1 (Lx , F0x ) −→ H 1 (Lx , Fx ) −→ H 1 (Lx , F x ) −→ · · · . 1

By 2) of Theorem 2.4 and H 0 (Lx , F x ) = 0, we know that Υ x is an injection. Corollary 3.7 1 1 Ker(Υx1 ) = Ker(j∗1 ) ∩ i∗1 (Ker(Υ0x )) ∼ ). = Ker(Υ0x

Proof. Suppose that ω ∈ H 1 (A, F ) satisfies Υx1 (ω) = 0, then 1

j ∗1 ◦ Υx1 (ω) = Υ x ◦ j∗1 (ω) = 0. 1

Since Υ x is an injection, we have j∗1 (ω) = 0. Hence we know that ω ∈ Ker(j∗1 ) = Im(i∗1 ) ∼ = H 1 (A, F0 ). 1 Since the left square in (14) commutes, we have ω ∈ Ker(j∗1 )∩i∗1 (Ker(Υ0x )). 1 Conversely, given any ω as above, there naturally holds Υx (ω) = 0. Since i∗1 is also an injection, we have the isomorphism in the expression.

We can of course directly regard H 1 (A, F0 ) = Ker(j∗1 ) ⊂ H 1 (A, F ) and 1 therefore Υ0x = Υx1 |H 1 (A,F0 ) . The above corollary says that to know what 1 1 Ker(Υx ) really is, it suffices to study Ker(Υ0x ). Theorem 3.8 H 1 (Lx , F0x ) = D1 (Lx , F0x ), and 1 Ker(Υx1 ) ∼ ) = [Ω]| Ω ∈ D1 (A, F0 ), Ω|Lx = 0 . = Ker(Υ0x

(15)

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Proof. By definition of the complex C k (Lx , F0x ), we have DF0x = 0 and hence H 1 (Lx , F0x ) = D1 (Lx , F0x ). Given any [Ω] ∈ H 1 (A, F0 ), where Ω ∈ D1 (A, F0 ), we have 1 Υ0x ([Ω]) = [Ω|Lx ] = Ω|Lx .

So we have the expression in (15). Now we have the second way expressing the kernel of the localization map. ∼ H 1 (M, F0 ). Theorem 3.9 Ker(Υ 1 ) = Im[ρ∗ (·)] = x

deR

Proof. We first point out that, in (15), an Ω ∈ D 1 (A, F0 ) satisfies Ω|Lx = 0, for some x ∈ M , if and only if Ω|Ly = 0 hold for all y ∈ M , i.e., or Ω ∈ D1 (A, F0 )0 . In fact, the first conclusion of Theorem 2.4 claims that the kernel of the localization map does not depend on the choice of the points: Ker(Υx1 ) = Ker(Υy1 ). So the set described in (15) does not depend on the choice of x. Combining Theorem 3.8 with these facts and the correspondence given by Lemma 3.6, we know that each element in the kernel of Υx1 must be the cohomology class of some ρ∗ θ ∈ D1 (A, F0 )0 . We restate the conclusions of Corollary 3.5, Theorem 3.8, and Theorem 3.9 in the following theorem. Theorem 3.10 Suppose that a transitive Lie algebroid (A, [ · , · ]A , ρ) has a f → representation on a vector bundle F → M , L : A → CDO(F ). Let p : M ! e M be a universal covering map. The pull back Lie algebroid A = p A has an e Write induced representation on Fe = p! F , denoted by L. F0 = {ν ∈ Fy |y ∈ M, Lu ν = 0, ∀u ∈ Ly } . The Lie algebroid A has an induced representation on F0 , also denoted by L. ¯ : T M → CDO(F0 ) be the reduced (flat) connection of F0 coming from Let L L. Then for each Ω ∈ D 1 (A, F ), x ∈ M , the following six statements are equivalent. 4

1) δx = Ω|Lx is a coboundary, i.e., ∃τ ∈ Fx , such that δx (u) = Lu τ,

∀u ∈ Lx .

4

2) δy = Ω|Ly is a coboundary, for every y ∈ M . e Fe) is a coboundary, i.e., there exists 3) The pull back cochain p1∗ Ω ∈ D1 (A, some µ e ∈ Γ (Fe ), such that e (ey,A) µ Ω(A) = L e,

∀A ∈ Ay .

f satisfies p(e Here ye ∈ M y ) = y. 4) There exists an Ω0 ∈ D1 (A, F0 ), Ω0 |Lx = 0, such that Ω and Ω0 are homologic.

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5) There exists an Ω0 ∈ D1 (A, F0 ), Ω0 |L = 0, such that Ω and Ω0 are homologic. 6) There exist θ ∈ D 1 (T M, F0 ) and µ ∈ Γ (F ), such that Ω = ρ∗ θ + Dµ. Corollary 3.11 For any x, y ∈ M , the image of localizations Υx1 and Υy1 are naturally isomorphic. That is, the isomorphism J in Diagram (5) naturally defines an isomorphism J: H 1 (Ly , Fy ) ∼ = H 1 (Lx , Fx ). Proof. Consider the commute diagram which follow from Diagram (5), 1

ImΥy m6 mmm m m mmm Υy1

H 1 (A, F )

J

RRR RRR RRR ( Υx1

ImΥx1

Since it is proved in Theorem 3.10 that Ker(Υx1 ) = Ker(Υy1 ), the map J in the above diagram must be an isomorphism and naturally defined.

4 Application of the localization theories for principal bundles and their associated bundles The remaining part of this paper is devoted to apply the preceding localization theories to that of principal bundles and their associated vector bundles. The idea originally appeared in ref. [1] and in what follows, we will recover some results in that text. We first recall basic facts about principal bundles and the associated bunπ dles. Let (P, →, M ; G) be a principal bundle with structure group G (a Lie group) on the base manifold M . We always assume that G freely acts on P to the right. The action of G on P naturally lifts to an action on T P . We denote the orbit of w ∈ Tp P by [w] and quotient manifold by TGP . Since this action is free, TGP admits a vector bundle structure with base M , and bundle projection q : [w] 7→ π(p). Sections of TGP can be regarded as vector fields on P which are G-invariant: Γ(

TP ) = {U ∈ X (P )|Up.g = Rg∗ Up , G

∀p ∈ P, g ∈ G} .

It follows that Γ ( TGP ) has an induced Lie bracket structure transferred from X (P ). Besides, the tangent map π∗ can also be transferred to TGP → T M .

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Thus, ( TGP , π∗ , M ) is a Lie algebroid which is transitive (known as the gauge algebroid ). The Atiyah sequence for this algebroid is as follows 0 →

T ⊥ P i T P π∗ −→ −→ T M → 0. G G

(16)

Here by T ⊥ P we denote the collection of all vertical vectors, which are in fact the set 4 d bxp = |t=0 p. exp tx; p ∈ P, x ∈ Lie(G) . dt If we also have a right action of G on a vector space V , then we obtain the associated bundle (by G acting diagonally on P × V ): F =

P ×V . G

Elements in F are of the form [p, v], for p ∈ P , v ∈ V . Most importantly, sections of F can be naturally regarded as V -valued, G-equivariant functions on P . I.e., Γ (F ) ∼ = C ∞,G (P, V ) = {µ ∈ C ∞ (P, V )|µ(p.g) = µ(p).g,

∀p ∈ P, g ∈ G} .

In fact, for any µ ∈ C ∞ (P, V ), it can be regarded as a V -valued function on P given by 4 µ(p) = v, such that [p, v] = µ(π(p)). And conversely, a V -valued, G-equivariant function µ ∈ C ∞,G (P, V ) corresponds to the section of F given by 4

µ(x) = [p, µ(p)],

by choosing an arbitary p ∈ π −1 (x),

∀x ∈ M.

The Lie group G has the canonical adjoint action on g, and hence G has a right action on g defined by 4

x.g = Adg−1 x,

∀x ∈ g, g ∈ G.

⊥

The adjoint Lie algebra bundle T GP of the gauge Lie algebroid in Sequence (16) is indeed the associated bundle P × Lie(G) . G In fact, given any g-valued G-equivariant function κ: P → g, which satisfies κ(p.g) = κ(p).g = Adg−1 κ(p), it corresponds to a vertical vector field on P :

∀p ∈ P, g ∈ G,

The Cohomology of Transitive Lie Algebroids 4

d = κ b|p = κ(p) p

d |t=0 p. exp tκ(p), dt

∀p ∈ P.

121

(17)

One can directly check that κ b is a G-invariant vector field on P . The ring of smooth functions C ∞ (M ) can also be regarded as G-invariant functions on P : C ∞ (M ) ∼ = π ∗ C ∞ (M ) = {f ∈ C ∞ (P )|f (p.g) = f (p),

∀p ∈ P, g ∈ G} .

There is a standard representation of the gauge algebroid by 4

LU (µ) = U (µ),

∀U ∈ Γ (

TP G

on F , defined

TP ), µ ∈ C ∞,G (P, V ). G

Let Ω 1,G (P, V ) denote the V -valued, G- equivariant 1-forms on P : Ω 1,G (P, V ) = {ω : T P → V ; ω|p (w).g = ω|p.g (Rg∗ w), ∀p ∈ P, g ∈ G, w ∈ Tp P } TP ∞,G = ω : T P → V ; ω(U ) ∈ C (P, V ), ∀U ∈ Γ ( ) . G Lemma 4.1 There is a canonical pull back morphism of C ∞ (M )-modules π ∗ : Γ (Hom(T M, F )) → Ω 1,G (P, V ),

ϑ 7→ π ∗ (ϑ),

where π ∗ (ϑ) :

w 7→ ϑ(π∗ w)(p),

∀w ∈ Tp P.

∗

Moreover, π is injective. Proof. To see that π ∗ (ϑ) is a G- equivariant 1-form, we suppose that for w ∈ Tp P , Rg∗ w ∈ Tp.g P , x = π(p) ∈ M and X = π∗ (w) = π∗ (Rg∗ w) ∈ Tx M, one has ϑ(X) = [p, v] = [p.g, v.g]. Then, π ∗ (ϑ)|p (w).g = ϑ(X)(p).g = v.g = ϑ(X)(p.g) = π ∗ (ϑ)|p.g (Rg∗ w). It is easy to see that π ∗ (ϑ) being a zero 1-form implies that ϑ is zero. We also introduce the notation Ω k,G (P, V ) (k > 1) to denote the V -valued G-equivariant k-forms on P . I.e., Ω k,G (P, V ) = ω : ∧k T P → V ; ω|p (w).g = ω|p.g (Rg∗ w), ∀p ∈ P, g ∈ G, w ∈ ∧k Tp P k ∞,G k TP = ω : ∧ T P → V ; ω(U ) ∈ C (P, V ), ∀U ∈ Γ (∧ ) . G

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These Ω k,G (P, V ) together with Ω 0,G (P, V ) = C ∞,G (P, V ) become a complex (over the ring C ∞ (M )), equipped with the usual exterior differential operator d. It is in fact isomorphic (as complexes) to C k ( TGP , F ). And in turn, we have k,G HdeR (P, V ) ∼ = Hk(

TP , F ). G

Definition 4.2 Given an arbitrary p ∈ P , the localization map for the gauge algebroid TGP and its associated vector bundle F = P ×V is a group morphism G k,G Υpk : HdeR (P, V ) → H k (Lie(G), V ).

It sends the cohomology class of ω ∈ Ω k,G (P, V ) to the cohomology class of ω b |p ∈ Dk (G, V ) which is defined by 4

ω b |p (x) = ω|p (bxp ),

∀x ∈ Lie(G).

Let Ge denote the subgroup of G which is the connected component of G containing the unit element e. Consider a sub vector space V0 = {v ∈ V |v.h = v,

∀h ∈ Ge } .

One is easy to prove that V0 is G-invariant. Hence G also has a right action on V0 . Lemma 4.3 For the representation of the gauge Lie algebroid P ×V G , P × V0 T ⊥P F0 = = H 0( , F ). G G

TP G

on F =

Proof. Let κ: P → g be a g-valued G-equivariant function which corresponds to a G-invariant vertical vector field κ b on P given by (17). Let µ ∈ C ∞,G (P, V ) be a V -valued, G-equivariant function which can also be regarded as an element of Γ (F ). At any p ∈ P , s ∈ R, let g = exp sκ(p). Then we have b p.g (µ) λ| d d = |t=0 µ(p.g. exp tκ(p.g)) = |t=0 µ(p).g. exp tAdg−1 κ(p) dt dt d d = |t=0 µ(p). exp tκ(p).g = |t=0 µ(p). exp tκ(p).g dt dt d d = |t=0 µ(p). exp(t + s)κ(p) = |t=s µ(p). exp tκ(p). dt dt Hence we know that µ ∈ Γ (F0 ) implies d µ(p) exp tκ(p) = 0, dt

i.e., µ(p) exp tκ(p) ≡ µ(p), ∀t ∈ R.

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Since κ is arbitrary and Ge is generated by elements of the form exp tx, x ∈ g, µ(p) is an element of V0 . This shows that the function µ takes values in V0 . ⊥ Conversely, if µ ∈ Γ (F0 ), then it obviously satisfies κ b(µ) = 0, ∀b κ ∈ T GP . Using Theorem 3.9, we have the following conclusion describing the kernel of the localization map Υp1 . Theorem 4.4 1 1 Ker(Υp1 ) = π ∗ HdeR (M, F0 ) ∼ (M, F0 ) . = HdeR k,G 1 Here π ∗ : HdeR (M, F0 ) → HdeR (P, V ) is given by

[θ] 7→ [π ∗ (θ)],

∀θ ∈ D1 (T M, F0 ),

and it is an injection. And one may restate the above theorem into the following form, analogue to that of Theorem 3.10. Theorem 4.5 With the preceding notations, let w ∈ Ω 1,G (P, V ) be a close 1-form. For any p ∈ P , we have five equivalent statements: 1) ω|Tp⊥ P is a coboundary, i.e., ∃v ∈ V such that ω(bxp ) =

d |t=0 v. exp tx, dt

∀x ∈ Lie(G).

2) ω|Tq⊥ P is a coboundary, for all q ∈ P . 3) There exists a closed 1-form ω0 ∈ Ω 1,G (P, V0 ), ω0 |Tp⊥ P = 0, such that ω0 and ω are homologic, i.e., ω = ω0 + dµ,

for some µ ∈ C ∞,G (P, V ).

4) There exists a closed 1-form ω0 ∈ Ω 1,G (P, V0 ), ω0 |T ⊥ P = 0, such that ω0 and ω are homologic. 5) For some closed 1-form θ ∈ D 1 (T M, F0 ) and µ ∈ C ∞,G (P, V ), ω = π ∗ θ + dµ. In particular, we conclude that: Corollary 4.6 If M is simply connected (or V0 = 0), then the localization map Υp1 is an injection. In other words, any one of the five statements in the above theorem implies that ω is a coboundary, i.e., ω = dµ for some µ ∈ C ∞,G (P, V ).

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5 Transitive Lie Bialgebroids In this section , we recall some results on the structure of transitive Lie bialgebroids in ref. [1] as an another application of our localization theory. • Lie bialgebroids. A Lie bialgebroid is a pair of Lie algebroids (A, A∗ ) satisfying the following compatibility condition d∗ [A, B]A = [d∗ A, B]A + [A, d∗ B]A , ∀A, B ∈ Γ (A),

(18)

where the differential d∗ on Γ (∧• A) comes from the Lie algebroid structure on A∗ (see ref. [9], [15] for more details). Of course, one can also denote a Lie bialgebroid by the pair (A, d∗ ), since the anchor ρ∗ : A∗ → T M and the Lie bracket [ · , · ]∗ on the dual bundle are defined by d∗ as follows: ρ∗∗ (df ) = d∗ f, ∀f ∈ C ∞ (M ) and for all A ∈ Γ (A), ξ, η ∈ Γ (A∗ ), h[ξ, η]∗ , Ai = ρ∗ (ξ)hη, Ai − ρ∗ (η)hξ, Ai − d∗ A(ξ, η). Again we suppose that the Lie algebroid A is transitive. Recall the adjoint representation of A on L defined in (3). In this paper, we also consider the adjoint representation of A on L ∧ L associated to that on L, and we write L2 for L ∧ L. In this case, Ω ∈ C 1 (A, L2 ), i.e., a bundle map from A to L2 , is a Lie algebroid 1-cocycle if and only if Ω[A, B]A = [Ω(A), B]A + [A, Ω(B)]A ,

∀A, B ∈ Γ (A).

(19)

Ω is a coboundary if Ω = [µ, · ]A for some µ ∈ Γ (L2 ). The structure of transitive Lie bialgebroids is studied in ref. [1]. We quote directly some of the conclusions in that text. Definition 5.1 For a transitive Lie algebroid (A, [ · , · ]A , ρ), given Λ ∈ Γ (∧2 A) and a bundle map Ω: A → L2 , the pair (Λ, Ω) is called A-compatible if Ω is a 1-cocycle and satisfies 1 [ [Λ, Λ]A + Ω(Λ), · ]A + Ω 2 = 0, 2

as a map Γ (A) → Γ (∧3 A).

(20)

Here Ω(Λ) and Ω 2 make sense by means of the extension of Ω as a derivation of the graded bundle, Ω: ∧k A → ∧k+1 A, k ≥ 0. For k = 0, it is zero. For k ≥ 1, it is defined by Ω(A1 ∧ · · · ∧ Ak ) =

k X

(−1)i+1 A1 ∧ · · · ∧ Ω(Ai ) ∧ · · · ∧ Ak ,

(21)

i=1

for all A1 ∧ · · · ∧ Ak ∈ Γ (∧k A). It is easy to see that if (Λ, Ω) is A-compatible, then so is the pair (Λ + ν, Ω − [ν, · ]A ), for any ν ∈ Γ (L2 ). Thus, two A-compatible pairs (Λ, Ω) and (Λ0 , Ω 0 ) are called equivalent, written (Λ, Ω) ∼ (Λ0 , Ω 0 ), if ∃ν ∈ Γ (L2 ), such that Λ0 = Λ + ν and Ω 0 = Ω − [ν, · ]A .

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Theorem 5.2 Let (A, [ · , · ]A , ρ) be a transitive Lie algebroid over M . Then there is a one-to-one correspondence between Lie bialgebroids (A, d ∗ ) and equivalence classes of A-compatible pairs (Λ, Ω) such that d∗ = [Λ, · ]A + Ω.

(22)

For a Lie bialgebra (g, g∗ ), it is obvious that one can take Λ = 0 and −Ω as the cobracket of g. Another special case is the following. Corollary 5.3 If Ω: A → L2 is a 1-cocycle and satisfies Ω 2 = 0, as a map A → ∧3 A, then (A, Ω) is a Lie bialgebroid. In this case, the anchor ρ∗ of A∗ is zero and A∗ is a bundle of Lie algebras whose bracket is defined by h[ξ, η]∗ , Ai = −hΩ(A), ξ ∧ ηi, for all ξ, η ∈ A∗ and A ∈ A. Corollary 5.4 Let (A, d∗ ) be a transitive Lie algebroid over M and suppose f → M be a covering. that d∗ = [Λ, · ]A + Ω is given as in (22). Let p : M ! Then the pull back bundle Ae = p A is also a transitive Lie algebroid. For the e and the pull back bundle map pull back section Λe ∈ Γ (A) e: Ω let

e2 , Ae → L

e · ] e + Ω. e de∗ = [Λ, A

e de∗ ) is also a Lie bialgebroid over M. f Then (A,

It is known that, for any section Λ ∈ Γ (∧2 A), one can define a bracket on Γ (A∗ ) by [ξ, η]Λ = LΛ# ξ η − LΛ# η ξ − d < Λ# ξ, η > . With the bracket defined above and anchor map 4

ρ∗ = ρ ◦ Λ # :

A∗ → T M,

the dual bundle A∗ becomes a Lie algebroid if and only if [X, [Λ, Λ]A ]A = 0, ∀X ∈ Γ (A) ([11], Theorem 2.1). In this situation, the induced differential on Γ (∧• A) has the form, d∗ = [Λ, · ]A , and clearly satisfies compatibility condition (18). The Lie bialgebroid arising in this way is called a coboundary (or exact) Lie bialgebroid [11]. In the particular case where [Λ, Λ]A = 0, the Lie bialgebroid is called triangular [16]. By our definition of A-compatible pairs, the pair corresponding to a coboundary Lie bialgebroid can be chosen to be (Λ, 0) or, equivalently, (A, A∗ ) is a coboundary Lie bialgebroid if and only if the second element of the Acompatible pair Ω ∈ C 1 (A, L2 ) is a coboundary. Therefore, to deal with coboundary Lie bialgebroids, one first needs to study the properties of Lie algebroid 1-cocycles.

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Z. Chen and Z.-J. Liu

Corollary 5.5 With the same assumptions as in Theorem 5.2, if Rank(A) = 1, then d∗ = 0. That is, A∗ admits trivial Lie algebroid structures. Corollary 5.6 With the same assumptions as in Theorem 5.2, if Rank(L) = 1, then d∗ = [Λ, · ]. That is, (A, d∗ ) is coboundary. The following theorem follows directly from 2) of Theorem 2.4. Theorem 5.7 Suppose that a transitive Lie algebroid A satisfies one of the following conditions 1) H 0 (g, g2 ) = 0, where g = Lx , for some x ∈ M ; 2) M is simply connected. 4

Then Ω ∈ C 1 (A, L2 ) is coboundary if and only if δx = Ω|g is coboundary. In particular, if H 1 (g, g2 ) = 0, any Lie bialgebroid (A, A∗ ) is coboundary. It is a well known result that for any nontrivial representation of a semisimple Lie algebra g on some vector space V , the cohomology groups H 0 (g, V ) and H 1 (g, V ) are both zero. So we conclude: Corollary 5.8 Let A be a transitive Lie algebroid and let g = Lx be the fiber type of L. If g is semi-simple and its adjoint representation on g2 is not trivial, then any Lie bialgebroid (A, A∗ ) is coboundary. We also have the following corollaries which follow from Theorem 3.10. Note that now L20 = ν ∈ L2y |y ∈ M, [u, ν]L = 0, ∀u ∈ Ly is a vector bundle over M which has a natural flat connection. Corollary 5.9 Suppose that a transitive Lie bialgebroid (A, d∗ ) satisfies H 1 (g, g2 ) = 0, where g = Lx for some x ∈ M . Then f → M , the pull back Lie bialgebroid (A, e de∗ ) 1) for a universal covering p: M given by Corollary 5.4 is coboundary; 2) the compatible pair corresponding to (A, d∗ ) can be chosen to be (Λ, ρ∗ θ), for some closed L20 -coefficient 1-form θ ∈ D 1 (T M, L20 ), where ρ∗ θ :

A 7→ θ(ρ(A)),

∀A ∈ A.

Note that in this case, Ω|L = ρ∗ θ|L is trivial, and hence Ω 2 = 0. So the compatible relation given in Equation (20) becomes 1 [ [Λ, Λ]A + (ρ∗ θ)yΛ, · ]A = 0, 2

as a map Γ (A) → Γ (∧3 A).

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References 1. Chen, Z., Liu, Z.-J., On transitive Lie bialgebroids and Poisson groupoids, Diff. Geom. Appl., 22(3): 253-274, 2005. 2. Chen, Z., Liu, Z.-J., The Localization of 1-Cohomology of Transitive Lie Algebroids, Science in China Ser. A, 49(2): 277-288, 2006. 3. Connes, A., Noncommutative geometry, Academic Press, San Diego, 1994. 4. M. Crainic, Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes, Commentarii Mathematici Helvetici (78)4: 681-721, 2003. 5. Evens, S., Lu, J.-H., Weinstein, A., Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford, Ser.2 50: 417-436, 1999. 6. Grabowski, J., Iglesia, D., Marrero, J., Padron, E., Urbanski, P., Poisson-Jacobi reduction of homogeneous tensors, J. Phys. A: Math. Gen., 37: 5383-5399, 2004. 7. Kapustin, A., Li, Y., Open string BRST cohomology for generalized complex branes, arXiv: hep-th /0501071, 2005. 8. Kobayashi, S., Nomizu, K., Foundations of differential geometry, Volume I, Interscience Publishers, 1963. 9. Kosmann-Schwarzbach, Y., Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math., 41: 153-165, 1995. 10. Kosmann-Schwarzbach, Y., Mackenzie, K., Differential operators and actions of Lie algebroids, Contemp Math, 315: 213-233, 2002. 11. Liu, Z.-J. and Xu, P., Exact Lie bialgebroids and Poisson groupoids, Geom. Funct. Anal., 6(1):138-145, 1996. 12. Liu, Z.-J. and Xu, P., The local structure of Lie bialgebroids, Lett. Math. Phys., 61:15-28, 2002. 13. Mackenzie. K., Lie groupoids and Lie algebroids in differential geometry, LMS Lecture Notes Series 124, Cambridge University Press, 1987. 14. Mackenzie, K., Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27: 97-147, 1995. 15. Mackenzie, K. and Xu, P., Lie bialgebraoids and Poisson groupoids. Duke Math. J., 73(2):415-452, 1994. 16. Mackenzie, K. and Xu, P., Integration of Lie bialgebroids, Topology., 39:445– 467, 2000. 17. Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan., 40:705-727, 1988. 18. Weinstein. A., Poisson geometry, Diff. Geom. Appl., 9: 213-238, 1998.

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Differential Equations and Schwarzian Derivatives Hajime SATO1 , Tetsuya OZAWA2 and Hiroshi SUZUKI3 1 2 3

Graduate School of Math., Nagoya Univ., [email protected] Mathematical Department, Meijo Univ., [email protected] Graduate School of Math., Nagoya Univ., [email protected]

1 Introduction There are two important non-commutative algebraic structures in geometry, the Lie algebra and the Grassmann algebra. Lie found that continuous family of transformations dominates geometrical structures and integrations of differential equations. He noticed that in the investigation of transformation groups, the analyses of infinitesimal transformations are fundamental. This leads to the notion of the Lie algebras and their representations. Many results are obtained since then on the theory of continuous groups. Grassmann introduced the exterior algebra which is often called the Grassmann algebra for his study of linear subspaces of vector spaces. E.Cartan extended it to the theory of exterior differential forms on smooth manifolds. Using this, he constructed the method of the moving frames and obtained many results on the investigations of differential equations. To know the actions of continuous transformation groups on the set of differential equations, the following picture [Figure 1] is helpful. Let G be a continuous group which in general is of infinite dimension. For example, G is the diffeomorphism group Diff(M ) of a finite dimensional manifold M or the contact diffeomorphism group ContDiff(M ) of a contact manifold M . Let M be the (infinite dimensional) set of all differential equations of a fixed type or the set of all structures of a fixed type on a manifold M. For example, let M be the set of all second order ordinary differential equations {y 00 = f (x, y, y 0 )} where y = y(x) is a function on R, or the set of all projective structures ProjStr(M ) on a finite dimensional manifold M. An important topic which has been extensively studied is the equivalence problem of differential equations or geometric structures. This is to decide whether one element in M is transformed to another element in M by an element of G. 129

130

Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

G y M µ0 orbit

µ O Fig. 1. image picture

Another problem which has been investigated is the quadratures of differential equations. A main method of the quadrature is as follows. For a differential equation µ ∈ M , find an element µ0 ∈ M and an element g ∈ G such that µ0 is an easily integrable equation and g(µ) = µ0 . If we have an explicit expression of the inverse element g −1 ∈ G, we can integrate µ0 to obtain solutions. Thus this method is formulated as follows. Given a natural action of G on M, determine the orbit G ·µ = {g(µ) | g ∈ G} for µ ∈ M. For µ ∈ G ·µ0 , find g ∈ G such that g(µ0 ) = µ. In other words, the point is to know the orbit space O and to determine the orbit decomposition of M by G. Define the isotropy group Gµ at µ ∈ M by Gµ = {g ∈ G | g(µ) = µ}. The interesting cases for our study are when Gµ are finite dimensional Lie groups for all µ ∈ M. Then an important problem is to know the orbit G ·µ0 for µ0 ∈ M such that the dimension of the isotropy group dim Gµ0 is of maximum. Then the orbit G · µ0 is of minimal dimension and very singular, which we call a minimal orbit. In studying the orbit decomposition of M, we can find interesting phenomena by comparing two different sets M1 and M2 for one G. In some cases, we have M1 and M2 consisting of different types of differential equations such that the orbits of Mi (i = 1, 2) are transversal to each other. Then we say that G spaces M1 and M2 are dual to each other. A discovery of such phenomena will be helpful in understanding of each type of differential equations.

Differential Equations and Schwarzian Derivatives

131

In this note, we study the actions of diffeomorphism groups and contact diffeomorphism groups on the space of differential equations. An important geometric idea to understand these problems is, roughly speaking ; a differential equation determines a geometric structure. For example, a second-order differential equation determines a projective structure so that the geodesics of the projective structure are the solutions of the second-order equation. Similarly, a third-order differential equation determines a projective contact structure if we consider the equivalence problem under contact transformations. To determine the geometric structures, we proceed as follows. Among equations of a given type, we write down the simplest equation in the jet bundle so that the solutions give a foliation of the jet bundle. From the jet space, we have two projections; one is to the base space and the other is to the space of the solutions of the equation. Thus the jet bundle is the total space of a double fibration. We can naturally compactify the three spaces of this double fibration so that each of them is a quotient space of a semi-simple Lie group divided by a Borel subgroup. Each quotient space has a special isotropy group which defines a G- structure of the homogeneous space (cf. [Ca1], [YY]). We write down explicitly the orbits of simple systems of differential equations by groups of diffeomorphisms or contact diffeomorphisms using multidimensional Schwarzian derivatives or contact Schwarzian derivatives. We give a definition of contact Schwarzian derivatives which is equivalent to that of Fox [Fo]. For the lowest dimensional case, the definition is given in [St1], [OS]. Further we give the system of linear partial differential equations using Schwarzian derivatives as coefficients such that the solutions produce a diffeomorphism or a contact diffeomorphism whose Schwarzian derivatives are given ones.

2 System of second-order PDE’s In this section, we consider the system of second-order partial differential equations with m independent variables and n dependent variables of normal form ∂y1 ∂y1 ∂yn ∂yn ∂ 2 yk = fijk (x1 , . . . , xm , y1 , . . . , yn , ,... ,..., ,... ) ∂xi ∂xj ∂x1 ∂xm ∂x1 ∂xm for all 1 ≤ i ≤ j ≤ m, 1 ≤ k ≤ n. The number of equations in (1) is equal to mn(m + 1)/2. Put x = (x1 , x2 , . . . , xm ),

y = (y1 , y2 , . . . , yn ),

(1)

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Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

pij =

∂yi (1 ≤ i ≤ n, 1 ≤ j ≤ m), ∂xj

p = (pij ).

Thus p = (pij ) is an n × m matrix whose (ij)-component is equal to pij . Then the system (1) is written as ∂ 2 yk = fijk (x, y, p) ∂xi ∂xj

for all 1 ≤ i ≤ j ≤ m, 1 ≤ k ≤ n.

For k ≥ 0 , we denote by J k (m, n) the set of all k-jets of mappings from K m to K n (cf. [Ya1] ). The system of second-order partial differential equations 2OPDE (1) corresponds to the exterior differential system I on J 1 (m, n) generated by the 1-forms (Pfaffian forms) dyi −

m X

(1 ≤ i ≤ n)

pij dxj

j=1

dpki −

m X

fijk dxj

(1 ≤ i ≤ m, 1 ≤ k ≤ n).

j=1

The null space I ⊥ of the Pfaffian system I is a distribution of dimension m generated by n

m

n

X XX ∂ ∂ ∂ + pij + fijk ∂xj ∂y ∂p i ki i=1 i=1

(1 ≤ j ≤ m).

k=1

We say that 2OPDE (1) is integrable if the m-dimensional distribution I ⊥ is completely integrable: [Γ (I ⊥ ), Γ (I ⊥ )] ⊂ Γ (I ⊥ ). In such case, the distribution I ⊥ defines an m-dimensional foliation on J 1 (m, n). The restriction of the volume form Ω = dx1 ∧ · · · ∧ dxm on each leaf is non-zero. Let π01 : J 1 (m, n) → J 0 (m, n) be the natural projection. It induces the linear map between the tangent spaces; (π01 )∗ : T J 1 (m, n) → T J 0 (m, n). The kernel Ker (π01 )∗ ⊂ T J 1 (m, n) is ∂ an mn-dimensional distribution generated by , (1 ≤ i ≤ n, 1 ≤ j ≤ m) . ∂pij This distribution Ker (π01 )∗ is the null space of the exterior differential system J generated by (m + n) 1-forms (Pfaffian forms) dyi (1 ≤ i ≤ n),

dxj (1 ≤ j ≤ m).

Thus we have two Pfaffian systems I and J on J 1 (m, n).

Differential Equations and Schwarzian Derivatives

133

On the jet space J 1 (m, n), we have the generalized contact forms dyi −

m X

pij dxj (1 ≤ i ≤ n)

j=1

whose null distribution D is of dimension m + mn. We have I ⊥ ⊕ J ⊥ = D. We say that a distribution on J 1 (m, n) (subbundle of T J 1 (m, n)) is isotropic if it is contained in the generalized contact distribution D. Thus we say that I and J are transverse isotropic Pfaffian systems on J 1 (m, n). Suppose that 2OPDE (1) is integrable. Let L be a leaf of the distribution I ⊥ . Since the volume form Ω = dx1 ∧ · · · ∧ dxm is non-zero on L, the leaf L projects down by π01 to an m-dimensional manifold in J 0 (m, n) which is expressed as π01 (L) = (x, y1 L (x), . . . , yn L (x)). Since the leaf L in J 1 (m, n) is an integral manifold of I, L is written as L = (x, y1 L (x), . . . , yn L (x), pij L (x)) ∂yi L (x) . ∂xj The set of functions (y1 L (x), . . . , yn L (x)) is a solution of 2OPDE (1). Thus we say that L in J 1 (m, n) is a prolongation of a solution. The codimension n+mn of a leaf L in J 1 (m, n) corresponds to the number of values (y1 L (x0 ), . . . , yn L (x0 ), pij L (x0 )), where pij L (x) =

where x0 is an arbitrary fixed point in Rm Thus the m-dimensional foliation on J 1 (m.n) defined by an integrable 2OPDE (1) projects down by π01 to an mn-dimensional parameter families of m-dimensional foliations on the (m+n)dimensional manifold J 0 (m, n). In the theory of webs, a p-web of dimension q means a p-tuple of foliations of dimension q which are pairwise transverse. Thus we may say that the set of solutions is an ∞mn web of dimension m on J 0 (m, n). Since foliation is 1-web, we may say that the set of prolongations of solutions is an isotropic 1-web on J 1 (m, n). We study the case when 2OPDE (1) is integrable. More precisely, we investigate the conditions where 2OPDE (1) is equivalent by a diffeomorphism of J 0 (m, n) to the most simple system ∂ 2 yk =0 ∂xi ∂xj

(2)

for all 1 ≤ i ≤ j ≤ m, 1 ≤ k ≤ n. We call this simple system the f lat system. Obviously the flat system is integrable.

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Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

3 Compactification of the flat system If a 2OPDE (1) is integrable, the m-dimensional distribution I ⊥ is completely integrable on J 1 (m, n). Each leaf is the prolongation of a solution of 2OPDE (1). Thus the space SOL of all solutions of (1) is an (n + mm)-dimensional space. The induced topology on SOLn+mn may not be Hausdorff. Corresponding to an integrable 2OPDE (1) , we have the following double fibration; (J 1 (m, n), 1-web) K mn

&K

.

(J 0 (m, n), ∞mn -web)

m

SOLn+mn

For the flat system (2), we can describe concretely each member of the double fibration. We can naturally compactify each member with a structure. We use the same notation for compactified members. Thus by a compactification, we have; J 0 (m, n) = Gr(m + n + 1, 1) = SL(m + n + 1, K)/H1 = Um+n+1 /Um+n × U1 , J 1 (m, n) = SL(m + n + 1, K)/H12 = Um+n+1 /Um × Un × U1 , SOL = SL(m + n + 1, K)/H2 = Um+n+1 /Um+1 × Un = Gr(m + 1, n). Here, when K = C, Ui = U (i). When K = R, Ui = SO(i). Thus corresponding to the flat system (2), we have the double fibration of compactified spaces; SL(m + n + 1, K)/H12 = J 1 (m, n) Gr(m, n)

& Gr(m + 1, 1)

.

J 0 (m, n) = Gr(m + n + 1, 1)

SOL = Gr(m + 1, n)

This corresponds to the following double fibration of Dynkin diagrams.

α1

αm+1

. α1

αm+1

αm+n

& αm+n

α1

αm+1

αm+n

Here, the black nodes in Dynkin diagram of a semisimple Lie algebra g over R decide a parabolic subalgebra p of g as follows (cf. [Ya2, 3.4]). Let

Differential Equations and Schwarzian Derivatives

135

Π = {αi , 1 ≤ i ≤ `} be the set of simple roots represented by the nodes in the Dynkin diagram and let Π1 be the subset represented by the black nodes. Let M=M+ ∪ M− be the set of non-zero roots consisting of positive and negative roots. We have the root space decomposition X g = g0 ⊕ gα ⊕ g−α , α∈M+

where g0 is the Cartan subalgebra. Associated with Π1 , for k ≥ 0, put ) ( ` X X + + ni (α) = k Mk = α = ni (α)αi ∈M i=1

and put p = g0 ⊕

X

αi ∈Π1

X

gα ⊕ g−α ⊕

gα .

α∈M+ k , k>0

α∈M+ 0

Let P ⊂ G be the Lie subgroup whose Lie algebra is equal to p. Then P is a parabolic subgroup of G. Put M = G/P. This is the space represented by the Dynkin diagram with black nodes. The tangent space of M at the base point is isomorphic to X m= g−α . Put g−k =

X

α∈M+ k , k>0

g−α . Then there exists µ > 0 and the vector space m has the

α∈M+ k

graded Lie algebra structure m = g−1 ⊕ · · · ⊕ g−µ . Especially, if all nodes are black, then M = G/B where B is the maximal solvable subgroup (called Borel subgroup) and M is diffeomorphic to the maximal compact subgroup K of G. If all nodes are white, p = g and M = {0}.

4 Orbit of diffeomorphisms Let K be R or C and let φ : K m+n → K m+n be a nondegenerate map (diffeomorphism) given by φ : z = (z1 , . . . , zm+n ) → (Z1 , . . . Zm+n ). According to Yoshida[Yo1], [Yo2] and Sasaki [Sa], (multi dimensional) k Schwarzian derivative Sij (φ) for 1 ≤ i, j, k ≤ ` = m + n is given by k Sij (φ) =

` ` 2 p 2 p X ∂ 2 Z p ∂z k 1 X ∂z q ∂z q k ∂ Z k ∂ Z δ , − + δ j ∂z i ∂z j ∂Z p ` + 1 p,q=1 i ∂z q ∂z j ∂Z p ∂z q ∂z i ∂Z p p=1

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Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

k k (cf. also Gunning[Gu], Kobayashi-Ochiai[KO]). Clearly Sij (φ) = Sji (φ) and they satisfy the canonical-form relation ` X

k Sik (φ) = 0 for i = 1, 2, . . . , `.

k=1

So the number of Schwarzian derivatives is equal to (` − 1)`(` + 2)/2. If we regard (z1 , z2 , . . . , zm+n ) = (x1 , . . . , xm , y1 , . . . , yn ), the diffeomorphism φ maps the flat system (2) to another system of partial differential equations ∂ 2 yk = f φ k ij (x, y, p) ∂xi ∂xj

for all 1 ≤ i ≤ j ≤ m, 1 ≤ k ≤ n,

(3)

where f φ k ij = f φ k ij (x, y, p) are smooth functions of m + n + mn variables. ∂ 2 yk The set of the systems of equations = f φ k ij (x, y, p) for all diffeo∂xi ∂xj ∂2y morphisms φ is the orbit of the system of equations = 0 by the ∂xi ∂xj diffeomorphism group Diff(Rm+n ). The determination of this set is a problem of the paper [St2] In the following Theorem 1, we write down explicitly the functions k f φ k ij (x, y, p) by using the Schwarzian derivatives Sij (φ). Remark 2 [ ∂ yk = f φ k ij (x, y, p) ∂xi ∂xj n+1 φ∈Diff(R

)

∂2y = 0 by Diff(Rm+n ) ∂xi ∂xj Put ` = m + n. Then φ ∈ Diff(R` ). The case for m = n = 1 is explained in [St1] and higher dimensional cases are left unsolved.

is the orbit of

Theorem 1. By the inverse diffeomorphism φ−1 , the system of 2O-PDE 2 ∂ Yk = 0 (1 ≤ i ≤ j ≤ m, 1 ≤ k ≤ n) ∂Xi ∂Xj is mapped to a system of 2O-PDE 2 ∂ yk = f φ k ij (x, y, p) ∂xi ∂xj where

(1 ≤ i ≤ j ≤ m, 1 ≤ k ≤ n)

Differential Equations and Schwarzian Derivatives

f φ k ij (x, y, p) = −Sim+k + j

m X

Sisj pks −

s=1

+

m X n X

n X

137

m+k m+k (Sm+t j pti + Si m+t ptj )

t=1

s s (Sm+t j pks pti + Si m+t pks ptj ) −

s=1 t=1

n X

m+k Sm+t m+s pti psj

s,t=1

+

m X

n X

s Sm+u m+t pks pui ptj .

s=1 u,t=1

This is an extension of the following result (cf. [St1]). Example 1. Let φ : K 2 → K 2 be a diffeomorphism expressed by φ(x, y) = (X(x, y), Y (x, y)). By the inverse diffeomorphism φ−1 , Y 00 (X) = 0 is mapped to y 00 (x) = −S12 1 + 3S11 1 y 0 − 3S22 2 (y 0 )2 + S21 2 (y 0 )3 .

Example 2. m = 1, n = 2. Let φ : K 3 = (x, y1 , y2 ) → K 3 = (x, y1 , y2 ) be a diffeomorphism. By the inverse diffeomorphism φ−1 , the flat system d2 yj = 0 (j = 1, 2) dx2 is mapped to the system with the following coefficients f φ 1 (x, y1 , y2 , p1 , p2 ) = −S12 1 − 2S12 3 p2 + (S11 1 − 2S12 2 ) p1 − S32 3 (p2 ) 2

2

2

2

+(S31 1 −2S32 2 ) p1 p2 +(2S11 2 −S22 2 ) (p1 ) +S31 3 p1 (p2 ) +2S31 2 (p1 ) p2 +S21 2 p1 3 f φ 2 (x, y1 , y2 , p1 , p2 ) = −S13 1 − 2S13 2 p1 + (S11 1 − 2S13 3 ) p2 − S23 2 (p1 ) 2

2

2

2 3

+(S21 1 −2S23 3 ) p1 p2 +(2S11 3 −S33 3 ) (p2 ) +S21 2 (p1 ) p2 +2S21 3 p1 (p2 ) +S31 3 (p2 ) .

138

Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

5 Construction of Diffeomorphisms By a set of Schwarzian derivative Skij = Skij (φ), 1 ≤ i, j, k ≤ ` for a diffeomorphism K ` → K ` , we have a system of linear partial differential equations di dj u − Skij dk u − S0ij u = 0 (4) ∂ . We often write ui for di u. By a straightforward calculation, ∂xi we know that this system of linear partial differential equation SLPD (4) is integrable if we put where di =

S0ij =

` ` ` X X 1 X k k k (Sik )j − (Skij )k + Sm Sm ik Smj − ij Smk . `−1 m=1 m=1

(5)

k=1

The space of solutions is of dim = ` + 1 (cf. [MSY]). We can reproduce the diffeomorphism φ from SLPD (4) with (5) whose coefficients are given by the Schwarzian derivatives of φ. Let u0 , . . . , u` be a set of linearly independent solutions. Then we have φ ≡ (u1 /u0 , . . . , u` /u0 )

(mod P GL(` + 1, K)).

Note that the group GL(` + 1, K) acts naturally on the space of solutions, and the group P GL(` + 1, K) acts projectively on K ` . Conversely, given a SLPD di dj u − pkij dk u − p0ij u = 0 with pkij = pkji ,

` X

(6)

pkik = 0 (canonical-form condition),

k=1

then this SLPD (6) is integrable if and only if  ` ` ` X X X   k k m k m k 1 0  = p (p ) − (p ) + p p − p p  j k ik ij ik mj ij mk ij `−1    m=1 m=1 k=1   ` `  X X t k k 0 k t (pkij )m + pij pmt + δm pij = (pmj )i + pmj pkit + δik p0mj   t=1 t=1    ` `  X X   0  ptij p0mt = (p0mj )i + ptmj p0it (pij )m + t=1

t=1

(7)

Differential Equations and Schwarzian Derivatives

139

for 1 ≤ i, j, k, m ≤ `. The collection pkij defines an affine connection on K ` . The set of equations (7) is just equivalent to the vanishing of its projective Weyl tensor Wmij k = Rmij k +

1 k (Rmj δik − Rij δm ) `−1

(` > 2) or projective Cotton tensor Wmij = (Rij )m − (Rmj )i +

X

Γija Rma −

X

Γmj aRia

a

a

(` = 2), which, in turn, is equivalent to the projective flatness of the affine connection. Here Rijk m and Rij respectively denotes the Riemann and the Ricci curvature tensors with respect to pkij .

6 System of third-order PDE’s In this section, we consider the system of third-order partial differential equations of m independent variables and one dependent variable ∂3y ∂2y ∂y ∂y ∂ 2 y ∂2y , = fijk (x1 , . . . , xm , y, ,..., , ,..., ) 2 ∂xi ∂xj ∂xk ∂x1 ∂xm ∂x1 ∂x1 ∂x2 ∂xm 2 (8) for all 1 ≤ i ≤ j ≤ k ≤ m. The number of equations in (8) is equal to m(m − 1)(m − 2)/6. Put x = (x1 , x2 , . . . , xm ),

pi =

∂y , ∂xi

qij =

∂2y ∂xi ∂xj

(1 ≤ i, j ≤ m),

and put p = (p1 , . . . , pm ), q = (qij ). Then q is a symmetric m × m matrix whose (ij)-component is equal to qij . The system (8) is written as ∂3y = fijk (x, y, p, q) ∂xi ∂xj ∂xk

for all 1 ≤ i ≤ j ≤ k ≤ m.

We have the spaces J 0 (m, 1) = {x, y}, J 1 (m, 1) = {x, y, p}, J 2 (m, 1) = {x, y, p, q}. The (2m + 1)-dimensional manifold J 1 (m, 1) = {x, y, p} = K 2m+1 has the Pm natural contact form η = dy − i=1 pi dxi . The (m2 + 5m + 2)/2-dimensional manifold J 2 (m, 1) = {x, y, p, q} = Pm 2 K (m +5m+2)/2 has the contact form η = dy − i=1 pi dxi and the natural higher order contact forms

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Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

θi = dpi −

m X

(1 ≤ i ≤ m).

qij dxj

j=1

Let D1 be the null space of η on J 2 (m, 1). Then D1 is a distribution of dimension (m2 + 5m)/2 (codimension one) on J 2 (m, 1). Let D2 be the null space of the Pfaffian system {η, θi (1 ≤ i ≤ m)} on J 2 (m, 1). Then D2 is a distribution of dimension (m2 + 3m)/2 (codimension m + 1) on J 2 (m, 1). We have [Γ (D2 ), Γ (D2 ] = Γ (D1 ),

[Γ (D2 ), Γ (D1 )] = Γ (T J 2 (m, 1)).

The system of the third-order partial differential equations 3OPDE (8) corresponds to the exterior differential system I on J 2 (m, 1) generated by the 1-forms η = dy −

m X

pk dxk ,

θi = dpi −

k=1

m X

qik dxk

(1 ≤ i ≤ m),

k=1

dqij −

m X

fijk dxk

(1 ≤ i ≤ j ≤ m).

k=1

The null space I ⊥ of the Pfaffian system I is a distribution of dimension m generated by m

X X ∂ ∂ ∂ ∂ + pk + qik + fijk ∂xk ∂y i=1 ∂pi ∂qij

(1 ≤ k ≤ m).

i≤j

Thus we have I ⊥ ⊂ D2 . We say that 3OPDE (8) is integrable if the m-dimensional distribution I ⊥ is completely integrable. In such case, the distribution I ⊥ defines an mdimensional foliation on J 2 (m, 1). The restriction of the volume form Ω = dx1 ∧ · · · ∧ dxm on each leaf is non-zero. Let π12 : J 2 (m, 1) → J 1 (m, 1),

π02 : J 2 (m, 1) → J 0 (m, 1)

be the natural projections, and let (π12 )∗ : T J 2 (m, 1) → T J 1 (m, 1), (π02 )∗ : T J 2 (m, 1) → T J 0 (m, 1) be the induced linear maps. The kernel Ker (π12 )∗ ⊂ T J 2 (m, n) is a m(m + 1)/2-dimensional distribution generated by ∂ , (1 ≤ i ≤ j ≤ m) . This distribution Ker (π01 )∗ is the null space of the ∂qij exterior differential system J generated by (2m + 1) 1-forms (Pfaffian forms) dxi (1 ≤ i ≤ m), dy, dpi (1 ≤ j ≤ m). Hence Ker (π01 )∗ is contained in the distribution D2 . We say that a distribution on J 2 (m, 1) is isotropic if it is contained in D2 . We have two Pfaffian systems I and J on J 2 (m, 1) such that

Differential Equations and Schwarzian Derivatives

141

I ⊥ ⊕ J ⊥ = D2 . Thus I and J are transverse isotropic Pfaffian systems on J 2 (m, 1) (cf. [SY]). Suppose that 3OPDE (8) is integrable. Let L be a leaf of the distribution I ⊥ . Since the volume form Ω = dx1 ∧ · · · ∧ dxm is non-zero on L, the leaf L projects down by π02 to an m-dimensional manifold in J 0 (m, 1) which is expressed as π01 (L) = (x, y L (x)). Since the leaf L in J 2 (m, n) is an integral manifold of I, L is written as ∂y L (x) ∂ 2 y L (x) L = x, y L (x), , . ∂xj ∂xi ∂xj The function y L (x) is a solution of 3OPDE (8). Thus L in J 2 (m, 1) is a second-order prolongation of a solution. On the other hand, π12 (L) = ∂y L (x) 1 L in J (m, 1) is a prolongation of a solution. x, y (x), ∂xj The codimension (m2 + 3m + 2)/2 of a leaf L in J 2 (m, 1) corresponds to the number of values ∂y L (x0 ) ∂ 2 y L (x0 ) L y (x0 ), , , ∂xj ∂xi ∂xj where x0 is an arbitrary fixed point in Rm Thus the m-dimensional foliation on J 2 (m, 1) defined by an integrable 3OPDE (8) projects down by π12 to an m(m + 1)/2-dimensional parameter families of m-dimensional foliations on the (2m + 1)-dimensional manifold J 1 (m, 1). Thus we may say that the set of prolongations of solutions is an ∞m(m+1)/2 web of dimension m on J 1 (m, 1). Since a foliation is a 1-web, we may say that the set of second-order prolongations of solutions is a 1-web on J 2 (m, 1). We study the case when 3OPDE (8) is integrable. More precisely, we investigate the conditions such that 3OPDE (8) is equivalent by a contact diffeomorphism of J 1 (m, 1) to the most simple system ∂3y =0 ∂xi ∂xj ∂xk

(9)

for all 1 ≤ i ≤ j ≤ k ≤ m. We call this simple system the f lat system. Obviously the flat system is integrable.

7 Compactification of the flat system for 3OPDE If a 3OPDE (8) is integrable, the m-dimensional distribution I ⊥ is completely integrable on J 2 (m, 1). Each leaf is the second-order prolongation of a solution

142

Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

of 3OPDE (8). Thus the space SOL of all solutions of (8) is an (m2 +3m+2)/22 dimensional space. The induced topology on SOL(m +3m+2)/2 may not be Hausdorff. For the flat system (9), we can describe concretely each member of the double fibration. We can naturally compactify each member with a structure. We use the same notation for compactified members. Thus by a compactification, we have; J 1 (m, 1) = Gr(2m + 2, 1) = Sp(m + 1, K)/H1 = Spm+1 /Spm × U1 , J 2 (m, 1) = Sp(m + 1, K)/H12 = Spm+1 /Um × U1 , SOL = Sp(m + 1, K)/H2 = Spm+1 /Um+1 . Here, when K = C, Ui = U (i) and Spi = Sp(i). When K = R, Ui = SO(i) and Spi = U (i). The space SOL = Spm+1 /Um+1 is isomorphic to the space of the set of all Lagrangian subspaces Lag(m + 1) of the symplectic vector space K m+1 . Thus corresponding to the flat system (9), we have the double fibration of compactified spaces; Sp(m + 1, K)/H12 = J 2 (m, 1) Lag(m)

.

J 1 (m, 1) = Gr(2m + 2, 1)

& Gr(m + 1, 1) SOL = Lag(m + 1)

This corresponds to the following double fibration of Dynkin diagrams with black nodes.

α1

α2

αm−1 αm

. α1

α2

αm−1 αm

αm+1

& αm+1

α1

α2

αm−1 αm

αm+1

8 Orbit of contact diffeomorphisms and Contact Schwarzian Derivatives Let ϕ : K 2m+1 → K 2m+1 be a contact diffeomorphism with a nonvanishing function f such that ϕ∗ η = f η. The contact diffeomorphism ϕ maps the ∂3y equations = 0 to another system of partial differential equations ∂xi ∂xj ∂xk

Differential Equations and Schwarzian Derivatives

∂3y = f ϕ ijk (x, y, p, q) ∂xi ∂xj ∂xk

143

for 1 ≤ i ≤ j ≤ k ≤ m.

Here f ϕ ijk are smooth functions of 2m + 1 + 21 m(m + 1) =

m2 +5m+2 2

variables.

k We define (multi-dimensional) contact Schwarzian derivatives Cij (ϕ) (Definition 2.1) which is equivalent to that of Fox [Fo]. For m = 1, the definition is given in [St1], [OS]. In the following Theorem 2, we write down explicitly the function k f ϕ ij (x, y, p, q) by using the contact Schwarzian derivatives Cij (ϕ). This extends a result of [St1] for the case of m = 1.

On the contact space K 2n+1 = {(x, y, p)}, the total differential dxdk is ∂ ∂ d = + pk . We use the following notation. For a function defined by dxk ∂xk ∂y A, the total differential of A is expressed as A;k =

dA = A xk + A y p k . dxk

Let ϕ : K 2n+1 → K 2n+1 be a contact transformation given by ϕ(x1 , . . . , xn , y, p1 , . . . , pn ) = (X1 , . . . , Xn , Y, P1 , . . . , Pn ). Corresponding to ϕ, define (n × n) matrices X; , Xp , P; , Pp by (X; )ij = Xi ;j ,

(Xp )ij =

∂Xi , ∂pj

(P; )ij = Pi ;j ,

(Pp )ij =

∂Pi , ∂pj

and define (2n) × (2n)-matrix Xϕ by X; Xp Xϕ = P; Pp

x; xP . For 1 ≤ r, s ≤ 2n, let αrs be p; pP the (r, s)-component of the matrix Xϕ ; αrs = (Xϕ )rs . Let βrs be the (r, s)component of the matrix Xϕ−1 ; βrs = (Xϕ−1 )rs . Then βst is equal to αst by replacing small letters x, p with capital letters X, P .

(cf. [Sto]). Then (Xϕ )−1 = Xϕ−1 =

For 1 ≤ r, s, t ≤ 2n, put ( αrs ;t if 1 ≤ t ≤ n , αrst = ∂αst if n ≤ t ≤ 2n ∂pt−n

µtr s =

2n X

βtu αurs

u=1

Then, for 1 ≤ i, j, k ≤ 2n, the contact Schwarzian derivatives Cikj = Cikj (ϕ) is defined by

144

Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI 2n

Cikj =

X 1 1 k µi j + µkj i − (δik µsj s + δik µss j + δjk µsi s + δjk µss i ). 2 2(2n + 1) s=1

Now we have the following theorem; Theorem 2. By the inverse contact diffeomorphism ϕ−1 , the system of 3OPDE ∂3Y = 0 (1 ≤ i ≤ j ≤ k ≤ m) ∂Xi ∂Xj ∂Xk is mapped to a system of 3O-PDE ∂3y = f ϕ ijk (x, y, p, q) ∂xi ∂xj ∂xk

(1 ≤ i ≤ j ≤ k ≤ m)

where f

ϕ

ijk (x, y, p, q)

=

−Cjm+i k

+

m X

Cj` k qi`

−

`=1

− −

m X

Cjm+i m+` q`k +

`=1 m X `,r=1

m X

m+i Cm+` k qj`

`=1 m X

m X

Cj` m+r qi` qrk +

`,r=1

`,r=1 m+i Cm+` m+r q`j qrk +

` Cm+r k qi` qrj

m X

s Cm+` m+r qis q`j qrk .

`,r,s=1

This is an extension of the following result ([St1],[OS]). Example 3. Let ϕ : K 3 → K 3 given by ϕ(x, y, p) = (X(x, y, p), Y (x, y, p), P (x, y, p)) be a contact diffeomorphism. By the inverse contact diffeomorphism ϕ−1 , Y 000 (X) = 0 is mapped to y 000 (x) = −C12 1 + 3C11 1 y 00 − 3C22 2 (y 00 )2 + C21 2 (y 00 )3 .

9 Construction of Contact Diffeomorphisms The space J 1 (m, 1) = K 2m+1 = {(x, y, p)} is a contact manifold and we may identify it with the (2m + 1)-dimensional Heisenberg group   1 p 1 . . . pm y       0 1 . . . 0 x1        .. . 2m+1 . .. ..  . H = .           0 0 . . . 1 xm    0 0 ... 0 1

Differential Equations and Schwarzian Derivatives

145

We have the Heisenberg vector fields vi (1 ≤ i ≤ 2m + 1) on K 2m+1 = H 2m+1 defined by  ∂ ∂ (1 ≤ i ≤ m),   ∂xi + pi ∂y ∂ (m + 1 ≤ i ≤ 2m), vi = ∂pi−m   ∂ (i = 2m + 1). ∂y such that [vi+n , vj ] = δij v2m+1 . Define the second-order differential operators vij (1 ≤ i, j ≤ 2m) by vij =

1 (vi ◦ vj + vj ◦ vi ). 2

Let φ : K 2m+1 → K 2m+1 be a contact transformation given by φ(x, y, p) = (X, Y, P). We define a function fφ on K 2m+1 by X X dY − Pi dXi = fφ (dy − pi dxi ). Using the contact Schwarzian derivatives Cikj = Cikj (φ), we define a system of linear second order partial differential equations for unknown function Z by vij (Z) =

2m X

Cikj vk (Z) + Ci0j Z,

(10)

i=k

for 1 ≤ i ≤ j ≤ 2m. By a straightforward calculation, we know that this system of equations SLPD (10) is integrable if and only if Ci0j = −

)vm+s − C`si Cj` s − C`m+s Cj` m+s (Cisj )vs + (Cim+s j i . 2 + δi s + δi m+s + δj s + δj m+s

(11)

Here, note that the right hand side does not depend on s (1 ≤ s ≤ m) for contact Schwarzian derivatives Cikj = Cikj (φ). In such case, the space of solutions is of dimension 2m + 2. A basis of solutions are given by the following functions, where we write (f = fφ ). f −1/2 X1 , . . . , f −1/2 Xm , f −1/2 P1 , . . . , f −1/2 Pm , f −1/2 , f −1/2 (Y −

X

Pi Xi ).

In the case K = R, f −1/2 should be changed to |f |−1/2 . We can reproduce the contact diffeomorphism φ from SLPD (10) with (11) whose coefficients are given by the Schwarzian derivatives of φ. Conversely, given a set of functions ckij which are symmetric with respect to i, j, k. For 1 5 i, j 5 2m, 1 5 s 5 m, put

146

Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

Cisj = cim+s j ,

Cim+s = −csi j . j

Let Ci0j (1 ≤ i, j ≤ 2m) be symmetric functions with respect i, j. Define the system of second order linear partial differential equations vij (Z) =

2m X

Cikj vk (Z) + Ci0j Z.

(12)

i=k

for 1 ≤ i ≤ j ≤ 2m. Note that this system is made of vector fields vi , which are non-commutative. We can write down explicitly the integrability conditions of this system of equations. For 1 ≤ i, j, k ≤ 2m, 1 ≤ s ≤ m, put c

0 s ` 0 s Rsi j k = −(δm+i j − δi m+j )Cm+s k − C` k Ci j − δs k Ci j − (Ci j )vk 0 s ` 0 s +(δm+i k − δi m+k )Cm+s j + C` j Ci k + δs j Ci k + (Ci k )vj 0 +2(δm+j k − δj m+k )Cm+s i,

c

m+s m+s ` 0 0 )v k Rim+s j k = (δm+i j − δi m+j )Cs k − C` k Ci j − δm+s k Ci j − (Ci j m+s ` 0 −(δm+i k − δi m+k )Cs0 j + C`m+s j Ci k + δm+s j Ci k + (Ci k )vj

−2(δm+j k − δj m+k )Cs0 i . Then we have the following; Theorem 3. For m ≥ 2, the system of linear partial differential equations (12) is integrable if and only if c

Rsi j k = 0

for

1 ≤ i, j, k, s ≤ 2m.

(13)

If (13) holds, we obtain that, for any 1 ≤ s ≤ m, the equation (11) holds. In the case m = 1, this condition is just equivalent to (11) with s = 1. When m = 1, for 1 ≤ j ≤ 2m, 1 ≤ s ≤ m, put Cj0 2m+1 =

2((Cj0 s )vm+s − (Cj0 m+s )vs + C`0m+s Cj` s − C`0s Cj` m+s ) 2 + δj s + δj m+s

(14)

and, for 1 ≤ i, j ≤ 2m, 1 ≤ s ≤ m, put c

0 0 ` s Rsi j 2m+1 = −(δm+i j − δi m+j )Cm+s 2m+1 − 2Cm+s ` Ci j − (Ci j )v2m+1 0 0 s 0 0 +2Ctsj Cm+t i + 2(Cm+s i )vj − 2Cm+t j Ct i + δs j Ci 2m+1 ,

c

m+s 0 0 ` Rim+s )v2m+1 j 2m+1 = (δm+i j − δi m+j )Cs 2m+1 + 2Cs ` Ci j − (Ci j m+s 0 0 0 0 +2Ctm+s Cm+t i − 2Cm+t j Ct i − 2(Cs i )vj + δm+s j Ci 2m+1 . j

Differential Equations and Schwarzian Derivatives

147

If m = 1, for the integrability of (12), we need additional conditions c Rsi j 2m+1 = 0 for 1 ≤ i, j, s ≤ 2m. In the case m ≥ 2, the conditions (13) imply that the right hand side of (14) does not depend on s and that c Rsi j 2m+1 = 0 for 1 ≤ i, j, s ≤ 2m. Thus for m ≥ 2, the additional conditions c Rsi j 2m+1 = 0 for 1 ≤ i, j, s ≤ 2m are redundant. For two functions A and B on K 2m+1 , let I(A, B) be the function K 2m+1 defined by I(A, B) =

m X i=1

1 1 vi (A)vm+i (B) − vi (B) + Av2m+1 B − v2m+1 B . 2 2

Obviously, I(B, A) = −I(A, B). By a simple calculation, we have the following; Proposition 1. Let A and B be two solutions of SLPD (12), then for 1 ≤ j ≤ 2m, vj (I(A, B)) = 0, and, consequently, I(A, B) is a constant. Thus I is an alternate bilinear form on the space of solutions of SLPD (12) with the integrability condition (13). A solution A is determined by the values A(x0 ), vj (A)(x0 ) (1 ≤ j ≤ 2m + 1), where x0 ∈ K 2m+1 is an arbitrary fixed point. It is easy to show that the alternate bilinear form I is non-degenerate and it defines a symplectic structure on the space of solutions of integrable SLPD (12). Let {uj } (0 ≤ j ≤ 2m + 1) be a symplectic basis of the solutions such that O Im+1 (I(ui , uj ))0≤i,j≤2m+1 = cJ, where J = −Im+1 O and c is a non-zero constant. Then, for 1 ≤ j ≤ 2m, we have vj (um+1 )u0 − vj (u0 )um+1 +

m X

(vj (um+k+1 )uk − vj (uk )um+k+1 ) = 2δj2m+1 .

k=1

Note that the group P Sp(2m + 2, K) acts naturally on the space of solutions. The choice of the basis is determined up to an action of P Sp(2m+2, K). For 1 ≤ i ≤ m, put ! m ui um+i+1 1 u2m+1 X uk um+k . (15) Xi = , Pi = , Y = + u0 u0 2 u0 u0 2 k=1

148

Hajime SATO, Tetsuya OZAWA and Hiroshi SUZUKI

Theorem 4. Suppose that the SLPD (12) satisfies the integrability condition (13). Let {uj } (0 ≤ j ≤ 2m + 1) be a symplectic basis of the solutions. Define functions (Xi , Y, Pi ) for 1 ≤ i ≤ m) by (15). Then the map U : (xi , y, pi ) 7→ (Xi , Y, Pi ) is a contact transformation such that the contact Schwarzian derivatives k k Cij (U ) is equal to the coefficients Cij of the system SLPD (12). For m = 1, the system of linear partial differential equations and its integrability conditions are studied in [St1], [OS]. For m ≥ 2, this is a result of D.Fox ([Fo]). Another straightforward proof is given in [OSS] along the lines of this note.

References [Ca1] Cartan, E., Sur les vari´et´es a ` connexion projective, Bull. Soc. math. France, 52, pp. 205–241 (1924). [Fo] Fox, D., Contact Schwarzian Derivatives, Nagoya. Math. J., 179, pp. 163–187 (2005). [Gu] Gunning, R., On uniformization of complex manifolds: the role of connections, Math. Notes No.22, Princeton, Princeton University Press, 1978. [KO] Kobayashi, S. and Ochiai, T., Holomorphic Projective Structures on Compact Complex Surfaces, Math. Ann. 249, pp. 75–94 (1980). [MSY] Matsumoto, K, Sasaki, T. and Yoshida, M., Recent progress of GaussSchwarz theory and related geometric Structures, Mem. Fac. Sci. Kyushu Univ., 47, pp. 283–381 (1993). [OS] Ozawa, T. and Sato, H., Contact Transformations and Their Schwarzian Derivatives, Advanced Studies in Pure Math.,37, pp. 337–366 (2002). [OSS] Ozawa, T., Sato, H. and Suzuki, H, Orbit of differential equations and Schwarzian derivatives, preprint. [Sa] Sasaki, T., Projective Differential Geometry and Linear Homogeneous Differential Equations, Rokko Lectures in Math. 5, Kobe, Kobe University, 1999. [St1] Sato, H., Schwarzian derivatives of contact diffeomorphisms, Lobachevskii J. of Math. , 4, pp. 89–98 (1999). [St2] Sato, H., Orbit decomposition of space of differential equations, UK-Japan Winter School 2004: Geometry and Analysis Towards Quantum Theory, Seminar on Mathematical Science, 30, pp. 78–88 (2004), Dep. of math., Keio Univ. [SY] Sato, H. and Yoshikawa, A. Y., Third order ordinary differential equations and Legendre connections, J. Math. Soc. Japan, 50, pp. 993–1013 (1998). [Sto] Stormark, O., Lie’s Structure Approach to PDE systems, Encyclopedia of mathematics and its Applications, v. 80, Cambridge Univ. Press, 2000. [Ya1] Yamaguchi, K., Geometrization of jet bundles, Hokkaido math. J., 12, pp. 27–40 (1983). [Ya2] Yamaguchi, K., Differential Systems Associates with Simple Graded Lie Algebras, Advanced Studies in Pure Math.,22, pp. 413–494 (1993).

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[YY] Yamaguchi, K and Yatsui T., Geometry of Higher Order Differential Equations of Finite Type Associated with Symmetric Spaces, Advanced Studies in Pure Math.,37, pp. 397–458 (2002). [Yo1] Yoshida, M., Fuchsian Differential Equations, Aspects of Mathematics, Vieweg, Braunschweig, 1987. [Yo2] Yoshida, M., Schwarzian program (in Japanese), Sˆ ugaku, 40, pp. 36–46 (1988).

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Deformation of Batalin-Vilkovisky Structures Noriaki IKEDA Department of Mathematical Sciences, Ritsumeikan University Kusatsu, Shiga 525-8577, Japan [email protected]

A Batalin-Vilkovisky formalism is a most general framework to construct consistent quantum field theories. Its mathematical structure is called a BatalinVilkovisky structure. First we explain the rather mathematical setting of a Batalin-Vilkovisky formalism. Next, we consider deformation theory of a Batalin-Vilkovisky structure. Especially, we consider deformation of topological sigma models in any dimension, which is closely related to deformation theories in mathematics, including deformation from commutative geometry to noncommutative geometry. We obtain a series of new nontrivial topological sigma models and we find these models have the Batalin-Vilkovisky structures based on a series of new algebroids.

1 Introduction Topological field theory is a powerful method to analyze geometry by means of a quantum field theory.[BBRT] A Batalin-Vilkovisky formalism [BV] is a most general systematic method to treat a consistent quantum field theory. So it is natural to treat topological field theory by means of a Batalin-Vilkovisky formalism. Deformation theory is one of the main topics in mathematics. On the other hand, deformation theory of a quantum field theory is proposed by [BBH][BH] in the physical context. Purposes of this deformation theory are to find a new gauge theory, or to prove a no-go theorem to construct a new gauge theory. Here we apply deformation theory to topological field theories, especially to topological sigma models. Our purpose is to construct many geometries by topological field theories, classify geometries as topological field theories, and unify many deformation theories as a topological field theories. Moreover we can analyze many deformation theories in mathematics as quantum field theories. In this article, we consider a topological sigma model. Let X and M be two manifolds. We denote by φ a (smooth) map from X to M . A sigma model is a 151

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quantum field theory constructed from a map φ (and other auxiliary fields). We analyze structures on M by the structures induced from X. A topological sigma model is a sigma model independent of metrics of X and M [W1]. The AKSZ formulation [AKSZ] of the Batalin-Vilkovisky formalism is a general framework to construct a topological sigma model by the BatalinVilkovisky method. This formulation is appropriate to analyze geometry by means of a topological sigma model. In this article, first we generalize the AKSZ formulation to a general n dimensional base manifold X and a target manifold with general gradings p. Next we discuss deformation of a Batalin-Vilkovisky structure with general gradings. We can construct various geometrical structures on M if we consider general n and general grading p. This formulation provides a clear method to unify and to classify many geometries as Batalin-Vilkovisky structures, and to analyze them as quantum field theories. This construction includes many new topological sigma models as special cases, for examples, the topological sigma model with a K¨ ahler structure (A model), with a complex structure (B model) [W2][AKSZ], with a symplectic structure [W1], with a Poisson structure (the Poisson sigma model) [II][SS], with a Courant algebroid structure [R1][I4][HP], with a twisted Poisson structure [KS][I5], with a Dirac structure [KSS], with a generalized complex structure [Z][I5][Pe], and so on.

2 AKSZ formulation of Batalin-Vilkovisky formalism on Graded Bundles We explain general setting of the AKSZ formulation [AKSZ] of the BatalinVilkovisky formalism for a general graded bundle in the rather mathematical context. Let M be a smooth manifold in d dimensions. We consider a supermanifold ΠT ∗ M . Mathematically, ΠT ∗ M , whose bosonic part is M , is defined as a cotangent bundle with reversed parity of the fiber. That is, the base manifold M has a Grassman even coordinate and the fiber of ΠT ∗ M has a Grassman odd coordinate. We can introduce a grading. The coordinate on the base manifold has grade zero and the coordinate on the fiber has grade one. This grading is called the total degrees. Similarly, we can define ΠT M for a tangent bundle T M . We can consider more general assignments for the degree of the fibers of T ∗ M or T M . For a nonnegative integer p, we define T ∗ [p]M , which is called a graded cotangent bundle. T ∗ [p]M is a cotangent bundle the degree of whose fiber is p. If p is odd, the fiber is Grassman odd, and if p is even, the fiber is Grassman even. The coordinate on the bass manifold has the total degree zero and the coordinate on the fiber has the total degree p. We define a graded tangent bundle T [p]M in the same way. For a general vector bundle E, a

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graded vector bundle E[p] is defined in a similar way. E[p] is a vector bundle with degree of the fiber being shifted by p. 1 We consider a Poisson manifold N with a Poisson bracket {∗, ∗}. If we can ˜ from N , then the Poisson structure {∗, ∗} shifts construct a graded manifold N ˜ The graded Poisson bracket to a graded Poisson structure by grading of N. is called an antibracket and denoted by (∗, ∗). (∗, ∗) is graded symmetric and satisfies the graded Leibniz rule and the graded Jacobi identity with respect to the grading of the manifold. The antibracket (∗, ∗) with total degree −n + 1 satisfies the following identities: (F, G) = −(−1)(|F |+1−n)(|G|+1−n)(G, F ), (F, GH) = (F, G)H + (−1)(|F |+1−n)|G|G(F, H), (F G, H) = F (G, H) + (−1)|G|(|H|+1−n) (F, H)G, (−1)(|F |+1−n)(|H|+1−n) (F, (G, H)) + cyclic permutations = 0,

(1)

˜ and |F |, |G| and |H| are the total where F, G and H are functions on N degrees of functions, respectively. The graded Poisson structure is also called the P-structure. Typical examples of a Poisson manifold N are the cotangent bundle T ∗ M and the vector bundle E ⊕ E ∗ . Both bundles play important roles in this paper. Another example is a vector bundle E with a Poisson structure on the fiber. We consider these three bundles. First we consider a cotangent bundle T ∗ M . Since T ∗ M has a natural symplectic structure, we can define a Poisson bracket induced from the natural symplectic structure. If we take a local coordinate φi on M and a local coordinate Bi of the fiber, we can define a Poisson bracket as follows: 2 {F, G} ≡ F

← − → − − ← − → ∂ ∂ ∂ ∂ G − F G, ∂φi ∂Bi ∂Bi ∂φi

(2)

← − → − where F and G are a function on T ∗ M , and ∂ /∂ϕ and ∂ /∂ϕ are the right and left differentiations with respect to ϕ, respectively. Next we shift the degree of fiber by p, and we consider the space T ∗ [p]M . The Poisson structure changes to a graded Poisson structure. The corresponding graded Poisson bracket is called the antibracket, (∗, ∗). Let φi be a local coordinate of M and B n−1,i a basis of the fiber of T ∗ [p]M . 3 An antibracket (∗, ∗) on a cotangent bundle T ∗ [p]M is represented as: (F, G) ≡ F 1 2 3

← − → ← − → − − ∂ ∂ ∂ ∂ G − F G. ∂B p,i ∂φi ∂φi ∂B p,i

(3)

Note that only the fiber is shifted and the base space is not shifted. We take Einstein’s summation notation. We use bold notations for local coordinates of graded (super)vector bundles, while we use nonbold notations for local coordinates of usual vector bundles.

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The total degree of the (∗, ∗) is −p. Next, we consider a vector bundle E ⊕ E ∗ . There is a natural Poisson structure on the fiber of E ⊕ E ∗ induced from natural paring between E and E ∗ . If we take a local coordinate Aa on the fiber of E and Ba on the fiber of E ∗ , we define {F, G} ≡ F

− ← − → ← − → − ∂ ∂ ∂ ∂ G − F G, ∂Aa ∂Ba ∂Ba ∂Aa

(4)

where F and G are functions on E ⊕ E ∗ . We shift the degrees of fibers of E and E ∗ to E[p] ⊕ E ∗ [q], where p and q are positive integers. The Poisson structure changes to a graded Poisson structure (∗, ∗). Let Ap a be a basis of the fiber of E[p] and B q,a a basis of the fiber of E ∗ [q]. An antibracket is represented as ← − → − → − ← − ∂ ∂ ∂ ∂ pq (F, G) ≡ F G − (−1) F G. ∂Ap a ∂B q,a ∂B q,a ∂Ap a

(5)

The total degree of the (∗, ∗) is −p − q. Next, we consider a vector bundle E with a Poisson structure on the fiber. If we shift the degree of the fiber of E to E[p], the Poisson structure changes to a graded Poisson structure (∗, ∗). Let Ap a be a basis of the fiber of E[p], An antibracket is represented as (F, G) ≡ F

→ − ← − ∂ ab ∂ k G, ∂Ap a ∂Ap b

(6)

where F and G are a function on E[p] and k ab is a nondegenerate constant bivector induced from a (graded) Poisson structure. The total degree of the antibracket (∗, ∗) is −2p. Next we define a Q-structure. A Q-structure is a function S on a super˜ which satisfies the classical master equation (S, S) = 0. S is called manifold N a Batalin-Vilkovisky action, or simply an action. We require that S satisfies the compatibility condition S(F, G) = (SF , G) + (−1)|F |+1 (F, SG),

(7)

where F and G are arbitrary functions and |F | is the total degree of F . (S, F ) = δF generates an infinitesimal transformation(a Hamiltonian flow). We call this a BRST transformation, which coincides with the gauge transformation of the theory. We define a (classical) Batalin-Vilkovisky structure as follows: Definition 1. If a structure on a supermanifold has P-structure and Qstructure, it is called a Batalin-Vilkovisky structure.

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3 Batalin-Vilkovisky Structures of Abelian Topological Sigma Models 3.1 BF case In this section, we consider Batalin-Vilkovisky structures of topological sigma models. Let X be a base manifold in n dimensions, with or without boundary, and M be a target manifold in d dimensions. We denote φ a smooth map from X to M . We consider a supermanifold ΠT X, whose bosonic part is X. ΠT X is defined as a tangent bundle with reversed parity of the fiber. We extend a smooth function φ to a function φ : ΠT X → M . φ is an element of ΠT ∗ X ⊗ M . The total degree defined in the previous section is a grading with respect to M . We introduce a nonnegative integer grading on ΠT ∗ X. A coordinate on a bass manifold is zero and a coordinate on the fiber is one. This grading is called the form degrees. We denote degF the form degree of a function F . ghF = |F | − degF is called the ghost number. First we consider a P-structure on T ∗ [p]M . It is natural to take p = n − 1 to construct a Batalin-Vilkovisky structure in a topological sigma model. In other words, the dimensions of X labels the total degree of a BatalinVilkovisky structure on the supermanifold T ∗ [p]M . We consider T ∗ [n − 1]M for an n-dimensional base manifold X. Let φi be a local coordinate expression of ΠT ∗ X ⊗ M , where i, j, k, · · · are indices of the local coordinate on M . Let B n−1,i be a basis of sections of ΠT ∗ X ⊗ φ∗ (T ∗ [n − 1]M ). According to the discussion in the previous section, we can define an antibracket (∗, ∗) on a cotangent bundle T ∗ [n − 1]M as ← − ← − → − → − ∂ ∂ ∂ ∂ (F, G) ≡ F G−F i ∂B i G, ∂B ∂φ n−1,i n−1,i ∂φ

(8)

where F and G are functions of φi and B n−1,i . We take a Darboux coordinate φi , B n−1,i , but this is for simplicity. We can also take more general coordinates. The total degree of the antibracket is −n + 1. If F and G are functionals of φi and B n−1,i , we understand an antibracket is defined as Z

← − → − ← − → − ∂ ∂ ∂ ∂ G−F F (F, G) ≡ i ∂B i G, ∂B ∂φ n−1,i n−1,i ∂φ X

(9)

where the integration over X is understood as that on the n-form part of the integrand. Through this article, we always understand an antibracket on two functionals in a similar manner and abbreviate this notation. Next we consider a P-structure on E ⊕ E ∗ . A natural assignment of the total degrees is given by the nonnegative integers p and q with p + q = n − 1. That is, we consider E[p]⊕E ∗ [n−p−1], where 1 ≤ p ≤ n−2. We can naturally

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construct a topological sigma model in this case. Let bxc be the floor function which gives the largest integer less than or equal to x. If b n2 c ≤ p ≤ n − 2, we identify E[p] ⊕ E ∗ [n − p − 1] with the dual bundle E ∗ [n − p − 1] ⊕ (E ∗ )∗ [p]. Therefore 1 ≤ p ≤ b n−1 2 c provides different structures of grading. Let Ap ap be a basis of sections of ΠT ∗ X ⊗ φ∗ (E[p]) and B n−p−1,ap a basis of the fiber of ΠT ∗ X ⊗ φ∗ (E ∗ [n − p − 1]). From (5), we can define an antibracket as → − → − ← − ← − ∂ ∂ ∂ ∂ np G − (−1) F G. (10) (F, G) ≡ F ∂Ap ap ∂B n−p−1,ap ∂B n−p−1,ap ∂Ap ap We want to consider various grading assignments for E ⊕ E ∗ . Because each assignment induces different Batalin-Vilkovisky structures. In order to consider all independent assignments, we define the following bundle. Let Ep n−1 be b n−1 2 c series of vector bundles, where 1 ≤ p ≤ b 2 c. We consider Ep [p] ⊕ ∗ Ep [n − p − 1] and consider a direct sum b n−1 2 c

X

Ep [p] ⊕ Ep∗ [n − p − 1]

(11)

p=1

And we define a P-structure on the graded vector bundle  n−1  b 2 c X  Ep [p] ⊕ Ep∗ [n − p − 1] ⊕ T ∗ [n − 1]M.

(12)

p=1

A local (Darboux) coordinate expression of the antibracket (·, ·) is a sum of (8) and (10): b n−1 2 c

(F, G) ≡

X p=0

F

→ − ← − ∂ ∂ a p ∂Ap ∂B n−p−1

G − (−1)np F ap

← − ∂ ∂B n−p−1

ap

→ − ∂ G. (13) ∂Ap ap

where the p = 0 component is the antibracket (8) on the graded cotangent bundle T ∗ [n−1]M and A0 a0 = φi . Note that all terms of the antibracket have the total degree −n + 1. We can confirm that the antibracket (13) satisfies the identity (1). We construct a Q-structure on the bundle (12). The simplest and natural action for a topological sigma model is Z

b n−1 2 c

S0 =

X p=0

(−1)

n−p

B n−p−1

ap dAp

ap

,

(14)

X

where d is an exterior differential on X. The total degree of d is 1 because the form degree is 1 and we assign the ghost number 0. The integration over

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X is understood as that over the n-form part (the form degree n part) of the integrand. This action is an analogue of the fundamental form θ = pi dq i for a symplectic form ω, which has ω = −dθ, therefore this action is directly derived from the P-structure on the graded bundle. The total degree of S0 is n. It is called an abelian BF theory in n dimensions. S0 defines a Q-structure, since we can easily confirm that S0 satisfies the classical master equation: (S0 , S0 ) = 0,

(15)

’Abelian’ means that the theory has a U (1) gauge symmetry. The BRST transformation (the gauge symmetry) is defined as δ0 Φ ≡ (S0 , Φ) = dΦ,

(16)

where Φ is an arbitrary sectionof a total bundle Pb n−1 2 c Ep [p] ⊕ Ep∗ [n − p − 1] ⊕ T ∗ [n − 1]M . δ02 = 0 is satisfied from p=1 (S0 , S0 ) = 0, which is consistent with d2 = 0. 3.2 Chern-Simons with BF case For a vector bundle E with a Poisson structure on the fiber, we can construct an another topological sigma model if n is odd. We consider a graded vector bundle E[q] for E, the degree of fiber of which is shifted by q. Let Aq aq be a basis of sections of ΠT ∗ X ⊗ φ∗ (E[q]). From the equation (6), we can define an antibracket as → − ← − ∂ ab ∂ k G, (17) (F, G) ≡ F ∂Aq a ∂Aq b If we take a nonnegative integer q = n−1 2 , then the total degrees of (8), (10) and (17) are all −n + 1. Thus we can consider a combined P-structure. with Ep [p] ⊕ Ep∗ [n − p − 1] We consider a direct sum of E[q] = E n−1 2   n−3 2 X n−1 ∗   Ep [p] ⊕ Ep [n − p − 1] ⊕ E , (18) 2 p=1 where we have absorbed Ep [p] ⊕ Ep∗ [n − p − 1] to E n−1 for p = n−1 2 2 . We can define a natural P-structure on the graded vector bundle P n−3 ∗ 2 ⊕ T ∗ [n − 1]M . A local (Darboux) E [p] ⊕ E [n − p − 1] ⊕ E n−1 p p p=1 2 coordinate expression of the antibracket (·, ·) is a sum of (8), (10) and (17): n−3

(F, G) ≡

← − → − ∂ ∂ ap ∂B ∂A p n−p−1 p=0 → − ← − ∂ ∂ k ab G, +F ∂Aq a ∂Aq b 2 X

G − (−1)np F

F

ap

← − ∂ ∂B n−p−1

ap

→ − ∂ G ∂Ap ap (19)

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where the p = 0 component is the antibracket on the graded cotangent bundle T ∗ [n − 1]M and A0 a0 = φi . q = n−1 2 is needed for all the terms to have the total degree −n + 1. The simplest and natural action S0 is Z

n−3

S0 =

2 X

Z

(−1)n−p

p=0

B n−p−1

ap dAp

ap

+

X

X

k ab A n−1 a dA n−1 b , (20) 2 2 2

The second term is called an abelian Chern-Simons theory in n dimensions. S0 defines a Q-structure, since we can easily confirm that S0 satisfies the classical master equation: (S0 , S0 ) = 0,

(21)

This action also has a U (1) gauge symmetry. The BRST transformation is δ0 Φ = (S0 , Φ) = dΦ,

(22)

where Φ is an arbitrary section of the total bundle.

4 Deformation In this section, we consider deformation of Batalin-Vilkovisky structures. Deformation means deformation of the Q-structure for a fixed P-structure. We consider local deformation from the settled point (14) or (20) of the moduli space. we consider the BF case PFirst of the bundle b n−1 ∗ 2 c E [p] ⊕ E [n − p − 1] ⊕ T ∗ [n − 1]M . The Q-structure to begin with p p p=0 is the equation (14). We deform this Batalin-Vilkovisky action S0 to S = S0 + gS1 ,

(23)

under the condition that S also satisfies the classical master equation: (S, S) = 0,

(24)

where g is a deformation parameter and S1 represents all the deformation terms, P n−1which are functionals onX and functions on b 2 c Ep [p] ⊕ Ep∗ [n − p − 1] ⊕ T ∗[n − 1]M . We require that S is the total p=0 degree n. It is equivalent that the ghost number is zero ghS = |S| − degS = 0. This condition is physically necessary, though we can relax this condition mathematically. If two deformations S and S 0 satisfy S 0 = S + (S0 , T ) = S + δ0 T for some functional T , the two Batalin-Vilkovisky structures are equivalent. Therefore the problem is to look for the total degree n cohomology

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class P n−1 b 2 c Ep [p] ⊕ Ep∗ [n − p − 1] ⊕ T ∗ [n − 1]M . H n ΠT ∗ X ⊗ p=0 Substituting S = S0 + gS1 to (24), we obtain g expansion (S0 , S0 ) + 2g(S0 , S1 ) + g 2 (S1 , S1 ) = 0,

(25)

The zero-th order of the equation (25), (S0 , S0 ) = 0, is already satisfied. The first order is (S0 , S1 ) = 0. Because of the equation (16), S1 is the integration of an arbitrary function F of all fundamental superfields Φ which are Ap ap or B qbq , and their derivatives dΦ: Z S1 = F (Φ, dΦ). (26) X

For simplicity, we assume that there is no boundary contribution to S, that is, integration of total derivative terms on X is always zero. This corresponds to assume that there is no obstruction to deformation. Then we can prove the following theorem: R Theorem 1. Assume there is no boundary contribution on X, i.e. X dG(Φ) = 0 R for any function G. If a monomial of F (Φ, dΦ) includes at least one dΦ, X F (Φ, dΦ) is δ0 -exact. R Pn−1 R Proof. We can assume that X F (Φ, dΦ) = p=0 X Fn−p−1 dGp , where Fn−p−1 are functions with the form degree n − p − 1 and Gp are functions with the form degree p. From (16), we obtain δ0 F0 = 0, δ0 Fn−p−1 = dFn−p−2 dFn = 0, δ0 G0 = 0, δ0 Gp = dGp−1

for −1 ≤ p ≤ n − 2,

for 1 ≤ p ≤ n,

dGn = 0.

(27)

For even p, adjoining two terms are combined as Fn−p−1 dGp + Fn−p−2 dGp+1 = (−1)n−p−1 δ0 (Fn−p−1 Gp+1 ) − (−1)n−p−1 d(Fn−p−2 Gp+1 )

(28)

using the relations (27). Thus integration of these two terms is δ0 -exact. Pn−1 R If n is even, S1 = p=0 X Fn−p−1 dGp has even numbers of terms, therefore we combine each two term like (28) and we can confirm that S1 is δ0 -exact. If n is odd, the last term F0 dGn−1 remains. However this term is δ0 -exact because F0 dGn−1 = δ0 (F0 Gn ). Therefore S1 is δ0 -exact. t u

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Therefore sinceR the nontrivial S1 cohomological class does not include d, we can take S1 = X F (Φ). Concretely we can express Z X Fp(1)···p(k),q(1)···q(l) ap(1) ···ap(k) bq(1) ···bq(l) (A0 a0 ) S1 = p(1),···,p(k),q(1),···,q(l) X ×Ap1 ap(1) · · · Apk ap(k) B q1 bq(1)

· · · B ql bq(l) ,

(29)

where Fp(1)···p(k),q(1)···q(l) ap(1) ···ap(k) bq(1) ···bq(l) (A0 a0 ) is a function of A0 a0 and p(r) 6= 0, q(s) 6= 0 for r = 1, · · · , k, s = 1, · · · , l. From the degS = n condition, Pk Pl we obtain α=0 |Apα ap(α) | + β=0 |B qβ bq(β) | = n. Second order of the equation (25) (S1 , S1 ) = 0,

(30)

imposes conditions on functions Fp(1)···p(k),q(1)···q(l) ap(1) ···ap(k) bq(1) ···bq(l) (A0 a0 ). These conditions determine the mathematical structure of a Batalin-Vilkovisky structure. We call the resulting field theory S = S0 + gS1 a nonlinear gauge theory in n dimensions. we consider the Chern-Simons with BF case PNext n−3 n−1 ∗ 2 . We make a similar discussion with p=1 Ep [p] ⊕ Ep [n − p − 1] ⊕ E 2 the BF case. We consider deformation of the Batalin-Vilkovisky action (20) to (23), S = S0 + gS1 , under the condition that S also satisfies the classical master equation (24). We obtain S1 by using Theorem 1. S1 has a similar expression with (29) but has different field contents. The second order of the equation (25) (S1 , S1 ) = 0, imposes conditions on functions Fp(1)···p(k),q(1)···q(l)

(31) ap(1) ···ap(k)

bq(1) ···bq(l)

(A0 a0 ).

5 Deformation in lower dimensions In this section, we concretely analyze the algebraic and geometric structure of deformation of a topological sigma model in lower dimensional X. We can easily find that we cannot obtain nontrivial deformation in case of the total degree n = 1, thus we cannot obtain a nontrivial structure. 5.1 n = 2 We analyze the algebraic structure of the total degree n = 2 topological sigma model. In two dimensions, the total graded bundle (12) is T ∗ [1]M = ΠT ∗ M . (23) under (29) is

Deformation of Batalin-Vilkovisky Structures

S = S0 + gS1 , Z S0 = B 1i dφi , X

Z S1 = Σ

1 ij f (φ)B 1i B 1j , 2

161

(32)

where i, j, · · · are indices of a local coordinate expressions on T ∗ [1]M . and we rewrite the notations as φi = A0 i and 12 f ij (φ) = F,11 ij (A0 ). This topological sigma model is known as the Poisson sigma model [II][SS]. If we substitute this S1 to the condition (30), we obtain the geometric structure of the action. We obtain the following identity on f ij : → − → − → − kl ∂ ij il ∂ jk jl ∂ f f +f f +f f ki = 0. (33) ∂φl ∂φl ∂φl If we restrict the identity on X (i.e. a ghost number zero sector), (33) reduces to f kl (φ)

∂f jk (φ) ∂f ki (φ) ∂f ij (φ) + f il (φ) + f jl (φ) = 0. l l ∂φ ∂φ ∂φl

(34)

Under the identity (33), −f ij defines a Poisson structure as {F (φ), G(φ)} ≡ −f ij (φ)

∂F ∂G , ∂φi ∂φj

(35)

on the space M . Conversely, if we consider the Poisson structure −f ij on M , which satisfies the identity (33), we can define the action (32) consistently. This Poisson bracket (35) is directly constructed if we restrict the derived bracket [Kos] of a Batalin-Vilkovisky structure: → − ← − ∂ ∂ ij f (φ) j G = ((S, F ), G), {F (φ), G(φ)} = −F i ∂φ ∂φ

(36)

to M and X, i.e. to the ghost number zero sector. The Batalin-Vilkovisky structure of this theory has the structure of a Lie algebroid T ∗ [1]M [LO][O], where M is a space of a (smooth) map from ΠT X to a target space M . A Lie algebroid is a generalization of a bundle of a Lie algebra over a base manifold M. A Lie algebroid over a manifold is a vector bundle E → M with a Lie algebra structure on the space of the sections Γ (E) defined by the Lie bracket [e1 , e2 ], e1 , e2 ∈ Γ (E) and a bundle map (the anchor) ρ : E → T M satisfying the following properties: 1, 2,

For any e1 , e2 ∈ Γ(E), [ρ(e1 ), ρ(e2 )] = ρ([e1 , e2 ]), For any e1 , e2 ∈ Γ(E), F ∈ C∞ (M), [e1 , F e2 ] = F [e1 , e2 ] + (ρ(e1 )F )e2 ,

(37)

We can derive the bracket of a Lie algebroid from the antibracket and the Batalin-Vilkovisky structure of the n = 2 theory. In our case, a vector bundle

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E is a cotangent bundle T ∗ [1]M. The Lie bracket of the two sections e1 and e2 is defined as a derived bracket of the antibracket by [e1 , e2 ] ≡ ((S, e1 ), e2 ),

(38)

and the anchor is defined by ρ(e)F (φ) ≡ (e, (S, F (φ))).

(39)

Then we can confirm [e1 , e2 ] = −[e2 , e1 ], from (e1 , e2 ) = 0 and the graded Jacobi identity of the antibracket. Similarly, a Lie algebroid conditions 1 and 2 on the bracket [·, ·] and the anchor map ρ are obtained from the classical master equation (S, S) = 0. From this derived bracket, we directly obtain the following “noncommutative” relation of the coordinates: [φi , φj ] = −f ij (φ). The anchor is a differentiation on functions of φ as → − ∂ ρ(φi )F (φ) = −f ij (φ) j F (φ). ∂φ

(40)

(41)

5.2 n = 3 BF case We analyze a nonlinear gauge theory in three dimensions. In this case, the theory defines the topological open 2-brane as a sigma model [I2][Pa]. We consider a supermanifold ΠT X whose ghost number zero part is a three-dimensional manifold X. A base space M is the space of a (smooth) map φ from ΠT X to a target space M . In n = 3, the total bundle (11) is a vector bundle E[1] ⊕ E ∗ [1]. From (12), we consider (E[1] ⊕ E ∗ [1]) ⊕ T ∗ [2]M . We obtain an antibracket (·, ·) on the space of sections of ΠT ∗ X ⊗ φ∗ (E[1] ⊕ E ∗ [1] ⊕ T ∗ [2]M ) if we set n = 3 in (13). In order to write down the Batalin-Vilkovisky action S, we take a local basis on ΠT ∗ X ⊗φ∗ (E[1]⊕E ∗ [1]) as A1 a , B 1 a , which are Darboux coordinates such that (A1 a , A1 b ) = (B 1a , B 1b ) = 0 and (A1 a , B 1b ) = δ a b . Moreover we introduce B 2i a section of ΠT ∗ X ⊕ φ∗ (T ∗ [2]M ). The total action (23) under (29) is S = S0 + gS1 , Z S0 = [−B 2i dφi + B 1a dA1 a ], X Z 1 S1 = [f1a i (φ)A1 a B 2i + f2ib (φ)B 2i B 1b + f3abc (φ)A1 a A1 b A1 c 3! X 1 1 + f4ab c (φ)A1 a A1 b B 1c + f5a bc (φ)A1 a B 1b B 1c 2 2 1 abc + f6 (φ)B 1a B 1b B 1c ], (42) 3!

Deformation of Batalin-Vilkovisky Structures

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1 f3abc = F111,abc , 12 f4ab c = F11,1ab c , where we set f1a i = F1,2a i , f2ib = F,21 ib , 3! 1 1 bc bc abc abc f = F , f = F , for clarity. The condition of the 1,11a ,111 2 5a 3! 6 classical master equation (30) imposes the following identities on six fi ’s, i = 1, · · · , 6[I2]:

f1e i f2 je + f2 ie f1e j = 0, ! ! → − → − ∂ ∂ i i j f1c f1b + f1b f1c j + f1e i f4bc e + f2 ie f3ebc = 0, − ∂φj ∂φj ! ! → − → − ∂ ∂ j ic jc i f1b f2 − f2 f1b + f1e i f5b ec − f2 ie f4eb c = 0, ∂φj ∂φj ! ! → − → − ∂ ∂ f2 ic + f2 jc f2 ib + f1e i f6ebc + f2 ie f5e bc = 0, −f2 jb ∂φj ∂φj ! ! → − → − ∂ ∂ j jd d −f1[a + f2 f4bc] f3abc + f4e[a d f4bc] e + f3e[ab f5c] de = 0, ∂φj ∂φj ! ! → − → − ∂ ∂ f5b] cd − f2 j[c f4ab d] −f1[a j ∂φj ∂φj +f3eab f6 ecd + f4e[a [d f5b] c]e + f4ab e f5e cd = 0, ! ! → − → − ∂ ∂ j[b bcd cd] j + f2 + f4ea [b f6 cd]e + f5e [bc f5a d]e = 0, f6 f5a −f1a ∂φj ∂φj ! → − ∂ bcd] j[a f6 + f6 e[ab f5e cd] = 0, −f2 ∂φj ! → − ∂ j −f1[a f3bcd] + f4[ab e f3cd]e = 0, (43) ∂φj where [· · ·] on the indices represents the antisymmetrization of them, e.g., Φ[ab] = Φab − Φba . If we restrict fields to X, (43) reduces to the following identities: f1e i f2 je + f2 ie f1e j = 0, ∂f1c i ∂f1b i j − + f f1c j + f1e i f4bc e + f2 ie f3ebc = 0, 1b ∂φj ∂φj i ∂f2 ic jc ∂f1b f1b j − f + f1e i f5b ec − f2 ie f4eb c = 0, 2 ∂φj ∂φj ib ∂f2 ic jc ∂f2 −f2 jb + f + f1e i f6ebc + f2 ie f5e bc = 0, 2 ∂φj ∂φj ∂f4bc] d ∂f3abc −f1[a j + f2 jd + f4e[a d f4bc] e + f3e[ab f5c] de = 0, ∂φj ∂φj

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Noriaki IKEDA

d] ∂f5b] cd j[c ∂f4ab − f + f3eab f6 ecd + f4e[a [d f5b] c]e + f4ab e f5e cd = 0, 2 ∂φj ∂φj cd] ∂f6 bcd j[b ∂f5a + f + f4ea [b f6 cd]e + f5e [bc f5a d]e = 0, −f1a j 2 ∂φj ∂φj ∂f6 bcd] −f2 j[a + f6 e[ab f5e cd] = 0, ∂φj ∂f3bcd] −f1[a j + f4[ab e f3cd]e = 0. (44) ∂φj

−f1[a j

The algebraic structure (43) (or (44)) is a Courant algebroid. A Courant algebroid has been introduced by Courant in order to analyze the Dirac structure as a generalization of a Lie algebra of vector fields on a vector bundle [C][LWX]. The Batalin-Vilkovisky structure on a Courant algebroid is first analyzed in [R1]. A Courant algebroid is a vector bundle E → M and has a nondegenerate symmetric bilinear form h· , ·i on the bundle, a bilinear operation ◦ on Γ (E) (the space of sections on E), and a bundle map (called the anchor) ρ : E → T M satisfying the following properties: 1, 2,

e1 ◦ (e2 ◦ e3 ) = (e1 ◦ e2 ) ◦ e3 + e2 ◦ (e1 ◦ e3 ), ρ(e1 ◦ e2 ) = [ρ(e1 ), ρ(e2 )],

3,

e1 ◦ F e2 = F (e1 ◦ e2 ) + (ρ(e1 )F )e2 , 1 e1 ◦ e2 = Dhe1 , e2 i, 2 ρ(e1 )he2 , e3 i = he1 ◦ e2 , e3 i + he2 , e1 ◦ e3 i,

4, 5,

(45)

where e1 , e2 and e3 are sections of E, and F is a function on M; D is a map from functions on M to Γ (E) and is defined by hDF , ei = ρ(e)F . In our nonlinear gauge theory, M is the space of a map φ from ΠT X to M and E is the space of sections of ΠT ∗ X ⊕ φ∗ (E[1] ⊕ E ∗ [1]). Let ea be a local basis of sections of E. We take ea = A1 a or B 1 a , which are a E[1] component or a E ∗ [1] component respectively. We define a graded symmetric bilinear form h· , ·i, a bilinear operation ◦, a bundle map ρ and D from the antibracket as follows: ea ◦ eb ≡ ((S, ea ), eb ), hea , eb i ≡ (ea , eb ), ρ(ea )F (φ) ≡ (ea , (S, F (φ))), D(∗) ≡ (S, ∗).

(46)

Then we can confirm that the classical master equation (S, S) = 0, which derive the identity (43) on structure functions f ’s, is equivalent to the conditions 1 to 5 of the equation (45). [R1][I4] We calculate the operations ◦ and ρ on the basis as follows:

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A1 a ◦ A1 b = −f5c ab (φ)A1 c − f6 abc (φ)B 1c , A1 a ◦ B 1b = −f4bc a (φ)A1 c + f5b ac (φ)B 1c , B 1a ◦ B 1b = −f3abc (φ)A1 c − f4ab c (φ)B 1c , ρ(A1 a )φi = −f2 ia (φ), ρ(B 1a )φi = −f1a i (φ).

(47)

This topological sigma model defines a Courant algebroid structure on the space E[1] ⊕ E ∗ [1]. We call this model as the Courant sigma model. Chern-Simons with BF case Since we can consider Chern-Simons with BF case if n is odd, In three dimension, we can construct another model. Let E be a vector bundle with a Poisson structure on the fiber. If we take n = 3 in the equation (18), we obtain E [1]. A P-structure is defined on the graded vector bundle E[1] ⊕ T ∗ [2]M by setting n = 3 in the equation (19). The abelian action is the n = 3 case in the action (20): Z −B 2i dφi +

S0 = X

k ab A1a dA1b , 2

(48)

and deformation is obtained as follows: S = S0 + gS1 , Z 1 a a b c i S1 = f1a (φ)A B i + f2abc (φ)A A A , 6 X

(49)

where we rewrite two structure functions f1a i = gF11,a i and 16 f2abc = gF30,abc for clarity. If we substitute (49) to the condition (30), we obtain the identities on the structure functions f1a i and f2abc as k ab f1a i f1b j = 0, ! ! → − → − ∂ ∂ j i i f1b f1c − f1c f1b j + k ef f1e i f2f bc = 0, ∂φj ∂φj ! → − → − → − → − ∂ ∂ ∂ ∂ f1d j f2abc − f1c j f2dab + f1b j f2cda − f1a j f2bcd ∂φj ∂φj ∂φj ∂φj +k ef (f2eab f2cdf + f2eac f2dbf + f2ead f2bcf ) = 0.

(50)

The identities (50) define a Courant algebroid. In the definition of the Courant algebroid, M is the space of a map φ from ΠT X to M and E is the space of sections of ΠT ∗ X ⊕ φ∗ (E[1]). We define a graded symmetric bilinear form h· , ·i, a bilinear operation ◦, a bundle map ρ and D by the antibracket with the same as the equation (46), Then we can confirm that

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the classical master equation (S, S) = 0, which derive the identity (50) on structure functions f ’s, is equivalent to the conditions 1 to 5 of a Courant algebroid (45). [I3] We take the basis of the section of the fiber ea = A1 a . We calculate the operations ◦ and ρ on the basis as follows: Aa ◦ Ab = −k ac k bd f2cde (φ)Ae , hAa , Ab i = k ab , ρ(Aa )φi = −f1c i (φ)k ac .

(51)

All deformations of a topological sigma model on the space E[1] have a Courant algebroid structure. 5.3 General n In n dimensions, the equation (30), (S1 , S1 ) = 0, impose an algebroid structure Pb n−1 c on the space (11), p=12 Ep [p] ⊕ Ep∗ [n − p − 1]. The algebroid structure is derived from the Batalin-Vilkovisky structure (S1 , S1 ) = 0 of nonlinear gauge theories. Now we obtain an infinite series of algebroids labeled by n. We call this algebroid an n-algebroid. In the previous section, we have found that the n = 2 case defines a Lie algebroid on T ∗ M and the n = 3 case defines a Courant algebroid on E ⊕ E ∗ . For n ≥ 4 cases, we can easily calculate algebraic relations but characterization of algebroid structures is still unknown. Higher order generalization has also been discussed in [Se]. In the Chern-Simons case, the equation an alge P n−3 (30), (S1 , S1 ) = 0 impose n−1 ∗ 2 . broid structure on the space (18), p=1 Ep [p] ⊕ Ep [n − p − 1] ⊕ E 2 In the previous section, we have found that the n = 3 case defines a Courant algebroid on E.

6 Quantum Version of Deformation In the previous sections, we have considered a classical BV structure. In this section, we discuss quantum version of deformation of a Batalin-Vilkovisky structure. In this section, we discuss the BF case. We can make a similar discussion in the Chern-Simons with BF case. In order to quantize a gauge theory, we must fix the gauge. Gauge fixing is carried out by adding a gauge fixing term SGF to the classical action S.[GPS] The gauge fixed quantum action is Sq = S + SGF . We need the BV Laplacian. The BV Laplacian is defined as follows: b n−1 2 c

∆F ≡

X p=0

→ − → − ∂ ∂ F a p ∂Ap ∂B n−p−1,ap

(52)

Deformation of Batalin-Vilkovisky Structures

167

The BV Laplacian satisfies the following identity: ∆(F · G) = (∆F )G + (−1)(n+1)|F | (F, G) + (−1)|F | F ∆G.

(53)

In order for the generating functional to be gauge invariant in the quantum sense, the following quantum master equation is required: (Sq , Sq ) − 2i¯ h∆Sq = 0,

(54)

for the quantum action Sq . In our n-algebroid topological sigma model, two terms are independently satisfied, i.e. ∆Sq = 0 and (Sq , Sq ) = 0. O is called an observable if an operator O satisfies the following equation: (Sq , O) − i¯ h∆O = 0.

(55)

Generally, there are two kinds of observables. One is the integration of a local function F on the boundary ∂X. Let Xr ⊂ ∂X be a r-cycle on the boundary ∂X. If F has the form degree r, the integration of F on the r cycle Xr : Z O= F (Φ) (56) Xr

is nontrivial and satisfies (55). Another observable is constructed from A0 a . We consider a function F of a A0 and restrict F on the boundary, OF ≡ F (A0 a )|∂X . We can confirm that (0) the form degree zero part OF of OF is an local observable with ghost number zero on the boundary. The generating functional is defined by the path integral as Z[Ok ] =

Z [ n−1 2 ] Y

i

DAp DB n−p−1 e h¯ (Sq +

P r

Jr O r )

,

(57)

p=0

where DAp DB n−p−1 is a path integral measure and Jk are source fields and Ok are observables and h ¯ = g. We consider n = 2 case. Let X be a two-dimensional disc. Note that classical deformation derives a Poisson structure on T ∗ M in n = 2 case. On the other hand, quantum deformation derives the deformation quantization on a Poisson manifold M . [Kon] The correlation function of two local observables (0) (0) Of and Og derives the Kontsevich’s star product formula [CF] on a Poisson manifold: Z i (0) f ∗ g(x) = DφDB 1 Of (φ(1))Og(0) (φ(0))e h¯ Sq , (58) φ(∞)=x

where φ = A0 and 0, 1, ∞ are three distinct points at the boundary ∂X.

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Noriaki IKEDA

If we calculate the same correlation function for S0 , we obtain the usual product of functions f and g: Z i (0) f (x)g(x) = (59) DφDB 1 Of (φ(1))Og(0) (φ(0))e h¯ S0q , φ(∞)=x

where S0q = S0 + SGF . Therefore quantum deformation in n = 2 is equivalent to the star deformation on C ∞ (M ). We can generalize this discussion to higher orders. Deformation S0 → S derives a generalization of the star deformation to higher dimensions as follows: mk [O1 , O2 , · · · , Ok ] =

Z [ n−1 2 ] Y

i

DAp DB n−p−1 O1 O2 · · · Ok e h¯ Sq ,

(60)

p=0

under the appropriate regularization and the boundary conditions, where S is the deformation (23) of the abelian topological sigma model and Or ’s are two kinds of observables at the boundary. The correlation functions satisfy the Ward-Takahashi identity derived from the gauge symmetry: Z [ n−1 2 ] Y

i DAp DB n−p−1 ∆ Oe h¯ Sq = 0,

(61)

p=0

which leads a quantum geometric structure on the space of correlation functions.

7 Summary and Outlook We have discussed deformation of Batalin-Vilkovisky structures of topological sigma models in n dimensions. We have constructed general theory of most general deformation in general n dimensions. We have analyzed structures in the case of n = 2 and 3 in detail. In n = 2 deformation of a BV structure produces a Lie algebroid structure, and in n = 3, deformation produces a Courant algebroid structure. For n ≥ 4, characterization of n-algebroids obtained by deformation of topological sigma models is still unknown and an open problem. We have also discussed quantum version of deformation. For n = 2 case, the deformation on the disc X is equivalent to the deformation quantization on a Poisson manifold M . In n = 2, there are two special important cases of deformations. They are A-model and B-model. [W2] There are many investigations to analyze quantum moduli. For reviews, see [BCOV], [HKKPTVVZ] and references therein. n = 3 quantum deformation is analyzed in [HM]. For general n ≥ 4, quantum structures are unknown. If we analyze higher n cases, we will obtain interesting mathematical and physical structures.

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In this article, we assume the p and n − p − 1 are nonnegative integers in E[p] ⊕ E ∗ [n − p − 1], where we identify the p = 0 bundle with a cotangent bundle T ∗ [n − 1]M . We will be able to generalize our discussions to negative integers p and n − p − 1. A special case has been analyzed in [BM][II2]. We have to analyze all moduli, i.e. we should consider the Kodaira-Spencer theory of Batalin-Vilkovisky structures.

References [AKSZ]

M. Alexandrov, M. Kontsevich, A. Schwartz and O. Zaboronsky: The Geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys., A 12, 1405(1997), hep-th/9502010 [BBH] G. Barnich, F. Brandt and M. Henneaux: Local BRST Cohomology In The Antifield Formalism. 1. General Theorems, Commun. Math. Phys. 174, 57 (1995), hep-th/9405109; For a review, M. Henneaux: hep-th/9712226. [BBRT] D. Birmingham, M. Blau, M. Rakowski and G. Thompson: Topological field theory. Phys. Rept., 209, 129 (1991) [BCOV] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165, 311 (1994), hep-th/9309140 [BG] G. Barnich and M. Grigoriev, BRST extension of the non-linear unfolded formalism, hep-th/0504119 [BH] G. Barnich and M. Henneaux: Consistent Couplings Between Fields With A Gauge Freedom And Deformations Of The Master Equation, Phys. Lett. B 311, 123 (1993), hep-th/9304057 [BM] I. Batalin and R. Marnelius: Superfield algorithms for topological field theories, hep-th/0110140 [BV] I. A. Batalin and G. A. Vilkovisky: Gauge Algebra And Quantization, Phys. Lett. B 102, 27 (1981); Quantization Of Gauge Theories With Linearly Dependent Generators, Phys. Rev. D 28, 2567 (1983) , [Erratumibid. D 30, 508 (1984) ] [C] T. Courant, Dirac manifolds: Trans. A. M. S. 319, 631 (1990) [CF] A. S. Cattaneo and G. Felder: A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212, 591 (2000), math.qa/9902090 [GD] M. A. Grigoriev and P. H. Damgaard, Superfield BRST charge and the master action, Phys. Lett. B 474, 323 (2000), hep-th/9911092 [GPS] For a review, J. Gomis, J. Paris and S. Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept. 259 (1995) 1, hep-th/9412228 [HM] C. M. Hofman and W. K. Ma: Deformations of closed strings and topological open membranes, JHEP 0106, 033 (2001), hep-th/0102201 [HP] C. Hofman and J. S. Park, Topological open membranes, hep-th/0209148 [HT92] M. Henneaux, C. Teitelboim: Quantization of Gauge Systems, Princeton University (1992) [HKKPTVVZ] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow: Mirror symmetry, volume 1 of Clay Mathematics Monographs, American Mathematical Society, Providence, RI, 2003.

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[II2] [I1] [I2]

[I3] [I4] [I5] [Kon] [Kos] [KS] [KSS] [LO] [LWX] [O] [Pa] [Pe] [R1]

[Sc] [SS] [Se] [W1] [W2]

Noriaki IKEDA N. Ikeda and K. -I. Izawa: Gauge theory based on quadratic Lie algebras and 2-d gravity with dynamical torsion, Prog. Theor. Phys. 89, 1077(1993); General form of dilaton gravity and nonlinear gauge theory, Prog. Theor. Phys. 90, 237 (1993) ; For a review, N. Ikeda: Two-dimensional gravity and nonlinear gauge theory, Ann. Phys., 235, 435 (1994), hep-th/9312059 N. Ikeda and K. I. Izawa: Dimensional reduction of nonlinear gauge theories, JHEP 0409, 030 (2004), hep-th/0407243 N. Ikeda: A deformation of three dimensional BF theory. JHEP 0011, 009(2000), hep-th/0010096 N. Ikeda: Deformation of BF theories, topological open membrane and a generalization of the star deformation, JHEP 0107, 037(2001) , hepth/0105286 N. Ikeda: Chern-Simons gauge theory coupled with BF theory, Int. J. Mod. Phys. A 18, 2689 (2003), hep-th/0203043 N. Ikeda: Topological field theories and geometry of Batalin-Vilkovisky algebras, JHEP 0210, 076 (2002), hep-th/0209042 N. Ikeda, Three dimensional topological field theory induced from generalized complex structure, hep-th/0412140 M. Kontsevich: Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157 (2003), q-alg/9709040 Y. Kosmann-Schwarzbach: Derived brackets, Lett. Math. Phys. 69, 61 (2004), math.dg/0312524 C. Klimcik and T. Strobl, WZW-Poisson manifolds, J. Geom. Phys. 43, 341 (2002), math.sg/0104189 A. Kotov, P. Schaller and T. Strobl, Dirac sigma models, Commun. Math. Phys. 260, 455 (2005), hep-th/0411112 A. M. Levin and M. A. Olshanetsky: Hamiltonian algebroid symmetries in W-gravity and Poisson sigma-model, hep-th/0010043 Z. J. Liu, A. Weinstein and P. Xu: Manin Triples for Lie Bialgebroids, J. Differential Geom., 45, 547-574, (1997), dg-ga/9611001 M. A. Olshanetsky: Lie algebroids as gauge symmetries in topological field theories, hep-th/0201164 J. Park: Topological open p-branes, Symplectic geometry and mirror symmetry, 311-384, Seoul (2000) hep-th/0012141. V. Pestun, Topological strings in generalized complex space, hepth/0603145. D. Roytenberg: Courant algebroids, derived brackets and even symplectic supermanifolds, math.qa/0112152; On the structure of graded symplectic supermanifolds and Courant algebroids, math.sg/0203110 A. Schwarz: Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys. 155, 249 (1993), hep-th/9205088 P. Schaller and T. Strobl: Poisson structure induced (topological) field theories, Mod. Phys. Lett., A9, 3129 (1994), hep-th/9405110 P. Severa, Some title containing the words ”homotopy” and ”symplectic”, e.g. this one, math.SG/0105080 E. Witten. Topological sigma models. Commun. Math. Phys., 118, 411, (1988) E. Witten, Mirror manifolds and topological field theory, hep-th/9112056

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R. Zucchini, A sigma model field theoretic realization of Hitchin’s generalized complex geometry, JHEP 0411, 045 (2004), hep-th/0409181; Generalized complex geometry, generalized branes and the Hitchin sigma model, JHEP 0503, 022 (2005), hep-th/0501062; A topological sigma model of biKaehler geometry, JHEP 0601, 041 (2006), hep-th/0511144

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Part II

DEFORMED FIELD THEORY AND SOLUTIONS

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Noncommutative Solitons Olaf Lechtenfeld Institut f¨ ur Theoretische Physik, Universit¨ at Hannover, Appelstraße 2, D–30167 Hannover, Germany [email protected]

1 Introduction Noncommutative geometry is a possible framework for extending our current description of nature towards a unification of gravity with quantum physics. In particular, motivated by findings in string theory, field theories defined on Moyal-deformed spacetimes (or brane world-volumes) have attracted considerable interest. For reviews, see [KS02, DN02, Sza03]. In modern gauge theory, a central role is played by nonperturbative objects, such as instantons, vortices and monopoles. These solitonic classical field configurations usually arise in a BPS sector (or even integrable sector) of the theory. This fact allows for their explicit construction and admits rather detailed investigations of their dynamics. It is then natural to ask how much of these beautiful results survives the Moyal deformation and carries over to the noncommutative realm. Studying classical solutions of noncommutative field theories is also important to establish the solitonic nature of D-branes in string theory (see the reviews [Har01, Ham03, Sza05]). It turns out that already scalar field theories, when Moyal deformed, have a much richer spectrum of soliton solutions than their commutative counterparts. The simplest case in point are noncommutative scalar field theories in one time and two space dimensions. Therefore, in these lectures I will concentrate on models with one or two real or with one complex scalar field. In the latter case my prime example is the abelian unitary sigma model and its Grassmannian subsectors. By adding a WZW-like term to the action it is extended to the noncommutative Ward model [War88, War90, IZ98a, LP01a], which is integrable in 1+2 dimensions and features exact multi-soliton configurations. As another virtue this model can be reduced to various lower-dimensional integrable systems such as the sine-Gordon theory [LMPPT05]. After covering the basics in the beginning of these lectures, I will present firstly static and secondly moving abelian sigma-model solitons, with spacespace and with time-space noncommutativity. Multi-soliton configurations 175

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Olaf Lechtenfeld

and their scattering behavior will make a brief appearance. The nonabelian generalization is sketched in a U(2) example. Next, I shall compute the full moduli-space metric for the abelian Ward model and discuss its adiabatic two-soliton dynamics. A linear stability analysis for a prominent class of static U(1) solitons follows, with an identification of their moduli. In the last part of the lectures, I dimensionally reduce to 1+1 dimensions and find noncommutative instantons as well as solitons. The latter require also an algebraic reduction, from U(2) to U(1)×U(1), which produces an integrable noncommutative sine-Gordon model. Its classical kink and tree-level meson dynamics will close the lectures. The material of these lectures is taken from the papers [LP01a, LP01b, DLP05, CS05, KLP06]. I’d like to add that the topics presented here are by no means exhaustive. I have deliberately left out important issues such as noncommutative vortices, monopoles and instantons, the role of the Seiberg-Witten map, quantum aspects such as renormalization, or non-Moyal spaces like fuzzy spheres and quantum groups. The noncommutative extension of integrable systems technology (ADHM, twistor methods, dressing, Riemann-Hilbert problem etc.) is also missing. Finally, I did not touch the embedding in the framework of string theory. Any of these themes requires lectures on its own.

2 Beating Derrick’s theorem 2.1 Solitons in d = 1+2 scalar field theory I consider a real scalar field φ living at time t on a plane with complex coordinates z, z¯. The standard action Z S0 = dt d2 z 21 φ˙ 2 − ∂z φ ∂z¯φ − V (φ) (1) depends on a polynomial potential V of which I specify V (φ) ≥ 0,

V (φ0 ) = 0

and

V 0 (φ) = v

Y

(φ − φi ).

(2)

i

In this situation, Derrick’s theorem states that the only non-singular static solutions to the equation of motion are the ground states φ = φ0 , allowing for degeneracy. The argument is strikingly simple [Der64]: Assume you have b z¯). Then by scaling I define a family found a static solution φ(z, b z , z¯ ) φbλ (z, z¯) := φ( λ λ

(3)

of static configurations, which must extremize the energy at λ=1. However, over a time interval T , I find that E(λ) := −S0 [φbλ ]/T = λ0 Egrad + λ2 Epot ,

(4)

which is extremal at λ=1 only for Epot = 0, implying Egrad = 0 and φb1 = φ0 as well.

Noncommutative Solitons

177

2.2 Noncommutative deformation Let me deform space (but not time) noncommutatively, by replacing the ordinary product of functions with a so-called star product, which is noncommutative but associative. This step introduces a dimensionful parameter θ into the model, which I use to define dimensionless coordinates a, a ¯ via √ √ z = 2θ a and z¯ = 2θ a ¯. (5) For static configurations, the energy functional then becomes Z Z θ→∞ Eθ = d2 a |∂a φ|2? + 2θ V? (φ) −→ 2θ d2 a V? (φ),

(6)

where the subscript ‘?’ signifies star-product multiplication. In the large-θ limit, the stationarity equation obviously becomes b = v(φ−φ b 0 ) ? (φ−φ b 1 ) ? · · · ? (φ−φ b n ). 0 = V?0 (φ) (7) Due to the noncommutativity (you may alternatively think of φb as a matrix) this equation has many more solutions than just φb = φi = const, namely X X φb = U ? φi P i ? U † with Pi ? Pj = δij Pj and Pi = i

i

(8) featuring a resolution of the identity into a complete set of (star-)projectors {Pi } and an arbitrary star-product unitary U . The energy of these solutions comes out as Z X X b θ→∞ Eθ [φ] −→ 2θ V (φi ) d2 a Pi = 2πθ V (φi ) trPi , (9) i

R

i6=0

where I defined the ‘trace’ via trP = π d2 a P . Clearly, the moduli space U(∞) . For of the large-θ solutions (8) is the infinite-dimensional coset Q U(rankP i) i finite values of θ, the effect of the gradient term in the action lifts this infinite degeneracy and destabilizes most solutions. 2.3 Moyal star product Specifying to the Moyal star product, I shall from now on use ← → ← → (f ? g)(z, z¯) = f (z, z¯) exp θ ( ∂ z ∂ z¯ − ∂ z¯ ∂ z ) g(z, z¯) = f g + θ (∂z f ∂z¯g − ∂z¯f ∂z g) + . . .

(10)

= f g + total derivatives with a constant noncommutativity parameter θ ∈ R+ . The most important properties of this product are Z Z (f ? g) ? h = f ? (g ? h), d2 z f ?g = d2 z f g, [z , z¯]? = 2 θ. (11)

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Olaf Lechtenfeld

2.4 Fock-space realization A very practical way to realize the Moyal-deformed algebra of functions on R 2 is by operators acting on a Hilbert space H. This realization is provided by the Moyal-Weyl map between functions f and operators F , i.e. f (z, z¯), ? ↔ F (a, a† ), · . (12) For the coordinate functions I take √ 2θ a such that z ↔

[ a , a† ] =

The concrete translation prescriptions read h √ √ i ¯ F = Weyl-order f 2θ a, 2θ a and and derivatives and integrals become algebraic: √ √ 2θ ∂z f ↔ −[a† , F ], 2θ ∂z¯f ↔ [a, F ],

f = F?

(13)

.

√z , √z¯ 2θ 2θ

, (14)

∫d2 z f = 2πθ trF (15)

where the trace runs over the oscillator Fock space H with basis |ni =

√1 n!

(a† )n |0i

for

n ∈ N0

and

a |0i = 0.

(16)

3 d = 0+2 sigma model 3.1 U? (1) sigma model in d = 0+2θ To be specific, let me turn to the simplest noncommutative sigma model in the 2d plane, i.e. the abelian sigma model, φ ∈ U? (1)

⇐⇒

φ ? φ† =

.

(17)

Naively, it looks like a commutative U(∞) sigma model. Restricting to static fields, the action or, rather, the energy functional is Z 2 E = 2 d2 z ∂z φ† ∂z¯φ = 2π tr [a, Φ] , (18) which yields the equation of motion (I drop the hats on Φ) 0 = Φ† a , [a† , Φ] − a† , [a , Φ† ] Φ =: Φ† ∆Φ − ∆Φ† Φ,

(19)

thereby defining the laplacian. This model possesses an ISO(2) isometry: the Euclidean group of rigid motions a , a† 7−→ eiϑ (a+α) , e−iϑ (a† +¯ α) for α ∈ C and ϑ ∈ R/2πZ (20)

Noncommutative Solitons

179

induces the global field transformations Φ

7−→

eiϑ ad(a

†

a) α ad(a† )−α ¯ ad(a)

e

Φ =: R(ϑ) D(α) Φ D(α)† R(ϑ)† . (21)

The unitary transformation acts on the vacuum state as R(ϑ) D(α) |0i = eiϑ a

†

a αa† −αa ¯

e

|0i =: |eiϑ αi

(22)

and produces coherent states. Furthermore, the model enjoys a global phase invariance under Φ 7−→ eiγ0 Φ. (23) 3.2 Grassmannian subsectors There exist unitary fields which are hermitian at the same time. The intersection of both properties yields idempotent fields, Φ† = Φ

Φ2 =

⇐⇒

(24)

,

and defines hermitian projectors 1 2(

−Φ) = P = P 2

⇐⇒

Φ =

− 2P.

(25)

The set of all such projectors decomposes into Grassmannian submanifolds, Φ ∈ Gr(r, H) =

U(H) U(imP ) × U(kerP )

with r = rankP = 0, 1, 2, . . . .

(26) A restriction of the configuration space to some Gr(r, H) defines a Grassmannian sigma model embedded in the U? (1) model. Quite generally, any projector of rank r can be represented as −1 P = |T i hT |T i hT | with |T i = |T1 i, |T2 i, . . . , |Tr i , (27) where the column vector hT | is the hermitian conjugate of |T i, and hT |T i stands for the r×r matrix of scalar products hTi |Tj i. In the rank-one case, this simplifies to |T i ∈ CP (H) = CP ∞ ,

i.e. |T i ' |T i Γ

for Γ ∈ C∗ .

(28)

If finite, the rank r, which labels the Grassmannian subsectors, is also the value taken by the topological charge 1 ∫ d2 z φ ? ∂[z φ ? ∂z¯] φ = tr P a ( −P ) a† P − P a† ( −P ) a P , (29) Q = 8π which may be compared to the energy 2 1 = tr P a ( −P ) a† P + P a† ( −P ) a P . 8π E = tr [a, P ]

(30)

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Olaf Lechtenfeld

3.3 BPS configurations In a given Grassmannian the energy is bounded from below by a BPS argument: 1 8π E

= Q + tr(F † F + F F † ) ≥ Q

F = ( −P )aP.

with

(31)

For finite-rank projectors1 I have Q = trP and hence EBPS = 8πtrP . The energy is minimized when the projector obeys the BPS equation ( −P ) a P = 0

⇐⇒

a : imP ,→ imP,

(32)

which is equivalent to a |T i = |T i Γ

for some r×r matrix Γ,

(33)

meaning that |T i spans an a-stable subspace. By a basis change inside imP one can generically diagonalize 2 Γ → diag(α1 , α2 , . . . , αr ),

(34)

whence BPS solutions are just coherent states |T i =

|α1 i, |α2 i, . . . , |αr i

with

|αi i = eαi a

†

−α ¯i a

|0i.

(35)

The corresponding projector reads P =

r X

|αi i hα. |α. i

−1 ij

hαj | = U

i,j=1

r−1 X

|kihk| U † ,

(36)

k=0

where U is a unitary which in general does not commute with a. To develop the intuition, I display the Moyal-Weyl image of the basic operators 1

2

|αihβ|

↔

2eiκ e− 2 |α−β| e−(z−

|kihk|

↔

2 Lk ( 2zθz¯ ) e−zz¯/θ ,

√ √ 2θα)(¯ z − 2θβ)/θ

and

(37) (38)

where Lk denotes the kth Laguerre polynomial. Obviously, P is related to a superposition of gaussians in the Moyal plane. Note that the gaussians are singular for θ → 0.

1 2

For finite-corank projectors, Q = −tr( −P ) and E ≥ −Q. In general I must allow for confluent eigenvalues, which produce Jordan cells. For each cell, the multiple state |αi gets replaced with the collection |αi, a† |αi, . . . .

Noncommutative Solitons

181

4 d = 1+2 sigma model 4.1 d = 1+2 Yang-Mills-Higgs and Ward model At this stage I’d like to bring back the time dimension, but return to the commutative situation (θ=0) for a while. The sigma model of the previous section extends to 1+2 dimensions in more than one way, but only a particular generalization yields an integrable theory, the so-called Ward model [War88, War90, IZ98a]. Interestingly, its equation of motion follows from specializing the Yang-Mills-Higgs equations: The latter are implied by the Bogomolnyi equations 1 abc (∂[b Ac] 2ε

+ A[b Ac] ) = ∂ a H + [Aa , H]

with

a, b, c ∈ {t, x, y}, (39)

where the Yang-Mills potential Aa and the Higgs field H take values in the Lie algebra of U(n) for definiteness. A light-cone gauge and ansatz of the form At = A y =

1 † 2 φ (∂t

+ ∂y )φ

and

Ax = −H =

1 † 2 φ ∂x φ

(40)

yields a Yang-type Ward equation for the prepotential φ ∈ U(n), (η ab + kc εcab )∂a (φ† ∂b φ) = 0

∂x (φ† ∂x φ) − ∂v (φ† ∂u φ) = 0, (41)

⇐⇒

introducing the metric (ηab ) = diag(−1, +1, +1), a fixed vector (kc ) = (0, 1, 0) and the light-cone coordinates u =

1 2 (t

+ y)

and

v =

1 2 (t

− y).

(42)

4.2 Commutative Ward solitons Due to the appearance of the fixed vector k, the ‘Poincar´e group’ ISO(1,2) is broken to the translations times the y-boosts. This is the price to pay for integrability. The existence of a Lax formulation, a linear system, B¨ acklund transformations etc. suggest the existence of multi-solitons in this theory, which indeed can be constructed by classical means. Rather than directly integrating the Ward equation (41), multi-solitons require solving only firstorder equations and so in a way are second-stage BPS solutions of the YangMills-Higgs system. The U(n)-valued one-soliton configurations reads φ = ( −P ) + µµ¯ P

for µ ∈ C \ R,

(43)

with a hermitian projector P = T

1 T †T

T†

(44)

subject to ( −P ) (¯ µ∂x − ∂u ) P = 0 = ( −P ) (¯ µ∂v − ∂x ) P.

(45)

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Olaf Lechtenfeld

It turns out that each finite-rank P yields a soliton with constant velocity (vx , vy ) and energy E given by q µ+µ 1−vx2 −vy2 ¯ µ¯ µ−1 (vx , vy ) = − and E = , 8π trP µ¯ µ + 1 µ¯ µ+1 1 − vy2 (46) making obvious the Lorentz symmetry breaking (see also figure 1).

anti−solitons

µ v=0

i

vy = 0 v2=1 −1

vx = 0

1

−i solitons

v=0

Fig. 1. Soliton velocities in the µ plane

4.3 Co-moving coordinates Since one-soliton configurations are lumps moving with constant velocity, I can pass to their rest frame via a linear coordinate transformation (u, v, x) 7→ (w, w, ¯ s) given by w ¯ = ν¯ µ u + µ1 v + x , s = . . . (47) w = ν µ ¯ u + µ1¯ v + x , with ν ∈ C to be chosen later and s not needed. The transformation degener→2 ates for µ ∈ R ↔ v = 1, as is seen in the map for the partials, 1 µ¯ µ 2iµ¯ µ ∂w = ν1 (µ−¯ ∂u = ν µ ¯ ∂w + ν¯µ ∂w¯ − µ−¯ µ ) 2 µ ∂u + µ ∂ v − 2 ∂ x , µ ∂s , ∂w¯ = ∂s =

µ¯ µ 1 ν¯ (µ−¯ µ )2 −i µ−¯ µ

1 µ ¯

∂u + µ ¯ ∂v − 2 ∂x ,

∂u + µ¯ µ ∂v − (µ+¯ µ) ∂x ,

∂v =

ν µ ¯

∂w +

ν¯ µ

∂w¯ −

∂x = ν ∂w + ν¯ ∂w¯ −

In the co-moving coordinates, the BPS conditions (45) reduce to

2i µ−¯ µ

∂s ,

i(µ+µ) ¯ µ−¯ µ

∂s . (48)

Noncommutative Solitons

( −P ) ∂w¯ P = 0 = ( −P ) ∂s P.

183

(49)

The static case is recovered at vx = v y = 0

⇐⇒

µ = −i

=⇒

w = ν (x + iy) ,

s = t.

(50)

The spacetime picture is visualized in figure 2.

s

solito

n

t

y’

w y − v>

x’

z

x Fig. 2. Spacetime picture of co-moving coordinates

4.4 Time-space versus space-space deformation Now I set out to Moyal-deform the Ward model. In contrast to the static sigma model, two distinct possibilities appear, namely space-space or timespace noncommutativity:3 q →2 [x , y]? = iθ =⇒ [w, w] ¯ ? ∝ 2θ ν ν¯ 1− v , (51) → → → → [t , n · r ]? = iθ =⇒ [w, w] ¯ ? ∝ 2θ ν ν¯ | n × v |, →

→

where r = (x, y) and n = (nx , ny ) = const in the xy plane. It is apparent → → that the time-space deformation becomes singular when v k n , including the → static case v = 0 ! Hence, soliton motion purely in the deformed direction yields commutative rest-frame coordinates. In all other cases, (w, w) ¯ decribes a standard Moyal plane, and each rest-frame-static BPS projector, ∂s P = 0

and

( −P ) ∂w¯ P = 0,

(52)

gives a soliton solution. In the rest frame, the original type of deformation is no longer relevant. 3

I do not discuss light-like deformations here.

184

Olaf Lechtenfeld

4.5 U? (1) Ward solitons The Moyal-Weyl map associates to the co-moving coordinate w an annihilation operator c via √ (53) w ↔ 2θ c. Now I adjust the free parameter ν such that [w, w] ¯ ? = 2θ

[ c , c† ] =

⇐⇒

(54)

.

The BPS condition (49) becomes ( −P ) c P = 0

P (w, w) ¯ = |T i hT |T i−1 hT |,

for projectors

(55)

and it is solved in the abelian case by |T i =

|T1 i, |T2 i, . . . , |Tr i

|Ti i = eαi c

with

†

−α ¯i c

|vi,

(56)

where |vi is the ‘co-moving vacuum’ defined by c |vi = 0.

(57)

One finds that the soliton velocity and energy are θ independent, hence the commutative relations (46) still apply. Like in the static case, the U? (1) soli→ →0 tons have no commutative limit. A change of velocity, v → v , is effected by an ISU(1,1) squeezing transformation c = S(t) c 0 S(t)†

and

|vi = S(t) |v0 i,

(58)

and so all co-moving vacua |vi are obtained from |0i in this fashion. For the simplest case, a moving rank-one soliton, one gets Φ = eαc ↔

†

−αc ¯

¯ † − (1− µµ¯ )|vihv| eαc−αc

φ = 1 − (1− µµ¯ ) 2 e−|w−

√ 2θα|2 /θ

(59)

.

Remembering that w = w(z, z¯, t) one encounters a squeezed gaussian roaming the Moyal plane. 4.6 Ward multi-solitons Integrability allows me to proceed beyond the one-soliton sector. The dressing method, for example, allows for the construction of multi-solitons (with relative motion). More concretely, a U? (1) m-soliton configuration is built from (k) k=1,...,m (rows of) states |Ti i i=1,...,r parametrized by k

(µ1 , . . . , µm )

⇐⇒

→

→

( v1 , . . . , v m )

(60)

Noncommutative Solitons

185

and rk ×rk matrices Γ (k) in eigenvalue equations ck |T (k) i = |T (k) i Γ (k)

with

→

→

ck = S( v k , t) a S( vk , t)† .

(61)

In a basis diagonalizing Γ (k) the solution reads k †

(k)

k

|Ti i = |αki , µk , ti := eαi ck −α¯ i ck |vk i

with

→

|vk i = S(v k , t) |0i (62)

such that ck |vk i = 0. The two-soliton with r1 = r2 = 1 provides the simplest example: µ¯11 µ ¯ 22 µ ¯12 1 ¯ µ µ¯µ¯21 |2ih1| (63) Φ = − 1−µ|σ| 2 µ ¯1 |1ih1| + µ ¯ 2 |2ih2| − σµ µ ¯1 |1ih2| − σ 2 with the abbreviations |ki ≡ |αk , µk , ti,

σ ≡ h1|2i,

µij ≡ µi −¯ µj ,

µ≡

µ11 µ22 µ12 µ21 .

(64)

Because of the no-force property familiar to integrable models, the energy is additive: E[Φ] = E(µ1 ) + E(µ2 ), (65) with E(µ) given in (46) and trPi = ri = 1. The two lumps distort each other’s shape but escape the overlap region as if each one had been alone. 4.7 Ward soliton scattering It follows that abelian Ward multi-solitons are squeezed gaussian lumps mov→ ing with different but constant velocities vk in the Moyal plane. The large-time asymptotics of these configurations shows no scattering for pairwise distinct velocities. However, in coinciding-velocity limits there appear new types of multi-solitons with novel time dependence. This kind of behavior extends to the nonabelian case. Moreover, U? (n>1) multi-solitons (with zero asymptotic relative velocity) as well as soliton-antisoliton configurations [LP01b, Wol02] can be made to scatter at rational angles πq in this manner. In addition, breather-like ring-shaped bound states are found as well. Unfortunately, for U? (1) only the latter kind of configurations appear in the coinciding-velocity limits, hence true scattering solutions are absent for abelian solitons. 4.8 U? (n) Ward solitons For completeness, let me briefly illustrate how the generalization to the nonabelian case works. I restrict myself to a comparison of U? (1) with U? (2) static one-solitons. Since the nonabelian BPS projectors have infinite rank, it is convenient to switch from states |T i to operators Tb: U? (1) :

|T i =: Tb |Hi

with

|Hi ≡

|0i |1i |2i . . . ,

(66)

186

Olaf Lechtenfeld

which implies that Tb = P here. In contrast, U? (2) :

|T i =

0 0 ...

0

|Hi

|0i |1i . . . |r−1i ∅

! =

Sr Pr

! |Hi =: Tb |Hi (67)

with ∅ ≡ 0 0 . . . and the standard rank-r projector and shift operator, Pr =

r−1 X

|nihn|

n=0

and

Sr =

∞ X

|n−rihn|,

(68)

n=r

respectively. The U? (2) operator Tb in (67) can be written as a (slightly singular) limit of a regular expression: r 1 µ→0 Sr a p (69) −→ U (0) Tb = Tb = U (µ) Tb = Pr µ a†r ar +µ¯ µ with a particular unitary transformation U (µ). This transformation relates the projectors smoothly as   r ! a a†r a1r +µ¯µ a†r ar a†r aµr¯+µ¯µ H 0H . U (µ) (70) U † (µ) =  µ¯ µ µ †r 0H P r a †r r †r r a a +µ¯ µ a a +µ¯ µ Note that for the construction of P I can drop the square root in (69) as r effecting a basis change in imP and use Tb = aµ . For U? (2), the BPS condition (33) generalizes to a 0 |T i = |T i Γ ⇐⇒ a Tb = Tb Γb (71) ∞×∞ 0 a for some operator Γb. Choosing Γb = a, the BPS equation reduces to the holomorphicity condition [ a , Tb ] = 0, (72) which is indeed obeyed by the solution above. By inspection, the nonabelian Ward solitons smoothly approach their commutative cousins for θ → 0.

5 Moduli space dynamics 5.1 Manton’s paradigm A qualitative understanding of soliton scattering can be achieved for small relative velocity via the adiabatic or moduli-space dynamics invented by Manton [Man82, MS04]. This approach approximates the exact scattering configuration of m rank-one solitons by a time sequence of static m-lump solub z¯; α). For the U? (1) sigma model the latter are constructed from (35) tions φ(z,

Noncommutative Solitons

187

for r = m. Thereby one introduces a time dependence for the moduli α ≡ {αi }, which is determined by extremizing the action on the moduli space Mr 3 α. Being a functional of finitely many moduli αi (t), this action describes the motion of a point particle in Mr , equipped with a metric gij (α) and a magnetic field Ai (α). Hence, the scattering of r slowly moving rank-one solitons is well described by a geodesic trajectory in Mr , possibly with magnetic forcing. Since the U? (1) moduli are the spatial locations of the individual quasi-static lumps, the geodesic in Mr may be viewed as trajectories of the various lumps in the common Moyal plane, modulo permutation symmetry. Manton posits that b z, z¯) ≈ φ(z, b z¯; α(t)) =: φα , φ(t, (73) b z, z¯) with dynamics for α(t). Quite generally, thus replacing dynamics for φ(t, starting from an action of the type Z S[φ] = dt d2 z 21 φ˙ 2 + C? (φ, φ0 ) φ˙ − W? (φ, φ0 ) with φ0 ≡ (∂z φ, ∂z¯φ), (74) I am instructed to compute Smod [α] := S[φα ] Z = dt 21 {∫ (∂α φα )2 } α˙ 2 + {∫ C? (φα , φ0α ) ∂α φα } α˙ − ∫ W? (φα , φ0α ) Z =:

dt

1

˙ 2 gαα (α) α

2

+ Aα (α) α˙ − U (α)

(75) and read off the metric g, magnetic field F = dA and potential U on the moduli space. 5.2 Ward model metric The Ward equation (41) follows from the action [IZ98b] Z 1 S[φ] = 2 dt dx dy trU(n) φ˙ † φ˙ − ∂x φ† ∂x φ − ∂y φ† ∂y φ + WZW term, (76) both for commutative and noncommutative unitary fields. Let me impose the space-space Moyal deformation, choose the abelian case (n=1), pass to the operator formulation, insert the static solution b = Φ − 2 P {α` } with trP = r (77) into S[Φ] and integrate over the Moyal plane, i.e. perform the trace over R R H. Then, the gradient term in (76) contributes with − dt E[φα ] = −8πr dt 1 to Smod and can be dropped. More importantly, the WZW term yields

188

Olaf Lechtenfeld

Aα = ∂ α Ω

=⇒

F = 0,

(78)

hence it too can be ignored and fails to produce a magnetic forcing (see also [DM05])! It remains to find the metric gαi αj ({α` }) on the moduli space Mr = Cr /Sr = Ccenter-of-mass × Mrel

with

Mrel ' Cr−1 , (79)

which is the configuration space of r identical bosons on the Moyal plane. The result is Z Z Smod = 4πθ dt trH P˙ 2 = 8πθ dt trimP hT |T i−1 hT˙ | −P |T˙ i (80) with |T i = |α1 i, |α2 i, . . . , |αr i |T˙ i ≡ ∂t |T i = a† |T i Γ˙ −

and

† 1 2 |T i (Γ Γ )˙

where Γ = diag({α` }). (81)

It is not hard to see that the metric hiding in (80) is K¨ ahler, with the K¨ ahler potential K given by X 1 |αi |2 + tr ln hαi |αj i = tr ln eα¯ i αj , (82) 8πθ K = i

which makes the permutation symmetry manifest. This K¨ ahler structure is the natural one, induced from the embedding Grassmannian Gr(r, H), enjoys a cluster decomposition property and allows for easy separation of the free center-of-mass motion. In the coinciding limits αi → αj , coordinate singularities appear which, however, may be removed by a gauge transformation of K or, equivalently, by passing to permutation invariant coordinates (see also [LRU00, HLRU01, GHS03]). 5.3 Adiabatic two-soliton scattering Let me be explicit for the simplest case of m = r = 2. The moduli space M2 of rank-two BPS projectors is parametrized by {α, β} ' {β, α} ∈ C2 /S2 , hence Mrel ' C but curved. The static two-lump configuration derived from (36) reads Φ =

−

2 |αihα| + |βihβ| − σ|αihβ| − σ¯ |βihα| 2 1 − |σ|

with σ = hα|βi, (83)

and the corresponding K¨ ahler potential becomes [LRU00] 2 + 12 |α−β|2 + ln 1 − e−|α−β| (84) with center-of-mass separation. Introducing the lump distance via α−β = r eiϕ and putting α+β = 0, the relative K¨ ahler potential has the limits

1 8πθ K

= |α|2 + |β|2 + ln(1−|σ|2 ) =

2 1 2 |α+β|

Noncommutative Solitons 2

2

+ O(r8 ), (85) revealing asymptotic flat space for r → ∞ but a conical singularity with an opening angle of 4π at r = 0. The ensueing metric takes the conformally flat form ds2 = 4πθ grr (r) dr2 + r2 dϕ2 (86) 1 8πθ Krel

= 12 r2 − e−r + O(e−2r )

1 8πθ Krel

and

= ln r2 +

189

1 4 24 r

with the conformal factor 2

grr (r) =

1 − e−2r − 2r2 e−r (1 − e−r2 )2

2

=

sinh r2 − r2 r2 r6 ≈ − + O(r10 ) (87) 2 cosh r − 1 3 90

displayed in figure 3.

grr

0.8

0.6

0.4

0.2

Fig. 3. Conformal factor of two-soliton metric

In terms of the symmetric coordinate ρ eiγ = σ := (α−β)2 = r2 e2iϕ

(88)

the metric desingularizes, ds2 = 4πθ

√ grr ( ρ) 4ρ

dρ2 +ρ2 dγ 2

= 4πθ

ρ2 4 1 12 − 360 +O(ρ )

dρ2 +ρ2 dγ 2

(89)

which is smooth at the origin. Head-on scattering of two lumps corresponds to a single radial trajectory in Mrel , which in the smooth coordinate σ must pass straight through the origin. In the ‘doubled coordinate’ α−β, I then see two straight trajectories with 90◦ scattering off the singularity in the Moyal plane. Increasing the impact factor, the scattering angle decreases from π2 to 0. Moreover, due to the absence of potential and magnetic field, the scattering angle depends only on the impact parameter and not separately on kinietc energy and angular momentum. A comparison of this moduli-space motion with exact two-soliton dynamics has recently been performed in [KLP06].

190

Olaf Lechtenfeld

6 Stability analysis 6.1 Fluctuation Hessian So far I have investigated the soliton dynamics purely on the classical level. As a first step towards quantization, let me now turn to fluctuations around the classical solutions. More concretely, I shall consider perturbations of the 2d static noncommutative sigma-model solitons encountered earlier. This task has two applications: First, it is relevant for the semiclassical evaluation of the Euclidean path integral, revealing potential quantum instabilities of the two-dimensional model. Second, it yields the (infinitesimal) time evolution of fluctuations around the static multi-soliton in the time-extended threedimensional theory, indicating classical instabilities if they are present. More concretely, any static perturbation of a classical configuration can be taken as (part of the) Cauchy data for a classical time evolution, and any negative eigenvalue of the quadratic fluctuation operator will give rise to an exponential runaway behavior, at least within the linear response regime. Furthermore, fluctuation zero modes are expected to belong to moduli perturbations of the classical configuration under consideration. The current knowledge on the effect of quantum fluctuations is summarized in [Zak89]. For a linear stability analysis, I must study the U? (n) energy functional (18) for a perturbation φ of a background Φ, E[Φ+φ] = E[Φ] + δE[Φ, φ] + δ 2 E[Φ, φ] + . . . , where the φ-linear term δE vanishes for classical backgrounds, and δ 2 E[Φ, φ] = 2π tr φ† ∆φ − φ† (Φ∆Φ† ) φ =: 2π tr φ† H φ

(90) (91)

defines the Hessian operator H[Φ] which acts in the space of fluctuations φ. For a given static soliton Φ, the goal is to determine the spectrum of H, at least the negative part and the zero modes. To this end, a decomposition of {φ} into H-invariant subspaces is essential. A natural segmentation is φimP φGrP on H = imP ⊕ kerP. (92) φ = φGrP φkerP Here φGrP is hermitian and keeps me inside the Grassmannian of Φ, while φimP and φkerP are anti-hermitian and lead away from the Grassmannian. Even though this structure is not H-invariant, it decomposes the energy, δ 2 E[Φ, φ] = δ 2 E[Φ, φimP +φkerP ] + δ 2 E[Φ, φGrP ].

(93)

Without further assumptions about Φ it is difficult to identify H-invariant subspaces. Let me adopt the basis (16) in H. Then, for backgrounds diagonal in this basis, an H-invariant decomposition is φ =

∞ X

φ(k) ,

k=0

where φ(k) denotes the kth diagonal plus its transpose.

(94)

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191

6.2 Diagonal U? (1) soliton: fluctuation spectrum Once more I specialize to the abelian sigma model, where each static soliton is essentially a coherent-state projector (36) labelled by r complex numbers α i . Although all these backgrounds (for fixed r) are degenerate in energy, their fluctuation spectra differ unless related by ISO(2) rigid motion in the Moyal plane. Presently, the fluctuation analysis is technically feasible only for the special backgrounds where all αi coalesce. Translating the common value to the origin, this amounts to the diagonal abelian background Φr =

− 2

r−1 X n=0

|nihn| = diag(−1, −1, . . . , −1, +1, +1, . . . ). {z } |

(95)

r times

In this case, the decomposition (94) applies and yields three qualitatively different types of fluctuation subspaces carrying the following characteristic spectra of H:   (k) spec(HGrP ) = {0 < λ1 < · · · < λr }         (k) k>r ‘very off-diagonal’ spec(HimP ) = ∅         (k) spec(HkerP ) = R+  (k) spec(HGrP ) = {0 = λ1 < λ2 < · · · < λk }     1≤k≤r

k=0

(k)

   

    

spec(HimP ) = {0 = λk+1 < λk+2 < · · · < λr } (k) spec(HkerP )

= R+

 (0)  spec(HGrP )=∅

 





(0)

spec(HimP +kerP ) = R≥0 ∪ {λ− < 0}

   

‘slightly off-diagonal’

‘diagonal’

(96) These findings are visualized for r=4 in figure 4, with the following legend: double line solid segment dashed line

= b = b = b

negative eigenvalue zero eigenvalue admissible zero mode

−→ −→ −→

single instability (2r−1)C moduli phase modulus

Figures 5 and 6 show a numerical spectrum of the H (k) with cut-off size 30, also for the background Φ4 . Here, the legend is: boxes stars crosses circles

= b = b = b = b

Gr(P ) eigenvalues imP eigenvalues (k6=0) kerP modes (k6=0) diagonal modes (k=0)

−→ −→ −→ −→

# = min(r, k) # = max(r−k, 0) R+ continuum R≥0 ∪ {λ− }

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Olaf Lechtenfeld

Fig. 4. Decomposition of perturbation around Φ4

15 12.5 10 7.5 5 2.5

Fig. 5. Discrete spectrum of H for Φ4

60

50

40

30

20

10

Fig. 6. Continuous spectrum of H for Φ4

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193

6.3 Single negative eigenvalue The numerical analysis for abelian diagonal backgrounds Φr revealed a single negative eigenvalue λ− among the diagonal fluctuations. It is found by diagonalizing the k=0 part of the Hessian,   1 −1 −1 3 −2      −2 5 −3     .. ..   . −3 .     . (0) .  , . 2r−3 −r+1 (97) (Hm` ) =     −r+1 −1 −r      −r +1 −r−1   .    −r−1 2r+3 . .   .. .. . . where I have emphasized in boldface the entries modified by the background. The result is indeed that spec(H (0) ) = {λ− } ∪ [0, ∞),

(98)

where λ− is computed as the unique negative zero of the determinant 1 Z ∞ −x Ir−1,r−1 (λ) − 2r Ir−1,r (λ) e dx Lk (x) Ll (x) with Ik,l (λ) := 1 x−λ 0 Ir,r−1 (λ) Ir,r (λ) − 2r

(99) being variants of the integral logarithm. The r complex zero eigenvalues of HGrP arise from turning on the location moduli αi of (35), while the r−1 complex zero eigenvalues of HimP point at non-Grassmannian classical solutions. Since H (0) is not non-negative, δ 2 E[Φ, φ(0) ] may vanish even if H (0) φ(0) 6= 0. 6.4 Instability in unitary sigma model The fluctuations φGrP are tangent to GrP ≡ Gr(r, H) and cannot lower the energy, as the BPS argument (31) had assured me from the beginning. Therefore, all solitons of Grassmannian sigma models are stable. On the other hand, an unstable mode of H occurred in imP ⊕kerP , indicating a possibility to continuously lower the energy E = 8πr of Φr along a path starting perpendicular to GrP . Indeed, there exists a general argument for any static soliton Φ = −2P inside the unitary sigma model, commutative or noncommutative. It goes as follows. Given a projector inclusion Pe ⊂ P (including Pe = 0), i.e. a ‘smaller’ projector Pe of rank re < r. Then, the path [Zak89] e

Φ(s) = ei s (P −P ) ( −2P ) =

− (1+ei s )P − (1−ei s )Pe

(100)

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Olaf Lechtenfeld

connecting

Φ(0) = Φ = −2P

to

e = −2Pe Φ(π) = Φ

(101)

interpolates between static solitons in different Grassmannians inside U(H). Please note that the tangent vector (∂s Φ)(0) = −i(P −Pe ) is not an eigenmode of the Hessian. A quick calculation gives the energy along the path, 1 8π E[Φ(s)]

=

r+e r 2

+

r−e r 2 cos s

= r cos2

s 2

+ re sin2 2s .

(102)

For nonabelian noncommutative solitons the argument persists, with the topoe replacing r and re. Therefore, all solitons in unitary logical charges Q and Q sigma models eventually decay to the ‘vacua’ Q = 0, which belong to the constant (nonabelian) projectors.

7 d = 1+1 sine-Gordon solitons 7.1 Reduction to d = (1+1)θ : instantons In the remaining part of this lecture I look at the reduction from 1+2 to 1+1 dimensions, with the goal to generate new noncommutative solitons. However, naive reduction of the Ward solitons is not possible. Due to shape invariance, ∂s = 0, the one-soliton sector is already two-dimensional (in the rest-frame) but with Euclidean signature: ∂s = 0

↔

∂u + µ¯ µ ∂v − (µ+¯ µ) ∂x = 0

∂x = ν ∂w + ν¯ ∂w¯ , (103) hence I cannot simply put ∂x = 0 without killing the soliton entirely. Instead, the x dependence may be eliminated by taking the snapshot φ(x=0, y, t). Then, ∂s = 0 maps the remaining ty plane to the ww¯ plane as illustrated in figure 7. Because for vx 6=0 the soliton worldline pierces the xy plane as shown t

↔

s

x’ −> v

x

Fig. 7. Action of reduction ∂s = 0

in figure 8, the x=0 slice of the soliton is just an instanton!

Noncommutative Solitons

195

y

l so

ito

n

instanton

− v>

x

Fig. 8. x=0 instanton snapshot of soliton

7.2 d = 1+1 sigma model metric Due to the x-derivatives in the Ward equation (41) the snapshot φ(x=0, y, t) will not satisfy this equation. Using (103) I find that instead it obeys the equation (1− µµ¯ ) ∂w (φ† ∂w¯ φ) − (1− µµ¯ ) ∂w¯ (φ† ∂w φ) = 0, (104) which is an extended sigma-model equation in 1+1 dimensions due to (w, w) ¯ ∼ (t, y). Comparison with (h(ij) + b[ij] ) ∂i (φ† ∂j φ) = 0 yields the metric  tt h  hyt

hty hyy

for i, j ∈ {t, y}

  1 | µ −µ|2 | µ1 |2 − |µ|2 µ¯ µ    = (µ+¯ µ)2 | 1 |2 − |µ|2 | 1 +µ|2 µ µ

(105)



(106)

and the magnetic field btt

bty

byt

byy

! =

0 1 −1 0

! .

(107)

The notation suggests a Minkowski signature, but a short computation says that µ 2 ≥ 0, (108) det(hij ) = Im Re µ hence the metric is Euclidean! Indeed, this very fact permits the Fock-space realization of the Moyal deformation, which follows.

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Olaf Lechtenfeld

7.3 Moyal deformation in d = 1+1 In the present case I have no choice but to employ the time-space deformation [t, y]? = iθ

=⇒

[w, w] ¯ ? = 2θ

for µ ∈ / R or i R.

As before, I realize this algebra via the Moyal-Weyl correspondence √ w ↔ 2θ c such that [ c , c† ] =

(109)

(110)

on the standard Fock space H. In this way, the moving U? (1) soliton (59) becomes a gaussian instanton in the d = 1+1 U? (1) sigma model, after reexpressing w = w(y, t). The only exception occurs for vx = 0 (⇔ µ ∈ i R), i.e. motion in y direction only, because (51) then implies that [w, w] ¯ ? = 0. In fact, (47) shows (for x=0) that w ¯ ∼ w in this case, the rest frame degenerates •• to one dimension and there is no room left for a Heisenberg algebra. ∠ _ 7.4 Reduction to d = (1+1)θ : solitons So far, my attempts to construct noncommutative solitons in 1+1 dimensions by reducing such solitons in a d=1+2 model have failed. The lesson to learn is that the dimensional reduction must occur along a spatial symmetry direction of the d=1+2 configuration, i.e. along its worldvolume. In other words, the starting configuration should be spatially extended, or a d=1+2 noncommutative wave! Luckily, such wave solutions exist in the nonabelian Ward model [Lee89, Bie02]. Let me warm up with the commutative case and the sigma-model group of U(2). The Ward-model wave solutions Φ(u, v, x) dimensionally reduce to d=1+1 WZW solitons g(u, v) via Φ(u, v, x) = E eiα x σ1 g(u, v) e−iα x σ1 E †

for g(u, v) ∈ U(2)

(111)

and a constant 2×2 matrix E. The Ward equation for Φ descends to ∂v (g † ∂u g) + α2 (σ1 g † σ1 g − g † σ1 g σ1 ) = 0.

(112)

In a second step, I algebraically reduce g from U(2) to being U(1)-valued, allowing for an angle parametrization, i

g = e 2 σ3 φ .

(113)

The algebra of the Pauli matrices then simplifies (112) to ∂v ∂u φ + 4α2 sin φ = 0 which is nothing but the familiar sine-Gordon equation!

(114)

Noncommutative Solitons

197

7.5 Integrable noncommutative sine-Gordon model Now I introduce the time-space Moyal deformation [t, y]? = iθ

[u, v]? = − 2i θ.

⇐⇒

(115)

The sine-Gordon kink must move in the y direction, which (we have learned) forbids a Heisenberg algebra (note the i above). Thus, no Fock-space formulation exists and I must content myself with the star product. Recalling the dimensional reduction (111) and (112) I must now solve ∂v (g † ? ∂u g) + α2 (σ1 g † ? σ1 g − g † σ1 ? g σ1 ) = 0.

(116)

The algebraic reduction U(2) → U(1) turns out to be too restrictive. In the i commutative case, the overall U(1) phase factor e 2 ρ of g decouples in (112), so I could have started directly with g ∈ SU(2) instead. In the noncommutative case, in contrast, this does not happen, and I am forced to begin with U? (2). Thus, I should not prematurely drop the overall phase and algebraically reduce g to U? (1) × U? (1), i

g(u, v) = e?2

ρ(u,v)

i

? e?2

σ3 ϕ(u,v)

(117)

.

With this, the 2×2 matrix equation (116) turns into the scalar pair − 2i ϕ

∂ v e?

i 2ϕ

∂ v e? ?

i

? ∂u e?2

ϕ

− i ϕ ∂ u e? 2

with the abbreviation

i −iϕ ϕ + 2iα2 sin? ϕ = −∂v e? 2 ? R ? e?2

2

i 2ϕ

− 2iα sin? ϕ = −∂v e? ? R ? − 2i ρ

R = e?

i

? ∂u e?2

− i ϕ e? 2

ρ

(118)

(119)

carrying the second angle ρ. For me, (118) are the noncommutative sineGordon (NCSG) equations. As a check, take the limit θ → 0, which indeed yields ∂v ∂u ρ = 0 and ∂v ∂u ϕ + 4α2 sin ϕ = 0. (120) 7.6 Noncommutative sine-Gordon kinks As an application I’d like to construct the deformed multi-kink solutions to the NCSG equations (118), e.g. via the associated linear system. First, consider the one-kink configuration, which obtains from the wave solution of the U? (2) Ward model by choosing x=0

as well as

µ = ip ∈ iR

Consequently, the co-moving coordinate becomes

=⇒

ν = 1.

(121)

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Olaf Lechtenfeld

=: −i η. w = µ ¯u + µ1¯ v = −i (p u + 1p v) = −i √y−vt 1−v2

(122)

The BPS solution of the reduced Ward equation (116) is g = σ3 ( −2P )

with projector

P = T?

1 T † ?T

? T †,

(123)

where the 2×1 matrix function T (η) is subject to (∂η + α σ3 ) T (η) = 0.

(124)

Modulo adjusting the integration constant and (irrelevant) scaling factor, the general solution reads T =

e−αη

i eαη tanh 2αη

g =

i cosh 2αη

−2αη e −i 1 P = , 2 cosh 2αη i e+2αη

=⇒

i cosh 2αη

!

tanh 2αη

!

= E

i 2

i 2

ρ

e? ? e ? 0

ϕ

0 i 2

ρ

e? ?

−i ϕ e? 2

(125)

! E †.

7.7 One-kink configuration Since the expressions above depend on u and v only in the rest-frame combination η, it is clear that the deformation becomes irrelevant here, and the one- kink sector is commutative, effectively θ = 0 and ρ = 0. With E = √12 11 −11 the latest equation is solved by cos ϕ2 = tanh 2αη

tan ϕ4 = e−2αη (126) 2 which is precisely the standard sine-Gordon kink with velocity v = 1−p . 1+p2 With hindsight this was to be expected, since a one-soliton configuration in 1+1 dimensions depends on a single (real) co-moving coordinate. The deformation should reappear, however, in multi-soliton solutions. For instance, breather and two-soliton configurations seem to get deformed since pairs of rest-frame coordinates are subject to p [ηi , ηk ]? = −i θ (vi −vk ) (1−vi2 )(1−vk2 ). (127) and

sin ϕ2 =

1 cosh 2αη

=⇒

7.8 Tree-level scattering of elementary quanta Finally, it is of interest to investigate the quantum structure of noncommutative integrable theories, i.e. take into account the field excitations above the classical configurations. In my noncommutative sine-Gordon model (118) the elementary quanta are ϕ and ρ, and the Feynman rules for their scattering do get Moyal deformed. For illustrative purposes I concentrate on the ϕϕ → ϕϕ scattering amplitude in the vacuum sector. The kinematics of this process is

Noncommutative Solitons

k1 = (E, p) ,

k2 = (E, −p) ,

k3 = (−E, p) ,

k4 = (−E, −p),

199

(128)

subject to the mass-shell condition E 2 − p2 = 4α2 . The action (which I did not present here) is non-polynomial; it contains hϕϕρi,

hρρρi,

hϕϕϕϕi,

hϕϕρρi,

hρρρρi

(129)

as elementary three- and four-point interaction vertices. Denoting ϕ propagators by solid lines and ρ propagators by dashed ones, there are the following four contributions to the ϕϕ → ϕϕ amplitude at tree level: 1

1 2

2

= − 2i p2 sin2 (θEp)

= 2iα2 cos2 (θEp)

4 3

1

4

2

3

1

2

= 2i E 2 sin2 (θEp) 3

4

= 0. 4

3

Taken together this means that Aϕϕ→ϕϕ = 2iα2

(130)

is causal. I can show that all other 2 → 2 tree amplitudes vanish. Hence, any θ dependence seems to cancel in the tree-level S-matrix! Furthermore, it can be established that there is no tree-level particle production in this model, just like in the commutative case. Although at tree-level I still probe only the classical structure of the theory, the absence of a deformation until this point is conspicuous: Could it be that the time-space noncommutativity in the sine-Gordon system is a fake, to be undone by a field redefinition? With this provoking question I close the lecture.

References [KS02]

[DN02] [Sza03] [Har01]

Konechny, A., Schwarz, A.S.: Introduction to matrix theory and noncommutative geometry, Phys. Rept., 360, 353 (2002) [hep-th/0012145, hep-th/0107251] Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory, Rev. Mod. Phys., 73, 977 (2002) [hep-th/0106048] Szabo, R.J.: Quantum field theory on noncommutative spaces, Phys. Rept., 378, 207 (2003) [hep-th/0109162] Harvey, J.A.: Komaba lectures on noncommutative solitons and D-branes, hep-th/0102076

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[Ham03] [Sza05] [War88]

Hamanaka, M.: Noncommutative solitons and D-branes, hep-th/0303256 Szabo, R.J.: D-branes in noncommutative field theory, hep-th/0512054 Ward, R.S.: Soliton solutions in an integrable chiral model in 2+1 dimensions, J. Math. Phys., 29, 386 (1988) [War90] Ward, R.S.: Classical solutions of the chiral model, unitons, and holomorphic vector bundles, Commun. Math. Phys., 128, 319 (1990) [IZ98a] Ioannidou, T.A., Zakrzewski, W.J.: Solutions of the modified chiral model in 2+1 dimensions, J. Math. Phys., 39, 2693 (1998) [hep-th/9802122] [LP01a] Lechtenfeld, O., Popov, A.D.: Noncommutative multi-solitons in 2+1 dimensions, JHEP, 0111, 040 (2001) [hep-th/0106213] [LP01b] Lechtenfeld, O., Popov, A.D.: Scattering of noncommutative solitons in 2+1 dimensions, Phys. Lett. B, 523 178 (2001) [hep-th/0108118] [LMPPT05] Lechtenfeld, O., Mazzanti, L., Penati, S., Popov, A.D., Tamassia, L.: Integrable noncommutative sine-Gordon model, Nucl. Phys. B, 705, 477 (2005) [hep-th/0406065] [DLP05] Domrin, A.V., Lechtenfeld, O., Petersen, S.: Sigma-model solitons in the noncommutative plane: Construction and stability analysis, JHEP, 0503, 045 (2005) [hep-th/0412001] [CS05] Chu, C.S., Lechtenfeld, O.: Time-space noncommutative abelian solitons, Phys. Lett. B, 625, 145 (2005) [hep-th/0507062] [KLP06] Klawunn, M., Lechtenfeld, O., Petersen, S.: Moduli-space dynamics of noncommutative abelian sigma-model solitons, hep-th/0604219 [Der64] Derrick, G.H.: Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., 5, 1252 (1964) [Wol02] Wolf, M.: Soliton antisoliton scattering configurations in a noncommutative sigma model in 2+1 dimensions, JHEP, 0206, 055 (2002) [hep-th/0204185] [Man82] Manton, N.S.: A remark on the scattering of BPS monopoles, Phys. Lett. B, 110, 54 (1982) [MS04] Manton, N.S., Sutcliffe, P.: Topological solitons. Cambridge University Press (2004) [IZ98b] Ioannidou, T.A., Zakrzewski, W.J.: Lagrangian formulation of the general modified chiral model, Phys. Lett. A, 249, 303 (1998) [hep-th/9802177] [DM05] Dunajski, M., Manton, N.S.: Reduced dynamics of Ward solitons, Nonlinearity, 18 1677 (2005) [hep-th/0411068] [LRU00] Lindstr¨ om, U., Roˇcek, M., von Unge, R.: Non-commutative soliton scattering, JHEP, 0012 004 (2000) [hep-th/0008108] [HLRU01] Hadasz, L., Lindstr¨ om, U., Roˇcek, M., von Unge, R.: Noncommutative multisolitons: Moduli spaces, quantization, finite theta effects and stability, JHEP, 0106 040 (2001) [hep-th/0104017] [GHS03] Gopakumar, R., Headrick, M., Spradlin, M.: On noncommutative multisolitons, Commun. Math. Phys., 233 355 (2003) [hep-th/0103256] [Zak89] Zakrzewski, W.J.: Low dimensional sigma models. Adam Hilger (1989) [Lee89] Leese, R.: Extended wave solutions in an integrable chiral model in 2+1 dimensions, J. Math. Phys., 30, 2072 (1989) [Bie02] Bieling, S.: Interaction of noncommutative plane waves in 2+1 dimensions, J. Phys. A, 35, 6281 (2002) [hep-th/0203269]

Non-anti-commutative Deformation of Complex Geometry Sergei V. Ketov Department of Physics, Tokyo Metropolitan University, Hachioji-shi, Tokyo 192–0397, Japan; [email protected]

Abstract In this talk I review the well known relation existing between extended supersymmetry and complex geometry in the non-linear sigma-models, and then briefly discuss some recent developments related to the introduction of the non-anti-commutativity in the context of the supersymmetric non-linear sigma-models formulated in extended superspace. This contribution is suitable for both physicists and mathematicians interesting in the interplay between geometry, supersymmetry and non(anti)commutativity.

1 Introduction Being a theoretical physicist, one gets used to mathematical tools, whose role in modern theoretical high-energy physics is indispensable and indisputable. Especially when the experimental base is limited or does not exist, the use of advanced mathematics to get new insights in physics is particularly popular. So it is no surprise that mathematical knowledge of theoretical physicists is quite high. However, the way of dealing with mathematics in theoretical physics is different from that commonly used by mathematicians, and even the motivation and goals are different, as is also well known. I would like to draw attention to another, less known fact that physical considerations may sometimes lead to new mathematics, or rediscovery of some famous mathematical facts. In my contribution to this workshop, aimed towards more cooperation and understanding between physicists and mathematicians, I would like to explain how investigation of supersymmetry in field theory of the non-linear sigma-models might have led to rediscovery of complex geometry and related mathematical tools. In addition, I would like to explain how introducing more fundamental structure (namely, non-anti-commutativity in superspace) leads 201

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Sergei V. Ketov

to a new deformation of complex geometry, whose geometrical significance is yet to be understood. The paper is organized as follows. In sect.2 the basic notions of the nonlinear sigma-models are introduced. My presentation is ‘minimal’ on purpose, without going into details and/or many generalizations that might be easily added. I just summarize the basic ideas. In sect. 3, I introduce a simple superspace, and review the known relation between extended supersymmetry and complex geometry in the non-linear sigma-models, by getting all basic notions of complex geometry from a single and straightforward fieldtheoretical calculation. In sect. 4 some extended superspace techniques, making extended supersymmetry to be manifest, are briefly discussed. In sect. 5 some more superspace structure is added by introducing the notion of NonAnti-Commutativity (NAC) or ‘quantum superspace’, and its impact on the non-linear sigma-model target space is calculated. The simplest non-trivial explicit example of the NAC-deformed CP (1) metric is given in sect. 6. Our conclusion is sect. 7.

2 Non-linear sigma-models The Non-Linear Sigma-Model (NLSM) is a scalar field theory whose (multicomponent) scalar field φa (xµ ) is defined in a d-dimensional ‘spacetime’ or a ’worldvolume’ parametrized by local coordinates {xµ }, µ = 1, 2, . . . , d. The fields φa take their values in a D-dimensional Riemannian manifold M , called the NLSM target space, a = 1, 2, . . . , D. The NLSM field values φa can thus be considered as a set of (local) coordinates in M , whose metric is fielddependent. The NLSM format is the very general field-theoretical concept whose geometrical nature is the main reason for many useful applications of NLSM in field theory, string theory, condensed matter physics and mathematics (see e.g., the book [1] for much more). We assume the NLSM spacetime or worldvolume to be flat Euclidean space Rd , for simplicity, so that the NLSM action is supposed to be invariant under translations (with generators Pµ ) and rotations (with generators Mµν ) in Rd . Let ds2 = gab (φ)dφa dφb be a metric in M . Then a generic NLSM action is given by Z 1 Sbos.dbφ] = dd x L(∂µ φ, φ) , L = gab (φ)δ µν ∂µ φa ∂ν φb + m2 V (φ) , (1) 2 where summation over repeated indices is always implied. The function V (φ) is called a scalar potential in field theory with a mass parameter m. The higher derivatives of the field φ are not allowed in the Lagrangian L, with the notable exception of d = 2 where an extra (Wess-Zumino) term may be added to eq. (1): 1 LWZ = bab (φ)εµν ∂µ φa ∂ν φb . (2) 2

Non-anti-commutative Deformation of Complex Geometry

203

The 2-form B = bab (φ)dφa ∧ dφb in eq. (2), is called a torsion potential in M , by the reason to be explained in the next sect. 1.3. In string theory, it is called a B-field (or a Kalb-Ramond field).

3 Supersymmetric NLSM There are two different ways to supersymmetrize the NLSM: either in the worldvolume, or in the target space. Here we only discuss the worldvolume supersymmetrization of NLSM, in the case of even d. 1 Then adding supersymmetry amounts to the extension of the Euclidean space motion group SO(d) × T d to a supergroup, with the key superalgebra relation ¯ • }+ = 2σ µ • Pµ δ i j , {Qiα , Q βj

(3)

αβ

¯ are chiral and anti-chiral where the additional fermionic supercharges Q and Q spinors of SO(d), repectively, in the fundamental representation of the internal U (N ) symmetry group, denoted by latin indices i, j = 1, 2, . . . , N . The N here is a number of supersymmetries, so that the N > 1 supersymmetry is called the extended one. The chiral σ-matrices in eq. (3) obey Clifford algebra in d dimensions. As regards the NLSM, it is not difficult to demonstrate by using only group-theoretical arguments that d ≤ 6 [1], and when d=2 then N ≤ 4 [2]. The model-independent technology for a construction of off-shell manifestly supersymmetric field theories is called superspace. To give an example, let us consider the simplest case of the N = 1 supersymmetric NLSM in d = 2. The two-dimensional complex coordinates z and z¯ can be extended by the anti-commuting (Grassmann) fermionic (spinor) coordinates θ and θ¯ to ¯ Tensor functions in superspace are called superform a superspace (z, z¯, θ, θ). fields. A superfield is always equivalent to a supermultiplet of the usual fields, e.g. ¯ = φ + θψ + θ¯ψ¯ + θθF ¯ Φ(z, z¯, θ, θ) , (4) in terms of the bosonic field components φ(z, z¯) and F (z, z¯), and the fermionic ¯ z¯). field components ψ(z, z¯) and ψ(z, The supercharges can be easily realized in superspace as the differential operators ∂ ¯ = ∂ − θ¯∂¯ , Q= − θ∂ , and Q (5) ∂θ ∂ θ¯ where we have introduced the notation ∂ = ∂z and ∂¯ = ∂z¯. It is not difficult to check that the covariant derivatives 1

In string theory, the world-sheet supersymmetrization is known as the NeveuSchwarz-Ramond (NSR) approach, whereas the target space supersymmetrization is called the Green-Schwarz (GS) approach.

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D=

∂ + θ∂ , ∂θ

¯ = ∂ + θ¯∂¯ , D ∂ θ¯

∂

and

∂¯ ,

(6)

all (anti)commute with the supercharges (5) indeed, which allows us to use them freely in the covariant superspace action. Then the unique N=1 supersymmetric extension of the bosonic NLSM action (1) is easily constructed in N=1 superspace as follows (we ignore a scalar potential here): Z ¯ b, S1 = d2 xd2 θ (gab + bab ) DΦa DΦ (7) where both gab (Φ) and bab (Φ) are now functions of the superfields Φa of eq. (4). The component fields F appear to be non-propagating (they are called to be auxiliary), since they satisfy merely algebraic equations of motions. They are supposed to be substituted by solutions to their ’equations of motion’. Having evaluated the Berezin integral in eq. (7), one gets eq. (1) as the only purely bosonic contribution that is modified by the fermionic terms, namely, by a sum of the covariant Dirac term and the quartic fermionic interaction whose field-dependent couplings are given by the curvature tensor with torsion. The B-field, in fact, enters the field action S1 only via its curl (= torsion in M ) (8) T a bc = − 32 g ad b[bc,d] , that, in its turn, enters the action only via the connections (in M ) a a Γ ±bc = ± T a bc . bc

(9)

By construction the two-dimensional action S1 is invariant under the N=1 supersymmetry transformations ¯ a, δsusy Φa = εQΦa + ε¯QΦ with QΦa | = ψ a

and

¯ a = ψ¯a , QΦ

(10) (11)

¯ independent) part of a superfield, where | denotes the leading (i.e. θ- and θwhile ε and ε¯ are the infinitesimal fermionic (Grassmann) N=1 supersymmetry transformation parameters. Eq. (11) can serve as the definition of the fermionic superpartners of the bosonic NLSM field φa . It is not difficult to generalize the NLSM (7) by adding a scalar superpotential in superspace, Z Spot. = m d2 xd2 θ W (Φ) , (12) with a arbitrary real function W (Φ) and a mass parameter m. It gives rise to the scalar potential

Non-anti-commutative Deformation of Complex Geometry

205

1 2 ab m g (φ)∂a W (φ)∂b W (φ) , 2

(13)

V (φ) =

while it does not modify the NLSM kinetic terms, as is already clear from dimensional reasons. A generic two-dimensional NLSM with an arbitrary Riemannian target space M (and no scalar potential) can always be N=1 supersymmetrized, as in eq. (7). When M is a Lie group manifold, there is a preferred (groupinvariant) choice for its metric and torsion, while such NLSM is called a WessZumino-Novikov-Witten (WZNW) model [1]. One may also introduce the socalled gauged WZNW models with a homogeneous target space G/H, where H is a subgroup of G. In differential geometry, it corresponds to the quotient construction [1]. The next relevant question is: which restrictions on the NLSM target space M , in fact, imply more supersymmetry, i.e. N > 1 ? To answer that question, all one needs is to write down the most general Ansatz for the second supersymmetry transformation law (it follows by dimensional reasons) in terms of the N=1 superfields as ¯ b, δ2 Φ = ηJ a b (Φ)DΦb + η¯J¯a b (Φ)DΦ

(14)

and then impose the invariance condition δ 2 S1 = 0 .

(15)

In equation (14), the η and η¯ are the infinitesimal parameters of the second supersymmetry, while J a b (Φ) and J¯a b (Φ) are some tensor functions to be fixed by eq. (15). It is straightforward (though tedious) to check that the condition (14) amounts to the following restrictions (see e.g., ref. [3]): a − ¯a ∇+ c J b = ∇c J b = 0 ,

and gbc J c a = −gac J c b ,

gbc J¯c a = −gac J¯c b .

(16) (17)

In addition, one gets the standard (on-shell) N=2 supersymmetry algebra (3) provided that (see e.g., ref. [3]) ¯ J] ¯ =0, J 2 = J¯2 = −1 and N a bc [J, J] = N a bc [J,

(18)

where we have introduced the Nijenhuis tensor N a bc [A, B] = Ad [b B a c],d + Aa d B d [b,c] + B d [b Aa c],d + B a d Ad [b,c] .

(19)

So we can already recognize (or re-discover) the basic notions of (almost) complex geometry, such as an (almost) complex structure, a hermitean metric, a covariantly constant (almost) complex structure, and an integrable complex structure (see e.g., ref. [4]). To be precise, we get the following theorem:

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a two-dimensional N=1 supersymmetric NLSM is actually (on-shell) N=2 supersymmetric, if and only if (1) it allows two (almost) complex structures, ¯ (2) the NLSM metric is hermitean with respect to each of them, and J and J, (3) each (almost) complex structure is covarianlty constant with respect to the asociated (±) connection, so that it is actually integrable. The integrability here means the existence of holomoprhic and antiholomorphic coordinates (i.e. the holomorphic transitions functions) after rewriting a complex structure to the diagonal form (with the eigenvalues i and −i). It should be noticed that the complex structures J and J¯ may not be ¯e 6= 0, because they are covariantly constant commuting with each other, dbJ, Jc with respect to the different connections in eq. (9), respectively. For instance, the mixed N=2 supersymmetry commutator ¯ c , ¯ea b DDΦ ¯ b + Γ b −cd DΦd DΦ dbδ(η), δ(¯ η )ceΦa = η η¯dbJ, Jc (20) is required to be vanishing by the N=2 supersymmetry algebra (3). It is ¯ b+ already true on-shell, i.e. when the NLSM equations of-motions, D DΦ b d ¯ c Γ −cd DΦ DΦ = 0, are satisfied, though it is also the case off-shell only if ¯e = 0. The complex structures J and J¯ may not therefore be simultanedbJ, Jc ously integrable, in general. If, however they do commute, then the existence of an off-shell N=2 extended superspace formulation of such N=2 NLSM with manifest N=2 supersymmetry is guaranteed.

4 NLSM in extended superspace To give the simplest example of the N=2 extended superspace in two dimensions (z, z¯), let’s introduce two (Grassmann) fermionic coordinates for each chirality, i.e. (z, θ + , θ− ) and (¯ z , θ¯+ , θ¯− ), and then the N=2 supercharges Q± ¯ ± , the N=2 superspace covariant derivatives D± and D ¯ ± , and N=2 and Q scalar superfields Φi (z, z¯, θ+ , θ− , θ¯+ , θ¯− ), like in the N=1 case (see the previous section), where now i = 1, 2, . . . , m. However, there is the immediate problem: a general (unconstrained) N=2 scalar superfield has a physical vector field component that is not suitable for the NLSM. The simplest way to remedy that problem is to use the (off-shell) N=2 chiral and anti-chiral superfields, subject to the constraints ¯ ±Φ = 0 D

and

D± Φ¯ = 0 ,

(21)

respectively. Their most general NLSM action is then given by Z Z Z ¯ + m d2 xd2 θ W (Φ) + m d2 xd2 θ¯ W ¯ (Φ) ¯ , (22) S2 = d2 xd2 θd2 θ¯ K(Φ, Φ) ¯ and a holomorphic in terms of a non-holomorphic kinetic potential K(Φ, Φ) superpotential W (Φ).

Non-anti-commutative Deformation of Complex Geometry

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A simple straightforward calculation of the NLSM metric out of eq. (22) reveals a K¨ ahler metric gi¯j = ∂i ∂¯j K with the K¨ ahler potential K, and no torsion. Therefore, K¨ ahler complex geometry could have been also re-discovered from the N=2 supersymmetric NLSM. When one adds the so-called twisted chiral N=2 superfields, subject to the following off-shell N=2 superspace constraints: ¯ + Φ˜ = D− Φ˜ = 0 and D ¯ − Φ˜ = D+ Φ˜ = 0 , D their most general N=2 superspace action, Z ˜ + obvious superpotential terms , ¯ Φ, ˜ Φ) S2,T = d2 xd2 θd2 θ¯ K(Φ, Φ,

(23)

(24)

would give rise to a non-trivial torsion too, though with the commuting com¯e = 0 [5]. Actually, the exchange Φ ↔ Φ ˜ corresponds to plex structures, dbJ, Jc the T-duality in string theory. ¯e 6= 0, one To get the most general N=2 supersymmetric NLSM with dbJ, Jc ˆ has to add the so-called semi-chiral (reducible) N=2 superfields, Φˆ and Φ, subject to the off-shell N=2 superspace constraints [6, 3] ¯ +D ¯ − Φˆ = 0 D

and

D+ D− Φˆ = 0 .

(25)

When asking for even more supersymmetry in a two-dimensional supersymmetric NLSM, one gets three linearly independent (almost) complex structures of each chirality, obeying a quaternionic algebra (see e.g., ref. [1]), ±(B)c

Ja±(A)b Jb

= −δ AB δac + εABC Ja±(C)c ,

where A, B, C = 1, 2, 3 , (26)

¯ They all must be covariantly constant, and similarly for J. ∇± J ± = 0 ,

(27)

respectively. In particular, N=3 supersymmetry implies N=4 supersymmetry. Unfortunately, a geometrical description of the two-dimensional N=4 supersymmetric NLSM with torsion is still incomplete. In the case of the vanishing torsion, an N=1 supersymmetric NLSM is, in fact, N=4 supersymmetric if and only if its target space is hyper-K¨ ahler (see e.g., ref. [1] for more details). When a supersymmetric NLSM in question is, in fact, a (gauged) WZNW model, then its N=4 supersymmetry implies that its target space must be a product of Wolf spaces [7, 3]. The Wolf space can be associated with any simple Lie group G. Let ψ be the highest weight root of G, and let (Eψ± , H) be the generators of the su(2)ψ subalgebra of Lie algebra of G (say, in Chevalley basis), associated with ψ. Then the Wolf space is given by the coset W olf space =

G , H⊥ ⊗ SU (2)ψ

(28)

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where we have introduced the SU (2)ψ Lie group of the Lie algebra su(2)ψ and the centralizer H⊥ of the SU (2)ψ in G. An efficient off-shell N=4 superspace description of all two-dimensional N=4 supersymmetric NLSMs does not exist, though the use of harmonic superspace [8] with the infinite number of auxiliary fields may be useful for describing a large class of the manifstly N=4 supersymmetric NLSM. It is also worth mentioning that the chiral generators of supersymmetry in two dimensions are independent, so that it is possible to have an unequal number of ’left’ and ‘right’ supersymmetries. It is called ’heterotic’ or (p, q) supersymmetry. 2 It is always possible to construct the minimal or (1/2, 0) supersymmetric extention of any NLSM. A generic (1/2, 1) supersymmetric NLSM can be formulated in (1/2, 1) superspace. Less is known about other (p, q) supersymmetric NLSM with n = 3, 4. Finally, there is a simple relation between extended supersymmetry and higher (d > 2) dimensions, which is just based on the representation theory of spinors and Clifford algebras in various dimensions. A supersymmetric NLSM can first be formulated in six or four dimensions, and then it can be rewritten to lower dimensions, by simply restricting all its fields to be dependent upon lower number of their worldvolume coordinates (this procedure is called dimensional reduction). The manifestly supersymmetric formulation of a higher-dimensional supersymmetric NLSM often requires the use of sophisticated (constrained) superfields [1]. In quantum theory, only two-dimensional NLSM are renormalizable, while their higher-dimensional counterparts are not. The same is true for the supersymmetric NLSM [1].

5 Non-anticommutative deformation of four-dimensional supersymmetric NLSM Non-Anti-Commutativity (NAC) or quantum superspace [9] is a natural extension of the ordinary superspace, when the fermionic superspace coordinates are assumed to obey a Clifford algebra instead of being Grassmann (i.e. anticommutative) variables [10]. The non-anticommutativity naturally arises in the D3-brane superworldvolume, in the type-IIB constant Ramond-Ramond type background, in superstring theory [11]. In four dimensions, the NAC deformation is given by {θα , θβ }∗ = C αβ , (29) where C αβ can be identified with a constant self-dual gravi-photon background [11]. The remaining N=1 superspace coordinates in the chiral basis (y µ = ¯ µ, ν = 1, 2, 3, 4 and α, β, . . . = 1, 2) can still (anti)commute, xµ + iθσ µ θ, •

•

•

•

dby µ , y ν ec = {θ¯α , θ¯β } = {θα , θ¯β } = dby µ , θα ec = dby µ , θ¯α ec = 0 . 2

It is conventional to set p + q = 2n.

(30)

Non-anti-commutative Deformation of Complex Geometry

209

provided we begin with a four-dimensional Euclidean 3 worldvolume having the coordinates xµ . A supersymmetric field theory in the NAC superspace was extensively studied in the recent past, soon after the pioneering paper [13]. The choice (30) preserves locality in a NAC-deformed field theory. The C αβ 6= 0 explicitly break the four-dimensional Euclidean invariance. The NAC nature of θ’s can be fully taken into account by using the (associative, but non-commutatvive) Moyal-Weyl-type (star) product of superfields,   ← αβ ∂ ∂ C  g(θ) , f (θ) ∗ g(θ) = f (θ) exp − (31) 2 ∂θα ∂θβ which respects the N=1 superspace chirality. nomial in the deformation parameter ,

4

The star product (31) is poly-

C αβ ∂f ∂g ∂2f ∂2g − det C 2 2 , α β 2 ∂θ ∂θ ∂θ ∂θ

f (θ) ∗ g(θ) = f g + (−1)degf

(32)

where we have used the identity det C = 12 εαγ εβδ C αβ C γδ ,

(33)

and the notation

∂ ∂ ∂2 = 14 εαβ α β . (34) ∂θ2 ∂θ ∂θ We also use the following book-keeping notation for 2-component spinors: θχ = θα χα ,

•

θ¯χ ¯ = θ¯• χ ¯α , α

θ 2 = θ α θα ,

•

θ¯2 = θ¯• θ¯α . α

(35)

The spinorial indices are raised and lowered by the use of two-dimensional Levi-Civita symbols. Grassmann integration amounts to Grassmann differentiation. The anti-chiral covariant derivative in the chiral superspace basis is • ¯ • = −∂/∂ θ¯α . The field component expansion of a chiral superfield Φ reads D α

Φ(y, θ) = φ(y) +

√

2θχ(y) + θ2 M (y) .

(36)

An anti-chiral superfield Φ in the chiral basis is given by √ ¯ θ) ¯ = φ(y) ¯ + 2θ¯χ(y) ¯ Φ(y µ − 2iθσ µ θ, ¯ + θ¯2 M(y) √ √ ¯ µ φ(y) ¯ ¯ ¯ θ¯2 − i 2σ µ θ∂ + θ2 θ¯2 4φ(y) , + 2θ iσ µ ∂µ χ(y) (37) where 4 = ∂µ ∂µ . The bars over fields serve to distinguish between the ‘left’ and ‘right’ components that are truly independent in Euclidean space. 3 4

The Atiyah-Ward space-time of signature (+, +, −, −) is also possible [12]. We use the left derivatives as a default, the right ones are explicitly indicated.

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The non-anticommutativity Cαβ 6= 0 also explicitly breaks half of the original N=1 supersymmetry [13]. Only the chiral subalgebra generated by the chiral supercharges (in the chiral basis) Qα = ∂/∂θα is preserved, with {Qα , Qβ }∗ = 0, thus defining what is now called N=1/2 supersymmetry. The use of the NAC-deformed superspace allows one to keep N=1/2 supersymmetry manifest. The N=1/2 supersymmetry transformation laws of the chiral and anti-chiral superfield components in eqs. (36) and (37) are as follows: √ √ δφ = 2εα χα , δχα = 2εα M , δM = 0 , (38) and δ φ¯ = 0 ,

• • √ δχ ¯α = −i 2(˜ σµ )αβ εβ ∂µ φ¯ ,

• √ ¯ = −i 2∂µ χ δM ¯ • (˜ σµ )αβ εβ ,

α

(39)

respectively, where we have introduced the N=1/2 supersymmetry (chiral) parameter εα . To the end of this section, we are going to demonstrate that, in the case of the supersymmetric NLSM, its NAC superworldvolume gives rise to the induced smearing or fuzzyness in the NLSM target space [14, 15]. Here we follow ref. [15] where the most general four-dimensinal supersymmetric NLSM with an arbitrary scalar potential in the NAC superspace was considered (without any gauge fields), with the action Z Z Z Z ¯ ¯ S[Φ, Φ] = d4 y d2 θd2 θ¯ K(Φi , Φj ) + d2 θ W (Φi ) + d2 θ¯ W (Φj ) . (40) This action is completely specified by the K¨ ahler superpotential K(Φ, Φ), the scalar superpotential W (Φ), and the anti-chiral superpotential W (Φ), in terms of some number n of chiral and anti-chiral superfields, i, ¯j = 1, 2, . . . , n. In Euclidean superspace the functions W (Φ) and W (Φ) are independent upon each other. The NAC-deromed action is formally obtained by replacing all superfield profucts in eq. (40) by their star products (31). The NAC-deformed extension of eq. (40) in four dimensions after a ‘Seiberg-Witten map’ (i.e. after an explicit computation of all star products) was found in a closed form (i.e. in terms of finite functions) in refs. [15, 16, 17]. Our four-dimensional results are in agreement with the results of ref. [14] in the case of the NAC-deformed N=2 supersymmetric two-dimensionl NLSM, after dimensional reduction to two dimensions. ¯ We use the following notation valid for any function F (φ, φ): ∂ s+t F , (41) ∂φi1 ∂φi2 · · · φis ∂ φ¯p¯1 ∂ φ¯p¯2 · · · ∂ φ¯p¯t R and the Grassmann integral normalisation d2 θ θ2 = 1. The actual deformation parameter, in the case of the NAC-deformed field theory (40), appears to be F,i1 i2 ···is p¯1 p¯2 ···¯pt =

Non-anti-commutative Deformation of Complex Geometry

√ c = − det C ,

211

(42)

where we have used the definition [13] det C = 21 εαγ εβδ C αβ C γδ .

(43)

As a result, unlike the case of the NAC-deformed supersymmetric gauge theories [13], the NAC-deformation of the NLSM field theory (40) appears to be invariant under Euclidean translations and rotations. A simple non-perturbative formula, describing an arbitrary NAC-deformed scalar superpotential V depending upon a single chiral superfield Φ, was found in ref. [16], Z 1 [V (φ + cM ) − V (φ − cM )] d2 θ V∗ (Φ) = 2c (44) χ2 − [V,φ (φ + cM ) − V,φ (φ − cM )] . 4cM The NAC-deformation in the single superfield case thus gives rise to the split of the scalar potential, which is controlled by the auxiliary field M . When using an elementary identity Z +1 ∂ f (x + a) − f (x − a) = a dξ f (x + ξa) , (45) ∂x −1 valid for any function f , we can rewrite eq. (44) to the equivalent form [14] Z +1 Z +1 Z 1 2 ∂2 ∂ 1 2 dξ V (φ + ξcM ) . dξ V (φ + ξcM ) − χ d θ V∗ (Φ) = 2 M ∂φ −1 4 ∂φ2 −1 (46) Similarly, in the case of several chiral superfields, one finds [14] Z ∂ ∂2 1 d2 θ V∗ (ΦI ) = 21 M I I Ve (φ, M ) − (χI χJ ) I J Ve (φ, M ) (47) ∂φ 4 ∂φ ∂φ in terms of the auxiliary pre-potential Z +1 Ve (φ, M ) = dξ V (φI + ξcM I ) .

(48)

−1

Hence the NAC-deformation of a generic scalar superpotential V results in its smearing or fuzziness controlled by the auxiliary fields M I . A calculation of the NAC deformed K¨ ahler potential Z Z Z ¯ (49) d4 y Lkin. ≡ d4 y d2 θd2 θ¯ K(Φi , Φj )∗ can be reduced to eqs. (44) or (47), when using a chiral reduction in superspace, with the following result [17]:

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Lkin. =

1 i 2 M Y,i

+ 21 ∂ µ φ¯p¯∂µ φ¯q¯Z,p¯ ¯q +

1 1 i j ¯p¯ 2 4φ Z,p¯ − 4 (χ χ )Y,ij

1 i µ − 12 i(χi σ µ χ ¯p¯)∂µ φ¯q¯Z,ip¯ ¯p¯)Z,ip¯ , ¯q − 2 i(χ σ ∂µ χ

(50)

where we have introduced the (component) smeared K¨ ahler pre-potential Z +1 ¯ ¯ M) = Z(φ, φ, dξ K ξ with K ξ ≡ K(φi + ξcM i , φ¯j ) , (51) −1

as well as the extra (auxiliary) pre-potential [14] Z +1 h i p¯ q¯ ¯p¯ ξ ¯ M, M) ¯ =M ¯ p¯Z,p¯− 1 (χ Y (φ, φ, ¯ χ ¯ )Z +c dξξ ∂ µ φ¯p¯∂µ φ¯q¯K,ξp¯ ,p¯ ¯q ¯q + 4φ K,p¯ 2 −1

(52) It is not difficult to check that eq. (50) does reduce to the standard (K¨ ahler) N=1 supersymmetric NLSM (cf. sect. 4) in the limit c → 0. Also, in the case ¯¯j , there is no deformation at of a free (bilinear) K¨ ahler potential K = δi¯j Φi Φ all. ¯ (Φ) ¯ ∗ imply, via The NAC-deformed scalar superpotentials W (Φ)∗ and W eqs. (47) and (48), that the following component terms are to be added to eq. (50): p¯ q¯ ¯ f ,i − 1 (χi χj )W f ,ij + M ¯ p¯W ¯ ,p¯ − 1 (χ Lpot. = 12 M i W ¯ )W,p¯q¯ , 4 2 ¯ χ

where we have introduced the smeared scalar pre-potential [14] Z +1 f (φ, M ) = W dξ W (φi + ξcM i ) .

(53)

(54)

−1

The anti-chiral superpotential terms are inert under the NAC-deformation. The ξ-integrations in the equations above represent the smearing effects. However, the smearing is merely apparent in the case of a single chiral superfield, which gives rise to the splitting (44) only. This can also be directly demonstrated from eq. (50) when using the identity (45) together with the related identity [17] Z +1 Z +1 ∂ f (x + a) + f (x − a) = dξ f (x + ξa) + a dξ ξf (x + ξa) . (55) ∂x −1 −1 The single superfield case thus appears to be special, so that a sum of eq. (50) and (53) can be rewritten to the bosonic contribution [17] ¯ µ φ¯ K ¯ ¯(φ + cM, φ) ¯ + K ¯ ¯(φ − cM, φ) ¯ Lbos. = + 1 ∂ µ φ∂ 2

+

1 ¯ 2 4φ

,φφ

,φφ

¯ + K ¯(φ − cM, φ) ¯ K,φ¯(φ + cM, φ) ,φ

¯ M ¯ − K ¯(φ − cM, φ) ¯ K,φ¯(φ + cM, φ) ,φ 2c ¯ 1 ¯ ∂W , + [W (φ + cM ) − W (φ − cM )] + M 2c ∂ φ¯

+

(56)

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213

supplemented by the following fermionic terms [17]: Lferm. = − − − − − + − + + −

1 2 ¯ − K ¯ ¯(φ − cM, φ) ¯ χ ¯ K,φ¯φ¯ (φ + cM, φ) ,φφ 4c i ¯ − K ¯ ¯(φ − cM, φ) ¯ (χσ µ χ)∂ ¯ µ φ¯ K,φ¯φ¯(φ + cM, φ) ,φφ 2cM i ¯ − K ¯(φ − cM, φ) ¯ (χσ µ ∂µ χ) ¯ K,φ¯(φ + cM, φ) ,φ 2cM ¯ M ¯ − K ¯(φ − cM, φ) ¯ χ2 K,φφ¯(φ + cM, φ) ,φφ 4cM 1 2 µ ¯ ¯ ¯ + K ¯ ¯(φ − cM, φ) ¯ χ ∂ φ∂µ φ K,φφ¯φ¯(φ + cM, φ) ,φφφ 4M 1 ¯ µ φ¯ K ¯ ¯(φ + cM, φ) ¯ − K ¯ ¯ (φ − cM, φ) ¯ χ2 ∂ µ φ∂ ,φφ ,φ φ 2 4cM 1 2 ¯ ¯ + K ¯(φ − cM, φ) ¯ χ 4φ K,φφ¯(φ + cM, φ) ,φφ 4M 1 ¯ − K ¯(φ − cM, φ) ¯ χ2 4φ¯ K,φ¯(φ + cM, φ) ,φ 2 4cM 1 2 2 ¯ − K ¯ ¯ (φ − cM, φ) ¯ χ χ ¯ K,φφ¯φ¯(φ + cM, φ) ,φφφ 8cM 1 2 ¯ ,φ¯φ¯ . ¯2 W χ [W,φ (φ + cM ) − W,φ (φ − cM )] − 21 χ 4cM

(57)

¯ p¯ enter the action (50) linearly (as LaThe anti-chiral auxiliary fields M grange multipliers), while their algebraic equations of motion, 1 1 i j i ¯ 2 M Z,ip¯ − 4 (χ χ )Z,ij p¯ + W,p¯

=0,

(58)

¯ 5 are the non-linear set of equations on the auxiliary fields M i = M i (φ, φ). As a result, the bosonic scalar potential in components is given by [17] ¯ = 1 M iW f ,i Vscalar (φ, φ) . (59) 2 ¯ M =M (φ,φ)

Some comments are in order. The NAC-deformation just described is only possible in Euclidean superspace where the chiral and anti-chiral spinors are truly independent. The NAC-deformed NLSM is completely specified by a K¨ ahler function ¯ a chiral function W (Φ), an anti-chiral function W ¯ (Φ) ¯ and a constant K(Φ, Φ), deformation parameter c. As a matter of fact, we didn’t really use the constancy of c, so our results are valid even for any coordinate-dependent NAC deformation with c(y). 5

Equation (58) is not a linear system because the function Z is M -dependent.

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Solving for the auxiliary fields in eq. (50) represents not only a technical but also a conceptual problem because of the smearing effects described by the ξ-integrations. To bring the kinetic terms in eqs. (50) or (56) to the standard NLSM form (i.e. without the second order derivatives), one has to integrate by parts that leads to the appearance of the derivatives of the auxiliary fields. This implies that one has to solve eq. (58) before integration by parts. Let ¯ be a solution to eq. (58), and let’s ignore fermions for simM i = M i (φ, φ) plicity (χiα = χ ¯p¯• = 0). Substituting the auxiliary field solution back to the α

Lagrangian (50) and integrating by parts yield ¯ = − 1 (∂µ φ¯p¯∂µ φq ) Lkin. (φ, φ) 2

Z

+1 −1

j +c2 ξ 2 M i K,ξpij ¯ M,q

−

1 ¯p¯ ¯q¯ 2 (∂µ φ ∂µ φ )

Z

h ξ i i ξ dξ K,ξpq ¯ + 2cξM,q K,pi ¯ + cξM K,piq ¯

i

+1 −1

h i j i 2 2 i ξ dξ 2cξK,ξpi M + c ξ M K M . ,¯ q ,¯ q ¯ ,pij ¯

(60) It is now apparent that the NAC-deformation does not preserve the original K¨ ahler geometry of eq. (40), though the absence of (∂µ φ)2 terms and the particular structure of various contributions to eq. (60) are quite remarkable. The action (60) takes the form of a generic NLSM, being merely dependent upon mixed derivatives of the K¨ ahler function, so that the original K¨ ahler gauge invariance of eq. (40), ¯ → K(φ, φ) ¯ + f (φ) + f¯(φ) ¯ , K(φ, φ)

(61)

¯ φ) ¯ is preserved. See ref. [16] for with arbitrary gauge functions f (φ) and f( more discussion about elimination of the auxiliary fields. As a result, the NAC deformation of the NLSM (40) amounts to the nonK¨ ahlerian and non-Hermitian deformation of the original K¨ ahlerian and Hermitian structures, which is controlled by the auxiliary field solution to eq. (58). In the case of a single chiral superfield, the deformed NLSM metric can be read off from the following kinetic terms [16]: ¯ µ φ) ∂ ∂ K(φ + cM (φ, φ), ¯ φ) ¯ + K(φ − cM (φ, φ), ¯ φ) ¯ ¯ = − 1 (∂µ φ∂ Lkin. (φ, φ) 2 ¯ ∂φ ∂ φ ¯ ¯ µ φ) ∂ cK,φ (φ + cM, φ) ¯ − cK,φ (φ − cM, φ) ¯ ∂M (φ, φ) + 12 (∂µ φ∂ ∂φ ∂ φ¯ ¯ µ φ) ¯ cK ¯(φ + cM (φ, φ), ¯ φ) ¯ − cK ¯(φ − cM (φ, φ), ¯ φ) ¯ − 21 (∂µ φ∂ ,φφ ,φφ ×

¯ ∂M (φ, φ) . ∂ φ¯ (62)

Non-anti-commutative Deformation of Complex Geometry

215

In the case of several superfields, the deformed NLSM can be read off from eq. (60), when assuming all the ξ-integrations to be performed with the auxiliary fields considered as spectators. We thus find a new (NAC) mechanism of deformation of complex geometry in the supersymmetric NLSM target space, by using a non-vanishing anti-holomorphic scalar potential W (Φ), because elimination of the auxiliary ¯ via their algebraic equations of motion in the NAC deformed fields M and M NLSM results in the deformed bosonic K¨ ahler potential depending upon C 0 and W (Φ). This feature is specific to the NAC deformation, because a scalar potential does not affect a K¨ ahler potential in the usual (undeformed) NLSM.

6 Example: NAC-deformed CP (1) model Let’s consider the simplest non-trivial example provided by a four-dimensional supersymmetric CP (1) NLSM with the (undeformed) K¨ ahler, Hermitian and symmetric target space characterized by the K¨ ahler potential ¯ = α ln(1 + κ−2 φφ) ¯ , K(φ, φ)

(63)

where two dimensional constants, α and κ, have been introduced, and with an arbitrary anti-holomorphic scalar superpotential W (Φ). An explicit solution to the auxiliary field equation (58) in this case reads [16] q ¯ + κ−2 φφ) ¯W ¯ ,φ¯ 2 α − α2 + 2cφ(1 M= , (64) ¯ ¯ 2c2 κ−2 φ¯2 W ,φ ¯ A straightforward calculation ¯ ,φ¯ = ∂ W ¯ /∂ φ. where we have used the notation W yields the following deformed NLSM kinetic terms [16]: ¯ µ φ¯ , Lkin. = −gφφ ∂µ φ∂µ φ − 2g ¯∂µ φ∂µ φ¯ − g ¯ ¯∂µ φ∂ φφ

φφ

(65)

where g

¯ φφ

=

q −α +

¯ ,φ¯ )2 −ακ−2 c2 φ¯2 (W q , 2 2 2 −2 2 −2 ¯ ¯ ¯ ¯ ¯ ¯ α + 2cφ(1 + κ φφ)W,φ¯ α + 2cφ(1 + κ φφ)W,φ¯

gφφ = 0 , g¯ ¯ = φφ

q α−

¯W ¯ ,φ¯ −2α−1 c2 (1 + κ−2 φφ) 2 q ¯ + κ−2 φφ) ¯W ¯ + κ−2 φφ) ¯W ¯ ,φ¯ ¯ ,φ¯ 2 α2 + 2cφ(1 α2 + 2cφ(1

¯ ¯ ,φ¯)3 (1 + κ−2 φφ) × 4c2 φ¯2 (W q ¯ + κ−2 φφ) ¯W ¯ ,φ¯ 2 (2W ¯ ,φ¯ + φ¯W ¯ ,φ¯φ¯ ) . + α α − α2 + 2cφ(1 (66)

216

Sergei V. Ketov

It is worth noticing that det g = −(g ¯)2 . The most apparent feature φφ

gφφ = 0 is also valid in the case of a generic NAC-deformed NLSM (in the given parametrization).

7 Conclusion Our approach to the NAC-deformed NLSM is very general. The NAC deformation (i.e. smearing or fuzziness) of the NLSM K¨ ahler potential is controlled by the auxiliary fields M i entering the deformed K¨ ahler potential in the highly non-linear way. Both locality and Euclidean invariance are preserved, while no higher derivatives appear in the deformed NLSM action. One should distinguish between the NAC-deformation and N=1/2 supersymmetry. Though the NAC-deformation we considered is N=1/2 supersymmetric, the former is stronger than the latter. When requiring merely N=1/2 supersymmetry of a four-dimensional NLSM, it would give rise to much weaker restrictions on the NLSM target space. It is still the open question how to describe the NAC deformation of the NLSM metric in purely geometrical terms. This investigation was supported in part by the Japanese Society for Promotion of Science (JSPS). I am grateful to K. Ito, O. Lechtenfeld, P. Bowknegt and S. Watamura for useful discussions during the workshop.

References 1. Ketov S.V.: Quantum Non-Linear Sigma Models. Springer-Verlag, Berlin Heidelberg New-York (2000) 2. Spindel, P., Sevrin, A., Troost, W., van Proeyen, A.: Extended Supersymmetric Sigma Models on Group Manifolds. Nucl. Phys. B308, 662–698 (1988), and B311, 465–492 (1988) 3. Sevrin, A., Troost J.: Off-shell Formulation of N=2 Non-linear Sigma-models. Nucl. Phys. B492, 623–646 (1997) 4. Huybrechts D.: Complex Geometry. Springer-Verlag, Berlin Heidelberg NewYork (2004) 5. Gates, S.J. Jr., Hull, C., Roˇcek, M.: Twisted Multiplets and New Supersymmetric Non-linear Sigma Models. Nucl. Phys. B248, 157–186 (1984) 6. Buscher, T., Lindstr¨ om, U., Roˇcek, M.: New Supersymmetric Sigma-models with Wess-Zumino Terms. Phys. Lett. B202, 94–98 (1988) 7. Gates, S.J. Jr., Ketov, S.V.: No N=4 Strings on Wolf Spaces. Phys. Rev. D52, 2278–2293 (1995) 8. Galperin, A., Ivanov, E., Ogievetsky, V., Sokatchev, E.: Harmonic Superspace. Cambridge University Press (2001) 9. Brink, L., Schwarz J.: Quantum Superspace. Phys.Lett. B100, 310–312 (1981) 10. Klemm, D., Penati, S., Tamassia, L.: Non(anti)commutative Superspace. Classical and Quantum Grav. 20, 2905–2916 (2003)

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11. Ooguri, H., Vafa C.: Gravity induced C Deformation. Adv. Theor. Math. Phys. 7, 405–417 (2004) 12. Gates, S.J. Jr., Ketov, S.V., Nishino, H.: Self-dual Supersymmetry and Supergravity in Atiyah-Ward Space-time. Nucl. Phys. B716, 149–210 (1993) 13. Seiberg, N.: Noncommutative Superspace, N=1/2 Supersymmetry and String Theory. JHEP 0306, 010 (2003) 14. Alvarez-Gaume, L., Vazquez-Mozo, M.: On Non-anti-commutative N=2 Sigmamodels in Two Dimensions. JHEP 0504, 007 (2005) 15. Hatanaka, T., Ketov, S.V., Kobayashi, Y., Sasaki, S.: N=1/2 Supersymmetric Four-dimensional Non-linear σ-models from Non-anti-commutative Superspace. Nucl. Phys. B726, 481–493 (2005) 16. Hatanaka, T., Ketov, S.V., Kobayashi, Y., Sasaki, S.: Non-anti-commutative Deformation of Effective Potentials in Supersymmetric Gauge Theories. Nucl. Phys. B716, 88–104 (2005) 17. Hatanaka, T., Ketov, S.V., Sasaki, S.: Summing up Non-anti-commutative K¨ ahler Potential. Phys. Lett. B619, 352–358 (2005).

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Seiberg-Witten Monople and Young Diagrams Akifumi Sako Department of Mathematics, Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan [email protected]

1 @Introduction The N = 2 supersymmetric gauge theory and related subjects have been studied by using nonperturbative approach [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. One big break through is given by Nekrasov [11, 12, 13] After them, many relating developments in N ≥ 2 Super Yang-Mills theories and string theories. One of the key ideas is the localization formula that is very strong tool. We can use the formula when fixed points of some groups action associated with symmetries are isolated. In this paper, we study the fixed points of the Seiberg-Witten monopole theory on noncommutative (N.C.) R4 . Its partition function is calculated by localization formula if its fixed points are isolated. The Seiberg-Witten duality conjecture implies that the preopotential on the patch of moduli space around the massless U (1) monopole theory is spread to the prepotential on the other patch that is low energy effective potential of SU (2) gauge theory. That is why, if we can get the result of the SeibergWitten monopole theory by the localization formula then we get the proof of the Seiberg-Witten duality. This is our motivation. In this article we report that N.C. cohomological field theories do not depend on N.C. parameter and this fact relates with dimensional reduction. For example, the partition function of N = 4 supersymmetric U (1) gauge theory is calculated by using D0-brane theory. Also, we show that the fixed points of the Seiberg-Witten theory is isolated and they are classified by Young diagrams. This article is based on the joint work with Toshiya Suzuki [14, 15].

2 N.C. Cohomological field theory We consider only N.C.R4 . Let N.C. parameter be iθ µν = [xµ , xν ]. At first, let us see the fact that the cohomological field theory is invariant under θ deformation [16, 17, 18]. Actions of cohomological field theories are given by BRS exact terms. The partition functions of the cohomological field theory 219

220

Akifumi Sako

is invariant under any infinitesimal operation, Z ˆ ˆ 0 = ±δ 0 δ, ˆ S = δV δδ Z 0 δ Zθ = DφDψDχDH Z = DφDψDχDH

variation that commute with the BRS

Z Dˆ δ − dx δV exp (−Sθ ) Z 0 ˆ δ − δ V exp (−Sθ ) = 0. 0

Here δˆ is a BRS operator and S is an action of some cohomological field theory defined by a gauge fermion V . Above equations show that the partition function of the cohomological field theory is invariant under infinitesimal deformation generated by δ 0 . Let introduce θ-deformation ; δθ θµν = θµν + δθµν .

(1)

If δθ commute with BRS operator, then the partition function is invariant under the θ-deformation. We can see that this statement is right for the N ≥ 2 supersymmetric N.C. Yang-Mills theory[14, 15]. Let us study what occur when the theory is invariant under the θ deformation. Here, we treat Moyal space that is defined by the Moyal product defined as

∗

θ

i← − − → := exp{ ∂µ θµν ∂ν } . 2

(2)

Let change variables as x µ xµ → x 0 = √ t dx =

√

∗

1 tdx0 , ∂µ = √ ∂µ0 , t

θ

−θ − → i← = exp{ ∂µ0 ∂ν0 } . 2 t

After this, do the θ-deformation θ → θ0 = tθ,

∗

θ0

− − → i← = exp{ ∂µ0 θ ∂ν0 } 2

From this procedure, we can interpret θ-deformation = Rescaling without ∗

(3)

That is to say Z Sθ 0 ∼

√ D 0D t dx L(

∗, √1t ∂x∂

0ν

),

(4)

Seiberg-Witten Monople and Young Diagrams

221

where ∗ is fixed under θ-deformation. Let’s consider “θ → ∞ ”= “t → ∞”. We find that kinetic terms (∂µ including terms) vanish. Using this phenomena, some calculations become easy. Indeed we can see such example. The Euler number of the moduli space of GMS soliton is calculated in [16, 17, 18] and it is equivalent to the partition function of balanced scalar cohomological field theory based on noncommutative plane. The moduli space of GMS soliton for real scalar field φ is Mm = {φ| bm φ ∗θ (φ − v1 ) ∗θ (φ − v2 ) · · · ∗θ (φ − vm ) = 0, bm > 0}. (5) Here, vi ∈ R. When the solutions of the algebraic equation x(x − v1 )(x − v2 ) · · · (x − vm ) = 0 are not degenerate, the Euler number of the space Mm is given by 1 : m is even number (6) χm = 0 : m is odd number. The partition function (Euler number of the moduli space) is invariant under the θ deformation. So, we can evaluate this partition function in the commutative limit, too, and this calculation is also easy. On the other hand, the case of large θ limit is a dimensional reduction to zero dimension. This is the theme of the next section.

3 Universality of Partition Functions Consider N.C.R2D . Let noncommutative parameters be (θ µν ) = ⊕θi 2i−1, 2i . As we saw in the previous section, in θ i → ∞, terms with derivative operators ∂x2i ∗ := −i(θi )−1 [x2i−1 , ∗] and −∂x2i−1 := −i(θi )−1 [x2i , ∗] become irrelevant in lagrangians. Therefore, if the theory is invariant under the θ-deformation, terms including ∂x2i or ∂x2i−1 can be removed. In the following, we consider only such cases. Operator expression of the N.C.R2D is constructed by using Fock basis. An arbitrary operator is expressed as X X n1 ···nD ˆ= O ··· Om |n1 , · · · , nD i hm1 , · · · , mD | . (7) 1 ···mD n1 ,m1

nD ,mD

This expression implies that N.C.field theories on the N.C.R2D are described by ∞ dimensional matrix models. Let us consider a matrix model given by n1 ···nD the Lagrangian L(O) where we take Om is a variable of path integration 1 ···mD and L(O) is a polynomial of O. Then we cannot distinguish the other model whose dynamical variables are n ···n

n

···n

i+1 D Om11 ···mi−1 i−1 mi+1 ···mD |n1 ,···,ni−1 ,ni+1 ,···,nD i hm1 ,···,mi−1 ,mi+1 ,···,mD |

222

Akifumi Sako

n1 ···nD |n1 , · · · , nD i hm1 , · · · , mD | , befrom the model whose variables are Om 1 ···mD cause both of them are ∞ dimensional matrices. That is why, in θi → ∞, there is no ∂zi or ∂z¯i and it is impossible to distinguish dynamical variables living in R2D from variables in R2D−2 . In other words, θi → ∞ is equivalent to the dimensional reduction corresponding to x2i−1 and x2i directions.

Claim Let Z2D and hOi2D be a partition function and VEV of O of a CohFT in N.C.R2D with D ≥ 1 s.t. δθ Z2D = 0 and δθ hOi2D = 0. Let Z2D−2 and hOi2D−2 be the partition function and VEV of O of a CohFT in N.C. R2D−2 , where they are given by dimensional reduction of Z2D and hOi2D . Then, Z2D = Z2D−2 , hOi2D = hOi2D−2 ,

(8)

i.e. the partition function of such theories do not change under dimensional reduction from 2D to 2D − 2. In this article, we do not manage topological terms to avoid difficulties to interpret them in dimensional reduced models. For example, as we will see in next section supersymmetric Yang-Mills theories are invariant under θ-deformation. From this fact, the following partition function of supersymmetric Yang-Mills theories on N.C. R2D are equivalent: 8dim 6dim 4dim 2dim 0dim ZN =2 = ZN =2 = ZN =4 = ZN =8 = Z∗∗∗ ,

(9)

Idim where ZN =J is a partition function of the N = J supersymmetric Yang-Mills theory in N.C. RI with arbitrary gauge group. Similarly, we get 4dim 2dim 0dim ZN =2 = ZN =4 = Z∗∗∗ .

(10)

4 Z of N =4 Super U (1) N.C.Theory As a well known fact, 0(+1)-dimension reduced Yang-Mills theory is D0-brane effective theory. From applying the above process (θ → ∞), we can expect that Z of N =4 supersymmetric U (1) theory is estimated by D0-brain theory and some correction from traceless part; Z[N = 4SU SY U (1)] = Z[∞D0 − brane] × Z[f inite] because D0-brane theory is given by 0-dim. reduction of N =1 10-dim YangMills theory. Detail estimations of this are in [14]. Final result is given by Z=

X 1 π2 = ζ(2) = . 2 d 6

d∈N

(11)

Seiberg-Witten Monople and Young Diagrams

223

To extend this technique to the N = 2, d = 4 series like (10), it is natural to investigate the Seiberg-Witten model on N.C.R4 . We will show that the fixing points are isolated and we can apply the localization formula to get the partition functions in the following of this article.

5 @N = 2 SUSY Gauge Theory on N.C.R4 At first, we denote the set up the model of N = 2 supersymmetric gauge theory on N.C. R4 . SO(4) rotation of Euclidian space is locally isomorphic to SU (2)L × SU (2)R . N = 2 supersymmetric theory has SU (2)I R-symmetry. (R-symmetry is a symmetry ¯ αi of fermionic coordinate rotation.) The supersymmetric generators Qαi , Q ˙ have the indices i = 1, 2 for the R-symmetry. Totally N = 2 supersymmetric theory has following symmetry; H = SU (2)L × SU (2)R × SU (2)I .

(12)

The supersymmetric gauge multiplet is ψ1

Aµ

ψ2 .

(13)

φ Here ψ 1 , ψ 2 and ψ¯1 ,ψ¯1 are the Weyl spinors and their CPT conjugate. φ and φ¯ are scalar fields. Their quantum number of H are assigned as ψ 1 = (1/2, 0, 1/2), ψ 2 = (1/2, 0, 1/2), φ = (0, 0, 0), ψ¯1 = (0, 1/2, 1/2), ψ¯2 = (0, 1/2, 1/2), φ¯ = (0, 0, 0).

(14)

The action functional is given by L=

a a µαα − 41 Fµν Faµν − iψ¯αi ¯ ˙ Dµ ψαa i − Dµ φ¯a Dµ φa ˙ σ ¯ ψαi ]a − √i ψ¯α˙ ia [φ, ψ¯α˙ i ]a − 1 [φ, ¯ φ]2 , . − √i2 ψ αia [φ, 2 2

(15) (16)

The supersymmetric transformations with parameter ξ are given by δAµ = iξ αi σµαα˙ ψ¯α˙ i − iψ αi σµαα˙ ξ¯α˙ i , √ µ i β ˙ ¯αi ¯ αi, δψα i = σ µν + [φ, φ]ξ α ξβ Fµν + 2iσαα˙ Dµ φξ .. . Let us introduce topological twist. We use a diagonal subgroup SU (2)R0 in SU (2)R × SU (2)I .

224

Akifumi Sako

K 0 := SU (2)L × SU (2)R0 .

(17)

Then a combination of spinors whose quantum number of H are (1/2, 0, 1/2)⊕ (0, 1/2, 1/2) has quantum number (1/2, 1/2) ⊕ (0, 1) ⊕ (0, 0) of K 0 . Note that ˙ ¯ (0, 0) is scalar and Q = αi Qαi ˙ is the BRS operator, that is the Fermionic scalar transformation generator. The BRS operator in this case is interpreted as an equivariant derivative operator as we will see soon. Similarly, spinor fields are twisted as 1 1 1 1 ψ i , ( , 0, ) → ψµ , ( , ) 2 2 2 2 1 1 i ¯ ψ , (0, , ) → χµν ⊕ η , (0, 1) ⊕ (0, 0) . 2 2

(18)

Their BRS transformations are given as ˆ µ δA ˆ µν δχ

= iψµ ,

= Hµν , ˆ δHµν = i[φ, χµν ],

ˆ µ δψ δˆφ¯

= −Dµ φ,

ˆ = 0, δφ

= iη, ˆ ¯ . δη = [φ, φ]

(19)

Next step, we consider a hypermultiplet. We introduce the Weyl fermions ψq and ψq˜† , and complex scalar fields q and q˜† ; ψq q˜† .

q ψq†˜

Their supersymmetric transformations are given by √ √ δq i = − 2ξ αi ψqα + 2ξ¯α˙ i ψ¯qα˜˙ , √ ¯ i ξαi , δψqα = − 2iσαµα˙ Dµ q i ξ¯α˙ i − 2φq √ ˙ σ µαα δ ψ¯α˙ = − 2i¯ Dµ q i ξαi + 2φq i ξ¯α˙ i , q˜

(20)

Let gauge representation of this Hypermultiplet be a fundamental representation. After topological twisting, their BRS transformations are given by ˆ α˙ = −ψ¯α˙ , δq ˆ † = −ψ¯qα˙ , δq q˜ α˙ α ˙ α˙ ˆ ¯ ˆ δ ψq˜ = −iφq , δ ψ¯qα˙ = iqα†˙ φ.

(21)

Using above fields, let us define the Seiberg-Witten monopole equations. The action with the hypermultiplet are defined by ˆ S = k − δΨ

(22)

Seiberg-Witten Monople and Young Diagrams

where k is instanton number k=

1 8π 2

225

Z tr(FA ∧ FA ),

(23)

and Ψ is a gauge fermion; a a †α †α †α Ψ = −χµνa + {H+µν − s+µν } − χq {Hqα − sα } − {Hq − s }χqα ¯ a η a + Dµ φ¯a ψ µa − (−iq † φ)ψ ¯ α˙ − ψ † (iφq ¯ α˙ ) . +i[φ, φ] α ˙

q

q α˙

(24)

Here sµν (A, q, q † ) = Fa+µν + q † σ ¯ µν Ta q. sα (A, q) = σαµα˙ Dµ q α˙ = (6 Dq)α . After integration of H+µν and Hq , Bosonic action is Z √ 1 1 SB = d4x g |sµν |2 + |sα |2 + · · · . 4 2

(25)

(26)

The BPS eqs. are given by sµν (A, q, q † ) = 0 , sα (A, q) = 0 .

(27)

These equations are called Seiberg-Witten monopole equations. In the following, we investigate this theory whose gauge group is U (1) and defined space is N.C. R4 with [xµ , xν ] = iθµν . Here, θµν is an antisymmetric matrix  0  −θ1 µν (θ ) =   0 0

called N.C. parameter.  θ1 0 0 0 0 0  . 0 0 θ2  0 −θ2 0

(28)

(29)

We only use operator formalisms here, so all the fields R are operators act−1 ν ing on H. ∂µ is written by −iθµν [x , ∗] ≡ [∂ˆµ , ∗] and d2D x is written by det(θ)1/2 T rH . In the Fock space representation, fields are expressed as X Aµ = Aµ nm11nm2 2 |n1 , n2 ihm1 , m2 |, X ψµ = ψµ nm11nm2 2 |n1 , n2 ihm1 , m2 | , etc. Therefore, the BRS transformations are expressed as

226

Akifumi Sako

ˆ µ n1 n2 = ψ µ n1 n2 , δA m1 m2 m1 m2

ˆ µ n1 n2 = (Dµ φ)n1 n2 , . . . , δψ m1 m2 m1 m2

(30)

−1 ν where Dµ ∗ := [∂ˆµ + iAµ , ∗ ] with ∂ˆµ := −iθµν x . The action functional without the topological term is defined as follows;

S = T rH L(Aµ , . . . ; ∂ˆzi , ∂ˆz¯i ) ˆ . = T rH trδΨ

(31)

Let prove the invariance under the θ-deformation, here. Let us change the dynamical variables as 1 1 1 ˜¯ 1 1 η → η˜, q → √ q˜, Aµ → √ A˜µ , ψµ → √ ψ˜µ , φ¯ → φ, θ θ θ θ θ 1 1 1 + ˜ + , φ → φ˜ ψq → √ ψ˜q , ˜+ , Hµν → H χ+ µν → χ θ µν θ µν θ 1 † 1˜ 1 † χq → χq , χq → χ˜q , Hq → Hq . θ θ θ Note that this changing does not cause nontrivial Jacobian from the path integral measure because of the BRS symmetry. Then, the action is 1 ˜ S θ2

1 L(Aµ , . . . ; ∂ˆzi , ∂ˆz¯i ) → 2 L(A˜µ , . . . ; −a†i , ai ) . (32) θ √ S depends on θ because ∂zi = − θ−1 [a†i , ]. In contrast, S˜ does not depend on θ, because all θ parameters are factorized out. S→

,

From the discussion in section 2, the BRS symmetry proves that the Z is invariant under the deformation of θ ; ˜ =0, δθ Z = −2(δθ)θ −3 hSi R where Z = DADψ . . . exp (−S[A, ψ, . . .]). This fact implies that Z can be determined by its dimensional reduced model. The dimensional reduced version of the Seiberg-Witten eqs. are P+µνρτ [Aρ , Aτ ]a + q¯ σ µν q † = 0 , σ µ Aµ q = 0 ,

(33) (34)

√ where P+µνρτ is a selfdual projection operator. Using q+ := (q1˙ + q2˙ )/ 2 and √ q− := (q1˙ − q2˙ )/ 2, the eqs. (33) are rewritten as ADHM eqs. : ∗t ∗t [Az1 , A†z1 ] + [Az2 , A†z2 ] + q− q− − q + q+ =0,

[Az1 , Az2 ] +

∗t q − q+

=0.

(35) (36)

Seiberg-Witten Monople and Young Diagrams

227

6 D-brane interpretation The eqs. in the previous section are important even if we consider a finite size matrix model. Let’s consider the relations between dimensional reduction of Seiberg-Witten monopole eqs. and D-brane picture. The second order action of N brane N anti-brane system is given by Z 1 ¯ ) (N)µν ¯ (N (N ) (N )µν ¯2 . tr Fµν F + Fµν F + Dµ φDµ φ + (τ 2 − φφ) 4 (N )

¯) (N

¯

¯

Here Fµν ,Fµν are the curvature of A(N ) , A(N ) , and A(N ) , A(N ) are connec¯ tions corresponding to the open strings on D-brane and D-brane, respectively. Up to topological terms, we can rewrite this as Z n 1 ¯ ) (N ¯ (N ) (N ) (N tr Fµν F )µν + |Fz1 z¯1 + Fz2 z¯2 + (φφ¯ − τ )|2 2 o 2 +|Fz1 z2 | + |Dz¯¯1 φ|2 + |Dz¯¯2 φ|2 . ¯) (N

Case of Aµ

= 0, the E.O.M. is given as (N )

(N )

∗t Fz1 z¯1 + Fz2 z¯2 + q− q− =ζ , (N )

Fz 1 z 2 = 0 , Dz¯1 q− = 0 ,

(37) (38)

Dz¯2 q− = 0 ,

(39)

where we replace φ by q− . These are nothing but Seiberg-Witten eqs. under q+ = 0. This is the case that we will see later. Therefore we can understand ¯ configuration. that the solution of (34) is realized as the D3 − D3

7 Deformed BRS Operators To get the isolated fixed points, let us deform the BRS transformation to δ 2 Azi = δψzi = i[Azi , φ] − ii Azi ,

(40)

β˙

δ 2 qα˙ = δψqα˙ = −iφqα˙ + MR α˙ qβ˙ + iqα˙ b, 2 †α˙

δ q

= δψq

where β˙

MR α˙ = and

†α˙

†α˙

= q iφ − MR 0 i+ i+ 0

α˙

β˙ q

, + =

†β˙

†α˙

− ibq ,

1 + 2 , 2

(41) (42)

(43)

228

Akifumi Sako

   b= 



b1

  . 

b2 ..

.

(44)

bN 2

This deformation is made to make δ be the Lie derivative including group of global symmetries that generated by the above action with the parameter i , bi . Here we have to recall the fact that our BRS operator is interpreted as the equivariant derivative operator. In fact, ˜

SU (2),SU (2)

G G δ 2 = δ(−φ) + δ(b) + δ(1 ,2 )

,

(45)

that is to say δ 2 is given by the Lie derivatives. We do not change the gauge fermion Ψ , so the bosonic BPS equations do not change: ¯z2 z¯2 )q † − iζ1 = 0, σz1 z¯1 + σ µR ≡ i([Az1 , Az¯1 ] + [Az2 , Az¯2 ]) + q(¯ µC ≡ i[Az1 , Az2 ] + q¯ σz1 z2 q † = 0, (46) Dirac : (Az1 σ z1 + Az¯1 σ z¯1 + Az2 σ z2 + Az¯2 σ z¯2 )q = 0. For the later convenience, we introduce constant back ground ζ, here. The fixed point equations of the deformed BRS transformations for ψµ and ψq are i[Azi , φ] − ii Azi = 0, β˙

−iφqα˙ + MR α˙ qβ˙ + iqα˙ b = 0.

(47) (48)

The contributions to path integrals are given by neighborhood of the solutions of these equations.

8 Solutions In this section, we solve above eqs. (46)-(48). In the following of this article, we study only finite size matrix model given by the model in section 5 reduced to zero dimension and truncated into finite size matrices, for simplicity. At first, let us diagonalize φ by U (N ) gauge symmetry, φ = diag.(φ1 , φ2 , · · · , φN ).

(49)

Furthermore we assume φI does not degenerate. This is ensured from KempfNess Theorem, when we study above fixed points equations. µ−1 (0)/G * ) Closed GC orbit → non-degeneracy . (47) and (48) are fixed point equations for some torus actions. From (47) we see immediately that if and only if

Seiberg-Witten Monople and Young Diagrams

φJ − φ I = i , A zi

IJ

229

(50)

could be non-zero. Also from (48) we see that if and only if, (±)

φI = b J ± + ≡ b J ,

(51)

q1˙ IJ and q2˙ IJ could be non-zero. From these facts and small calculations, we get the following lemma. Lemma 1. If µR = 0 and Fixed Point Eqs. have a solution, then φI takes (n1 ,n2 ) , given by any of ϕ (−) [bI

]

ϕ Each {ϕ

(−)

(n1 ,n2 ) (−)

[bI

(−)

(n1 ,n2 ) [bI

]

= bI

]

+ n 1 1 + n 2 2

n1 , n2 ∈ Z .

,

(52)

} assign some graphs G[b(−) ] . In Fig.1, the origin corresponds I

Fig. 1. G[ˆxI ]

to the eigenvalue ϕ to eigen values ϕ

(0,0) (−)

[bI ] (n1 ,n2 ) (−)

[bI

]

Fig. 2. G[ˆxI ] (−)

= bI , and other lattice points (n1 , n2 ) correspond

.

For given a set of G[b(−) ] , φ is written as I

   M  φ=  I  

ϕ



(n1 ,n2 ) (−) [bI ]

ϕ

    .   

(n01 ,n02 ) (−) [bI ]

ϕ

00 (n00 1 ,n2 ) (−)

[bI

]

..

.

We suppose that eigen values are arranged by order, ϕ

(n1 ,n2 ) (−) [bI ]

<ϕ

(n01 ,n02 ) (−) [bI ]

<ϕ

00 (n00 1 ,n2 ) (−)

[bI

]

< ··· .

The I-th block matrix corresponds to the graph G[b(−) ] , and each diagonal I component in the I-th block corresponds to a lattice point in G[b(−) ] . I

230

Akifumi Sako

Azi takes a similar block structure,  A zi =

M  · · · Az i 

.. .

(I,(n1 ,n2 )),(I,(m1 ,m2 ))

.. .

I

  ··· ,

A non-trivial components of Azi are A z1

(I,(n1 ,n2 )),(I,(n1 −1,n2 ))

, A z2

(I,(n1 ,n2 )),(I,(n1 ,n2 −1))

.

We can interpret these nontrivial components from a graphical view point. : Az1 ’s non-trivial component : Az2 ’s non-trivial component See Fig.2. Non-trivial components of qα˙ are q1˙ q1˙

(I,(0,0)),J

= −q2˙

(I,(0,0)),J

6= 0 , for I, J, s.t. φI = bJ + + ,

(I,(1,1)),J

= +q2˙

(I,(1,1)),J

6= 0 , for I, J, s.t. φI = bJ − + .

These are corresponding with the origin of Fig.2. From now on, we suppose ζ > 0.

(53)

After straightforward calculation, we obtain the following theorem. Theorem 1. Let G[bI ] be a graph defined from fixed point φ, Azi of the torus action. The following three conditions are necessary for µR = 0 to have a solution. (1) G[bI ] consists of one connected part. (2) G[bI ] includes the origin (0, 0). (3) All points (n1 , n2 ) in G[bI ] must be in n1 ≤ 0 , n2 ≤ 0. Also we introduce such a map I, that I : { l | l = 1, · · · , M } → { I | I = 1, · · · , N } , M ≤ N, and there is one to one correspondence between the index I and (l, (n1 , n2 )). For given GI(l) ≡ G[b(−) ] , non-trivial components of Azi are I(l)

A z1

{l,(n1 −1,n2 )}{l,(n1 ,n2 )}

,

(n1 − 1, n2 ), (n1 , n2 ) ∈ GI(l) ,

(54)

A z2

{l,(n1 ,n2 −1)}{l,(n1 ,n2 )}

,

(n1 , n2 − 1), (n1 , n2 ) ∈ GI(l) .

(55)

and Also non-trivial components of qα˙ are q1˙

I={l,(0,0)},J=I(l)

= −q2˙

I={l,(0,0)},J=I(l)

.

(56)

Seiberg-Witten Monople and Young Diagrams

231

Fig. 3. GI(l)

For the non-trivial eqs , µR = 0 and µC = 0 are reduced to n +Az1 {l,(n1 ,n2 )},{l,(n1 +1,n2 )} Az¯1 {l,(n1 +1,n2 )},{l,(n1 ,n2 )} n

o

−Az¯1 {l,(n1 ,n2 )},{l,(n1 −1,n2 )} Az1 {l,(n1 −1,n2 )},{l,(n1 ,n2 )} +Az2 {l,(n1 ,n2 )},{l,(n1 ,n2 +1)} Az¯2 {l,(n1 ,n2 +1)},{l,(n1 ,n2 )}

o

−Az¯2 {l,(n1 ,n2 )},{l,(n1 ,n2 −1)} Az2 {l,(n1 ,n2 −1)},{l,(n1 ,n2 )} +2q1˙ {l,(n1 ,n2 )},J q1∗T ˙ J,{l,(n1 ,n2 )} and

= ζ,

n Az1 {l,(n1 ,n2 )},{l,(n1 +1,n2 )} Az2 {l,(n1 +1,n2 )},{l,(n1 +1,n2 +1)}

(57) o

−Az2 {l,(n1 ,n2 )},{l,(n1 ,n2 +1)} Az1 {l,(n1 ,n2 +1)},{l,(n1 +1,n2 +1)} = 0. On the other hand, we get following theorem for the Dirac equation. Theorem 2. If Azi , q, q † , φ satisfy the equations µR = 0 , µC = 0 and (47),(48) then the Dirac eq. gives no constraint. From this theorem, we can conclude that fixed points we are investigating are some special case of usual ADHM matrix model. Let us give graphical interpretations of nontrivial undetermined components. • Az1 {l,(n1 ,n2 )}{l,(n1 +1,n2 )} = “ ← ” connecting (n1 , n2 ) and (n1 + 1, n2 ) in GI(l) . Number of Az1 : ]Az1 = 2× Number of “←”. • Az2 {l,(n1 ,n2 )}{l,(n1 ,n2 +1)} = “↓” connecting (n1 , n2 ) and (n1 , n2 + 1) in GI(l) . Number of Az2 : ]Az2 = 2× Number of “↓”. • q1˙ I={l,(0,0)}J=I(l) origin (0, 0) in GI(l) . Number of q : ]q = 2.

232

Akifumi Sako

Here ]Azi and ]q denote numbers of undetermined real valued components. Then the total number of undetermined variables is

Fig. 4. Az1

Fig. 5. Az2

Fig. 6. q1˙

]Az1 + ]Az2 + ]q. On the other hand, graphical meanings of µR = 0 , µC = 0 and the residual U (1)N are as follows. * points in GI(l) . • “Nontrivial equations in µR = 0 ” ) Number of nontrivial “µR = 0” : ]µR = Number of points in the graph GI(l) . • “Nontrivial equations in µC = 0” * ) hook connecting (n1 , n2 ) and (n1 + 1, n2 + 1), Number of nontrivial “µC = 0” : ]µR = 2× Number hooks. • Each U (1) in U (1)N * ) each point (n1 , n2 ) in GI(l) . Number of U (1) : ]U (1) = Number of points.

Fig. 7. µR

Fig. 8. µC

Fig. 9. U (1)

Here ]µR , ]µC and ]U (1) are numbers of real constraint equations. Then the total number of constraints is ]µR + ]µC + ]U (1). Now we will prove the next theorem. Theorem 3. If and only if GI(l) is a Young diagram, µR = 0 , µC = 0 , (47) and (48) have a solution, and the solution is an isolated one.

Seiberg-Witten Monople and Young Diagrams

233

(proof) Consider a diagram GI(l) as a quadranglation of a 2 dimensional surface. Recall the well-known formula for the Euler number χ of 2 dimensional surfaces, χ = 2 − 2h − b = # {points} − # {edges} + # {faces},

(58)

where h = ] handles , b = ] boundaries . Then we obtain, χ = 1 = # {points} − # {edges} + # {faces}.

(59)

Notice that #

{points} = ](µR ) = ]U (1),

and #

{edges} =

]Az1 + ]Az2 . 2

(60) (61)

Also one can understand that #

]µC . (62) 2 is a Young diagram. (See Fig.11.)

{faces} ≤

The equation in (62) holds when GI(l)

Fig. 10. Non-Young diagram and Young diagram

(] of components) − (] of eqs.) = (]Az1 + ]Az2 + ]q) − (]µR + ]µC + ]U (1)) = 2# {edges} + 2 − 2# {points} − ]µC ≤ −2 # {points} − # {edges} + # {faces} + 2 = −2 + 2 = 0.

(63)

From this, we find that if and only if GI(l) is a Young diagram, there is a solution, and that the solution is a isolated one.

234

Akifumi Sako

9 Localization Theorem The localization theorem is a powerful tool for calculations of path integrals of cohomological field theories. Let δˆ be the deformed BRS transformation operator. S is given by a BRS exact functional ; ˆ (φ, B, F). S = δΨ (64) B, F denote the BRS doublet and B, F is a set of bosonic fields and a set of fermionic fields, respectively. The localization theorem of the path integral formulation [19] is expressed as Q Z Z Y N Dφ ˜ I6=J (φI − φJ ) −δΨ Z= = DBDFe . (65) dφI U (N ) SdetL I=1

φI are the eigenvalues of φ, and SdetL is SdetL = Sdet

∂(Q)B ∂(Q)B ∂F ∂B ∂(Q)F ∂(Q)F ∂F ∂B

! ,

(66)

where (Q)B and (Q)F are defined by ∂ ∂ δˆ = (Q)B + (Q)F . ∂B ∂F

(67)

Note that this expression is analogue of d˜ = d + iX ,

(68)

where X is a vector defining the Lie derivative LX Using this formula for our model, the result is given as Z=

Z Y N I=1

Y I6=J

dφI

Y

(φI − φJ )

I6=J

N Y (1 + 2 ){−(φI − bI )2 + 2− } 1 2 {−(φI − bI )2 + 2+ }

(69)

I=1

{(φI − φJ )2 − 42+ }1/2 {−(φI − bJ )2 + 2− } , {−(φI − bJ )2 + 2+ }{(φI − φJ )2 − 21 }1/2 {(φI − φJ )2 − 22 }1/2

where − = (1 + 2 )/2. We can perform the calculation by the other way that is given by [20]. The result is as follows. (Q Q Q ˜ ˜ XX ˜ I s∈Y˜(F (s, l; I) + 1 )(F (s, l; I) + 2 ) l Z= Q Q Q l ˜ ˜ ˜ s∈Y˜l (E(s, l; l) + 1 )(E(s, l; l) − 2 ) l l {Yl } I ) 1 ×Q Q , (70) Q ˜ ˜ ˜ s∈Y˜ I6∈I F (s, l; I)(F (s, l; I) + 1 + 2 ) l l

Seiberg-Witten Monople and Young Diagrams

235

where E(s, ˜l; l) := bI(l) − bI(˜l) + 1 h(s) − 2 v(s) F (s, ˜l; I) := bI − bI(˜l) + 1 i(s) − 2 j(s) 0 v(s) := νi,l − j˜l , h(s) := νj,˜l − i˜l , 0 where νi,l is the length of the i-th column of a Young diagram Yl , and νj,˜l is the length of the j-th row of another Young diagram Y˜l . s is located at (i, j). In Fig.11, v(s) and h(s) are number of white circles and black circles,

Fig. 11. Young diagram Y˜l and Yl

respectively. We can get concrete expression of small N matrix model from (69) or (70). In general, (70) is easer than (69) as N grows.

10 Conclusion We have investigated the noncommutative cohomological field theories. We saw that they are invariant under the noncommutative parameter deformation and their kinetic terms become irrelevant in the large noncommutative parameter limit. Then some kind of dimensional reduction occur in the limit. This fact meant that there are some kind of universal classes of supersymmetric gauge theory. As an example, we can get the partition function of N = 4 d = 4 supersymmetric gauge theory by using 0 dimensional theory. Another example is universal class including N = 2 d = 4 supersymmetric gauge theory. In particular, we have studied Seiberg-Witten monopole model closely and clarified the following facts. •The solutions of fixed point eqs. are classified, where the fixed points equations are given by θ → ∞ limit of the theory on N.C. R4 . • Fixed point equations are reduced to ADHM eqs. In other words, the Dirac equation is trivial under the other conditions. Then it is known that Young diagrams classify the solutions [20]. We gave a new proof by making correspondence between Fields, Equations and points, edges and faces. Furthermore,

236

Akifumi Sako

we showed that the solutions are given by isolated points. • Then we found that we can perform the path integral to get the path integral by the localization formula. Indeed, we did calculation of the toy model of the same type Lagrangian of the θ → ∞ N.C. Seiberg-Witten monopole model. “Toy model” means that the calculated model is reduced to the finite size matrix model. The true matrix model of N.C. Seiberg-Witten monopole model is constructed as an infinite dimensional matrix model. The calculation for this model is left for a future work.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19; ibid. B431 (1994) 484. N. Dorey, V. V. Khoze and M. P. Mattis, Phys. Rev. D54 (1996) 2921. F. Fucito and G. Travaglini, Phys. Rev. D55 (1997) 1099. M. Matone, Phys. Lett. B 357 (1995) 342, arXiv:hep-th/9506102. N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “The calculus of many instantons”, arXiv:hep-th/0206063. D. Bellisai, F. Fucito, A. Tanzini and G. Travaglini, Phys. Lett. B 480 (2000) 365, arXiv:hep-th/0002110; JHEP 0007 (2000) 017, arXiv:hep-th/0003272. U. Bruzzo, F. Fucito, A. Tanzini and G. Travaglini, Nucl. Phys. B 611 (2001) 205, arXiv:hep-th/0008225. N. Dorey, T. J. Hollowood and V. V. Khoze, JHEP 0103 (2001) 040, arXiv:hepth/0011247. T. J. Hollowood, JHEP 0203 (2002) 038, arXiv:hep-th/0201075; Nucl. Phys. B 639, 66 (2002), arXiv:hep-th/0202197. G. W. Moore, N. Nekrasov and S. Shatashvili, Commun. Math. Phys. 209 (2000) 97, arXiv:hep-th/9712241. N. A. Nekrasov, arXiv:hep-th/0206161. A. Losev, N. Nekrasov and S. L. Shatashvili, Nucl. Phys. B 534 (1998) 549, arXiv:hep-th/9711108. N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238

14. A.Sako and T. Suzuki, Partition functions of Supersymmetric Gauge Theories in Noncommutative R2D and their Unified Perspective, hep-th/0503214. 15. Akifumi Sako and Toshiya Suzuki, Dimensional Reduction of Seiberg-Witten Monopole Equations, N=2 Noncommutative Supersymmetric Field Theories and Young Diagrams hep-th/0511085 16. A. Sako, S-I. Kuroki and T. Ishikawa, Noncommutative Cohomological Field Theory and GMS soliton, J.Math.Phys.43(2002)872-896, hep-th/0107033. 17. A. Sako, S-I. Kuroki and T. Ishikawa, Noncommutative-shift invariant field theory, proceeding of 10th Tohwa International Symposium on String Theory, (AIP conference proceedings 607, 340). 18. A.Sako, Noncommutative Cohomological Field Theories and Topological Aspects of Matrix models, NONCOMMUTATIVE GEOMETRY AND PHYSICS,p321355, World Scientific, hep-th/0312120.

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19. U. Bruzzo and F. Fucito, Superlocalization Formulas and Supersymmetric YangMills Theories, Nucl. Phys. B 678 (2004) 638, math-ph/0310036. 20. H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, AMS University Lecture Series vol. 18 (1999).

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Instanton Counting, Two Dimensional Yang-Mills Theory and Topological Strings Kazutoshi Ohta Theoretical Physics Laboratory, The Institute of Physical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0198, JAPAN [email protected]

1 Introduction String theory is a candidate for the unification of all matter and forces in our universe, but we do not fully understand the whole of the theory. Especially, if we would like to treat gravity quantum mechanically, we can not use tools of the differential geometry since quantum fluctuations of the geometry may drastically change the short distance structure. A usual perturbative formulation of string theory does not resolve this problem. So we need intrinsic tools in order to describe the quantum theory of gravity. One possible way is to define the geometries (manifolds) in string theory without a perturbative differential geometrical picture using only with an algebraic or discrete description. Some examples are known as the noncommutative geometry (see e.g.[Con94]). To import the idea of the noncommutative geometry into string theory, we have to reformulate theory non-perturbatively without using the differential geometrical language. Fortunately, some non-perturbative formulations are proposed as some kinds of the matrix models [BFSS97, IKKT97]. It has been already pointed out that the matrix models and non-commutative geometry are closely related with each other, but it is still unclear how these make quantized geometry including the gravity. To treat the non-perturbative effects in gauge and string theory, solitons like instantons, monopoles and D-branes play very important roles. Especially, Bogomol’nyi, Prasad and Sommerfeld (BPS) states make problems on non-perturbative dynamics exactly solvable if the theories possess the supersymmetry. The solitons in the supersymmetric gauge theory or D-branes in the string theory give various exact results when the system preserves a sufficiently large number of supercharges; nevertheless it is still difficult to calculate non-perturbative physical quantities in general at all orders, even in the supersymmetric theory.

239

240

Kazutoshi Ohta

However, a novel idea that I would like to explain in this article appeared recently, that is, a statistical mechanical counting of the BPS states. To obtain one of non-perturbative corrections to the effective theory, we need some integrals over the moduli space of solitons as we will see later. Nekrasov replaced the moduli space integrals with statistical partition function summations in a very ingenious way [Nek02, NO03]. These summations are defined over every sets of Young diagrams which is associated with representations of a unitary or symmetric group. So we can see the close relation to the free fermion or melting crystal system [ORV03]. The effective action which contains all instanton corrections can be obtained from a free energy of the statistical partition function and it also includes all corrections from a graviphoton background field surprisingly. In addition, Nekrasov’s partition function is closely related not only to 4 dimensional supersymmetric gauge theory but also to lower (2 or 3) dimensional gauge theory without supersymmetry and topological string amplitude on some Calabi-Yau manifolds [GV98, INOV03, MMO05, MO06]. This fact can be understood by considering a realization of the supersymmetric gauge theory via D-branes in the superstring theory and using duality relations between the various systems. In this article, I would like to explain the instanton counting by Nekrasov and the related various systems I have mentioned above. There we will find that an essential structure of Nekrasov’s partition function is a discretization of the random matrix model. The picture of the discrete matrix model is very useful to see the relations between various systems. I will discuss duality relations through this discrete matrix model structure.

2 Instanton Counting It is known that the prepotential of 4 dimensional N =2 (8 supercharges) SU (r) gauge theory has an instanton expansion F(al ) =

∞ X

Fk (al )Λ2rk ,

l = 1, . . . , r,

(1)

k=0

where k is the instanton number and Λ is a QCD scale. The coefficients of the expansion Fk (al ) are holomorphic functions with respect to the moduli parameters of vacua al (vacuum expectation value of the adjoint scalar field in the vector multiplet of N =2 theory). We will call the contribution from zero-instanton (k = 0) as a perturbative part Fpert ≡ F0 in the following. The problem here is to determine the coefficients Fk (al ), but Seiberg and Witten have shown that this prepotential can be determined “in principle” by solving a class of differential equations associated with a genus r − 1 algebraic curve (Seiberg-Witten curve), which also determines the moduli space of vacua. However, it is a very difficult problem practically to calculate the

Instanton Counting, 2D YM Theory and Topological Strings

k(1)5 k(1)4 k(1)3 k(1)2 k(1)1

k(r)7 k(r)6 k(r)5 k(r)4 k(r)3 k(r)2 k(r)1

k(2)4 k(2)3 k(2)2 k(2)1 Y1

241

Y2

Yr

Fig. 1. An example of a set of r Young diagrams. We express the number of boxes (l) in the i-th row of the l-th Young diagram by ki .

general k instanton contributions to the prepotential from the Seiberg-Witten curve. Primarily, to calculate the instanton contribution to the effective theory, we have to expand the fields around a non-trivial instanton background and perform path integrals over the fluctuations. The coefficient in the instanton expansion of the prepotential is, roughly speaking, proportional to the “volume” of the instanton moduli space. The difficulty of the problem essentially comes from the evaluation of the volume of the instanton moduli space, since the moduli space generally includes some singularities. So we need a suitable regularization in order to obtain the well-defined volume of the moduli space. Nekrasov gave a regularization in physical and mathematical manner and calculated the volume of the instanton moduli space exactly [Nek02, NO03]. He introduced the so-called Ω-background, which is equivalent to the graviphoton background in our context, and performed the integral over the moduli space to obtain the volume as a finite summation over fixed points on gauge transformation under the Ω-background by using the localization theorem (Duistermaat-Heckman formula), which is known in the mathematical literature. The fix points of the k instanton moduli space in SU (r) gauge theory are classified by r sets of Young diagrams, Y = (Y1 , Y2 , . . . , Yr ), whose total number of boxes is equal to k. (See Fig. 1.) The contribution to the prepotential is given by the free energy of the partition function ZNekrasov (al ; ) = Zpert (al ; ) ×

X

Y

{Y} (l,i)6=(n,j)

(l)

(n)

al − an + (ki − kj + j − i) al − an + (j − i)

! Λ2r|Y| , (2)

where Zpert (al ; ) is a contribution from the perturbative part, |Y| stands (l) for the total number of boxes, and ki denotes the number of boxes of the i-th row in the l-th Young diagram Yl . In addition, the parameter in the formula (2), which does not appear in the original supersymmetric gauge

242

Kazutoshi Ohta

theory, is introduced by the Ω-background in order to regularize the moduli space integral. The free energy of the above partition function (2) diverges in the limit of → 0, which reflects the fact that the original moduli space integral requires a regularization. But the divergence is up to the order 2 and if we consider the limit F(al ) ≡ lim 2 log ZNekrasov (al ; ), →0

(3)

it always converges. This limit gives, in fact, the exact effective prepotential including the full contribution from the instantons in the original supersymmetric gauge theory, which we would like to know. Thus we find that the leading term of the expansion of the free energy in represents the contribution without graviphoton background, but it is believed that the sub-leading terms represent the graviphoton background contributions to the prepotential. It is very interesting that the problem on the non-perturbative calculation in quantum gauge theory supersedes the statistical problem of the partition function over the sets of Young diagrams. Using Nekrasov’s formula, we can calculate any instanton contribution to the prepotential in more systematic way than the Seiberg-Witten theory.

3 Superstring Perspective and Discrete Matrix Model In this section, we would like to consider Nekrasov’s instanton calculation from the point of view of superstring theory. First of all, let us consider a realization of the 4 dimensional N =2 supersymmetric gauge theory in type IIB superstring theory, where D5-branes wraps a 2-cycle in an asymptotically locally Euclidean (ALE) space. The 6 dimensional world-volume of the D5brane is compactified by the 2-cycle. So we have 4 dimensional gauge theory on the residual flat space-time directions. A part of supersymmetries is broken by the existence of D-branes and ALE space. In this embedding, D5-branes have 2 (1 complex) dimensional transverse directions along which D5-brane can move freely. These transverse directions correspond to the flat direction of 4 dimensional N =2 theory and the positions of D5-branes represent the vev of the vector multiplet adjoint scalar. Due to the compactification of the D5brane, the gauge coupling of the 4 dimensional theory is proportional to the area of the 2-cycle A. So we expect that the k instanton contribution includes A a factor of e− gs k , where gs is the string coupling constant. Therefore, we can regard the k instanton contribution as a contribution from k (Euclidean) D1-branes wrapping on 2-cycles in the superstring picture. In general, the Euclidean D1-branes make BPS bound states with D(-1)-branes. So, in order to obtain the k instanton contribution, we need to count the number of the BPS D1-brane bound states wrapping on 2-cycle. In fact, the entropy factor of this multiplicity of the D1-D(-1) bound states is the coefficient of the instanton expansion of the prepotential and is proportional to the (regularized) volume of the moduli space.

Instanton Counting, 2D YM Theory and Topological Strings

243

Let us make the following considerations with the aim to evaluate the number of the D1-D(-1) bound states. If we use the idea of the large N reduction [EK82] and turn on a strong B-field along the 4 dimensional space-time where the supersymmetric gauge theory exits, we can equivalently treat the D5-brane as a large number of D1-branes wrapping on the 2-cycle. Therefore, we should statistically count D1-D(-1) brane bound states in a heat bath, which is a large number of D1-branes coming from D5-brane originally1 . First of all, to perform the above evaluation, we have to make an effective theory on the D1-branes, but it must be topologically twisted since the worldvolume of the D1-branes does not contain the time-direction (Euclidean Dbrane) [BVS96]. So the effective theory on the D1-branes is 2 dimensional topologically twisted U (N ) gauge theory with the large number N . Secondary, if of D(-1)-brane and a D1-brane can be expressed by R R we2 notice that charges dx Tr ΦF and 21 dx2 Tr Φ2 , respectively, where Φ represents the adjoint scalar field along the flat direction, the grand canonical ensemble of the Dbrane bound states is given by a vev in the 2 dimensional topological field theory Z 1 µ 2 2 Z = exp − . (4) dx Tr (ΦF + Φ ) gs S 2 2 top. 1/gs and µ/gs are chemical potentials for D(-1)-branes and D1-branes. Note that the original prepotential is recovered as the free energy of the partition function (4) in the limit of N → ∞. The partition function (4) reduces to the partition function of the ordinary (non-supersymmetric and non-topological) bosonic 2 dimensional gauge theory by using again the idea of the localization Z Z µ 1 (5) dx2 Tr (ΦF + Φ2 ) , Z = DADΦ exp − gs S 2 2 after integrating out all fermionic degrees of freedom [Wit92]. In this particular case, if we integrate out Φ too, the partition function (5) coincides with the bosonic 2 dimensional Yang-Mills theory. Therefore, the problem on the calculation of the prepotential of the supersymmetric theory including the instanton contributions reduces to the evaluation of the large N limit of the U (N ) bosonic two dimensional gauge theory without supersymmetry! Now, it is well known that this 2 dimensional Yang-Mills partition function can be rewritten as a statistical partition function [Mig75]. Indeed, if we perform the path integral of the partition function (5) by using the standard quantum field theory method, we finally obtain the following representation of the partition function [BT93] X Y µgs A PN 2 i=1 ni , ZDMM = N ! (gs ni − gs nj )2 e− 2 (6) n1 >n2 >···>nN i<j

1

To obtain the net contribution to the prepotential, we need to exclude the contribution from the large number of the D1-branes. As we will see, the contribution from the infinite D1-branes corresponds to the perturbative part.

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where the summation is done over integer sequences {ni } which satisfy a strongly decreasing condition n1 > n2 > · · · > nN . Observing the partition function (6), we find that it is a discretization of eigenvalues of the N ×N Hermite random matrix model with a quadratic potential ZMM =

Z Y N

dλi

Y i<j

i=1

µA

(λi − λj )2 e− 2gs

PN

i=1

λ2i

.

(7)

Comparing with the matrix model partition function, we find that the eigenvalues λi are quantized in units gs by gs ni in the 2 dimensional gauge theory partition function and the integrals over eigenvalues are replaced by summations. In this sense, we will call the partition function of the 2 dimensional Yang-Mills theory by a “discrete matrix model” in the following. Moreover, we find that the partition sum of the discrete matrix model is done over the sets of Young diagrams in the same way as Nekrasov’s partition function. The summation in (6) is over the integer sets which satisfy the strongly decreasing condition n1 > n 2 > · · · > n N ,

(8)

but if we set ni = ki − i + c, where c is an arbitrary constant, we can regard the summation as a summation of integer sequences which satisfy a weakly decreasing condition k1 ≥ k 2 ≥ · · · ≥ k N , (9) including equalities. So we can identify the integers {ki } with the number of rows in Young diagrams associated with the representation R of U (N ) group. Using this correspondence, if we rewrite the Vandermonde determinant and the quadratic potential in the discrete matrix model, they become the dimension of the representation R and second Casimir C2 (R), respectively. Therefore, by a suitable normalization, the partition function of the discrete matrix model (2 dimensional Yang-Mills theory) reduces to X µgs A (dim R)2 e− 2 C2 (R) . ZYM2 = (10) R

This is nothing but the original form of the 2 dimensional partition function given by Migdal.

4 Large N Limit Now, our purpose is to calculate the coefficients of the instanton expansion of the prepotential from counting the multiplicity of D1-branes. This can be obtained from the large N limit of the discrete matrix model or 2 dimensional Yang-Mills theory as we have seen. Next we would like to consider the

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large N limit. Unfortunately, however, we obtain the prepotential only for 4 dimensional N =2 “U (1)” gauge theory, which is not so interesting from the point of view of gauge theory dynamics, if we use the quadratic potential like in (6). To obtain the prepotential for the general SU (r) theory, we need to extend the quadratic potential to (r+1)-th order potential W (Φ) which has r critical points. We will generally consider the (r+1)-th order potential in the following. Let us first consider dynamics of eigenvalues in the discrete matrix model with the generic potential. Here we introduce the density of the eigenvalues similar to the ordinary continuous matrix model ρ(x) =

N 1 X δ(x − gs ni ). N i=1

(11)

Using this eigenvalue density, the partition function (6) can be written as X e−S[ρ(x)] , (12) ZDMM = N ! {ρ(x)}

where Z Z N S[ρ(x)] = −N 2 − dxdy log |x − y|ρ(x)ρ(y) + dxW (x)ρ(x), gs

(13)

is an effective action adopting the effect from the Vandermonde determinant. If we look at the above effective action, we find that the Vandermonde determinant part works as a repulsive force between the eigenvalues and the potential provides an attractive force to the critical points2 . (See Fig. 2.) This fact means that the Vandermonde determinant represents the effect of the exclusion principle of free fermions if we identify the eigenvalues with the levels of states of the free fermions. In fact, the eigenvalues, which is a strongly decreasing integer sequence, can not take the same value with each other and so the densest configuration is a continuous integer sequence. The boundaries of the integer sequence, which are separated by fillings and vacancies of the eigenvalues, correspond to Fermi surfaces. For finite N , there exist the Fermi surfaces for both sides of each bunch of eigenvalues. We will call the densest configuration the “ground state”. Contribution from the ground state comes from the heat bath which is originally the contribution from the large N reduction of the D5-branes, and it will become the perturbative part of the 4 dimensional supersymmetric gauge theory. Therefore the net contribution 2

Strictly speaking, the force from the potential depends on a sign of the second derivative of the potential whether it is attractive or repulsive. However we think here that the eigenvalues are analytically continued and gather to any critical points in spite of the sign of the second derivative like in Dijkgraaf-Vafa theory [DV02].

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nF

nF

Fig. 2. A distribution of the eigenvalues in the discrete matrix model with a quadratic potential. At the position of black dots, there exists the eigenvalues and white dots stand for vacant eigenvalues. nF represents the two Fermi surfaces. Using the so-called Maya/Young diagram correspondence, we can build the Young diagram above the eigenvalues. In the slant Young diagram, the right-down and right-up edges correspond to the black and white dots of the eigenvalues, respectively.

from the non-perturbative instanton correction is given by the residual part of the partition function divided by the ground state contribution. First of all, we would like to consider the contribution to the free energy (prepotential), but it is useful to introduce the following difference operator ∆gs f (x) ≡ f (x + gs ) − f (x),

(14)

since the eigenvalues take the discrete integer numbers. Using this difference operator, we can write the Vandermonde determinant part in the effective action (13) as Z Z 2 2 dxdyγ(x − y|gs )∆gs ρ(x)∆gs ρ(y), (15) N − dxdy log |x − y|ρ(x)ρ(y) = N where γ(x) satisfies ∆gs ∆−gs γ(x|gs ) = log x

(16)

The solution to this difference equation is very important later and a solution is given by γ(x|gs ) = − log Γ2 (x|gs ), (17) where Γ2 (x|gs ) is the so-called Barnes’ double gamma function. The double gamma function is an extension of the ordinary gamma function, but it also has an asymptotic expansion like the Stirling’s formula

Instanton Counting, 2D YM Theory and Topological Strings

γ(x|gs ) =

1 gs2

247

∞ g 2g−2 X B2g 1 1 2 3 s . (18) x log x − x2 − log x+ 2 4 12 2g(2g − 2) x g=2

As I mentioned before, this function relates to the perturbative part of the 4 dimensional supersymmetric gauge theory. At the same time, the asymptotic expansion (18) can be regarded as a genus expansion of closed string theory since powers of gs are proportional to the Euler number of closed Riemann surfaces. In fact, the function γ(x|gs ) is closely related to amplitudes of the topological B-model or c=1 non-critical string theory at self-dual radius. Originally, we have been considering the open string theory as gauge theory on the D-branes, but finally we have the closed string amplitude. So this model is a realization of the open/closed string duality. Now let us consider the large N limit after the above preparations. Here we assume that the eigenvalues gather around the r critical points of the potential. In particular, the ground state configuration is the situation where the eigenvalues in each bunch are closely P packed, and the distance between two Fermi surfaces is gs Ni , where Ni ( i Ni = N ) is the number of eigenvalues in each bunch and a sufficiently large number. So, if we consider the large N limit (the large Ni limit at the same time), the two Fermi surfaces in each bunch are separated far away and we can ignore effects between them. In addition, if we tune the coefficients of the potential expanded in terms of the order of Φ, the partition function of the discrete matrix model is decomposed into two parts coming from each Fermi surface (chiral decomposition) ZDMM ' |ZNekrasov (al ; gs )|2 ,

(19)

where ZNekrasov (al ; gs ) is Nekrasov’s partition function which appeared in (2), the Ω-background parameter is identified with the string coupling constant gs , and the moduli parameters of vacua al are determined by the positions of the critical points of the potential. The perturbative part in Nekrasov’s partition function Zpert (al ; gs ) is represented by the renormalized part from the ground state P Y Zpert (al ; gs ) = e l6=n γ(al −an ) = Γ2−1 (al − an |gs ), (20) l6=n

using the double gamma function. To see the contribution to the perturbative part of the prepotential, we should use the asymptotic expansion (18) and especially the term proportional to 1/gs2 is the contribution to the gauge theory without the graviphoton corrections. The residual terms in the gs expansion are regarded as the contribution from the graviphoton, the same as in Nekrasov’s formula. It is interesting that the perturbative calculation in gauge theory can be expressed in terms of the asymptotic expansion of the extended double gamma function.

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5 Topological M-theory and Non-critical M-theory In this section, we would like to extend the previous discussions. It is basically an extension of the calculation of the prepotential in the 4 dimensional gauge theory to 5 dimensional theory compactified on S 1 . There will appear some interesting structures of topological or non-critical M-theory, which include all non-perturbative corrections of the topological or non-critical string theory, behind the extension. To extend the discrete matrix model partition function, we need ideas of “T-duality” in string theory and matrix model. Here the T-duality will apply to the Vandermonde determinant part in the (discrete) matrix model, but recalling that the Vandermonde determinant originally comes from the path integral measure of the adjoint scalar field (and its adjoint action), the T-duality means replacing the adjoint action with a covariant derivative. We are now considering that the T-dual direction is compactified on S 1 . So there exists the effect from infinitely many Kaluza-Klein modes. If we sum up all Kaluza-Klein modes in the Vandermonde determinant, we obtain the T-dualized measure ∞ Y Y

Y n (i + λi − λj )2 ' β i<j n=−∞ i<j

1 sinh β(λi − λj ) β

2 .

(21)

This measure is trigonometrically extended and equivalent with a measure of the unitary matrix model. In the discrete matrix model formulation, we have to discretize the above measure with a suitable potential. So we obtain the partition function. # " 2 Y1 X gs µA X 2 ni . Zq-DMM = sinh β(gs ni − gs nj ) exp − β 2 n >n >···>n i<j i 1

2

N

(22) We can understand this model as a q-deformation of the Vandermonde measure, which corresponds to the q-deformed dimension of the representation R. So it is called the q-deformed 2 dimensional Yang-Mills theory [AOSV05]. It is also known that the continuum limit of the q-deformed discrete matrix model is equivalent to the partition function of 3 dimensional (bosonic) Chern-Simons theory on S 3 [AKMV04]. We can follow the same analysis as the discrete matrix model, but the different point is the difference equation (16), which is now trigonometrically extended or q-deformed ∆gs ∆−gs γ˜(x|gs ; β) = log sinh βx.

(23)

Solution to this trigonometric difference equation is also expressed by using an extended gamma function. Here we need the triple sine function, which is made by gluing the triple gamma functions together. The gluing is an analog of the fact that the ordinary sine function can be expressed in terms of the product

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249

of the gamma function and its reflection such as sin πx = πΓ −1 (x)Γ −1 (1 − x). If we use the triple sine function S3 (x|gs ; β), the above γ˜ (x|gs ; β) is written by its logarithm as γ˜ (x|gs ; β) = log S3 (x|gs ; β) Z ∞ iπ dt e−xt = B3,3 (x|gs ; β) − , − iπ 2 gs t 6 β t) −∞ t 4 sinh (1 − e 2

(24)

where B3,3 (x|gs ; β) is a 3rd order Bernoulli polynomial in x. Similar to the prepotential in the 4 dimensional gauge theory, which is expressed in terms of the double gamma function, we can express the perturbative part of the prepotential in the 5 dimensional supersymmetric gauge theory on S 1 using the triple sine function. Actually, the 3rd order polynomial in (24) is a characteristic of the 5 dimensional prepotential and the residues from the integral part give non-perturbative corrections proportional to e−2βnx (n ∈ Z). In addition, the function γ˜ (x|gs ; β) has a genus expansion in gs the same as the former case. This expansion now relates to the topological A-model amplitude. If we recall that γ(x|gs ) represents the topological B-model amplitude, the two models are related by the T-duality as expected. The discussion above is sufficient to see the relation between the ordinary topological string amplitudes, but we can find a more interesting feature if we examine the properties of the triple sine function. These are the residues in the integral of (24). If we evaluate more carefully the residues in the integral, we 2πix notice that they include terms proportional to e− gs n very non-trivially. The 2πix terms proportional to e− gs n vanish if we act with the difference operator. So these terms do not affect the difference equation (23) which we would like to 2πix consider first. However, the terms proportional to e− gs n naturally appear in the triple sine function and can be regarded as the non-perturbative effects in the topological string theory. So far, there is no proof that all non-perturbative corrections reduce to the triple sine function, but it strongly suggests the existence of the topological M-theory which gives all non-perturbative corrections from the point of view of the duality in string theory, since the radius β of S 1 and string coupling constant gs are symmetrically included in the triple sine function. On the other hand, the function γ(x|gs ) or double gamma function, which appears in the perturbative part of the 4 dimensional N =2 supersymmetric gauge theory as we pointed out in the previous section, are closely related to the c=1 non-critical string amplitude. The target space of the c=1 noncritical string is 2 dimensional, but the above uplift by the duality relation also suggests the existence of the non-critical M-theory [AK05, HK05] defined on higher 3 dimensional target space. There may appear q-deformed 2 dimensional Yang-Mills theory as a natural extension of the gamma function amplitudes. Finally, I would like to comment on connections to the non-perturbative formulation of M(atrix) theory by Banks, Fischler, Shenker and Susskind

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Kazutoshi Ohta

(BFSS) [BFSS97] and type IIB string theory by Ishibashi, Kawai, Kitazawa and Tsuchiya (IKKT) [IKKT97]. The original BFSS and IKKT matrix model are considered to represent the 11 dimensional M-theory and 10 dimensional critical type IIB string theory, respectively. Now let us consider a non-perturbative formulation of the 7 dimensional topological M-theory or 6 dimensional topological B-model, where the number of matrices is reduced from the original matrix model formulation. Namely, we consider a partition function of a matrix quantum mechanics for five time-dependent k×k Hermite matrices Z R − 1 β dtTr( 12 Dt Y i Dt Yi + 14 [Y i ,Y j ]2 +··· ) (i, j = 1, . . . , 5), Z˜k = DY DΨ e g2 0

(25)

or a 0 dimensional matrix model with six k×k Hermite matrices Z ¯ Γ µ [Xµ ,Ψ ]) − 1 Tr 1 [X ,X ]2 + 1 Ψ Zk = DXDΨ e g2 ( 4 µ ν 2 (µ, ν = 1, . . . , 6).

(26)

Then the partition functions are exactly evaluated since the theories we are considering are topologically twisted, and we find that the partition functions of the discrete matrix model ZDMM or its q-deformation Zq-DMM are generating functions of the above size k partition function in the large N limit. Therefore, roughly speaking, the partition functions are related as X Z˜k q k |2 , (27) Zq-DMM ' | k

ZDMM ' |

X

Z k q k |2 ,

(28)

k

after the chiral decomposition in the large N limit3 . So the matrix model (25) or (26) actually describes the closed topological strings, since the partition functions give the amplitudes of the topological A/B-models on Calabi-Yau manifold as we have seen before. This is a non-trivial realization of the philosophy in [BFSS97] and [IKKT97]. Note also that the target space of the original matrix model is drastically changed to a different closed string target space. To summarize, if we truncate full string/M-theory to 7 or 6 dimensional topological string theory, or to 3 or 2 non-critical string theory, these are very nice models to show exactly the large N transition, open/closed string duality, counting BPS states, exact duality including all non-perturbative correction, and so on, since these models possess crucial integrable structures. So we can expect to understand essential properties of the whole string/M-theory if we investigate in detail these integrable models, which pick up good properties from the original string/M-theory. 3

To put it more precisely, we need to introduce extra matrices corresponding to the hypermuletiplets in order to control the positions of the eigenvalues of the matrices and obtain the moduli parameter dependence.

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References [AK05]

Sergei Yu. Alexandrov and Ivan K. Kostov. Time-dependent backgrounds of 2d string theory: Non-perturbative effects. JHEP, 02:023, 2005. [AKMV04] Mina Aganagic, Albrecht Klemm, Marcos Marino, and Cumrun Vafa. Matrix model as a mirror of Chern-Simons theory. JHEP, 02:010, 2004. [AOSV05] Mina Aganagic, Hirosi Ooguri, Natalia Saulina, and Cumrun Vafa. Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys., B715:304–348, 2005. [BFSS97] Tom Banks, W. Fischler, S. H. Shenker, and Leonard Susskind. M theory as a matrix model: A conjecture. Phys. Rev., D55:5112–5128, 1997. [BT93] Matthias Blau and George Thompson. Lectures on 2-d gauge theories: Topological aspects and path integral techniques. Trieste HEP Cosmol., 0175:244, 1993. [BVS96] M. Bershadsky, C. Vafa, and V. Sadov. D-branes and topological field theories. Nucl. Phys., B463:420–434, 1996. [Con94] A. Connes. Noncommutative Geometry. Academic Press, 1994. [DV02] Robbert Dijkgraaf and Cumrun Vafa. Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys., B644:3–20, 2002. [EK82] Tohru Eguchi and Hikaru Kawai. Reduction of dynamical degrees of freedom in the large n gauge theory. Phys. Rev. Lett., 48:1063, 1982. [GV98] Rajesh Gopakumar and Cumrun Vafa. Topological gravity as large N topological gauge theory. Adv. Theor. Math. Phys., 2:413–442, 1998. [HK05] Petr Horava and Cynthia A. Keeler. Noncritical M-theory in 2+1 dimensions as a nonrelativistic fermi liquid. hep-th/0508024. [IKKT97] N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya. A large-N reduced model as superstring. Nucl. Phys., B498:467–491, 1997. [INOV03] Amer Iqbal, Nikita Nekrasov, Andrei Okounkov, and Cumrun Vafa. Quantum foam and topological strings. hep-th/0312022. [Mig75] A. A. Migdal. Recursion equations in gauge field theories. Sov. Phys. JETP, 42:413, 1975. [MMO05] Toshihiro Matsuo, So Matsuura, and Kazutoshi Ohta. Large N limit of 2d Yang-Mills theory and instanton counting. JHEP, 03:027, 2005. [MO06] So Matsuura and Kazutoshi Ohta. Localization on the D-brane, twodimensional gauge theory and matrix models. Phys. Rev., D73:046006, 2006. [Nek02] Nikita A. Nekrasov. Seiberg-Witten prepotential from instanton counting. hep-th/0206161. [NO03] Nikita Nekrasov and Andrei Okounkov. Seiberg-Witten theory and random partitions. hep-th/0306238. [ORV03] Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa. Quantum Calabi-Yau and classical crystals. hep-th/0309208 [Wit92] Edward Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9:303–368, 1992.

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Instantons in Non(anti)commutative Gauge Theory via Deformed ADHM Construction Takeo Araki, Tatsuhiko Takashima and Satoshi Watamura Department of Physics Graduate School of Science Tohoku University Aoba-ku, Sendai 980-8578, Japan

Abstract We generalize the differential algebra on superspace to non(anti)commutative superspace by defining the deformed wedge product. Then, we formulate a non(anti)commutative version of the super ADHM construction which gives deformed instantons in N = 1/2 super Yang-Mills theory with U(n) gauge group.

1 Introduction It has been found that supersymmetric gauge theory defined on a kind of deformed superspace, called non(anti)commutative superspace, arises in superstring theory as a low energy effective theory on D-branes in the presence of constant graviphoton field strength [1]-[3]. In non(anti)commutative space, anticommutators of Grassmann coordinates become non-vanishing. Such a deformation of (Euclidean) four dimensional N = 1 super Yang-Mills (SYM) theory has been formulated by Seiberg [2] and it is sometimes called N = 1/2 SYM theory. It was argued by Imaanpur [4] that the anti-self-dual (ASD) instanton equations should be modified in the N = 1/2 SYM theory with self-dual (SD) non(anti)-commutativity. Solutions to those equations (deformed ASD instantons) have been studied by many authors [4]-[6]. It is well known that in the ordinary theory the instanton configurations of the gauge field can be obtained by the ADHM construction [7]. The authors of ref. [6] have studied string amplitudes in the presence of D(−1)-D3 branes with the background RR field strength and derived constraint equations for the string modes ending on D(−1)-branes, which are the ADHM constraints for the deformed ASD instantons. We show that we can obtain these constraints in the purely field 253

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Takeo Araki,

Tatsuhiko Takashima

and Satoshi Watamura

theoretic context, formulating a non(anti)commutative version of a superfield extension of the ADHM construction initiated by Semikhatov and Volovich [8]. We follow the notation and conventions in refs. [9, 10].

2 N = 1/2 SYM theory We briefly describe the non(anti)commutative deformation of N = 1 superspace and N = 1/2 SYM theory [2]. The non(anti)commutative deformation of N = 1 superspace is given by introducing non(anti)commutativity of the product of N = 1 superfields. This deformation is realized by the following Moyal type star product: f ∗ g = f exp(P )g,

1 ←− −→ P = − Qα C αβ Qβ , 2

(1)

where f and g are N = 1 superfields and Qα is the (chiral) supersymmetry generator. C αβ is the non-anticommutativity parameter and is symmetric: C αβ = C βα . The above star product gives the following relations among the chiral coordinates (y µ , θα , θ¯α˙ ): {θα , θβ }∗ = C αβ , [y µ , · ]∗ = 0, [θ¯α˙ , · }∗ = 0. Turning on such a deformation, the original action formulated in the N = 1 superfield formalism is deformed by the star product. The deformed N = 1 SYM theory has N = 1/2 supersymmetry, so that they are called N = 1/2 SYM theory. The action of N = 1/2 SYM theory is given by Z Z Z 1 4 2 α 2¯ ¯ α˙ ¯ S= (2) d x d θtrW ∗ W + d θtr W ∗ W α α˙ 16N g 2 where 1 ¯ ¯ α˙ −V Wα = − D e∗ ∗ Dα eV∗ , α˙ D 4

¯ α˙ = 1 Dα Dα eV∗ ∗ D ¯ α˙ e−V , W ∗ 4

n

(3)

z }| { ≡ and V ∗ · · · ∗ V . Here V = V a T a with V a the vector superfields and T the hermitian generators which are normalized as tr[T a T b ] = N δ ab . We may redefine the component fields of V in the WZ gauge such that the component gauge transformation becomes canonical (the same as the undeformed case). In [2], such a field redefinition is found and then the component action becomes Z h 1 1 4 ¯ σ µ Dµ λ + 1 D 2 S= tr d x − v µν vµν − iλ¯ 4N g 2 4 2 1 2 ¯ ¯ 2i i µν ¯ ¯ (4) − C vµν λλ + |C| (λλ) , 2 8 eV∗ a

P

1 n n!

where C µν ≡ C αβ (σ µν )α γ εβγ and |C|2 ≡ C µν Cµν .

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255

From the component action, we can see that the equations for SD instantons are unchanged compared to the undeformed case. Therefore, the SD instanton solutions are not affected by the deformation. On the other hand, the equations for ASD instantons should be modified. The action can be rewritten as [4] Z i h 1 2 1 i 1 2 1 µν SD µ 4 ¯ ¯ ¯ S= v + −i λ¯ σ D λ+ d x − λ λ tr C D + v v ˜ µ µν µν , (5) 4N g 2 2 µν 2 2 4 where v˜µν ≡ 12 εµνρσ vρσ . From this expression, we can see that configurations which satisfies the equations of motion and is connected to the ASD instantons when turning off the deformation are the solutions to the following deformed ASD instanton equations [4]: i SD ¯λ ¯ = 0, λ = 0, Dµ σ µ λ ¯ = 0, D = 0. vµν + Cµν λ 2

(6)

3 Differential algebra in the deformed superspace We take a geometrical approach to formulate the deformed super ADHM construction by generalizing the differential algebra: we extend the star product between superfields to the differential forms in superspace. The principle of our construction of the deformed differential algebra is that the operators Qα appearing in the star product are identified with the generators of supertranslation. As a result, the star product of differential forms is defined according to the representations of supersymmetry they belong to. Since the 1-form bases eA are supertranslation invariant, the action of Qα on eA is naturally defined as Qα (eA ) = 0. Then for a 1-form ω = eA ωA , it holds that Qα (ω) = (−)|A| eA Qα (ωA ). Using this action of Qα , we define the deformed wedge product of a p-form ωp and a q-form ωq as ∗ 1 ←− αβ −→ ωp ∧ ωq ≡ ωp ∧ exp − Qα C Qβ ωq , (7) 2 ← − → − where Q ( Q ) acts on ωp (ωq ) from the right (left) and the normal wedge product is taken for the resulting (transformed) differential forms. Note that ←− ω Qα = (−)|ω| Qα (ω). Hereafter we will suppress the wedge symbols. In the eA -basis, the product of the p- and q-form is simply given by the star product of the coefficients: ωp ∗ ωq = (−)(|A1 |+···+|Aq |)(|B1 |+···+|Bq |) eA1 · · · eAp eB1 · · · eBq ×(ωpAp ...A1 ∗ ωq Bq ...B1 ),

(8)

The exterior derivative d is defined as a map from a p-form to a p + 1-form by using the basis eA :

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Takeo Araki,

Tatsuhiko Takashima

and Satoshi Watamura

dωp = eA1 · · · eAp eB DB ωpAp ...A1 p X + (−1)|Ar+1 |+···+|Ap | eA1 · · · deAr · · · eAp ωpAp ...A1

(9)

r−1

where deA is the same as the undeformed one. Let us mention about the relation of the above approach to the Hopf algebra approach taken in ref.[11, 12, 13]. As we described, the principle of the extention of the star product to the differential algebra is to identify the differential operator QA as the generator of the supersymmetry. This identification then defines the twist [14] of the supersymmetry algebra, which is a Hopf algebra including Poincare algebra. On the other hand the transformation of 1-forms is defined naturally by the action of QA as a Lie derivative and the action on the general forms follows by requireing the graded Leibniz rule, i.e., being the (graded) module algebra. This also leads to the twisted wedge product defined in eq.(7). We see that the deformed differential algebra defined above is consistent with the N = 1/2 SYM theory described in the previous section, in the sense that the curvature 2-from superfield will correctly reproduce the field strength ¯ α˙ in (3) (after imposing appropriate constraints as in the superfield Wα and W undeformed case [15]) based on the deformed differential algebra. Given a connection 1-form superfield φ, the curvature superfields FAB are obtained as the coefficient functions of the 2-form superfield F constructed in a standard way: F = dφ + φ ∗ φ. Therefore, we find the curvature superfields FAB as FAB = DA φB − (−)|A||B| DB φA − [φA , φB }∗ + TAB C φC ,

(10)

where TAB C is the torsion defined by deC = 21 eA eB TBA C with non-vanishing µ elements given by Tαβ˙ µ = Tβα = 2iσαµβ˙ . The proper constraints for the ˙ curvature superfields to give the N = 1/2 SYM theory turn out to be Fαβ = 0, Fα˙ β˙ = 0, Fαβ˙ = 0,

(11)

where the curvature superfields are given by (10) (see [15] for the undeformed case). We refer to these constraints as the Yang-Mills constraints. These constraints are solved in a parallel way to the undeformed case and the invariant action with respect to super- and gauge symmetry can be constructed which coincides with the action S given in (2). Therefore, imposing the Yang-Mills constraints (11), the N = 1/2 SYM theory can be correctly reproduced in a geometrical way based on the deformed differential algebra.

4 Review of the N = 1 super ADHM construction Before describing the deformed version, we briefly review the N = 1 super ADHM construction which was initiated by Semikhatov and Volovich [8]. Here we follow ref. [9].

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The U(n) (or SU(n)) k instanton configurations can be given by the ADHM construction [7]. Define ∆α (x) such as ∆α (x) = aα + xαα˙ bα˙

(12)

where aα and bα˙ are constant k × (n + 2k) matrices and xαα˙ ≡ ixµ σαµα˙ . We assume that ∆α has maximal rank everywhere except for a finite set of points. ˙ Its hermitian conjugate ∆†α ≡ (∆α )† is given by ∆†α (x) = a†α + b†β˙ xβα . Then the gauge field vµ is given by vµ = −2iv † ∂µ v, where v is the set of the normalized zero modes of ∆α : ∆α v = 0, v † v = 1n . For later use we define f as the inverse matrix of the quantity f −1 ≡ 21 ∆α ∆†α . In the superfield formalism, the ASD super instanton equations are equivalent to the following super ASD condition [8]: Fµα˙ = 0,

?Fµν = −Fµν .

(13)

Note that the latter equation follows from the former as long as the 2-form F satisfies the Bianchi identities and the (undeformed) Yang-Mills constraints. The super ADHM construction gives the solutions to (13) [9]. Define a superfield extension of ∆α (x): ˆα = ∆α (y) + θα M, ∆

(14)

where ∆α (y) is the zero dimensional Dirac operator in the ordinary ADHM construction with replacing xµ by the chiral coordinate y µ = xµ + iθσ µ θ¯ and M is a k × (n + 2k) fermionic matrix which includes the fermionic moduli. We suppose that ∆ˆα has a maximal rank almost everywhere as in the ordinary ADHM construction. Its ‡-conjugate [9] ∆ˆ‡α is found to be ∆ˆ‡α = ∆†α (y) + θα M† . As ∆ˆα has n zero modes we collect them in a matrix superfield vˆ[n+2k]×[n] and require that vˆ satisfies the normalization condition: ∆ˆα vˆ = 0, vˆ‡ vˆ = 1. (Its ‡-conjugate vˆ‡ satisfies vˆ‡ ∆ˆ‡α = 0.) Then the connection 1-form superfield φ is given by φ = −ˆ v ‡ dˆ v.

(15)

where d is exterior derivative of superspace. The connection φ defines the curvature ˆ α β d∆ˆβ vˆ, (16) F = dφ + φφ = vˆ‡ d∆ˆ‡α K ˆ βγ = ˆ α β is defined such that K ˆ −1 α β K ˆ −1 α β ≡ ∆ˆα ∆ˆ‡β and K where K ˆ −1 β γ = δ γ 1k . The curvature superfield Fµν becomes ASD if K ˆ satˆ αβ K K α ‡β β ˆ ˆ isfies ∆α ∆ ∝ δα and thus ˆ −1 α β = δαβ fˆ−1 K

(17)

where fˆ−1 ≡ 21 ∆ˆα ∆ˆ‡α is a k × k matrix superfield. There exists fˆ because ˆα has maximal rank. The above condition (17) leads we have assumed that ∆

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and Satoshi Watamura

to both the bosonic and fermionic ADHM constraints. When eq. (17) holds, i.e., the parameters in ∆ˆα are satisfying both bosonic and fermionic ADHM constraints, we can check that the above F satisfies the Yang-Mills constraints and the super ASD condition. To ensure the WZ gauge of the superfields obtained by the super ADHM ¯ α˙ vˆ = 0 and construction, we impose on the zero mode vˆ of ∆ˆα the conditions D vˆ‡ ∂θ∂α vˆ = 0. Then vˆ is determined as vˆ = v + θ γ (∆†γ f Mv) + θθ( 21 M† f Mv), and the connection φµ in (15) correctly gives the super instanton configuration in the WZ gauge: Its lowest component is the instanton gauge field and the θ-component is the fermion zero mode.

5 Deformed super ADHM construction The deformed super ASD condition is found to be of the same form as the super ASD condition (13) but the product replaced with the star product (1): Fµα˙ = 0,

?Fµν = −Fµν ,

(18)

where the curvature superfields FAB are given by eq.(10). We can prove the equivalence of the condition (18) and the deformed equations (6). One would expect that solutions to eq. (18) can be constructed by the super ADHM construction, replacing each product with the star product (1). For the deformed super ASD instantons, φµ in the WZ gauge becomes ¯ (y). This leads us to adopt ∆ˆα in our super ADHM φµ = − 2i vµ + iθσµ λ construction with the same form as before: ˆα = ∆α (y) + θα M. ∆

(19)

Then, according to the ‡-conjugation rules, we have ∆ˆ‡α = ∆‡α (y) + θα M‡ . We collect the n zero modes of ∆ˆ into a matrix form u ˆ[n+2k]×[n] and require ˆα ∗ u it to be normalized with respect to the star product: ∆ ˆ = 0, u ˆ‡ ∗ u ˆ = 1n . ˆ ∗α β (α, β = 1, 2) as the “inverse” matrices of K ˆ −1 α β ≡ Define k×k matrices K ∗ ˆ‡β such that K ˆ −1 α β ∗ K ˆ ∗β γ = K ˆ ∗α β ∗ K ˆ −1 β γ = δ γ 1k . Then we have a ∆ˆα ∗ ∆ ∗ ∗ α ˆ‡α ∗ K ˆ ∗α β ∗ ∆ˆβ . relation u ˆ∗u ˆ‡ = 1n+2k − ∆ With the use of the zero modes u ˆ of ∆ˆα , the connection φ is given by ‡ φ = −ˆ u ∗ dˆ u, and the curvature 2-form becomes ˆ ∗α β ∗ d∆ˆβ ∗ u F = dφ + φ ∗ φ = u ˆ‡ ∗ d∆ˆ‡α ∗ K ˆ

(20)

ˆ ∗α β ∗ DB} ∆ˆβ ∗ u which reads FAB = −ˆ u‡ ∗ D[A ∆ˆ‡α ∗ K ˆ, especially Fµν = † αα ‡ ˙ ˆ β β˙ ˆ∗ u ˆ ∗ bα˙ σ ¯[µ K∗α σν] β β˙ b ∗ u ˆ. Thus Fµν becomes ASD (see eq. (18)) if K commutes with the sigma matrices σµ : ˆα ∗ ∆ ˆ‡β = K ˆ −1 α β ∝ δ β . ∆ ∗ α

(21)

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259

ˆα is a Then we immediately find that Fα˙ β˙ = Fαβ˙ = 0 and Fµα˙ = 0, because ∆ chiral superfield. We can also check that Fαβ = 0 with the use of the constraint ˆα = εαβ (M + 4θ¯ ˙ bβ˙ ) and Dβ ∆ˆ‡α = δ α (M‡ + 4b† θ¯β˙ ), (21), the relations Dβ ∆ β β β˙ and the fact that Fαβ is symmetric with respect to α and β. Therefore, we have shown that the above described super ADHM construction gives curvature superfields that satisfy the Yang-Mills constraints (11) and the ASD conditions (18) if the condition (21) is imposed. ˆα ∗ ∆ ˆ‡β = ∆ˆα ∆ˆ‡β − 1 εαγ C γβ MM‡ , the requirement Since we can write ∆ 2 (21) leads to the following deformed bosonic ADHM constraint ∆α ∆‡β − 1 γβ MM‡ ∝ δαβ and the fermionic ADHM constraint ∆α M‡ + M∆‡ α = 2 εαγ C 0. These constraints agree with those in [6] obtained by considering string amplitudes. We can rewrite the deformed bosonic ADHM constraints in another form as follows. Let us denote ‡ ∆1 J[k]×[n] z¯2 1k + B2 ‡[k]×[k] z¯1 1k + B1 ‡[k]×[k] = , (22) ∆2 I[k]×[n] −z1 1k − B1[k]×[k] z2 1k + B2[k]×[k] where z1 ≡ y21˙ , z2 ≡ y22˙ and I ≡ ω2 , J ‡ ≡ ω1 , B1 ≡ a021˙ , B2 ≡ a022˙ . Then the bosonic ADHM constraints reads II ‡ − J ‡ J + [B1 , B1 ‡ ] + [B2 , B2 ‡ ] − C 12 MM‡ = 0, (23) 1 (24) IJ + [B2 , B1 ] − C 11 MM‡ = 0. 2 We can give an expression in terms of the ADHM data ∆α and M, of the general solution in the WZ gauge obtained by our construction, and have shown in [10] that it gives the known U(2) one instanton solution. In summary, we have correctly deformed the super ADHM construction to give solutions to the deformed ASD instantons in N = 1/2 SYM theory. We see that deformation terms emerge in the bosonic ADHM constraints (see also [6]), which are comparable with the U(1) terms due to space-space noncommutativity [16]. Our formulation reveals the geometrical meaning of those deformation terms as non(anti)commutativity of superspace. However, it needs a further study to clarify how those terms can be interpreted in the hyper-K¨ ahler quotient construction [17]. Acknowledgments: This research is partly supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 13640256, 13135202 and 14046201).

References 1. H. Ooguri and C. Vafa, Adv. Theor. Math. Phys. 7 (2003) 53, hep-th/0302109; Adv. Theor. Math. Phys. 7 (2004) 405, hep-th/0303063; J. de Boer, P. A. Grassi and P. van Nieuwenhuizen, hep-th/0302078.

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and Satoshi Watamura

2. N. Seiberg, JHEP 0306 (2003) 010, hep-th/0305248. 3. N. Berkovits and N. Seiberg, JHEP 0307 (2003) 010, hep-th/0306226. 4. A. Imaanpur, JHEP 0309 (2003) 077, hep-th/0308171; JHEP 0312 (2003) 009, hep-th/0311137. 5. P. A. Grassi, R. Ricci and D. Robles-Llana, JHEP 0407 (2004) 065, hepth/0311155; R. Britto, B. Feng, O. Lunin and S. J. Rey, Phys. Rev. D 69, 126004 (2004), hep-th/0311275; S. Giombi, R. Ricci, D. Robles-Llana and D. Trancanelli, JHEP 0510 (2005) 021, hep-th/0505077. 6. M. Billo, M. Frau, I. Pesando and A. Lerda, JHEP 0405 (2004) 023, hepth/0402160. 7. M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, Phys. Lett. A 65 (1978) 185. 8. A. M. Semikhatov, JETP Lett. 35 (1982) 560 [Pisma Zh. Eksp. Teor. Fiz. 35 (1982) 452]; Phys. Lett. B 120 (1983) 171; I. V. Volovich, Phys. Lett. B 123 (1983) 329; Theor. Math. Phys. 54 (1983) 55 [Teor. Mat. Fiz. 54 (1983) 89]. 9. T. Araki, T. Takashima and S. Watamura, JHEP 0508 (2005) 065, hepth/0506112. 10. T. Araki, T. Takashima and S. Watamura, JHEP 0512 (2005) 044, hepth/0510088. 11. M. Chaichan, P. Kulish, K. Nishijima, and A. Tureanu, Phys. Lett. B604 (2004) 98. 12. P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp and J. Wess, Class. Quant. Grav. 22 (2005) 3511. 13. Y. Kobayashi, Article in this book and references therein. 14. V. G. Drinfeld, Leningrad Math. J. 1 (1990) 1419;Alg. Anal. 1 N 6 (1989)114. 15. R. Grimm, M. Sohnius and J. Wess, Nucl. Phys. B 133 (1978) 275. 16. N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198 (1998) 689, hepth/980206. 17. N. J. Hitchin, A. Karlhede, U. Lindstr¨ om and M. Roˇcek, Commun. Math. Phys. 108 (1987) 535.

Noncommutative Deformation and Drinfel’d Twisted Symmetry Yoshishige Kobayashi Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan.

Abstract The Hopf algebraic construction of the non(anti)commutative superspace is discussed. In that context, Noncommutative (super)space arises as a representation of the modified Hopf algebra by the Drinfel’d twist. This construction gives the same result which is given by using the ordinary Moyal product, but has some advantages in considering the symmetry of the noncommutative theory. We construct the twisted super-Poincar´e algebra, and obtain the several non(anti)commutative superspaces.

1 Introduction Noncommutative geometry has become an important subject in theoretical physics, ever since its realization was found in superstring theory[1]. Spacetime noncommutativity is deeply related to the structure of spacetime, and it may appear as an evidence of the quantized spacetime, or as a result of the quantum correction of a gravitational theory. We expect that the well-known theories, e.g. the standard model, to be modified on noncommutative spacetime. However, if we consider such theories, we often lose some symmetries, because the noncommutativity parameters are introduced into the theory by hand. In the majority of cases the noncommutativity parameters are dimensionful, and they often break the symmetry of the original (commutative) theory. Lorentz symmetry is rather important in particle physics. This symmetry is an experimental fact, and no definite evidence of its violation has been found from experimental tests or observations. The trouble is that certain classes of spacetime noncommutativity break Lorentz symmetry. Let us consider the following commutator relation between spacetime coordinates, 261

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Yoshishige Kobayashi

[xµ , xν ] = iΘµν .

(1)

Here Θµν is a constant with antisymmetric indices. It is the most common noncommutativity which is extensively studied. This relation breaks Lorentz covariance of the theory, because no constant tensor is transformed in the Lorentz covariant way, except for the (Minkowski) metric and the Levi-Civita tensor εµνρσ . Usually it is explained by the argument that Θ µν is so small compared to our scale, that we cannot observe it. Therefore there is no inconsistency phenomenologically. But the explanation is not satisfactory theoretically, because an ordinary theory is built closely on the symmetry. One proposal for the problem is suggested[2]. In [2], the authors argued that Hopf algebra is a suitable framework for a description of noncommutative theory. In that context, we can think of a modified Lorentz symmetry, namely the twisted Lorentz symmetry, which is maintained on noncommutative space, even though the original Lorentz symmetry is broken indeed by the relation (1). On the other hand, another noncommutative geometry was obtained from superstring theory, that is the non-anticommutative superspace[3]. Superspace is a coordinate system which includes not only the ordinary spacetime coordinate but the anti-commutative, i.e. Grassmann number, coordinate. Supersymmetry, which is the symmetry between bosons and fermions, is naturally represented on superspace. It was revealed that in the graviphoton background, with taking a certain limit, the theory is described effectively by changing the anti-commutator relation, namely {θα , θβ } = C αβ 6= 0,

(2)

where θα is fermionic coordinates of superspace, α and β are spinor indices, and C αβ is a constant with symmetric indices. Like the spacetime noncommutativity, symmetry breaking occurs on noncommutative superspace. It is known that non-anticommutativity (2) breaks supersymmetry by half of the numbers of the degrees[3]. Our aim is to apply the Hopf algebraic method to a theory on superspace. Although superspace is different from ordinary spacetime owing to the anticommutative property, the Hopf algebra method can be applied with the applicable definitions. In this formulation, we can maintain the modified supersymmetry, namely the twisted supersymmetry. The next section is a very brief introduction to Hopf algebra and its twist. In section 3, following [2], we construct the twisted Poincar´e algebra. The twist operation changes the product of the representation space, consequently the spacetime coordinate becomes noncommutative. In section 4, we consider the twisted super-Poincar´e algebra on superspace. As the representations of the twisted super-Poincar´e symmetry, several types of non(anti)commutative superspaces are obtained.

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2 Hopf algebra In this section, we review Hopf algebra and the Drinfel’d twist, to construct a modified symmetry algebra. For more details on Hopf algebra and the related subject, see references[4, 5, 6]. A Hopf algebra H(H, m, i, ∆, , γ; K) is a linear space H over K together with the five linear maps on the tensor product of H, namely m,i,∆, and γ. K is some field or ring. These maps, the product m : H ⊗ H → H, the unit i : K → H, the coproduct ∆ : H → H ⊗ H, the counit : H → K and the antipode γ : H → H, satisfy the following conditions. m ◦ (id ⊗ m) = m ◦ (m ⊗ id), m ◦ (i ⊗ id) = id = m ◦ (id ⊗ i),

(3) (4)

(id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆, (id ⊗ ) ◦ ∆ = id = ( ⊗ id) ◦ ∆,

(5) (6)

m ◦ (γ ⊗ id) ◦ ∆ = i ⊗ = m ◦ (id ⊗ γ) ◦ ∆.

(7)

Here id stands for the identity map of H. Next, we make a stage on which the symmetry algebra acts. A H-module, or a representation space, V is a vector space over K, on which the Hopf algebra H acts as an endomorphism of V , H : V → V . The action of h ∈ H on v ∈ V , h : a → a0 , denoted by h . a = a0 , satisfies g . (h . v) = (gh) . v,

(8)

for ∀v ∈ V ,∀g, h ∈ H. We consider the case that V is an algebra. To be compatible with the Hopf algebra H, we require the relations, h . (v · w) = h . m(v ⊗ w) = m ◦ ∆(h) . (v ⊗ w) = m(h(1) . v ⊗ h(2) . w).

(9)

In the last line we use Sweedler’s notation, X (i) (i) ∆(h) = h1 ⊗ h2 = h(1) ⊗ h(2) ,

(10)

i

The Drinfel’d twist is a systematic procedure to make another Hopf algebra from a given Hopf algebra by Drinfel’d[7]. Consider an invertible biproduct element F ∈ H ⊗H, here we call it a twist element. Using F, the new coproduct and antipode are defined as follows. ∆t (h) = F∆(h)F −1 , γt (h) = U γ(h)U

−1

.

The twist element should satisfy the following two relations,

(11) (12)

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F12 (∆ ⊗ id)F = F23 (id ⊗ ∆)F, ˆ = (id ⊗ )F. ( ⊗ id)F = 1

(13) (14)

The symbol F12 is used to represent the action of F on the first and second element, namely F12 = F ⊗ id = F[1] ⊗ F[2] ⊗ id, (15) and similar for F23 . Here we use Sweedler’s notation again, X (i) (i) F= F1 ⊗ F2 ≡ F[1] ⊗ F[2] .

(16)

i −1 −1 U and U −1 are given by U = F[1] γ(F[2] ) and U −1 = γ(F[1] )F[2] respectively, −1 −1 −1 where F ≡ F[1] ⊗ F[2] . Eq.(13) and (14) are crucial for the twist operation. Once the relations are satisfied, the algebra Ht (H, m, i, ∆t , , γt ; K) also satisfies Eq.(3)-(7), then Ht becomes a new Hopf algebra.

3 Twisted Poincar´ e symmetry Following [2], we consider the Poincar´e algebra as the Lie algebra of the Lorentz symmetry and the translation symmetry. But this procedure also works for a general symmetry algebra. The Poincar´e algebra P consists of the translational generators Pµ and Lorentz generators Mµν , which satisfy the following commutator relations, [Pµ , Pν ] = 0 [Mµν , Pρ ] = −iηρµ Pν + iηρν Pµ , [Mµν , Mρσ ] = iηνρ Mµσ − iηµρ Mνσ − iηνσ Mµρ + iηµσ Mνρ ,

(17)

where Greek characters are the indices of spacetime coordinate. The universal enveloping Poincar´e algebra U(P) become a Hopf algebra with the following definitions. m(g ⊗ h) = gh, ˆ i(k) = k 1,

(18) (19)

ˆ+1 ˆ ⊗ g, ∆(g) = g ⊗ 1 (g) = 0,

(20) (21)

γ(g) = −g,

(22)

ˆ is used to represent the unit element for all g, h ∈ P. Here the hatted symbol 1 H, to distinguish it from the unit of the field K. In addition, ˆ =1 ˆ ⊗ 1, ˆ ∆(1) ˆ (1) = 1, ˆ = 1, ˆ γ(1)

(23) (24) (25)

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ˆ For algebraic consistency, we require hold for 1. ∆(hg) = ∆(h)∆(g), (hg) = (h)(g),

(26) (27)

γ(hg) = γ(g)γ(h).

(28)

Using Eq. (26)-(28), the definitions (19)-(22) are extended to the whole U(P). Next we will twist the the Poincar´e algebra by an appropriate twist element. Eq.(13) and (14) are nontrivial equations, though, the twist of a special form satisfies easily the equations. Let us consider the following F.   X F = exp  cij hi ⊗ hj  . (29) i,j

Here cij is some constant, and hi is an element in H. We assume that all hi commute with each other, namely [hi , hj ] = 0 ∀i, j. That means that all hi make some Abelian subsector in H. In fact, F satisfies Eq.(13). For more general twist, we will see it in the later section. The right-hand side of Eq.(29) is written in a formal expansion series, ˆ⊗1 ˆ + cij hi ⊗ hj + 1 (ckl hk ⊗ hl )(cmn hm ⊗ hn ) + · · · . F =1 2!

(30)

The counit condition (14) is satisfied clearly, because the second term and the subsequent terms are all annihilated by Eq.(21). Twisting the Hopf algebra does not change the algebra, but accompanies the modification of the multiplication rule in the representation space. The product is replaced by the star product, which is defined as v ? w ≡ m(F −1 . v ⊗ w).

(31)

Then the algebra of linear space V with the star product becomes consistent with the twisted algebra, i.e. V is the correct representation of the twisted Hopf algebra. Note that this product is associative, and obviously reduced to the ordinary product in the limit cij → 0. In the coordinate representation, Pµ and Mµν are written as derivative operators on the coordinate. Pµ = i∂µ , Mµν = i(xµ ∂ν − xν ∂µ ).

(32)

The idea is that the translation generator Pµ , which makes the Abelian subsector in the Poincar´e algebra, is written by i∂µ . The twist, i µν F P P = exp Θ Pµ ⊗ P ν , (33) 2

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is an appropriate twist element. In the coordinate representation, F P P is written as i (34) F P P = exp − Θµν ∂µ ⊗ ∂ν . 2 The above expression is very similar to the Moyal product, and in fact it works like that exactly. The product of spacetime coordinates is changed in such a way that xµ ? xν = m((F P P )−1 . xµ ⊗ xν ) i ρσ µ ν = m exp − Θ Pρ ⊗ Pσ . (x ⊗ x ) 2 i ρσ µ ν = m exp Θ ∂ρ ⊗ ∂σ (x ⊗ x ) 2 i ρσ µ ν µ ν = m x ⊗ x + Θ δρ ⊗ δ σ 2 i = xµ · xν + Θµν , 2

(35)

and therefore the spacetime coordinate becomes noncommutative, [xµ , xν ]? ≡ xµ ? xν − xν ? xµ = iΘµν 6= 0.

(36)

Now we turn to the modification of the symmetry algebra on such noncommutative representation space. While the coproduct of Pµ is same as before, the coproduct of Mµν is modified by the twist, P ˆ+1 ˆ ⊗ Mµν ∆P (Mµν ) = Mµν ⊗ 1 t 1 − Θρσ [(ηρµ Pν − ηρν Pµ ) ⊗ Pσ + Pρ ⊗ (ησµ Pν − ησν Pµ )] . 2 (37)

Consequently, the action of the Lorentz generator Mµν on the product in the representation space is changed. For example, the commutator is transformed such that −1 Mµν [xρ , xσ ]? = Mµν ◦ m (F PP ) xρ ⊗ xσ − xσ ⊗ xρ −1 = m (F PP ) ∆t (Mµν ) . xρ ⊗ xσ − xσ ⊗ xρ n h −1 ˆ+1 ˆ ⊗ (ixµ ∂ν − ixν ∂µ ) (ixµ ∂ν − ixν ∂µ ) ⊗ 1 = m (F PP ) 1 µ0 ν 0 + Θ (ηµ0 µ ∂ν − ηµ0 ν ∂µ ) ⊗ ∂ν 0 + ∂µ0 ⊗ (ην 0 µ ∂ν ) − ην 0 ν ∂µ ) 2 i xρ ⊗ x σ − x σ ⊗ x ρ = 0.

(38)

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So the noncommutativity parameter Θ µν = [xρ , xσ ]? is annihilated by Mµν , which is a desirable behavior because Θ µν is constant. The product xµ xν in ordinary commutative spacetime is associated with x{µ ? xν} ≡ 12 (xµ ? xν + xν ? xµ ). Its transformation property is the same as xµ xν . A brief calculation shows that Mµν x{ρ ? xσ} = iηνρ x{µ ? xσ} − iηνσ x{ρ ? xµ} − iηµρ x{ν ? xσ} + iηµσ x{ν ? xρ} (39) Like above, despite the product changed by the star, the twisted symmetry generator acts on the noncommutative space like the original symmetry on commutative space. So once we find an appropriate twist, the twisted symmetry is guaranteed by construction for any theory on noncommutative space which this procedure can achieve.

4 Super-Poincar´ e algebra and Non(anti)commutative Superspace The Hopf algebraic method is useful for constructing a noncommutative theory. We apply it to a supersymmetric theory on non(anti)commutative superspace[8]. The extension to a supersymmetric case is almost straightforward, but some equipments are needed. • To obtain superspace noncommutativity, super-Poincar´e algebra is used as a symmetry algebra, instead of the Poincar´e algebra. • Since the super-Poincar´e algebra contains fermionic generators, we should use Z2 graded Hopf algebra. The non(anti)commutative superspace is realized as the representation of the twisted super-Poincar´e algebra. Because the fermionic generators anti-commute, the definition of the product of the Hopf algebra H should be modified with an appropriate sign flip. (a ⊗ b)(c ⊗ d) = (−1)|b||c| (ac ⊗ bd). a, b ∈ H Here |a| stands for the fermionic character of the element a, 0 if a is fermionic |a| = 1 if a is bosonic.

(40)

(41)

The Z2 grading is extended to all order of the product, so as to change a sign whenever fermionic elements jump over each other. For instance, (a ⊗ b ⊗ c)(d ⊗ e ⊗ f ) = (−1)|c|(|d|+|e|)+|b||d|(ad ⊗ be ⊗ cf ).

(42)

268

Yoshishige Kobayashi

In addition, we allow not only complex numbers, but also Grassmann number ring to be the base K. Therefore, for consistency, we impose the anticommutative property of a fermionic number λ ∈ K with a fermionic element hi ∈ H, λh1 ⊗ h2 ⊗ h3 = (−1)|λ||h1 | h1 λ ⊗ h2 ⊗ h3 = (−1)|λ||h1 | h1 ⊗ λh2 ⊗ h3 = (−1)|λ|(|h1 |+|h2 |) h1 ⊗ h2 ⊗ λh3 .

(43)

With these definitions, the twist equation (13) of the extended version can be proved. Let us think the twist element which has the form, F = exp cij Gi ⊗ G0j . (44) Here cij is a constant, which can be a Grassmann number as well as a cnumber. G and G0 are generators, which can be either bosonic or fermionic. Assume that all G commute or anti-commute with each other, namely [Gi , Gj ]± ≡ Gi Gj − (−1)|Gi ||Gj | Gj Gi = 0

(45)

The left hand side(LHS) of the twist equation is expanded, F12 (∆0 ⊗ id)F

ˆ ⊗ Gi ⊗ G0j (46) ˆ exp cij Gi ⊗ 1 ˆ ⊗ G0j + 1 = exp cij Gi ⊗ G0j ⊗ 1

We consider the following commutator, ij ˆ ckl (Gk ⊗ 1 ˆ ⊗ G0l + 1 ˆ ⊗ Gk ⊗ G0l ) c Gi ⊗ G0j ⊗ 1, n kl 0 0 = cij ckl (−1)|c |(|Gi |+|Gj |)+|Gj ||Gk | Gi Gk ⊗ G0j ⊗ G0l o kl 0 +(−1)|c |(|Gi |+|Gj |) Gi ⊗ G0j Gk ⊗ G0l n ij 0 0 0 − ckl cij (−1)|c |(|Gl |+|Gk |)+(|Gi |+|Gj |)|Gl | Gk Gi ⊗ G0j ⊗ G0l o ij 0 0 0 +(−1)(|c |+|Gi |)(|Gk |+|Gl |)+|Gj ||Gl | Gi ⊗ Gk G0j ⊗ G0l = 0.

(47)

From the Baker-Campbell-Hausdorff formula, the right hand side(RHS) of Eq.(46) can be rewritten as ˆ + Gi ⊗ 1 ˆ ⊗ G0j + 1 ˆ ⊗ Gi ⊗ G0j . (48) (46) = exp cij Gi ⊗ G0j ⊗ 1 In a similar way, the RHS of Eq.(13) is F23 (id ⊗ ∆0 )F (49) ˆ ⊗ Gi ⊗ G0j exp cij (Gi ⊗ G0j ⊗ 1 ˆ + Gi ⊗ 1 ˆ ⊗ G0j ) = exp cij 1 ˆ ⊗ Gi ⊗ G0j + Gi ⊗ G0j ⊗ 1 ˆ + Gi ⊗ 1 ˆ ⊗ G0j = exp cij 1 (50)

Noncommutative Deformation and Drinfel’d Twisted Symmetry

269

Therefore, Eq.(13) is proved. The commutator relations of N = 1 super-Poincar´e algebra are [Pµ , Pν ] = 0, [Mµν , Mρσ ] = iηνρ Mµσ − iηµρ Mνσ − iηνσ Mµρ + iηµσ Mνρ , [Mµν , Pρ ] = −iηρµ Pν + iηρν Pµ , ¯ α˙ ] = 0, [Pµ , Qα ] = 0, [Pµ , Q α˙ ¯ β˙ ¯ α˙ ] = i (¯ , [Mµν , Q σµν ) β˙ Q

β

[Mµν , Qα ] = i (σµν )α Qβ , ¯ ˙ } = 2σ µ Pµ . {Qα , Q

(51)

αβ˙

β

The generators are written as differential operators on superspace, Pµ = i∂µ , β

Mµν = i(xµ ∂ν − xν ∂µ ) − iθα (σµν )α

∂ ∂ α˙ − iθ¯α˙ (¯ σµν ) β˙ ¯ , ∂θβ ∂ θβ˙

∂ ˙ − σαµβ˙ θ¯β ∂µ , ∂θα ¯ α˙ = −i ∂ + θβ σ µ ∂µ . Q β α˙ ∂ θ¯α˙ Qα = i

(52)

¯ α˙ ) make an In super-Poincar´e algebra, Pµ and Qα (otherwise Pµ and Q Abelian subalgebra. So any linear combination of the biproduct which includes only P and Q works as a twist, but not all combinations are meaningful and consistent with the noncommutativity. We take the following kinds of twists. • Q-Q Twist

F

QQ

1 = exp − C αβ Qα ⊗ Qβ 2

(53)

F QQ causes the desired non-anticommutative superspace, {θα , θβ }? = C αβ ,

(54)

[xµ , xν ]? =

˙ C αβ σαµγ˙ σβν δ˙ θ¯γ˙ θ¯δ ,

(55)

[xµ , θα ]? =

˙ −iC αβ σβµβ˙ θ¯β ,

(56)

the above results agree with [3]1 . • P -Q Twist i µα = exp λ (Pµ ⊗ Qα − Qα ⊗ Pµ ) . 2

F 1

PQ

Except for that they used the chiral coordinate base.

(57)

270

Yoshishige Kobayashi ˙

˙

[xµ , xν ]? = λµα σαν β˙ θ¯β − λνα σαµβ˙ θ¯β , [xµ , θα ]? = iλµα , α β θ , θ ? = 0.

(58)

In this twist, the noncommutativity parameter λµα is not an ordinary c-number, but a Grassmann number. This type of noncommutativity between spacetime coordinate and fermionic coordinate of the superspace has been discussed in the literature[9]. • Mixed Twist Of course, the linear combination of the previous three twists is a proper twist. i 1 i µν Θ Pµ ⊗ Pν + λµα (Pµ ⊗ Qα − Qα ⊗ Pµ ) − C αβ Qα ⊗ Qβ . 2 2 2 (59) The noncommutativity also result in the linear combination,

F mix = exp

˙

˙

˙

[xµ , xν ]? = iΘµν + C αβ σαµγ˙ σβν δ˙ θ¯γ˙ θ¯δ + λµα σαν β˙ θ¯β − λνα σαµβ˙ θ¯β , ˙

[xµ , θα ]? = iλµα − iC αβ σβµβ˙ θ¯β , α β θ , θ ? = C αβ .

(60)

¯ α˙ is Note that under the above twists, the coproduct of Mµν and Q changed, however we do not write down the explicit form here. They modify the action of the product of representation space. That gives the appropriate transformation relations of superspace coordinates on the corresponding non(anti)commutative superspace, in the twisted supersymmetric way.

5 Conclusion We constructed the Drinfel’d twisted (N = 1) super-Poincar´e algebra by using Hopf algebra and the Drinfel’d twist operation. Several appropriate twist elements have been found, and the non(anti)commutative superspace is obtained as a representation space of the twisted algebra. It provides quite the same non(anti)commutative superspace which has been formulated by the Moyal product in a physical context previously. The crucial difference between the two procedures is the reinterpretation of the origin of noncommutativity. In the twist sense, noncommutativity is a consequence from the modification of the Hopf coalgebra. The structure of the algebra itself is not modified. As a result, the Hopf algebra construction always keeps the twisted symmetry. Here only N = 1 supersymmetry is treated, but the twist for the extended supersymmetry case had been discussed[8][10].

Noncommutative Deformation and Drinfel’d Twisted Symmetry

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Hopf algebra and the twist method is systematic. Once we find the desired twist element, the procedure is almost automatic, and it gives the way to construct the representation space without ambiguity. It is hoped that this research contributes to understanding of symmetry on noncommutative space and provides new insights.

References 1. N. Seiberg and E. Witten, “String Theory and Noncommutative Geometry,” JHEP 9909, 032(1999)[hep-th/9908142]. 2. M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, “On a LorentzInvariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT,” Phys. Lett. B604, 98 (2004)[hep-th/0408069]. 3. N. Seiberg, “Noncommutative superspace, N = 1/2 supersymmetry, field theory and string theory,” JHEP 0306, 010 (2003) [hep-th/0305248]. 4. J. Fuchs, ”Affine Lie Algebras and Quantum Groups,” Cambridge University Press(1992), Cambridge. 5. V. Chari and A. Pressley, “A Guide to Quantum Groups,” Cambridge University Press(1994), Cambridge. 6. S. Majid, “Foundations of Quantum Group Theory,” Cambridge University Press(1995), Cambridge. 7. V. G. Drinfel’d, “Quasi-Hopf Algebras,” Leningrad Math. J. 1, 6, 1419(1990). 8. Y. Kobayashi and S. Sasaki, “Lorentz invariant and supersymmetric interpretation of noncommutative quantum field theory,” Int. J. Mod. Phys. A 2071757188(2005)[hep-th/0410164]. 9. D. Klemm, S. Penati and L. Tamassia “Non(anti)commutative superspace,” Class. Quant. Grav. 20, 2905-2916(2003)[hep-th/0104190]; J. de Boer, P. A. Grassi and P. van Nieuwenhuizen “Non-commutative superspace from string theory,” Phys. Lett. B574, 98-104(2003)[hep-th/0302078]; Y. Kobayashi and S. Sasaki, “Nonlocal Wess-Zumino model on nilpotent noncommutative superspace,” Phys. Rev. D 72065015(2005)[hep-th/0505011]. 10. B. M. Zupnik, “Twist-deformed supersymmetries in non-anticommutative superspaces,” Phys. Lett. B627, 208-216(2005)[hep-th/0506043]; M. Ihl and C. Saemann, “Drinfeld-twisted supersymmetry and non-anticommutative superspace,” JHEP 0601, 065 (2006)[arXiv:hep-th/0506057].

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\(2) and Twisted Affine Lie superalgebra gl(2|2) k Conformal Field Theory Xiang-Mao Ding1 , Gui-Dong Wang1,2 , and Shi-Kun Wang1,3 1

2 3

Institute of Applied Mathematics, Academy of Mathematics and Systems Science Chinese Academy of Sciences, P.O.Box 2734, Beijing 100080, China Graduate School of the Chinese Academy of Sciences KLMM, AMSS, CAS, Beijing 100080, China [email protected]; [email protected]; [email protected].

Abstract In this paper, the differential operators realization of Lie superalgebra gl(2|2)(2) is given with coherent state methods. According to the differential operators realization of the Lie superalgebra, the free field realization of the affine (2) \ Kac-Moody superalgebra gl(2|2) k is given. The primary field of the superalgebra is also given. It is all know that the primary field corresponding to the highest weight representation of algebra. In our case, we obtain sixteen series[5], for the complexity of the expression and the limit of the space, we given a little more simple but more important primary fields. This corresponds to the highest weight representation of twisted Lie superalgebra psl(2|2)(2). Partially Supported by NKBRPC(2004CB31800,2006CB805905) and NNSFC(] 10231050; 10375087)

Keyword: superalgebra, differential operator realization, free field realization, primary field, conformal field theory

1 Introduction It is well known that the conformal group in two dimensions is the set of all analytic maps, where the group multiplication is the composition of maps. Obviously, this set is infinite dimensional, and the generators satisfy the communication relations of the Virasoro algebra. In fact, the algebraic theory of conformal field theory is the vertex algebra theory that was brought forward by Borcherds[1]. And the Chiral algebra that was put forward by V.G.Drinfeld is 273

274

Xiang-Mao Ding, Gui-Dong Wang, and Shi-Kun Wang

to use the algebraic geometry to study the conformal field theory. So the study of conformal field theory becomes an important problem in mathematics[6]. There are many conformal field theories, in which, the more important ones are minimal model, Ising model, WZW-model(Wess-Zumino-Witten model)etc. From the mathematical viewpoint, the most fruitful and the broadest researched was the WZW-model. The WZW-model is a Sigma model with Wess-Zumino (WZ)-term[3]. In this model, affine algebra and Virasoro algebra were connected by Sugawara construction and the correlation functions of the model satisfy the KZ-equation. In general, the WZW-model was defined on a ordinary manifold. Goddard, Kent and Olive([2]) found that any conformal field theory can be obtained by WZW-models and their coset models. So it has been an important and interesting problem in the recent twenty years. If we define the model on a super-manifold, then we will obtain the supersymmetric WZW-model. Super-symmetry was an attempt to unify the boson and fermions, that is, it was used to describe the transformation of boson and fermions and viceversa. In mathematics, it corresponds to the Z2 −graded algebra. The elementary reference about Z2 −graded algebra is the paper of V.G.Kac([5]). The work in the ref.[5] mainly describes the finite dimensional Lie superalgebra, and the finite dimensional Lie superalgebra corresponding to the symmetry of super-symmetric quantum mechanics. If one wants to describe the super-symmetric quantum field theory, then one should extend the finite dimensional superalgebra to the symmetric algebra which has infinite generators, called affine superalgebra.Virasoro algebra and current algebras are important algebraic structures in Conformal Field Theories (CFTs)[4, 6, 7] and string theory[8]. In this paper, we will investigate its structure. It is well known, Heisenberg algebra is the most important tool in the study of finite dimensional algebra. The free field realization, which is an infinite dimensional generalization of the Heisenberg algebra, is a powerful tool in many physical applications. This approach was first given by Wakimoto for the simplest case [ [9], and extended to sl(n) [ in [10], and other simple Lie algebras in sl(2) k k [11]. For twisted algebra, the free field approach was first realized in [12, 13], and extended to higher rank in [14]. In [15, 16], the method was generalized to the Kac-Moody superalgebra, and the study of some important Lie superalgebras in subsequent work [17, 18, 19, 20]. Using this approach, the correlation function can be easily evaluated. Beside the applications mentioned above, new features will arise in the representation theory of the Lie superalgebra. For example, for Lie superalgebra there are two kinds of representations, which are named the typical representation and the atypical representation, respectively. There is no counter part for the atypical representation in the ordinary Lie algebra. In this paper, we will consider the CFT based on the twisted current \(2) at general level k. The important ingredient of CFT superalgebra gl(2|2) k is the primary field, or the highest weight representation of the Kac-Moody

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

275

(2)

\ , only the superalgebra. we think that for the twisted superalgebra gl(2|2) k atypical representation is surviving, and which are the chiral primary fields of Virasoro algebra with conformal dimension zero in the boundary theory. The paper is organized as follows. For conveniences, in section 2, we give some notations which will be used in the paper. In section 3, we will give the differential operators realization of the Lie superalgebra gl(2|2). In section 4, we derive the free field realization of the twisted Kac-Moody superalgebra \(2) , and construct a energy-momentum tensor of the current supergl(2|2) k algebra, which is CFT with zero central. Because of the complexity of the primary fields and the limit of space, in section 5, we only give the primary \(2) fields of Lie superalgebra of psl(2|2) k

2 Notations First of all, we will give the definition of Lie superalgebra gl(2|2). Take Eij , i, j = 1, 2, 3, 4 be the matrices with entry 1 at the i-th row and jth column, and zero elsewhere. Then gl(2|2) can be spanned by 2|2 matrices Eij , i, j = 1, 2, 3, 4, and its superalgebra structure is introduced by defining the Lie (anti-)bracket for any two matrices Eij and Ekl : [Eij , Ekl ] = δjk Eil − (−1)([i]+[j]+[k]+[l]) δil Ekj

(2.1)

where the Z2 -grading(superalgebra structure) is defined as [1] = [2] = 0, [3] = [4] = 1. It is well known that, unlike the ordinary Lie algebra, for a given Lie superalgebra, its Dynkin diagram is not unique, and not all of them have a finite order automorphism. For the given Lie superalgebra, if one can choose an appropriate Dynkin diagram, which has a finite order automorphism, the Lie superalgebra can be twisted by folding the Dynkin diagram. In the gl(2|2) case, there are three kinds of Dynkin diagrams, if we choose the following Dynkin diagram (for other choices, the process is similar):

⊗ ε1 − δ 1

g δ1 − δ 2

⊗ δ2 − ε 2

in which (εi |εj ) = δij , and (δi |δj ) = −δij . So that in the diagram ⊗ represent a null vector of fermionic type simple root. Now, we define an order-2 transformation on the Dynkin diagram τ : εi → −ε3−i ; δi → −δ3−i .

(2.2)

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Xiang-Mao Ding, Gui-Dong Wang, and Shi-Kun Wang

Obviously, the Dynkin diagram is invariant under the action of τ . Identically, we give the explicit action on the matrices as follows: τ (E11 ) = −E22 , τ (E22 ) = −E11 , τ (E33 ) = −E44 , τ (E44 ) = −E33 , τ (E13 ) = E42 , τ (E42 ) = E13 , τ (E34 ) = E34 , τ (E31 ) = −E24 , τ (E24 ) = −E31 , τ (E43 ) = E43 , τ (E14 ) = −E32 , τ (E32 ) = −E14 ,

(2.3)

τ (E12 ) = −E12 , τ (E41 ) = E23 , τ (E23 ) = E41 , τ (E21 ) = −E21 . It is easy to check that τ is an automorphism of the superalgebra gl(2|2) with order r = 2. So we obtain the twisted superalgebra gl(2|2), and we have gl(2|2) = gl(2|2)0 ⊕ gl(2|2)1 ,

(2.4)

in which gl(2|2)0 are spanned by 1 e1 = √ (E13 + E42 ), e2 = E34 , e3 = 2 H1 = E11 − E22 + E33 − E44 , 1 f1 = √ (E31 − E24 ), f2 = E43 , f3 = 2 H2 = E11 − E22 − E33 + E44 .

1 √ (E14 − E32 ), 2 1 √ (E41 + E23 ), 2

Here gl(2|2)0 is a fixed point sub-superalgebra under the automorphism. Obviously, this subalgebra is just the osp(2|2), or sl(2|1). If we twist the fixed subalgebra once more, we get its fixed subalgebra osp(2|1) [16]. So, twisting is an appropriate way to get a smaller algebra from a larger algebra. And the generators of gl(2|2)1 are 1 1 e1 = √ (E13 − E42 ), e2 = E12 , e3 = √ (E14 + E32 ), 2 2 H 1 = E11 + E22 + E33 + E44 , 1 1 f 1 = √ (E31 + E24 ), f 2 = E21 , f 3 = √ (E41 − E23 ), 2 2 H 2 = E11 + E22 − E33 − E44 . The (anti-)commutation relations of g(2|2) in this basis are as follows: [e1 , e2 ] = e3 , {e1 , f1 } =

1 H1 , {e1 , f3 } = f2 , [H2 , e1 ] = 2e1 , 2

1 (H1 − H2 ), [e2 , f3 ] = f1 , [H1 , e2 ] = 2e2 , 2 [H2 , e2 ] = −2e2 , [H1 , e3 ] = 2e3 , {e3 , f1 } = e2 , [e3 , f2 ] = e1 , (2.5) 1 {e3 , f3 } = H2 , [f1 , f2 ] = −f3 , [H2 , f1 ] = −2f1 , [H1 , f2 ] = −2f2 , 2 [H2 , f2 ] = 2f2 , [H1 , f3 ] = −2f3 , [e2 , f2 ] =

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

277

for ([gl(2|2)0 , gl(2|2)0 ]), and 1 H1 , [e1 , f 2 ] = −f3 , {e1 , f 3 } = f2 , 2 1 [H 2 , e1 ] = 2e1 , [e2 , f 1 ] = e3 , [e2 , f 2 ] = (H1 + H2 ), 2

{e1 , f 1 } =

[e2 , f 3 ] = −e1 , {e3 , f 1 } = e2 , [e3 , f 2 ] = f1 , {e3 , f 3 } =

1 H2 , 2

(2.6)

[H 2 , e3 ] = 2e3 , [H 2 , f 1 ] = −2f1 , [H 2 , f 3 ] = −2f3 , for ([gl(2|2)1 , gl(2|2)1 ]). At last 1 H 1 , [e1 , f 2 ] = f 3 , [H 2 , e1 ] = 2e1 , 2 [e2 , e1 ] = −e3 , [e2 , f 3 ] = f 1 , {e3 , e1 } = −e2 , [e3 , f 2 ] = −f 1 , 1 {e3 , f 3 } = H 1 , [H 2 , e3 ] = 2e3 , [H1 , e2 ] = 2e2 , [H1 , e3 ] = 2e3 , 2 [H1 , f 2 ] = −2f 2 , [H1 , f 3 ] = −2f 3 , [H2 , e1 ] = 2e1 , [H2 , e2 ] = 2e2 , 1 [H2 , f 1 ] = −2f 1 , [H2 , f 2 ] = −2f 2 , {f1 , e1 } = H 1 , [f1 , e2 ] = e3 , (2.7) 2 {f1 , f 3 } = −f 2 , [H 2 , f1 ] = −2f 1 , [f2 , e3 ] = −e1 , [f2 , f 1 ] = f 3 , 1 [f3 , e2 ] = −e1 , {f3 , e3 } = H 1 , {f3 , f 1 } = f 2 , [H 2 , f3 ] = −2f 3 . 2 {e1 , e3 } = e2 , {e1 , f 1 } =

for ([gl(2|2)0 , gl(2|2)1 ]). All of other relations are identically zero. So gl(2|2)0 and gl(2|2)1 satisfy [gl(2|2)i , gl(2|2)j ] ⊂ gl(2|2)(i+j)mod 2 .

(2.8)

For the twisted algebra gl(2|2) there is an endomorphism of any representation. For a given Killing form, we can construct the quadratic Casimir of gl(2|2) as C=

X

(−1)[j] Eij Eji

i,j

1 = H1 H2 − e 1 f 1 + f 1 e 1 − e 2 f 2 − f 2 e 2 − e 3 f 3 + f 3 e 3 2 1 + H 1 H 2 − e1 f 1 + f 1 e1 + e2 f 2 + f 2 e2 − e3 f 3 + f 3 e3 2

(2.9)

In fact, the quadratic Casimir is independent of the choice of the basis, it is useful to construct the energy-momentum operator.

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Xiang-Mao Ding, Gui-Dong Wang, and Shi-Kun Wang

3 Differential Operators realization of Lie superalgebra gl(2|2) To obtain a differential operators realization, we first construct a Fock space representation of gl(2|2). The Fock space is constructed by the actions of the lowering operators f1 , f2 , f3 f 1 , f 2 and f 3 on the highest weight state. Define the highest weight state |Λ > of the twisted algebra gl(2|2) by e1 |Λ >= e2 |Λ >= e3 |Λ >= e1 |Λ >= e2 |Λ >= e3 |Λ >= 0 Hi |Λ >= Λi |Λ >, H i |Λ >= Λi |Λ >, i = 1, 2.

(3.1)

Then the supergroup action of the operator eρ with vector ρ = θ 1 e1 + x 2 e2 + θ 3 e3 + θ 1 e1 + x 2 e2 + θ 3 e3 ,

(3.2)

on the highest state |Λ > generates a coherent state of the algebra, where x2 , x2 are bosonic coordinates satisfying [∂x2 , x2 ] = [∂x2 , x2 ] = 1, and θi , θj are fermionic coordinates obeying θi θj = −θj θi , θi θj = −θj θi and {∂θi , θj } = {∂θi , θ j } = δij , i, j = 1, 2, 3. This can be viewed as the super extension of the ordinary coherent state method. Please see [21] for more applications of coherent state method on representation theory and physics. Now we define T A eρ |Λ >= dT A eρ |Λ >,

(3.3)

where T A is any generator of gl(2|2) and dT A is the corresponding differential operator realization of the generator T A . By using the defining relations of twisted gl(2|2) and the Baker-Campbell-Hausdorff(BCH) formula, we obtain the differential operators representation of the twisted algebra. For positive simple roots, the expressions are very simple: 1 1 1 de1 = ∂θ1 − x2 ∂θ3 + θ3 ∂x2 + x2 θ1 ∂x2 2 2 12 1 1 1 de2 = ∂x2 + θ1 ∂θ3 + θ 1 ∂θ3 − θ1 θ1 ∂x2 2 2 6 1 de3 = ∂θ3 − θ1 ∂x2 2 1 1 1 de1 = ∂θ1 − x2 ∂θ3 − θ3 ∂x2 − x2 θ1 ∂x2 2 2 12 de2 = ∂x2 1 de3 = ∂θ3 + θ1 ∂x2 2 and the generators of the Cartan parts are DH1 = Λ1 − 2θ3 ∂θ3 − 2x2 ∂x2 − 2θ3 ∂θ3 − 2x2 ∂x2 DH2 = Λ2 − 2θ1 ∂θ1 + 2x2 ∂x2 − 2θ1 ∂θ1 − 2x2 ∂x2

(3.4)

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

DH 1 = Λ 1

279

(3.5)

DH 2 = Λ2 − 2θ1 ∂θ1 − 2θ3 ∂θ3 − 2θ1 ∂θ1 − 2θ3 ∂θ3 . But for the negative roots operators, the expressions of their realization are much more involved, 1 1 1 1 θ1 Λ1 + θ1 Λ1 + (θ3 − θ1 x2 )∂x2 − θ1 θ3 ∂θ3 2 2 2 2 1 1 1 1 − (θ1 x2 + θ1 θ3 θ1 )∂x2 − (θ1 θ3 + 2x2 − θ1 x2 θ1 )∂θ3 2 6 2 6 1 1 = x2 (Λ1 − Λ2 ) + θ3 ∂θ1 + θ 3 ∂θ1 − x2 (θ1 θ3 ∂x2 + θ3 θ1 ∂x2 ) 2 6 1 1 2 + x2 (θ1 ∂θ3 + θ1 ∂θ3 ) + x2 (θ1 ∂θ1 − 2x2 ∂x2 − θ3 ∂θ3 + θ1 ∂θ1 − θ3 ∂θ3 ) 4 2 1 1 1 1 = θ3 Λ2 + θ1 x2 (2Λ1 − Λ2 ) + ( θ3 + x2 θ1 )Λ1 + x2 ∂θ1 2 4 2 4 1 + [2θ1 θ3 ∂θ1 + 2x2 θ3 ∂x2 + (θ1 θ 3 − θ3 θ1 )∂θ1 − θ3 x2 ∂x2 − x2 x2 ∂θ3 ] 2 1 − [6θ1 x22 ∂x2 − 2θ1 x2 θ 1 ∂θ1 + (θ1 x2 x2 + 3θ1 θ3 θ3 )∂x2 + 3(θ1 x2 θ3 12 1 +x2 θ3 θ1 )∂θ3 ] + (4θ1 x22 θ1 ∂θ3 − 3θ1 x2 θ3 θ 1 ∂x2 ) 24 1 1 1 = θ1 Λ1 + θ1 Λ1 + x2 ∂θ3 + θ3 ∂x2 − θ1 θ 1 (x2 ∂θ3 − θ3 ∂x2 ) 2 2 12 1 − θ1 (x2 ∂x2 + θ3 ∂θ3 + x2 ∂x2 + θ3 ∂θ3 ) 2 1 1 1 = x2 (Λ1 + Λ2 ) − (θ1 θ 3 + θ3 θ1 )(Λ1 − Λ2 ) + (θ1 θ3 − θ 1 θ3 )Λ1 2 4 2 1 1 1 1 + θ1 x2 θ1 (2Λ1 − Λ2 ) − (θ3 θ3 − x2 θ3 θ1 − θ1 x2 θ3 + θ1 x22 θ1 )∂x2 6 2 2 4 1 1 1 −(θ1 x2 + θ1 θ3 θ1 )∂θ1 − (θ3 x2 + θ1 x2 θ3 θ 1 )∂θ3 − (θ 1 x2 − θ1 θ1 θ3 )∂θ1 2 12 2 1 1 −(x22 − θ1 θ3 θ1 θ3 )∂x2 − (x2 θ3 − θ1 x2 θ1 θ 3 )∂θ3 3 12 1 1 1 1 = θ3 Λ2 + x2 θ1 (2Λ1 − Λ2 ) + ( θ3 + θ1 x2 )Λ1 + θ 1 θ3 ∂θ1 2 4 2 4 1 1 1 1 +(−x2 + θ1 θ 3 − θ3 θ1 − θ1 x2 θ 1 )∂θ1 + (x2 θ3 − x22 θ1 )∂x2 2 2 6 2 1 1 1 1 +( x2 x2 + θ1 x2 θ3 + x2 θ3 θ1 − θ1 x22 θ 1 )∂θ3 2 4 4 6 1 1 1 1 +(− x2 θ3 + θ3 θ 1 θ3 − x2 θ 1 x2 + θ1 x2 θ 1 θ3 )∂x2 2 4 12 8

d f1 =

d f2

d f3

df

1

df 2

df

3

It is straightforward to check that the above differential operators satisfy the algebraic relations of the twisted algebra gl(2|2).

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4 Free Field realization of Lie superalgebra gl(2|2) With the help of the differential operator representation of finite Lie algebra g, we can find the Wakimoto realization of the corresponding Kac-Moody algebra gˆ. (2)

\ For gl(2|2) k Kac-Moody superalgebra, the free field realization for arbitrary k will be given in terms of sixteen fields, namely two bosonic β − γ pairs, four fermionic b−c type pairs and four free scalar fields φ. The free fields obey the following OPEs: 1 , z−w 1 , = −γ 2 (z)β 2 (w) = z−w δij = ψj+ (z)ψi (w) = , i, j = 1, 3, z−w δij + = ψ j (z)ψ i (w) = , i, j = 1, 3, z−w = φ1 (z)φ2 (w) = −4ln(z − w)

β2 (z)γ2 (w) = −γ2 (z)β2 (w) = β 2 (z)γ 2 (w) ψi (z)ψj+ (w) +

ψ i (z)ψ j (w) φ1 (z)φ2 (w)

and all other Operator Product Expansions (OPEs) are trivial. The conformal + weights of βi (z), β i (z), ψi+ (z), ψ i (z), ∂φi (z) and ∂φi (z) are 1, and 0 for else. (2)

\ The free field realization of the gl(2|2) k current superalgebra is obtained by substitution the differential operators with certain kinds of fields. Concretely, the relations are, xi → γi (z), ∂xi → βi (z), xi → γ i (z), ∂xi → β i (z), +

θi → ψi (z), ∂θi → ψi+ (z), θi → ψ i (z), ∂θi → ψ i (z), √ √ Λi → −k∂φi (z), Λi → −k∂φi (z). For simplicity, we denote the generating functions of the Kac-Moody superalgebra as T A (z), if the finite Lie superalgebra generator is DT A . The expressions for the positive and Cartan currents are straightforward, 1 1 1 e1 (z) = ψ1+ (z) − γ2 (z)ψ3+ (z) + ψ 3 (z)β 2 (z) + γ2 (z)ψ 1 (z)β 2 (z) 2 2 12 1 1 1 + + e2 (z) = β2 (z) + ψ1 (z)ψ3 (z) + ψ 1 (z)ψ 3 (z) − ψ1 (z)ψ1 (z)β 2 (z) 2 2 6 1 + e3 (z) = ψ3 (z) − ψ 1 (z)β 2 (z) 2

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

281

1 1 1 + + e1 (z) = ψ 1 (z) − γ2 (z)ψ 3 (z) − ψ3 (z)β 2 (z) − γ2 (z)ψ1 (z)β 2 (z) 2 2 12 e2 (z) = β 2 (z) 1 + e3 (z) = ψ 3 (z) + ψ1 (z)β 2 (z) 2 √ H1 (z) = −k∂φ1 (z) − 2ψ3 (z)ψ3+ (z) − 2γ 2 (z)β 2 (z) +

−2ψ3 (z)ψ3 (z) − 2γ2 (z)β2 (z) √ + H2 (z) = −k∂φ2 (z) − 2ψ1 (z)ψ1+ (z) − 2ψ1 (z)ψ 1 (z) +2γ2 (z)β2 (z) − 2γ 2 (z)β 2 (z) √ H 1 (z) = −k∂φ1 (z) √ + + H 2 (z) = −k∂φ2 (z) − 2ψ1 (z)ψ 1 (z) − 2ψ3 (z)ψ 3 (z) −2ψ1 (z)ψ1+ (z) − 2ψ3 (z)ψ3+ (z). For the negative currents, the results are not so naive. Additional terms are needed to satisfy the Kac-Moody algebra. The negative currents are, √ √ 1 1 ψ1 (z) −k∂φ1 (z) + ψ 1 (z) −k∂φ1 (z) 2 2 1 1 +(ψ3 (z) − ψ1 (z)γ2 (z))β2 (z) − ψ1 (z)ψ3 (z)ψ3+ (z) 2 2 1 1 − (ψ1 (z)γ 2 (z) + ψ1 (z)ψ3 (z)ψ 1 (z))β 2 (z) 2 6 1 1 + − (2γ 2 (z) + ψ1 (z)ψ 3 (z) − ψ1 (z)γ2 (z)ψ 1 (z))ψ 3 (z) + k∂ψ1 (z) 2 6 1√ + f2 (z) = −k(∂φ1 (z) − ∂φ2 (z))γ2 (z) + ψ3 (z)ψ1+ (z) + ψ 3 (z)ψ 1 (z) 2 1 + + γ22 (z)(ψ1 (z)ψ3+ (z) + ψ 1 (z)ψ 3 (z)) 4 1 + + γ2 (z)(ψ1 (z)ψ1+ (z) − 2γ2 (z)β2 (z) − ψ3 (z)ψ3+ (z) + ψ 1 (z)ψ 1 (z) 2 1 + −ψ3 (z)ψ 3 (z)) − γ2 (z)(ψ1 (z)ψ 3 (z)β 2 (z) + ψ3 (z)ψ1 (z)β 2 (z)) 6 −(k + 1)∂γ2 (z) √ √ 1 1 f3 (z) = ψ3 (z) −k∂φ2 (z) + ψ1 (z)γ2 (z) −k(2∂φ1 (z) − ∂φ2 (z)) 2 4 √ 1 1 + +( ψ 3 (z) + γ2 (z)ψ 1 (z)) −k∂φ1 (z) + γ 2 (z)ψ 1 (z) 2 4 1 + [2ψ1 (z)ψ3 (z)ψ1+ (z) + 2γ2 (z)ψ3 (z)β2 (z) + (ψ1 (z)ψ 3 (z) 2 + + −ψ3 (z)ψ 1 (z))ψ 1 (z) − ψ3 (z)γ 2 (z)β 2 (z) − γ2 (z)γ 2 (z)ψ 3 (z)] 1 + − [6ψ1 (z)γ22 (z)β2 (z) − 2ψ1 (z)γ2 (z)ψ 1 (z)ψ 1 (z) 12

f1 (z) =

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Xiang-Mao Ding, Gui-Dong Wang, and Shi-Kun Wang

+(ψ1 (z)γ2 (z)γ 2 (z) + 3ψ1 (z)ψ3 (z)ψ 3 (z))β 2 (z) +

+3(ψ1 (z)γ2 (z)ψ3 (z) + γ2 (z)ψ3 (z)ψ 1 (z))ψ 3 (z)] 1 + + (4ψ1 (z)γ22 (z)ψ 1 (z)ψ 3 (z) − 3ψ1 (z)γ2 (z)ψ3 (z)ψ 1 (z)β 2 (z)) 24 1 1 1 1 5 +(k + )∂ψ3 (z) + (k − )γ2 (z)∂ψ1 (z) − (k + )ψ1 (z)∂γ2 (z) 2 2 6 2 3 √ √ 1 1 f 1 (z) = ψ 1 (z) −k∂φ1 (z) + ψ1 (z) −k∂φ1 (z) + γ 2 (z)ψ3+ (z) 2 2 1 +ψ3 (z)β2 (z) − ψ1 (z)ψ 1 (z)(γ2 (z)ψ3+ (z) − ψ 3 (z)β 2 (z)) 12 1 + − ψ 1 (z)(γ2 (z)β2 (z) + ψ3 (z)ψ3+ (z) + γ 2 (z)β 2 (z) + ψ 3 (z)ψ 3 (z)) 2 +k∂ψ1 (z) √ 1 f 2 (z) = γ 2 (z) −k(∂φ1 (z) + ∂φ2 (z)) 2 √ 1 − (ψ1 (z)ψ 3 (z) + ψ3 (z)ψ 1 (z)) −k(∂φ1 (z) − ∂φ2 (z)) 4 √ 1 + (ψ1 (z)ψ3 (z) − ψ 1 (z)ψ 3 (z)) −k∂φ1 (z) 2 √ 1 + ψ1 (z)γ2 (z)ψ 1 (z) −k(2∂φ1 (z) − ∂φ2 (z)) 6 1 −(ψ1 (z)γ 2 (z) + ψ1 (z)ψ3 (z)ψ 1 (z))ψ1+ (z) 2 1 −(ψ3 (z)ψ 3 (z) − γ2 (z)ψ3 (z)ψ 1 (z) 2 1 1 − ψ1 (z)γ2 (z)ψ 3 (z) + ψ1 (z)γ22 (z)ψ 1 (z))β2 (z) 2 4 1 −(ψ3 (z)γ 2 (z) + ψ1 (z)γ2 (z)ψ3 (z)ψ 1 (z))ψ3+ (z) 12 1 + −(ψ 1 (z)γ 2 (z) − ψ1 (z)ψ 1 (z)ψ 3 (z))ψ 1 (z) 2 1 −(γ 22 (z) − ψ1 (z)ψ3 (z)ψ 1 (z)ψ 3 (z))β 2 (z) 3 1 + −(γ 2 (z)ψ 3 (z) − ψ1 (z)γ2 (z)ψ 1 (z)ψ 3 (z))ψ 3 (z) + k∂γ 2 (z) 12 1 1 1 1 + kψ 3 (z)∂ψ1 (z) − (k − )γ2 (z)ψ 1 (z)∂ψ1 (z) − kψ3 (z)∂ψ 1 (z) 2 6 2 2 1 1 1 + (k − )γ2 (z)ψ1 (z)∂ψ1 (z) − (k + 2)ψ1 (z)ψ 1 (z)∂γ2 (z) 6 2 3 1 1 − (k + 1)ψ 1 (z)∂ψ3 (z) + (k + 1)ψ1 (z)∂ψ3 (z) 2 2 √ √ 1 1 f 3 (z) = ψ 3 (z) −k∂φ2 (z) + γ2 (z)ψ 1 (z) −k(2∂φ1 (z) − ∂φ2 (z)) 2 4

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

283

√ 1 1 + +( ψ3 (z) + ψ1 (z)γ2 (z)) −k∂φ1 (z) + ψ 1 (z)ψ 3 (z)ψ 1 (z) 2 4 1 1 1 +(−γ 2 (z) + ψ1 (z)ψ 3 (z) − ψ3 (z)ψ 1 (z) − ψ1 (z)γ2 (z)ψ1 (z))ψ1+ (z) 2 2 6 1 1 +(γ2 (z)ψ 3 (z) − γ22 (z)ψ1 (z))β2 (z) + ( γ2 (z)γ 2 (z) 2 2 1 1 1 + ψ1 (z)γ2 (z)ψ 3 (z) + γ2 (z)ψ3 (z)ψ 1 (z) − ψ1 (z)γ22 (z)ψ 1 (z))ψ3+ (z) 4 4 6 1 1 1 +(− γ 2 (z)ψ3 (z) + ψ3 (z)ψ 1 (z)ψ 3 (z) − γ2 (z)ψ 1 (z)γ 2 (z) 2 4 12 1 1 + ψ1 (z)γ2 (z)ψ 1 (z)ψ 3 (z))β 2 (z) + (k + )∂ψ 3 (z) 8 2 1 5 1 1 + (k − )γ2 (z)∂ψ 1 (z) − (k + )ψ 1 (z)∂γ2 (z). 2 6 2 3 The energy-momentum tensor corresponding to the quadratic Casimir C is given by 1 X (−1)[j] Eij (z)Eji (z) : T (z) = : 2k ij 1 = − [∂φ1 (z)∂φ2 (z) + ∂φ1 (z)∂φ2 (z)] + β2 (z)∂γ2 (z) 4 + +β 2 (z)∂γ 2 (z) − ψ1+ (z)∂ψ1 (z) − ψ 1 (z)∂ψ 1 (z) 1 + −ψ3+ (z)∂ψ3 (z) − ψ 3 (z)∂ψ 3 (z) − √ ∂ 2 φ1 (z). 2 −k

(4.1)

One can easily check that, for tensor T (z) and all currents (here we denote them as J(z)), we have the following relations: 2T (w) ∂T (w) + , 2 (z − w) z−w J(w) ∂J(w) T (z)J(w) = + . 2 (z − w) z−w

T (z)T (w) =

(4.2) (2)

\ We can see that T (z) is the energy-momentum tensor of the gl(2|2) k current superalgebra, with zero central charge. In this sense, the energymomentum tensor is a spin 2 primary filed with respect to itself, and respect to this tensor, all currents are spin 1 primary fields.

5 Primary Field of the Superalgebra The representation of the algebra is an important problem in Lie theory and theoretical physics. In this section, we will give the primary field, it corresponds to the representation of the algebra. As mentioned above, for the

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complexity of the expression and the limit of the space, here we will just give the primary field that corresponds to the representation of the superalgebra psl(2|2)(2) . Let 1 V (z) = exp{ √ (β1 φ1 (z) + β2 φ2 (z) + β 1 φ1 (z) + β 2 φ2 (z))} k

(5.1)

in which β1 , β2 , β 1 and β 2 are four parameters. It is easy to see that the state V (z) is a highest state of the algebra. The conformal weight is 4=

2β2 (2β1 + 1) + 4β 1 β 2 k

(5.2)

2

If β22 = β 2 = 0, then the conformal weight 4 = 0 and the primary field corresponds to the atypical representation of the algebra. We will just consider this situation here. In this case, we obtained four linear independent series, 1. Bβn,m(z; 0) = β1 γ (z)β1 −n−2 γ2 (z)β1 −m−1 [4β1 γ 22 (z)γ2 (z) 3β1 − n 2 +(β1 − n)(β1 + m)γ 2 (z)γ2 (z)(ψ1 (z)ψ 3 (z) + ψ3 (z)ψ1 (z)) 1 − (β1 − n)(5β1 + 3m)γ 2 (z)γ22 (z)ψ1 (z)ψ1 (z) 6 +2(β1 − n)(β1 − m)γ 2 (z)ψ3 (z)ψ 3 (z) 1 + (β1 − n)(β1 − n − 1)(2β1 + m)γ2 (z)ψ3 (z)ψ 1 (z)ψ1 (z)ψ 3 (z)]V (z) 3 for this series we have β1 + m n,m−1 B (w; 0) z−w β 3β1 − n + 1 n−1,m Bβ (w; 0) f 2 (z)Bβn,m (w; 0) = − z−w f2 (z)Bβn,m (w; 0) =

2. f1 (z)Bβn,m (w; 0) =

1 F n,m (w; 0) z−w 1

in which F1n,m (w; 0) = β1 (β1 − m)γ 2 (w)β1 −n−1 γ2 (w)β1 −m−1 [−γ 2 (w)γ2 (w)ψ1 (w) + 2γ 2 (w)ψ3 (w) −(β1 − n)ψ1 (w)ψ3 (w)ψ 3 (w)

(5.3)

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

285

1 + (β1 − n)γ2 (w)ψ1 (w)ψ3 (w)ψ 1 (w)]V (w) 6 3. f 1 (z)Bβn,m (w; 0) =

1 n,m F 1 (w; 0) z−w

(5.4)

in which n,m

F1

(w; 0) = β1 (β1 − m)γ 2 (w)β1 −n−1 γ2 (w)β1 −m−1 [−γ 2 (w)γ2 (w)ψ 1 (w) + 2γ 2 (w)ψ 3 (w) 1 − (β1 − n)γ2 (w)ψ1 (w)ψ 1 (w)ψ 3 (w) 6 +(β1 − n)ψ 3 (w)ψ3 (w)ψ 1 (w)]V (w)

4. n,m

f1 (z)F 1

(w; 0) =

(β1 − m)(3β1 − n + 1) n−1,m+1 B1,1 (w; 0) z−w

(5.5)

in which n,m B1,1 (w; 0) =

β1 γ (w)β1 −n−2 γ2 (w)β1 −m−1 3β1 − n 2 [−2γ 22 (w)γ2 (w) + 2(β1 − m)γ 2 (w)ψ3 (w)ψ 3 (w) −(2β1 − m − n)γ 2 (w)γ2 (w)(ψ1 (w)ψ 3 (w) + ψ3 (w)ψ 1 (w)) 1 + (7β1 − 3m − 4n)γ 2 (w)γ22 (w)ψ1 (w)ψ 1 (w) 6 1 − (β1 − n − 1)(5β1 − 2m − 3n)γ2 (w)ψ1 (w)ψ 3 (w)ψ3 (w)ψ 1 (w)]V (w) 6 For this four series, the OPEs with rise operators and Cartan parts are as follows: for the serial Bβn,m (w; 0), we have 1 (β1 − n)(β1 + m) n+1,m−1 F1 (w; 0) z − w (3β1 − n)(β1 − m + 1) β1 − m n,m+1 e2 (z)Bβn,m (w; 0) = B (w; 0) z−w β

e1 (z)Bβn,m (w; 0) =

1 β1 − n n+1,m F (w; 0) z − w 3β1 − n 1 1 (β1 − n)(β1 + m) F n+1,m−1 (w; 0) e1 (z)Bβn,m (w; 0) = − z − w (3β1 − n)(β1 − m + 1) 1

e3 (z)Bβn,m (w; 0) =

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Xiang-Mao Ding, Gui-Dong Wang, and Shi-Kun Wang

1 (3β1 − n − 1)(β1 − n) n+1,m Bβ (w; 0) z−w 3β1 − n β1 − n n+1,m 1 F (w; 0) =− 3β1 − n z − w 1 2(2β1 − m − n) n,m =− Bβ (w; 0) z−w 2(2β1 + m − n) n,m Bβ (w; 0) =− z−w −4β 2 n,m B (w; 0) = z−w β −4β 1 n,m B (w; 0) = z−w β

e2 (z)Bβn,m (w; 0) = e3 (z)Bβn,m (w; 0) H1 (z)Bβn,m (w; 0) H2 (z)Bβn,m (w; 0) H 1 (z)Bβn,m (w; 0) H 2 (z)Bβn,m (w; 0)

and for the serial F1n,m (w; 0), we have

e1 (z)F1n,m (w; 0)

e2 (z)F1n,m (w; 0) e3 (z)F1n,m (w; 0) e2 (z)F1n,m (w; 0) f2 (z)F1n,m (w; 0) f 2 (z)F1n,m (w; 0)

−(2β1 − m − n) n,m Bβ (w; 0) z−w (β1 + m)(β1 − n) n,m + B13 (w; 0) z−w β1 − m n,m+1 F (w; 0) = z−w 1 (β1 − m)(β1 − n) n,m+1 β1 − m n,m+1 (w; 0) Bβ (w; 0) − B11 = z−w z−w β1 − n n+1,m = F (w; 0) z−w 1 1 (β1 + m)(β1 − m) n,m−1 = F1 (w; 0) z−w β1 − m + 1 3β1 − n + 1 n−1,m =− F1 (w; 0) z−w =

n,m

similarly for the serial F 1 n,m

e2 (z)F 1

n,m

e1 (z)F 1

n,m

e2 (z)F 1

n,m

e3 (z)F 1

n,m

f2 (z)F 1

β1 − m n,m+1 F (w; 0) z−w 1 (β1 − m) n,m (β1 − n)(β1 − m) n,m = − Bβ (w; 0) − B31 (w; 0) z−w z−w β1 − n n+1,m = F (w; 0) z−w 1 β1 − m n,m+1 (β1 − n)(β1 − m) n,m+1 = B (w; 0) − B11 (w; 0) z−w β z−w 1 (β1 + m)(β1 − m) n,m−1 = F1 (w; 0) z−w β1 − m + 1

(w; 0) = (w; 0) (w; 0) (w; 0) (w; 0)

(w; 0) we have

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k n,m

f 2 (z)F 1

(w; 0) = −

287

3β1 − n + 1 n−1,m F1 (w; 0) z−w

n,m finally for the bosonic series B11 (w; 0) we have n,m (β1 − m + 1)e1 (z)B11 (w; 0) = −

(2β1 − m − n) n+1,m−1 1 F1 (w; 0) 3β1 − n z−w

β1 − m − 1 n,m+1 B11 (w; 0) z−w 1 n+1,m n,m (w; 0) = F (w; 0) (3β1 − n)e3 (z)B11 z−w 1 (2β1 − m − n) n+1,m−1 1 n,m (β1 − m + 1)e1 (z)B11 (w; 0) = F1 (w; 0) 3β1 − n z−w 1 β1 − n − 1 n+1,m 1 n,m e2 (z)B11 (w; 0) = (w; 0) − B11 B n+1,m (w; 0) z−w z − w 3β1 − n β 1 n,m F n+1,m (w; 0) (w; 0) = − (3β1 − n)e3 (z)B11 z−w 1 1 n,m B n,m−1(w; 0) (w; 0) = − (β1 − m + 1)f2 (z)B11 z−w β (β1 + m − 1)(β1 − m) n,m−1 + B11 (w; 0) z−w 3β1 − n + 1 n−1,m n,m (w; 0) = − (w; 0) f 2 (z)B11 B11 z−w n,m e2 (z)B11 (w; 0) =

It is easy to see that when β1 6= 0, the representation is non trivial but infinite dimensional. The nontrivial finite dimensional representation will exist when β2 6= 0, but as we have mentioned before, in this case the expression will be very complex and we will not discuss it here.

Acknowledgments: X.-M. Ding thanks Prof. A. Bellen for his warm invitation and great help during his stay in Trieste, where the work was partially complected. Thanks also to Prof. G. Lindi for his kindness. This work is (partially) supported by Inistero degli Affari Esteri- Direzione Generale per la Promozione la Cooperazione Culturale, and by Istituto Nazionale di Alta Matematica, francesco severi(INdAM), Roma. The work is also financially supported partly by Natural Science Foundation of China through the grands No.10231050 and No.10375087.

References 1. R. Borcherds, Vertex algebras, Kac-Moody algebras and the Monster, Proc.Natl.Acad.Sci.USA, 83(1986)3068-3071.

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2. P. Goddard, A. Kent and D. Olive, Virasoro Algebras and Coset Space Models, Phys.Lett. 152B(1985)88. 3. E. Witten, Commun. Math. Phys. 92, (1984) 445. 4. V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd Edition, Cambridge Univ. Press, Cambridge, 1990. 5. V.G.Kac, Lie superalgebras, Adva. in Math. (1977),26: 8-96. 6. A.A.Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nucl. Phys. B241, (1984) 333. 7. Ph. Di Francesco, P. Mathieu, D. Senehal, Conformal Field Theory, Springer, Berlin, 1997. 8. J. Polchinski, String Theory, Cambridge Univ, Press,Cambridge, 1998. 9. M. Wakimoto, Commun. Math. Phys. 104, (1986) 605. 10. B.L. Feigin and E. Frenkel, Commun. Math. Phys. 128, (1990) 161. 11. J.L. Petersen, J. Rasmussen and M. Yu, Nucl. Phys. B502, (1997) 649. 12. M. Szczesny, Math. Res. Lett. 9, (2002) 433. 13. X.-M. Ding, M. D. Gould and Y.-Z. Zhang, Phys. Lett. B523, (2001) 367. 14. L. Feher and B.G. Pusztai, Nucl. Phys. B674, (2003) 509. 15. X.-M. Ding, M. D. Gould and Y.-Z. Zhang, Phys. Lett. (2003), A318: 354. 16. X.-M. Ding, M. D. Gould, C. J. Mewton and Y.-Z. Zhang, Jour. Phys. A 36,(2003) 7649. 17. Yao-Zhong Zhang, Coherent State Construction of Representations of osp(2|2) and Primary Fields of osp(2|2) Conformal Field Theory, Phys.Lett.A327(2004)442-451. 18. Yao-Zhong Zhang and Mark D. Gould, A Unified and Complete Construction of All Finite Dimensional Irreducible Representations of gl(2|2), J.Math.Phys.46(2005)013505. 19. Yao-Zhong Zhang, Super Coherent States, Boson-Fermion Realizations and Representations of Superalgebras, hep-th/0405066. 20. Yao-Zhong Zhang, Xin Liu and Wen-Li Yang, Primary Fields and Screening Current of gl(2|2) Non-unitary Conformal Field Theory, Nucl.Phys.B704(2005)510526. 21. A. Perelomov, ”Generalized Coherent States and Their Applications”, SpringerVerlag, 1986.

A Solution of Yang-Mills Equation on BdS Spacetime Xin’an Ren1,2 and Shikun Wang1,3 1

2 3

Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China Graduate school of Chinese Academy of Science, Beijing 100080, China KLMM, Academy of Mathematics and Systems Sciences,Chinese Academy of Sciences, Beijing 100080, China [email protected], [email protected]

Abstract Inspired by Prof. Lu’s lecture note [1], we will give the global solutions of Yang-Mills equation on de-Sitter spacetime with BHL metric, instead of the conformal metric. The project partially supported by NKBRPC(]2004CB31800, ]2006CB805905) and NNSFC(]10375087)

1 Introduction Recently, there are many authors [2-5] who have discussed the AdS/CF T correspondence. However, most of these discussions on AdS/CF T correspondence are based on the so-called Euclidean version of the AdS5 , or a 5dimensional unit ball B 5 . As was pointed out by Prof. Lu [6] the AdS/CF T correspondence of Lorentz version is the duality between a field theory on AdS5 and a conformal field theory on its boundary, the conformal space. It is well known that there are only three constant spacetimes, Minkowski, de-Sitter and anti-de-Sitter spacetimes with invariant groups being Poincare group, de-Sitter group SO(1, 4) and SO(2, 3). Besides various usual dS/AdS spacetimes with different metrics, there is a special one with most important properties analogue to Minkowski spacetime that should be paid attention to. It is the dS/AdS spacetime with Beltrami-Hua-Lu (BHL) metric, called BdS/BAdS spacetime [7,8]. Recent observations in cosmology show that our universe is in accelerated expansion [9]. This implies that the universe is probably asymptotically deSitter with positive cosmological constant Λ. So it is important to discuss the dS/CF T correspondence. However, first we should discuss the field theory on 289

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Xin’an Ren and Shikun Wang

de-Sitter spacetime. Prof. Lu has discussed the Yang-Mills equation on deSitter space with conformal flat metric in [1]. In this paper, we consider the BdS spacetime with BHL metric instead of conformal flat metric and find a global solution of Yang-Mills equation. This paper is organized as follows. In section 2, we give the definition of BdS spacetime with BHL metric in terms of classical domains and discuss the de-Sitter group. Yang-Mills equation on BdS is discussed and a global solution is given in section 3.

2 The Beltrami de-Sitter spacetime Originally, according the works of Professors Look, Tsou and Kuo, the deSitter spacetime can be regarded as the classical domain Dλ (1, 4), but here we firstly define the classical domains Dλ (m, n) = {X ∈ Rm×n |I − λXJX 0 > 0},

(1)

where J = {1, −1, · · · , −1} is diagonal matrix of order n and λ is a real constant. Let A, B, C, D be m × m, n × m, m × n, n × n matrices respectively and satisfy 0 I 0 A C I 0 A C , (2) = 0 − λJ B D 0 − λJ B D which means that AA0 − λCJC 0 = I,

AB 0 = λCJD0 ,

BB 0 − λDJD0 = −λJ,

(3)

λC 0 C − D0 JD = −J.

(4)

or λA0 A − B 0 JB = λI,

λA0 C = B 0 JD,

Therefore the transformation Y = (A + XB)−1 (C + XD)

(5)

must maps Dλ (m, n) one to one to itself because I − λY JY 0 = (A + XB)−1 (I − λXJX 0 )[(A + XB)−1 ]0 .

(6)

Moreover, there is in Dλ (m, n) a metric ds2 = tr[(I − λXJX 0 )−1 dX(J − λX 0 X)dX 0 ],

(7)

which is invariant under the transformation (2.5). Set X0 = −CD−1 , and the transformation can be written as

A Solution of Yang-Mills Equation on BdS Spacetime

Y = A−1 (I − λXJX00 )−1 (X − X0 )D.

291

(8)

From the last equation in (2.4), we have that D satisfies (DJD0 ) = (J − λX00 X0 )−1 and from the first and second equations in (2.4), it is not difficult to see that A satisfies (AA0 ) = (I − λX0 JX00 )−1 . (9) That is to say the transformation (2.5) can map every point X0 to Y = 0, which means Dλ (m, n) is homogeneous under the action of the group composed by transformation (2.8). In the case of Dλ (1, 4), X = (x0 , x1 , x2 , x3 ) and the condition (2.1) reduces to σ = σ(x, x) = 1 − ληij xi xj > 0, (10) where ηij (i, j = 0, 1, 2, 3) are elements in the diagonal matrix J . The metric in (2.7) reduces to ηjk ληjr ηks xr xs dX(J − λX 0 X)−1 dX 0 j k = g dx dx = ( + )dxj dxk , jk 1 − λXJX 0 σ σ2 (11) which is called the Beltrami-Hua-Lu (BHL) metric. Then λ is the Riemannian curvature of Dλ (1, 4). Moreover, by setting X0 = (a0 , a1 , a2 , a3 ), the transformation (2.8) has the form of ds2 =

y i = (σ(a, a))1/2

(xj − aj )Dji , σ(x, a)

(12)

where σ(x, a) = 1 − ληij xi aj and Dji satisfy ηpq Dip Djq = ηij +

ληir ηjs ar as . σ(a)

(13)

From the discussion above, we see that the de-Sitter group SO(1, 4) composed of the transformation (2.12) acts transitively on de-Sitter spacetime and leaves the BHL metric (2.11) invariant.

3 The connections and Yang-Mills equation on BdS spacetime In this section we will find the solutions of the Yang-Mills equation on BdS spacetime. First we compute the Christoffel symbol associated to the BHL metric ∂gki ∂gjk ∂gji 1 l + − , (1) = g li j k 2 ∂xk ∂xj ∂xi

292

Xin’an Ren and Shikun Wang

which is an gl(4, R) connection. By formula (2.11), we can see easily that g ij = σ(x, x)(η ij − λxi xj ),

(2)

and ηij ∂σ 2λxi xj ∂σ λ ∂ ∂gij =− 2 − + 2 k (xi xj ) ∂xk σ ∂xk σ 3 ∂xk σ ∂x 4λ2 xi xj xk 2ληij xk + ληik xj + ληjk xi + = 2 σ σ3 Moreover, by (3.1), we have 1 2ληij xk + ληik xj + ληjk xi 4λ2 xi xj xk l il i l = σ(η − λx x ) + j k 2 σ2 σ3 2ληik xj + ληij xk + ληjk xi 1 + σ(η il − λxi xl ) 2 σ2 2ληkj xi + ληik xj + ληji xk − (3) σ2 4λ2 xi xj xk 2ληij xk + 2ληik xj 1 + = σ(η il − λxi xl ) 2 σ2 σ3 2λ2 xl xj xk λ l 2λ2 xl xj xk 2λ2 xl xj xk (1 − σ) δj xk + δkl xj + = − − 2 σ σ σ σ2 λ l = δ xk + δkl xj , (4) σ j where δjl is the Kronecker symbol and xi = ηij xj . Let 3 X gjk dxj dxk = ηab ω a ω b ds2 =

(5)

j,k=0

be a Lorentz metric on M , where (ηab ) = {1, −1, −1, −1} is a diagonal matrix and (a) ω a = ej dxj , (a = 0, 1, 2, 3) (6) and

∂ , (a = 0, 1, 2, 3) (7) ∂xj are the Lorentz coframe and the dual frame respectively. It is not difficult to see that ληjk a k 1 (a) √ x x δja + ej = √ (8) σ σ+ σ Xa = ej(a)

and ej(a)

=

√

σ

δaj

ληab b j √ x x − 1+ σ

From the Christoffel symbol, we get a so(1, 3)- connection as following

(9)

A Solution of Yang-Mills Equation on BdS Spacetime (a)

∂ek(b)

(a)

l

293

ek(b) j k ∂xj 1 λxa xl √ = √ δka + × σ σ+ σ √ λδ k xj λ 2 xb xj xk λ σ √ ηbj xk + δjk xb + √ √ 2 − √b − σ 1+ σ σ(1 + σ) a k λ λx xl λxb xk l l a √ √ + δ j xk + δ k xj δ b − δl + σ σ+ σ 1+ σ √ λ λ √ xa xb xj = δba xj + σδja xb + σ 1+ σ √ λ a λ σ a λ λ 2 xa xb xj 1 a √ δ xb − √ ηbj x − √ √ − √ δ b xj − +√ σ σ 1+ σ j 1+ σ σ(1 + σ) λ √ δja xb − ηbj xa . = σ+ σ

a Γbj = ek

+ el

Moreover, according to Theorem. 2.4.2 in [10], associated to the so(1, 3)a connection Γbj , there is a sl(2, C)-connection 1 a Aj = η cb Γcj σa σb∗ , 4

σb∗

0

= ¯ σb ,

=

0 1 −1 0

(10)

,

where σ0 =

1 0 0 1

, σ1 =

0 1 1 0

, σ2 =

0 −i i 0

, σ3 =

1 0 0 −1

.

(11)

This connection is globally defined in BdS spacetime because there is spin structure on it, which is implied by Dirac’s paper [11]. We are now in a position to compute the connection 1 cb a η Γcj σa σb∗ 4 λ 1 cb a a √ δ xc − ηcj x σa σb∗ = η 4 σ+ σ j λ √ δja xb − δjb xa σa σb∗ . = 4(σ + σ)

Aj =

(12)

It remains to prove that Aj satisfies the Yang-Mills equation defined by ∂Fjk r r g kl Fjk;l ≡ g kl + [A , F ] − F − = 0, (13) F l jk rk jr j l k l ∂xl where Fjk =

∂Aj ∂Ak − + [Aj , Ak ] ∂xj ∂xk

(14)

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Xin’an Ren and Shikun Wang

and

r

is the Christoffel symbol of the BHL metric. j l According to (3.11), the elements of Aj are odd functions of xj , which means that [Aj (x)]|x=0 = 0. Hence all elements of Fjk are even functions of xj . It is not difficult to see that all elements of Fjk;l are odd functions of xj and consequently [Fjk;l (x)]x=0 = 0. Since BdS spacetime is transitive under the group SO(1, 4), for any point x0 of BdS there is at least a transformation (2.13) which transform x0 to y = 0. Since both gjk and Fjk;l are covariant under the transformation (2.13), ∂xp ∂xq ∂xr 0 = [Fjk;l ]y=0 = UT (x)Fpq;r UT (x)−1 j k , (15) ∂y ∂y ∂y l x=x0

which implies that [Fpq;r ]x=x0 = 0. Since x0 can be an arbitrary point of BdS spacetime, we have Fjk;l = 0 and obviously it satisfies the Yang-Mills equation. It should be noted that an su(2)-connection can also be obtained by using the Reduction Theorem of Connections. Moreover, since the BHL metric gij is a solution of Einstein equation with cosmological constant Λ = 3λ, we get in fact a solution (gij , Aj ) of the Einstein-Yang-Mills equation defined as follows Rjk − 12 Rgjk + Λgjk = 0 g kl Fjk;l = 0, where Rjk and R are Ricci tensor and scalar curvature of BHL metric, respectively.

Acknowledgements We would like to thank Professors H. Y. Guo and K. Wu for valuable discussions and comments. Special thanks are given to Prof. Q. K. Lu who gave us a long series of lectures on conformal space and de-Sitter space. The first author would also like to thank Professor H. W. Xu for his encouragements and assistances.

References 1. Q. K. Lu, Global solutions of Yang-Mills equation, preprint. 2. J. Maldacena, The large N superconformal field theory and supergravity, Adv. Theor. Math. Phys., 2, 1998, 231. 3. J. L. Petersen, Introduction to the Maldacena conjecture on AdS/CF T , NBIHE-99-05. 4. E. Witten, Anti-de-Sitter space and holography, Adv. Theor. Math. Phys., 2, 1998, 253. 5. E. Witten and S.-T. Yau, Connectedness of the boundary in the AdS/CF T correspondence, hep-th/9910245.

A Solution of Yang-Mills Equation on BdS Spacetime

295

6. Q. K. Lu, A note to a paper of Dirac in 1935, Lecture at Morningside Center, Beijing. 7. H. Y. Guo, C. G. Huang, Z. Xu and B. Zhou, On Beltrami model of de Sitter spacetime, hep-th/0311156. 8. K. H. Look, C. L. Tsou and H.Y. Kuo, The kinematic effects in the classical domains and the red-shift phenomena of extra-galactic objects”, Acta Physica Sinica. 23, 1974, 225. 9. M. Tegmark, et all, Arxiv:astro/0310723. 10. Q. K. Lu, Differential Geometry and its application to physics, Science Press, Beijing, 1982. 11. P. A. M., Dirac, The electron wave equation in de-Sitter space, Ann. Math., 36, 1935, 657.

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Solitonic Information Transmission in General Relativity Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, 55, Zhong Guan Cun Donglu, Beijing, China, 100080

Abstract An exact solution of the vacuum Einstein’s field equations is presented in which there exists a congruence of null geodesics whose shear behaves like a travelling wave of the KdV equation. On the basis of this exact solution, the feasibility of solitonic information transmission by exploiting the nonlinearity intrinsic to the Einstein field equations is discussed. Partially Supported by NKBRPC(2004CB31800,2006CB805905) and NNSFC(] 10231050; 10375087)

1 Introduction In electromagnetism, though the source free Maxwell equations are linear, the propagation of electromagnetic waves in a continuous media can display nonlinear behaviour due to the nonlinear structure of the media at the macroscopic level. When there is a balance between compression effect caused by nonlinearity and dispersive wave behaviour, a solitary wave occurs as a result of the interplay of these two competing factors (see for instance [1]). This is the mechanism underlying the formation of optical soliton propagating in an optical fibre. Optical soliton is highly stable under external interference and allows rapid and reliable information transmission, superior to conventional means of information transmission by means of cable. The present work aims to point out, in terms of an exact solution of the vacuum Einstein field equations, that instead of a continuous media, the nonlinear structure inherent in general relativity may also be exploited to achieve solitonic information transmission. The gravitational wave background itself acts as an effective continuous media to generate solitonic behaviour for light 297

298

Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

propagation. Though the example to be presented below is highly idealistic and quite remote from practical implementation, still at least in theory it is a feasibility allowed by Einstein’s theory of general relativity. Throughout the present work, notations for the Newman Penrose formalism will be adopted from that in [2]

2 Description of the exact solution Consider a spacetime manifold whose metric may be expressed as ds2 = 2dU dV − e2η cosh ω(dx2 + e2χ dy 2 ) − 2e2η+χ sinh ωdxdy

(1)

where ζ, η, ω, χ are functions of the null coordinates U, V in general , ∂/∂x, ∂/∂y are two commuting spacelike Killing vector fields. Commutativity of ∂/∂x, ∂/∂y implies the existence of a spacelike two-surface whose tangent space at each point is spanned by ∂/∂x, ∂/∂y. While ∂/∂U, ∂/∂V are the two null normals of the spacelike two-surface spanned by ∂/∂x, ∂/∂y. Given the metric in (1), a Newman-Penrose tetrad may be set up naturally as ∂ , ∂U ∂ na = , ∂V √ 1 ∂ ∂ ma = √ e−η cosh ω(eiφ − ie−χ ), ∂x ∂y 2 ∂ ∂ 1 −η √ a cosh ω(e−iφ + ie−χ ) m ¯ = √ e ∂x ∂y 2 la =

(2)

where sin φ = tanh ω. With the Newman-Penrose tetrad in (2) and the metric in (1), it may further be computed that the spin coefficients κ = κ0 = 0. This means that the two null normals ∂/∂U, ∂/∂V define in a very natural way two null geodesic congruences orthogonal to the two-surface generated by ∂/∂x, ∂/∂y, and U, V are affine parameters of the respective geodesic congruences. Suppose one of the two null geodesic congruences defined above (say the one defined by ∂/∂V ) is shearfree and subject further to the vanishing of the Ricci curvature of the metric in (1), then η, w, χ become functions of U only. Further, the Newman Penrose equations consisting of a system of coupled differential equations may be reduced drastically to a second-order linear ODE of Sturm-Liouville type (detailed derivation is in [3], see also [4]), given by d2 f + |σ|2 f = 0 dU 2 where

(3)

Solitonic Information Transmission in General Relativity

299

f = e2η+χ , σ is the shear of the null geodesic congruence defined by ∂/∂U and its modulus is denoted by |σ|. σ is further given by 1 1 ∂w ∂χ i σ = (1 + i sinh w) + + i tanh w 2 ∂u 2 cosh w ∂u with 1 |σ| = 4 2

(

2

cosh ω

∂χ ∂u

2

+

∂w ∂u

2 ) (4)

Exact solutions of the Einstein field equations may then be constructed from (f, η, χ) subject to (3) and (4). The metric may be written as ds2 = 2dU dV − f cosh ω(e−χ dx2 + eχ dy 2 ) − 2f sinh ωdxdy.

(5)

It describes a type N spacetime of a self interacting gravitational wave background. The Weyl curvature components of the metric are all zero except d ∂σ ∂χ 1 ∂ω Ψ0 = 2 σ+ ln f + i sinh ω − dU ∂U cosh ω ∂U ∂U Among the vacuum solutions of the Einstein field equations constructed this way, there is a class of exact solutions described by the ordered pair 1 1 1 , ; 1, tanh U , sech2 U , (6) (f, |σ|2 ) = F 2 2 2 where F is the hypergeometric function. We note that the Sturm Liouville equation defined by (f, |σ|2 ) also gives rise to a form of travelling wave of the KdV equation [5], with the wave profile of the KdV wave given by |σ|2 . As the KdV equation originates from the isospectral deformation of a Sturm-Liouville equation, it is perhaps not too surprising to find a solution of this kind, in view of the Sturm-Liouville equation in (3) derived from Newman-Penrose equations. Away from the possible singular points of the Weyl curvature caused by focusing, the shear of the congruence of null geodesics defined by ∂/∂U behaves like the a solitary wave of the KdV equation (see Fig. 1 below), this suggests the feasibility of solitonic information transmission in the spacetime background defined by the metric in (5), whose functional form is specified by (6) and (4). The information to be transmitted is encoded into the shear, say for instance in its maximum amplitude. Like in the case of optical solitons, the information is faithfully transmitted without any distortion due to the solitonic behaviour of the information carrier. The class of examples described above suggests that, in general relativity, if the gravitational field is manipulated in the right way, solitonic behaviour for

300

Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

light propagation is feasible. As a result, instead of an optical fibre, we may use a relativistic gravitational field as an effective continuous media for solitonic information transmission. Note also that we have actually described not a single but a class of spacetimes. Different choices of (f, η, χ) subject to (3) and (4) correspond to gravitational waves of different wave profile and polarisation. Our experience in the KdV equation also indicates that the solitonic behaviour described here is also likely to be stable.

I

I

V =0

U =0

1

Shear of the null geodesic congruence propagates like a KdV travelling wave

Figure 1: Propagation of the shear of the null geodesic congruence defined by ∂/∂U

References 1. Akira Hasegawa, Optical Solitons in Fibers (1989) (Springer-Verlag Press). 2. R. Penrose and W. Rindler, Spinors and Spacetime, Vol. 1, Camb. Univ. Press (1986) Chapter 4. 3. Z.Q. Kuang, Y.K.Lau and X.N.Wu, Gen. Rel. Grav. 31(9) (1999) 1327. 4. C .B. Liang, Gen. Rel. Grav. 27 (1995) 669. 5. P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Camb. Univ. Press (1989) Chapter 4.

1

Difference Discrete Geometry on Lattice Ke Wu1 , Wei-Zhong Zhao1 , and Han-Ying Guo2 1 2

Department of Mathematics, Capital Normal University, Beijing 100037, China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China

Abstract This is an extended version of our previous paper [1] on difference discrete connection and curvature on cubic lattice. Some new results about properties of discrete topological number on bundles with Abelian gauge group are added. Keyword: discrete connection, discrete curvature, noncommutative calculus, lattice gauge theory, discrete Lax pair

1 Introduction The discrete systems play a very important role in various fields, so they are widely studied in different branches. One of the successful ways to deal with quantum gauge field theory non-perturbatively is the lattice gauge theory in high energy physics, which has opened up new directions in both physics and mathematics in order to deal with the gauge potentials as connections in a discrete manner. Among the integrable systems, there are certain discrete integrable ones which can be obtained as the discrete counterparts of the continuous systems by means of the integrable discretization method. For structure preserving algorithms or geometric algorithms in computational mathematics, the continuous systems are discretized such that some important properties like symplectic or multisymplectic structure, or the symmetry of the systems, are preserved in a discrete manner. There are many important discrete systems, and some of them do not even have a proper or unique continuum limit. In this paper, we focus on the problem how to get the discrete counterparts in a systematic manner for such a kind of continuous systems maintaining their important properties like the gauge potentials as connections, the symplectic or multisymplectic structures, the Lax pairs, the symmetries and so on. A simple and direct method to get a discrete counterpart for a given continuous system such as an ODE or a PDE is to discretize the independent variable(s) and let the dependent variables become discrete correspondingly

301

302

Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

without specifying the specially chosen range. However, in most cases all important properties of the continuous system may be lost and the behaviour of the discrete systems are even hard to be controlled. For constructing the most (quarried??)/meaningful ones in all possible discretizations of the corresponding continuous systems, there is a working guide or a structure-preserving criterion: Namely, it is important to look at those discrete systems that preserve as much of the intrinsically important properties of the continuous system as possible (see, e.g. [2, 3, 4]). In the course of discretization, only a few of the most important properties, or ”structure” can be maintained. Thus, we have to select these “structures”, find their discrete counterparts and we have to know how to preserve them discretely at the lowest price to pay. There are, for example, two classes of conservation laws in canonical conservative mechanics. The first class is of phase-area conservation laws characterized by the symplectic preserving property. The other class is related to energy and all first integrals of the canonical equations. Thus, it is needed to know if it is possible to establish such a kind of discrete systems that they not only discretely preserving the “structure”, such as the symplectic structure, but also the energy conservation. And the question is whether it is possible to get these discrete systems by a discrete variational principle. In fact, as far as the discrete variation for the discrete mechanics is concerned, there are different approaches. In the usual approach (see, e.g. [5, 6]), only the discrete dependent variables are taken as the independent variational variables, while their differences are not. However, in the discrete variational principle proposed by two of the present authors and their collaborators recently [3, 4], the differences of the dependent variables as the discrete counterparts of derivatives are taken as the independent variational variables together with the discrete dependent variables themselves. Actually, this is just the discrete analogue of the variational principle for the continuous mechanics, where the derivatives of the dependent variables are dealt together with the independent variational variables. Thus, the difference discrete Legendre transformation can be made and the method can be applied to either discrete Lagrangian mechanics or its Hamiltonian counterpart, via the difference discrete Legendre transformation. The approach has been applied to the symplectic and multisymplectic algorithms in the both Lagrangian and Hamiltonian formalism. It has been also generalized to the case of variable steps in order to preserve the discrete energy in addition to the symplectic or the multisymplectic structure[4, 5, 7]. For the lattice gauge theory as the discrete counterpart of the gauge theory in continuous spacetime, the discrete gauge potentials, field strength and the action have been introduced in a manner almost completely different from (as least apparently) the ones in ordinary gauge fields or in the connection theory [8] in fibre bundle. Although the discrete connection theory has its own right (see, e.g. [9, 10]) as an application of the non-commutative geometry [11], is an important and interesting problem to investigate how the discrete

Difference Discrete Geometry on Lattice

303

gauge potential in the lattice gauge theory is introduced as a kind of discrete connection. Two of us with their collaborator had considered this issue in [12] very briefly in a way different from other relevant proposals (see, e.g. [13, 14]). In this paper, we study further the discrete connection and curvature on a regular lattice in a way similar to the connection and curvature on vector bundle. We define them in different but equivalent manner, find their gauge transformation properties, the Chern class in the Abelian case and related issues. We show that the discrete connection and curvature introduced here are completely equivalent to the ones in the lattice gauge theory on a regular lattice. We also apply these issues to the discrete integrable systems and show that their discrete Lax pairs and the discrete-curvature free conditions are certainly similar to their continuous cases. One of the key points in our approach is that the difference operators acting on the functions space over the regular lattice are still regarded as a kind of independent geometric objects and their dual should be the one forms, such that we can introduce the discrete tangent bundle over the lattice and the cotangent bundle as its dual, with the basis given by the difference operators and one-formes, respectively. They are just the discrete counterparts of the continuous case, where derivatives and the one-formes as their dual play the roles of the bases for the tangent bundle and the cotangent bundle, respectively. In order to do so, the non-commutative differential calculus on the function space over the lattice [12] has to be employed. Similarly, the discrete vector bundle over the regular lattice can also be set up. In the connection theory a la Cartan, the exterior differential of the basis of a vector space at a certain point should be expanded in terms of the basis, where the expanding coefficients are just the coefficients of the connection on the vector space. Since all counterparts of the basis, the exterior differential etc. are provided in the discrete vector bundle over the lattice, the discrete connection can also be introduced in a way similar to the one a la Cartan. This is our simple way to introduce the discrete connection. In the continuous case, there are several equivalence definitions for connection and curvature. Similarly, we introduce here some of the equivalent definitions for the discrete connection and curvature on the lattice. It should be noted that the definition of the discrete connection in terms of the decomposition of the tangent space of the discrete vector bundle may be written in a form without differences. Thus, it can be generalized to the case over the random lattices. The paper is organized as follows. In order to show the background and necessary preparation, we briefly recall the discrete mechanics and the noncommutative differential calculus on hypercubic lattice of high dimension in section 2 and section 3, respectively. In section 4, we discuss the discrete connection, curvature, Chern class and their gauge transformation properties. In section 5, we generalize one of the definitions for the discrete connection to the case of random lattice. Some applications to the lattice gauge theory and discrete integrable systems are given in section 6. We end with some remarks and discussions.

304

Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

2 Difference Discrete Mechanics In order to introduce the structure of a discrete bundle, it is useful to review the formulism of Lagrangian mechanics and the discrete mechanics as its discrete counterpart. Let t ∈ T be the time, M an n-dimensional configuration space, and we take here a vector space for simplicity. A particle moving on the configuration space is denoted in terms of its generalized coordinates as q i (t) ∈ M and its generalized velocities q˙i (t) = dq i (t)/dt as an element in tangent bundle T M of M . The space of all the possible paths of a particle moving in configuration space is an infinite dimensional space. The Lagrangian of a system is a functional defined on this space and denoted as L(q i (t), q˙i (t)), i = 1, · · · , n. For simplicity, the Lagrangian in our discussion is of first order and independent of t. The action functional is Z t2 i S([q (t)]; t1 , t2 ) = dtL(q i (t), q˙i (t)). (1) t1

Here q i (t) describes a curve Cab with end points a and b, ta = t1 , tb = t2 , along which the motion of the system may take place. For the difference discrete Lagrangian mechanics, let us consider the case that “time” t is difference discretized t ∈ T → tk ∈ TD = {(tk , tk+1 = tk + ∆tk = tk + h,

k ∈ Z)}

(2)

and the step-lengths ∆tk = h are equal to each other for simplicity, while the n-dimensional configuration space Mk at each moment tk , k ∈ Z, is still continuous and smooth enough. S Let N be the set of all nodes on TD with index set Ind(N ) = Z, M = k∈Z Mk the configuration space on TD that is at least pierce wisely smooth enough. At the moment tk , Nk be the set of nodes neighboring to tk . Let Ik the index set of nodes of Nk including tk . The coordinates of Mk are denoted by q i (tk ) = q i(k) , i = 1, · · · , n. T (Mk ) the tangent bundle of S Mk in the sense that difference at tk is its base, T ∗ (Mk ) its dual. Let Mk = S l∈Ik Ml be the union of configuration spaces Ml at tl , l ∈ Ik on Nk , T Mk = l∈Ik T Ml the union of tangent bundles on Mk , F (T Mk ) and F (T Mk ) the functional spaces on each of them respectively, etc.. Sometime, it is also necessary to include the links, plaquettes etc. as well as the dual lattice, like in the lattice gauge theory, the mid-point scheme in the symplectic algorithm and so on. In such cases, the notations and conceptions introduced here should be generalized correspondingly. The above considerations should also make sense for the vector bundle over either the 1-dimensional lattice TD or the higher dimensional lattice as a discrete base manifold. In the difference variational approach and the definition of the difference discrete connection, these notions may be used. Now the discrete Lagrangian LD (k) on F (T (Mk )) reads

Difference Discrete Geometry on Lattice

LD (k) = LD (q i(k) , ∆k q i(k) ),

305

(3)

with the difference ∆k q i(k) of q i(k) at tk defined by ∆k q i(k) :=

q i(k+1) − q i(k) 1 = (q i(k+1) − q i(k) ). tk+1 − tk h

The discrete action of the system is given by X h · LD (k) (q i(k) , ∆k q i(k) ). SD =

(4)

(5)

k∈Z

The discrete variation for q i(k) = q i (tk ) should be defined as i

δq i(k) := q 0 (tk ) − q i (tk ).

(6)

And the discrete variations for ∆k q i(k) are given by δ∆k q i(k) = ∆k δq i(k) .

(7)

Thus, the variations of the discrete Lagrangian can be calculated δLD (k) = [Lqi(k) ]δq i(k) + ∆k (pi (k+1) δq i(k) ),

(8)

where [Lqi(k) ] is the discrete Euler-Lagrange operator [Lqi(k) ] :=

∂LD (k) ∂LD (k−1) , −∆ i(k) ∂q ∂∆q i(k−1)

(9)

and pi (k) the discrete canonical conjugate momenta pi (k) :=

∂LD (k−1) . ∂∆q i(k−1)

And the variation of the discrete action can be written as X δSD = h[Lqi(k) ]δv q i(k) + ∆(pi (k+1) δv q i(k) ).

(10)

(11)

k

The variational principle requires δSD = 0, so the discrete Euler-Lagrange equations for q i(k) ’s follow as ∂LD (k) ∂LD (k−1) − ∆( ) = 0. ∂q i(k) ∂∆q i(k−1)

(12)

In order to transfer to the discrete Hamiltonian formalism, it is necessary to introduce the discrete canonical conjugate momenta according to the equation (10) and express the discrete Lagrangian by the discrete Hamiltonian via Legendre transformation

306

Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

HD (k) := pi (k+1) ∆t q i(k) − LD (k) . Thus, the discrete action can be expressed as X SD = h · (pi (k+1) ∆t q i(k) − HD (k) ).

(13)

(14)

k

Now, Hamilton’s principle requires δv SD = 0, and we obtain the discrete canonical equations for pi (k) ’s and q i(k) ’s ∆q i(k) =

∂HD (k) , ∂pi (k+1)

∆pi (k) = −

∂HD (k) . ∂q i(k)

(15)

As we mentioned, the advantage of the difference discrete variational principle is based on keeping the difference operator as a discrete derivative operator. It is also clear that this approach can be applied to the field theory as well and it can be generalized to the total discrete variation with variable step-lengths [4, 7]. Actually, this key point will also play a central role in our proposal for the discrete connection and curvature. In the usual discrete variation, however, the Q × Q is used to indicate the vector field on a discrete space and the difference has not been dealt with as an independent variable (see, e.g. [5], [15]). The corresponding discrete action is n−1 X S= h · L(qk , qk+1 ), (16) k=0

where the Lagrangian L(qk , qk+1 ) is a functional on Q × Q. This is also the central idea of the resent proposal to the disconnection in [10]. Namely, using the tensor product Q×Q to study discrete tangent spaces of Q. In other words, the tangent vector q(t) ˙ at tk is represented by a pair of nodes (qk , qk+1 ) without introducing the difference operator. Thus, the difference discrete Legendre transformation and the discrete Hamiltonian formalism via the transformation cannot be formulated. In this case it is expected that the groupoid formalism may be used and there is a possibility to understand some of the geometric meaning of discrete models [16].

3 Difference and Differential Form on Lattice In this section, we recall the application of the differential calculus on discrete group [13] to the hypercubic lattice [12]. Although the result is similar to the one in [14], the key point is different. In our approach, the shift operator is regarded as the generator of a discrete Abelian group in each direction of the high dimensional hypercubic lattice. For simplicity, we focus on the lattice with equal spacing h = 1. Thus, the dimension of vector fields or differential forms is equal to the number of the shift operators of the lattice.

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Let N and A be a lattice and the algebra of complex valued functions on N , respectively, then define the right and left shift operators Eµ , Eµ−1 at a node x ∈ N in the µ-direction by Eµ x = x + µ ˆ,

Eµ−1 x = x − µ ˆ,

(17)

and introduce a homeomorphism on the function space A, Eµ (f (x) · g(x)) = Eµ f (x) · Eµ g(x),

Eµ f (x) = f (x + µ ˆ),

f, g ∈ A, (18)

where (x − µ ˆ), x and (x + µ ˆ) are points on Nx and they are the nearest neighbors in the µ-direction, the dot denotes the multiplication in A. The tangent space at the node x of Nx is defined as T Nx := span{∆µ |x , µ = 1, · · · , n}, where the operator ∆µ is defined on the link between x and x + µ ˆ and its action on A is the differences in µ-th direction as, ∆µ f (x) := (Eµ − id)f = f (x + µ ˆ) − f (x).

(19)

The above difference operator is a discrete analogue of the basis ∂ν := ∂x∂ ν for a vector field X = X ν ∂ν in the continuous case. The action of a difference operator ∆µ in (19) on the functional space A satisfies the deformed Leibnitz rule ∆µ (f (x) · g(x)) = ∆µ f (x) · g(x + µ ˆ) + f (x)∆µ g(x).

(20)

For a given node x ∈ Nx , all ∆µ form a set of bases of the tangent space T Nx . Its dual space denoted as T ∗ Nx is a space of 1-forms with a set of bases dxµ defined on the link, too. They satisfy dxµ (∆ν ) = δνµ , which is also denoted as Ω 1 and Ω 0 = A like the continuous case. Thus, the tangent bundle and its dual cotangent bundle over N can be defined as [ [ T N := T Nx and T ∗ N := T ∗ Nx , (21) x∈N

x∈N

respectively. L n Let us construct the whole differential algebra Ω ∗ = Ω on T ∗ N as in n∈Z

continuous case. The exterior derivative operator dD : Ω k → Ω k+1 is defined as X dD ω = ∆α f dxα ∧ dxµ1 ∧ · · · ∧ dxµk ∈ Ω k+1 , (22) α

where ω = f dxµ1 ∧ · · · ∧ dxµk ∈ Ω k . When k = 0, Ω 0 = A, then d : Ω 0 → Ω 1 is given by

(23)

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Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

dD f =

X

∆α f dxα .

(24)

α

It is straightforward to prove that (a) : (dD f )(v) = v(f ), v ∈ T (N ), f ∈ Ω 0 , (b) : dD (ω ⊗ ω 0 ) = dD ω ⊗ ω 0 + (−1)degω ω ⊗ dD ω 0 ,

ω, ω 0 ∈ Ω ∗ ,

(25)

(c) : d2D = 0, provided that the following conditions hold (1) dxµ ∧ dxν = −dxν ∧ dxµ , (2) dD (dxµ ) = 0, (3) dxµ f = (Eµ f )dxµ , (no summation).

(26)

Thus, we set up a well defined differential algebra. Note that in order to do so the multiplication of functions and one-forms must be noncommutative.

4 Difference Discrete Connection and Curvature The discrete analogue of connection has been given by two of the present authors [12] and others (see, e.g. [14], [9], [10]). In this section we define the (difference) discrete connection in a simple way similar to that in the continuous case based on the noncommutative differential calculus introduced in the last section. As was mentioned, the key point is to replace the difference discrete exterior derivative by the covariant difference discrete derivative for the sections on a bundle. 4.1 Connection, Gauge Transformations and Holonomy Discrete Bundle and Discrete Sections Let P = P (M, G) be a principal bundle over an n-dimensional base manifold M isomorphic to Rn with a Lie group G as the structure group. Now consider its discrete counterpart in the following manner as a discrete principle bundle. Let the manifold M be discretized as Zn , i.e. the hypercubic lattice with equal spacing h = 1, and take such an M ' Zn (a lattice N ) as the discrete base space. For a node x ∈ N , there is a fiber Gx = π −1 (x) isomorphic to the Lie group G as the structure group. The union of all these fibers is called discrete principal bundle and denoted as Q(N, G): [ π −1 (x). (27) Q= x∈N

If the structure group G is a linear matrix group, for example, GL(m, R) or its subgroup, we may also define an associated vector bundle V =

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309

V (N, Rm , GL(m, R)) to the discrete principal bundle Q(N, GL(m, R)). For any x ∈ N there is a fiber Fx = π −1 (x) isomorphic to an m-dimensional linear space Rm with the right action of GL(m, R) on the Fx . The union of all these fibers is called a discrete vector bundle: [ V = π −1 (x). (28) x∈N

The discrete field on a discrete bundle is a section h(x) ∈ G on discrete principal bundle or a ψ j (x) ∈ Rm on the vector bundle for all x ∈ Nx . A section on discrete bundle is a map, [ G:N = Nx −→ Q = Q(N, G), G : x 7→ G(x). (29) Similarly, we can define the discrete section on a discrete vector bundle. The union of the all sections is denoted as Γ (Q) for the discrete principal bundle or as Γ (V ) for the discrete vector bundle, respectively. One example for the discrete vector bundle is the tangent bundle T N over an n-dimensional hypercubic lattice N in eq.(21). The base manifold of this bundle is the hypercubic lattice, the fiber π −1 (x) over x ∈ N is an ndimensional vector space with basis {∆µ , µ = 1, · · · , n}. Another example is its dual bundle T ∗ N , its fiber is also the n-dimensional vector space with basis {dxµ , µ = 1, · · · , n}. For the discrete vector bundle the section space Γ (V ) is also the vector space. The same for the Γ (T ∗ N ). Their tensor product is given by Γ (T ∗ N × V ) = Γ (T ∗ N ) × Γ (V ).

(30)

As was mentioned in the previous section, the one form basis dxµ in the discrete case is defined on a point x ∈ N but it links a point with another nearby point x + µ ˆ. Then the section of one forms in Γ (T ∗ N ) is generally defined on Nx , its structure is very different from the section in continuous case. So we call it a discrete section. The section in Γ (V ) may be the discrete section as in Γ (T N ) of tangent vector bundle T N . However there is another possibility. Namely, the section is defined on the node only as in the discrete principal G-bundle. The direct product here can be simply understood as the discrete counterpart of the direct product in the continuous case in the above manner. Definition of Connection and Gauge Transformations Definition : A difference discrete connection or covariant difference discrete derivative is the linear map D : Γ (V ) −→ Γ (T ∗ N × V ), which satisfies the following condition:

(31)

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Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

D(s1 + s2 ) = Ds1 + Ds2 , D(as) = dD a ⊗ s + aDs,

(32)

where s, s1 , s2 ∈ Γ (V ) and a ∈ A. This is almost the same as the definition of the connection or the covariant derivative in the continuous case, which is basically equivalent to the connection a la Cartan. Since our discussion on all geometric quantities are in the local sense, for the simplicity, we can choose the basis {sα , 1 ≤ α ≤ m} as the basis of the linear space Γ (V ) and {dxµ ⊗ sα , 1 ≤ α ≤ m, 1 ≤ µ ≤ n} as the basis of the sections space Γ (T ∗ N × V ). Then the covariant derivative Dsα should be the linear expansion on {dxµ ⊗ sα }. Hence, we can define it in the local sense as X X µ − (Bµ )α (33) − (B)α Dsβ = β dx ⊗ sα β ⊗ sα = α

α,µ

or simply Ds = −B · s,

(34)

µ

where B = Bµ dx is the local expression of a discrete connection 1-form. For the continuous case, the connection 1-form is valued on a Lie algebra. However the 1-form B = Bµ dxµ here is matrix valued. Since all 1-forms are defined on the links, the coefficients Bµ are also defined on the link (x, x + µ ˆ) and can be written as Bµ (x) = B(x, x + µ ˆ). (35) P α For any section S = α a sα , we have X DS = (dD aα · sα + aα · Dsα ) X = (4µ aα dxµ ⊗ sα − aα (Bµ )βα dxµ ⊗ sβ ) (36) X = (4µ aβ − aα (Bµ )βα )dxµ ⊗ sβ X = (DDµ aβ )dxµ ⊗ sβ , where DDµ aβ = 4µ aβ − aα (Bµ )βα is called the discrete covariant derivative of the vector aα and X X µ DD aα = DDµ aα dxµ = (4µ aα − aβ (Bµ )α β )dx

(37)

(38)

is the discrete exterior covariant derivative of the vector aα . On the space Γ (V ), we can choose another linear basis or perform a linear transformation of the basis, i.e., take the gauge transformation as follows sα 7−→ s˜α = g(x)βα · sβ ,

x ∈ N,

(39)

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311

where g(x)βα is the GL(m, R) gauge group valued function defined on the node x ∈ N . In order to keep section S being invariant, the coefficients aα of P the a general section S = aα sα should transform as aα 7−→ a ˜α = aβ · (g −1 (x))α β.

(40)

From the gauge invariance of S and DS, we can derive the gauge transformation property of DDµ aα , X β DS 7−→ (DDµ aγ )dxµ · (g −1 (x))α γ ⊗ g(x)α · sβ X (41) µ β = (DDµ aγ ) · (g −1 (x + µ ˆ))α γ dx ⊗ g(x)α · sβ , where the noncommutative commutation relation between function and 1form basis is used. Then the covariance of the covariant derivative follows DDµ aα 7−→ DDµ aγ · (g −1 (x + µ ˆ))α γ.

(42)

Thus, we get the gauge transformation property of the difference discrete connection 1-form as

or

Bµ (x)dxµ 7−→ g(x) · Bµ (x)dxµ · g −1 (x) + g(x) · dD g −1 (x),

(43)

Bµ (x) 7−→ g(x) · Bµ (x) · g −1 (x + µ ˆ)+g(x) · 4µ g −1 (x).

(44)

Together with the gauge transformation property of the coefficients of a vector field in (40), the gauge covariance of the derivative DDµ aγ is confirmed. Discrete Connection via Horizontal Tangent Vector For the vector bundle, there is another definition of connection. It is based on the decomposition of the total tangent vector of the bundle into the horizontal and vertical parts. Then the horizontal tangent vector invariant under the right operation of the structure group also defines a connection. In fact, the horizontal tangent vector is nothing but the covariant derivative. This definition can also be given in an analogous manner for the discrete case here. Let us consider the discrete vector bundle over a discrete base manifold N as a regular lattice with the fiber being a sufficiently smooth m-dimensional vector space Vx at x ∈ N , like the Vk used in section 2. As we discussed before, the basis of tangent space T Vx is Xα =

∂ , ∂aα

(1 ≤ α ≤ m),

(45)

where aα , 1 ≤ α ≤ m, are the coordinates of the fiber Vx . The basis of tangent vector on a discrete regular lattice is

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Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

4µ ,

(1 ≤ µ ≤ n).

(46)

Then the basis of the total tangent space of the discrete vector bundle is the union of (45) and (46). Similar to the continuous case, the vector space tangent to the fiber, i.e. the linear combination of basis in (45), is a vertical subspace of the total tangent space of the discrete vector bundle. Its complementary vectors of the vertical subspace in the total tangent of vector bundle are horizontal and constitute the horizontal subspace. The basis of horizontal subspace is as follows, β Xµ = 4µ − (Bµ )α βa

∂ , ∂aα

(1 ≤ µ ≤ n).

(47)

In comparison with the definition of difference discrete connection, it is easy to see that the horizontal vector is nothing but the covariant derivative in (37). This means that when we form the decomposition of tangent vector on the total bundle space we get the coefficients Bµ (x) of the discrete connection. For a given difference discrete connection or its coefficients Bµ (x), we can also get a decomposition of the total tangent vector space of bundle into vertical and horizontal parts sufficiently and necessarily. This shows that the difference discrete connection on a discrete vector bundle is equivalent to a decomposition of the total tangent vector space into vertical and horizontal subspaces as above. Similarly, we can define the basis of the dual space for the decomposition, i.e., the basis of the vertical and horizontal 1-form space, respectively, as β µ ω α = daα + (Bµ )α β a dx ,

(1 ≤ α ≤ m),

(48)

and ω µ = dxµ ,

(1 ≤ µ ≤ n).

(49)

They satisfy the following dual relation: ω α (Xβ ) = δβα ,

ω α (Xµ ) = 0,

ω µ (Xβ ) = 0,

ω µ (Xν ) = δνµ .

Covariant Derivative and Parallel Transport From the above discussions, we can get the difference discrete connection on discrete vector bundle through the definition of the absolute derivative or the horizontal tangent vector. Both lead to the difference discrete covariant derivative for the vectors, DDµ aβ = 4µ aβ − aα (Bµ )βα .

(50)

In terms of the relation between 4µ and Eµ , we obtain another expression for the covariant derivative as,

Difference Discrete Geometry on Lattice

DDµ aγ (x) = Eµ aγ (x) − aβ (x) · [(Bµ (x))γβ + δβγ ],

313

(51)

where δβα is the unit matrix. Then we obtain another expression for the coefficient of the discrete connection, Uµ (x) = [(Bµ (x))γβ + δβγ ],

(52)

which is an element of some group, for example the group GL(m, R) in our discussions, and connects the points x and x + µ ˆ. We can also call Uµ (x) the discrete GL(m, R)-connection and express it as Uµ (x) = U (x, x + µ ˆ).

(53)

From the second expression of the coefficient of connection and the definition of covariant derivative for the vectors, we obtain the parallel transport of the section of vector aβ , provided that its covariant derivative is zero

or Namely,

DDµ aβ = 4µ aβ − aα (Bµ )βα = 0

(54)

Eµ aγ (x) − aβ (x) · (Uµ (x))γβ = 0.

(55)

aγ (x + µ ˆ) = aβ (x) · (U (x, x + µ ˆ))γβ .

(56)

This means that the parallel transport of the section of vector aβ (x) along the path x 7→ x + µ ˆ is expressed as aβ (x) 7−→ aβ (x + µ ˆ) = aγ (x) · (U (x, x + µ ˆ))βγ .

(57)

It is shown that there is a parallel transport on the discrete bundle along the curve on discrete base manifold for a given discrete connection on discrete bundle. Due to the discrete connection coefficient U (x, x + µ ˆ) as a group element, there are the following group properties of U (x, x + µ ˆ) along the path decomposition of (x, x + µ ˆ + νˆ) into (x, x + µ ˆ) and (x + µ ˆ, x + µ ˆ + νˆ), U (x, x + µ ˆ) · U (x + µ ˆ, x + µ ˆ + νˆ) = U (x, x + µ ˆ + νˆ).

(58)

From the inverse of the parallel transport, it also follows that (U (x, x + µ ˆ))−1 = U (x + µ ˆ, x).

(59)

The coefficient of the discrete connection U (x, x + µ ˆ) determines the parallel transport not only on a vector bundle but also on a GL(m, R) principal bundle. Therefore, it is also called a discrete GL(m, R) connection.

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Parallel Transport on Discrete Principal Bundle The matrix structure group of a discrete vector bundle can be generalized to any Lie group G and the coefficients of a connection are the group G-valued with the right operations. Thus, we can get a difference discrete G-valued connection on a discrete principal G-bundle over the lattice. In this case, the concept of parallel transport can be extended to the discrete principal G-bundle. For a section h(x), the parallel transport with respect to a G-valued connection U (x, x + µ ˆ) reads h(x) 7−→ h(x + µ ˆ) = h(x) · U (x, x + µ ˆ).

(60)

All elements here belong to the Lie group G and the multiplication should be the group multiplication. The above equation can be expressed as h(x0 ) 7−→ h(x1 ) = h(x0 ) · U (x0 , x1 ),

(61)

h(x1 ) 7−→ h(x0 ) = h(x1 ) · U (x1 , x0 ),

(62)

U (x1 , x0 ) = U (x0 , x1 )−1 .

(63)

or which implies that

Similarly, the covariant derivative for the section h(x) can be given as DDµ h(x) = Eµ h(x) − h(x) · Uµ (x), and the covariant exterior derivative as X DD h(x) = DDµ h(x)dxµ = (Eµ h(x) − h(x) · Uµ (x))dxµ .

(64)

(65)

µ

If link (x, x + µ) ˆ appears as common boundary in two different faces, each of them could be understood as a local coordinate neighborhood and denoted as Uα , Uβ . Then the discrete connection U (x, x + µ ˆ) has different expressions U (α) (x, x + µ ˆ) and U (β) (x, x + µ ˆ) in Uα , Uβ respectively. They satisfy the following relations: −1 U (α) (x, x + µ ˆ) = gα,β (x) · U (β) (x, x + µ ˆ) · gα,β (x + µ ˆ),

(66)

which is nothing but the gauge transformation defined in (44), where gα,β (x) is the transition function between Uα , Uβ . If the link (x, x + µ ˆ) is the common boundary of three different faces or three local coordinate neighborhoods Uα , Uβ and Uγ . Then there will be three relations for the different expressions of discrete connection U (α) (x, x + µ ˆ), U (β) (x, x + µ ˆ) and U (γ) (x, x + µ ˆ) as follows, −1 U (α) (x, x + µ ˆ) = gα,β (x) · U (β) (x, x + µ ˆ) · gα,β (x + µ ˆ), −1 U (β) (x, x + µ ˆ) = gβ,γ (x) · U (γ) (x, x + µ ˆ) · gβ,γ (x + µ ˆ), (67) (γ) (α) −1 U (x, x + µ ˆ) = gγ,α (x) · U (x, x + µ ˆ) · gγ,α (x + µ ˆ),

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The consistent condition for those transformation relations require that the transition functions should satisfy the cocycle condition or consistent condition, gα,β (x) · gβ,γ (x) · gγ,α (x) = id. (68) 4.2 Difference Discrete Curvature, Bianchi Identity and Abelian Chern Class Difference Discrete Curvature Based on the definition of the difference discrete connection 1-forms on discrete vector bundle, we can define the curvatures 2-forms similar to the continuous case [8], F = dD B + B ∧ B. (69) If we assume F =

1 Fµν dxµ ∧ dxν , it follows that 2

Fµν (x) = 4µ Bν (x) − 4ν Bµ (x) + Bµ (x) · Bν (x + µ ˆ) − Bν (x) · Bµ (x + νˆ). (70) Under the continuous limit, it is easy to see that this curvature should be the same as the usual formula of curvature. Since the definition is in a similar formulation except for the non-commutative exterior differential calculus, it is also easy to check the covariance property of the curvature under the gauge transformation (43) or (44) as follows F (x) 7−→ Fe(x) = g(x) · F (x) · g −1 (x),

(71)

Fµν (x) 7−→ Feµν (x) = g(x) · Fµν (x) · g −1 (x + µ ˆ + νˆ).

(72)

or the covariance behavior of its components

It is important to see that from the non-commutative property of differential calculus on lattice the shifting operator appears in the covariance of discrete curvature. This is a main difference between the continuous case and the discrete one. And it may lead to more difficulties in the discussion of the gauge covariance and invariance property of tensors in the discrete case. Curvature via Holonomy In the continuous case, the curvature may naturally appear in the homolomy consideration. As the (difference) discrete counterpart, the (difference) discrete curvature may also be described based on the holonomy consideration. Let us consider the square of the exterior covariant derivatives

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Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

(DD )2 h(x) = DD DDµ h(x)dxµ X (Eµ h(x) − h(x) · Uµ (x))dxµ = DD X µ Eν (Eµ h(x) − h(x) · Uµ (x))dxµ ∧ dxν = µνX (Eµ h(x) − h(x) · Uµ (x))dxµ ∧ Uν (x))dxν − µνX Uµ (x)dxµ ∧ Uν (x)dxν = h(x) µν X = h(x) Uµ (x) · Uν (x + µ ˆ)dxµ ∧ dxν

(73)

µν

=

X 1 h(x) [Uµ (x) · Uν (x + µ ˆ) − Uν (x) · Uµ (x + νˆ)]dxµ ∧ dxν . 2 µν

This leads to another expression for the curvature 2-form with its coefficients X F = U2 = Uµ (x)dxµ ∧ Uν (x)dxν , µν (74) 1 ˆ) − Uν (x) · Uµ (x + νˆ)]. Fµν = [Uµ (x) · Uν (x + µ 2 The zero curvature condition F = 0 is

or

Uµ (x) · Uν (x + µ ˆ) = Uν (x) · Uµ (x + νˆ),

(75)

Uµ (x) · Uν (x + µ ˆ) · Uµ−1 (x + νˆ) · Uν−1 (x) = 1,

(76)

or U (x, x + µ ˆ) · U (x + µ ˆ, x + µ ˆ + νˆ) · U (x + µ ˆ + νˆ, x + νˆ) · U (x + νˆ, x) = 1. (77) The expressions in the last formula is nothing but the holonomy group in geometry or the plaquette variable in lattice gauge theory. We will discuss them in the section 6 and show also that zero curvature condition is just the integrability condition for a discrete integrable system. Bianchi Identity and Abelian Chern Class Similar to the Bianchi identity in differential geometry, we can also derive the Bianchi identity for the difference discrete curvature DD F = dD F − F ∧ B + B ∧ F = 0, or in its components

(78)

Difference Discrete Geometry on Lattice

317

ˆ = 0. (79) ελµν [4λ Fµν (x) − Fλµ (x) ∧ Bν (x + µ ˆ + νˆ) + Bλ (x) ∧ Fµν (x + λ)] For the Abelian case, one can define the following topological term as discrete Chern class [17], ck = F ∧ F ∧ · · · ∧ F,

(80)

which was used to discuss the chiral anomaly in the lattice gauge theory. The coefficient of the Abelian Chern class is εµ1 µ2 ···µ2k−1 µ2k Fµ1 µ2 (x)·Fµ3 µ4 (x+µˆ1 +µˆ2 ) · · · Fµ2k−1 µ2k (x+µˆ1 +µˆ2 +· · ·+µ2k−2 ˆ ) This equation first appeared in lattice gauge theory for the Abelian anomaly of chiral fermion in a quantum field theory [17].

5 Discrete Connection on G-Bundle over Random Lattice 5.1 Discrete Connection over Randam Lattice The definition of the discrete connection via the horizontal vector space in sect. 3 can be generalized to the one on a G-bundle Q(N, G) over a random lattice N . Let us consider the parallel transport of a section on such a G-bundle: h(x0 ) 7−→ h(x1 ) = h(x0 ) · U (x0 , x1 ),

(81)

where h(xj ) is the G-valued section defined on xj , j = 0, 1 and x0 , x1 are nearest neighbors. We can reexpress it equivalently as (x0 , h0 ) 7−→ (x1 , h1 ) = (x0 , h0 ) · U (x0 , x1 ),

(82)

where h0 = h(x0 ), h1 = h(x1 ) and right multiplication of U on the bundle acts only on the G-valued section h0 . It is easy to see that in these expressions there is no difference operator involved so that they could be make sense for the G-bundle over random lattice, if the discrete connection is properly introduced. On the other hand, if h0 and h1 satisfy eq.(81), it can be proved that the element (q0 , q1 ) = ((x0 , h0 ), (x1 , h1 )) ∈ Q × Q is just a horizontal vector on T Q, i.e. hor((x0 , h0 ), (x1 , h1 )) = ((x0 , h0 ), (x1 , h1 )), (83) where hor(∗, ∗) denotes the horizontal part of the (∗, ∗). In fact, this is almost the same as the one introduced in [10]. Thus, our definition for the discrete connection can be easily compared with the local expression A(x0 , x1 ) of the coefficients of a connection 1-form defined in [10]. Namely,

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U (x0 , x1 ) = A(x0 , x1 )−1 then

(84)

((x0 , h0 ), (x1 , h0 · A(x0 , x1 )−1 ))

is a horizontal vector. According to the formulation in [10], we get ((x0 , h0 ), (x1 , h0 · A(x0 , x1 )−1 )) = h0 · i(x0 ,e) (A(x0 , x1 )−1 ) · ((x0 , e), (x1 , e)) = h0 · hor((x0 , e), (x1 , e)) = h0 · hor((x0 , e), q1 )

(85)

= h0 · hor((x0 , e), h−1 0 q1 ) = hor((x0 , h0 ), q1 ), which means that the horizontal vector ((x0 , h0 ), (x1 , h0 · A(x0 , x1 )−1 )) is the horizontal part of any vector (q0 , q1 ) with q0 is fixed and q1 is any point on the fiber of π −1 (x1 ). According to the definition of A(x0 , x1 ), we have A(x0 , x1 ) = Ad (x0 , e, x1 , e) . From the gauge transformation property of U (x0 , x1 ), it follows that under the gauge transformation g(x) A(x0 , x1 ) 7−→ g(x1 ) · A(x0 , x1 ) · g −1 (x0 ).

(86)

Ad (x0 , g(x0 ); x1 , g(x1 )) = g(x1 )Ad (x0 , e, x1 , e)g −1 (x0 ).

(87)

This leads to

Thus, we recover the property of the connection 1-form defined in [10] Ad (gq0 , hq1 ) = hAd (q0 , q1 )g −1 .

(88)

As was shown above, our definition of discrete connection is equivalent to that in [10] in the case of the cubic lattice. However, our definition for the discrete curvature is only for the hypercubic lattice, since it is based on the noncommutative differential calculus. How to extend those results to the case of random lattice is under investigation. 5.2 Topological Number in Two Dimension After integration of the Chern class of a principal G-bundle over a base manifold one will get the Chern number, also called topological number. In the discrete case we have got some local geometric information such as connection and curvature and some global properties such as the Chern class of U (1)

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gauge theory. Can we get the similar Chern numbers in terms of discrete connections? Now we try to show the topological property of a discrete connection in the Abelian case using a simple example, i.e., to calculate its topological number in a two dimensional manifold which is topologically equivalent to the sphere and is discretized as a tetrahedron in the following picture, with notes A, B, C, D, links AB, BC, CA, · · · , and faces 4ABC, 4ABD, · · · , as follows:

A

D

B C

In the boundary of 4ABC there are three links and the holomomy expression F4ABC as F4ABC = U (A, B) · U (B, C) · U (C, A), and its logarithm ln F4ABC = ln U (A, B) + ln U (B, C) + ln U (C, A).

(89)

In this case the topological number is coming from the summation of ln F4ABC over the boundaries of all triangles, c=

1 X ln F4ABC . 2πi

(90)

4

It is not difficulty to see that the topological number c should be an integer and equal to c=

1 X (ln g4ABC,4ACD (A)+ln g4ACD,4ADB (A)+ln g4ADB,4ABC (A)), 2πi point

(91) where (ln g4ABC,4ACD (A) + ln g4ACD,4ADB (A) + ln g4ADB,4ABC (A)) is an integer since the consistency condition of gauge transformations as

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Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

g4ABC,4ACD (A) · g4ACD,4ADB (A) · g4ADB,4ABC (A) = 1. The discussion above can be easily used to the case of hypercubic lattice and for general case it will be given in forthcoming publications.

6 Applications 6.1 Lattice Gauge Theory and Difference Discrete Connection In the lattice gauge theory [18], the space-time is discretized as hypercubic lattice with equal spacing a in any direction in most cases. Suppose that Aµ is the gauge field or the connection on the continuous case. At each link on the lattice we introduce a discrete version of the path ordered product µ ˆ U (x, x + µ ˆ) ≡ Uµ (x) = eiaAµ (x+ 2 ) , (92) where µ ˆ is the vector in coordinate direction with length a and x is the point coordinate on the node of the hypercubic lattice which takes integer value only. The average field, which is denoted by Aµ (x + µ2ˆ ), is defined at the midpoint of the link (x, x + µ ˆ). Similarly, µ ˆ

ˆ, x). U (x, x − µ ˆ) ≡ U−µ (x) = e−iaAµ (x− 2 ) = U † (x − µ

(93)

If the connection Aµ is valued on the Lie algebra of SU (N ) with a hermitian basis, we have U † (x − µ ˆ, x) = U −1 (x − µ ˆ, x). The variable of a simplest Wilson loop called plaquette variable is expressed as the left side of (77), which is defined on the two dimensional square Wµν = Uµ (x) · Uν (x + µ ˆ) · Uµ† (x + νˆ) · Uν† (x).

(94)

It can be shown that the continue limit of Wµν is related to the Yang-Mills action a4 Re(1 − Wµν ) = Fµν F µν + O(a6 ) + · · · , 2 Im(Wµν ) = a2 Fµν + · · · . Therefore, plaquette variable Wµν should play some role of curvature in the discrete case as we discussed in previous section. However its continuous limit is related not only to the usual curvature but also to the Yang-Mills action as in the above expressions, there should be more geometric meaning in the theory of discrete connection and curvature than usual one.

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6.2 Geometric Meaning of Discrete Lax Pair In order to understand the discrete connection on discrete bundle, we first discuss some geometric meaning of the Lax pair and the discrete Lax pair. In fact, this gives one a solid motivation and some consideration for the study of discrete connections. Let us start with the concept of a Lax pair of an integral system in continuous two dimensional case with 1-dimension time and 1-dimension space as follows ∂x ψ = ψ · A x , (95) ∂t ψ = ψ · A t , where ψ is a vector and Ax , At are matrix valued. The consistent condition for this linear system is ∂x At − ∂t Ax + [Ax , At ] = 0.

(96)

Now let us discretize the 2-dimensional space-time as a square lattice, R2 → Z2 . The section field ψ(x) on the vector bundle becomes the field ψ(m, n) the functions depending on two discrete variable, i.e., two integer (m, n). Naively the discrete Lax pair may be written as 4x ψ(m, n) = ψ(m, n) · Ax (m, n), 4t ψ(m, n) = ψ(m, n) · At (m, n),

(97)

The derivatives ∂x and ∂t with respect to x and t are replaced by difference operators 4x and 4t , respectively. The consistent condition for the discrete Lax pair, i.e., 4x 4t ψ(m, n) = 4t 4x ψ(m, n) (98) leads to 4x At (m, n)−4t Ax (m, n)+Ax (m, n)At (m+1, n)−At(m, n)Ax (m, n+1) = 0. (99) Using the shift operator Ex and Et we can also rewrite the discrete Lax pair (97) as Ex ψ(m, n) = ψ(m, n) · [1 + Ax (m, n)] = ψ(m, n) · Ux (m, n) Et ψ(m, n) = ψ(m, n) · [1 + At (m, n)] = ψ(m, n) · Ut (m, n),

(100)

where Ut (m, n) = 1 + At (m, n),

Ux (m, n) = 1 + Ax (m, n).

(101)

On requiring the corresponding consistent condition Ex Et ψ(m, n) = Et Ex ψ(m, n), a straightforward calculation leads to

(102)

322

or

Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

Ux (m, n) · Ut (m + 1, n) = Ut (m, n) · Ux (m, n + 1),

(103)

Ux (m, n) · Ut (m + 1, n) · Ux−1 (m, n + 1) · Ut−1 (m, n) = 1.

(104)

If we use the relation of U and A in (101), we can derive the zero curvature condition (99) from (104). When we regard the quantities Ax , At , Ux and Ut as the discrete connections on discrete bundle, the equations (99) and (104) should be the zero curvature condition for these connections, and the left sides of (99) and (104) should be the extension of the curvature in the discrete case.

7 Remarks and Discussions As was mentioned previously, the study of discrete models is very important in both its own right as well as for its applications, although we mainly focus here on the discrete models as the discrete counterparts of the continuous cases. In order to get the discrete models that can keep the properties of continuous ones as much as possible, we may first consider those kinds of discrete models from their continuous counterparts with differences as discrete derivatives. These models can be given by replacing both, the continuous independent variables and their derivatives by their discrete independent variables as a regular lattice and their differences on the lattice, respectively. In general, for the discrete models on the regular lattices including the models just mentioned, it is natural to study first the properties of the function spaces on the lattices and the discrete bundles over the lattices both analytically and geometrically, such as discrete differential calculus, discrete metric, discrete Hodge operator, discrete connection and curvature, and so on. In doing so, we may follow a way similar to that in the continuous cases, as long as the differences are regarded as the discrete derivatives. In this paper, we have briefly reviewed the non-commutative differential calculus on hypercubic lattice, which have discussed by many groups. We have mainly introduced the (difference) discrete connections on discrete vector bundle in several manners, the parallel transport, the decomposition of the vector space into vertical and horizontal space, the covariant derivative on the section of vector bundle as well as the discrete curvature of the discrete connections. We have also studied their relation to the lattice gauge theory and applied to the Lax pairs for the discrete integrable systems. There are, of course, also many properties of the discrete models, which are apparently very different from the continuous ones. These should be investigated further. Although one of the definitions for the discrete connection can be extended to the case over the random lattice, however, for the discrete curvature on the random lattice in the lattice gauge theory, it is still open

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whether it can be defined, at least formally, in a way similar to the continuous case. This is also a very interesting question and from its investigation we expect to get more results on lattices with no-trivial topology. Another very important problem is how to get the topological classes with non-Abelian group. Needless to say, more attention should be payed to those questions. The work is partly supported by NKBRPC(2004CB318000), Beijing JiaoWei Key project (KZ200310028010) and NSF projects (10375087, 10375038, 90403018, 90503002). The authors would like to thank Morningside Center for Mathematics, CAS. Part of the work was done during the Workshop on Mathematical Physics there.

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