Lectures on Fuzzy and Fuzzy Susy Physics

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Author: A. P. Balachandran | S. Kurkcuoglu | S. Vaidya

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Lecturesh on Fuzzy and

Fuzzy SUSY Physics

This page intentionally left blank

Lectures on Fuzzy and

Fuzzy SUSY Physics

A. P. Balachandran Syracuse University, USA

S. Kürkçüog˘lu

Dublin Institute for Advanced Studies, Ireland

S. Vaidya

Indian Institute of Science, India

World Scientific NEW JERSEY

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LONDON

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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

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LECTURES ON FUZZY AND FUZZY SUSY PHYSICS 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-13 978-981-270-466-5 ISBN-10 981-270-466-3

Printed in Singapore.

Lakshmi - Lectures on Fuzzy.pmd

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Dedicated to Rafael Sorkin, our friend, teacher, a true and creative seeker of knowledge.

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Preface

One of us (Balachandran) gave a course of lectures on “Fuzzy Physics” during spring, 2002 for students of Syracuse and Brown Universities. The course which used video conferencing technology was also put on the websites [1]. Subsequently A.P. Balachandran, S. K¨ urk¸cu ¨oˇ glu and S.Vaidya decided to edit the material and publish them as lecture notes. The present book is the outcome of that effort. The recent interest in fuzzy physics begins from the work of Madore [2, 3] even though the basic physical and mathematical ideas are older and go back to Hoppe [4]. Many of the mathematical developments are based on the works of Kostant and Kirillov [5] and Berezin [6]. They emerge from the fundamental observation that coadjoint orbits of Lie groups are symplectic manifolds which can therefore be quantized under favorable circumstances. When that can be done, we get a quantum representation of the manifold. It is the fuzzy manifold for the underlying “classical manifold”. It is fuzzy because no precise localization of points thereon is possible. The fuzzy manifold approaches its classical version when the effective Planck’s constant of quantization goes to zero. Our interest will be in compact simple and semi-simple Lie groups for which coadjoint and adjoint orbits can be identified and are compact as well. In such a case these fuzzy manifold is a finite-dimensional matrix algebra on which the Lie group acts in simple ways. Such fuzzy spaces are therefore very simple and also retain the symmetries of their classical spaces. These are some of the reasons for their attraction. There are several reasons to study fuzzy manifolds. Our interest has its roots in quantum field theory (qft). Qft’s require regularization and the conventional nonperturbative regularization is lattice regularization. It has been extensively studied for over thirty years. It fails to preserve vii

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space-time symmetries of quantum fields. It also has problems in dealing with topological subtleties like instantons, and can deal with index theory and axial anomaly only approximately. Instead fuzzy physics does not have these problems. So it merits investigation as an alternative tool to investigate qft’s. A related positive feature of fuzzy physics, is its ability to deal with supersymmetry(SUSY) in a precise manner [7–10]. (See however,[11]). Fuzzy SUSY models are also finite-dimensional matrix models amenable to numerical work, so this is another reason for our attraction to this field. Interest in fuzzy physics need not just be utilitarian. Physicists have long speculated that space-time in the small has a discrete structure. Fuzzy space-time gives a very concrete and interesting method to model this speculation and test its consequences. There are many generic consequences of discrete space-time, like CPT and causality violations, and distortions of the Planck spectrum. Among these must be characteristic signals for fuzzy physics, but they remain to be identified. We are a part of a collaboration on fuzzy physics and noncommutative geometry. These lecture notes have been strongly influenced by our interactions with our colleagues in this collaboration. We thank them for discussions and ideas. The work of APB has been supported in part by DOE under contract number DE-FG02-85ER40231. The work of SK is supported by Irish Research Council for Science Engineering and Technology(IRCSET). These lecture notes are not exhaustive, and reflect the research interests of the authors. It is our hope that the interested reader will be able to learn about the topics we have not covered with the help of our citations.

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Contents

vii

Preface 1. Introduction

1

2. Fuzzy Spaces

5

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Fuzzy C2 . . . . . . . . . . . . . . . . . . . Fuzzy S 3 and Fuzzy S 2 . . . . . . . . . . . The Fuzzy Sphere SF2 . . . . . . . . . . . . Observables of SF2 . . . . . . . . . . . . . . Diagonalizing L2 . . . . . . . . . . . . . . . Scalar Fields on SF2 . . . . . . . . . . . . . . The Holstein-Primakoff Construction . . . . CP N and Fuzzy CP N . . . . . . . . . . . . The CP N Holstein-Primakoff Construction

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3. Star Products 3.1 3.2 3.3 3.4 3.5

5 5 6 8 9 9 10 11 14 17

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Coherent States . . . . . . . . . . . . . . . . The Coherent State or Voros ∗-Product on the Moyal Plane The Moyal ∗-Product on the Groenewold-Moyal Plane . . 3.4.1 The Weyl Map and the Weyl Symbol . . . . . . . Properties of ∗-Products . . . . . . . . . . . . . . . . . . . 3.5.1 Cyclic Invariance . . . . . . . . . . . . . . . . . . 3.5.2 A Special Identity for the Weyl Star . . . . . . . . 3.5.3 Equivalence of ∗C and ∗W . . . . . . . . . . . . . ix

17 19 20 23 23 24 24 25 25

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3.6

3.5.4 Integration and Tracial States . . 3.5.5 The θ-Expansion . . . . . . . . . . The ∗-Products for the Fuzzy Sphere . . . 3.6.1 The Coherent State ∗-Product ∗C 3.6.2 The Weyl ∗-Product ∗W . . . . .

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4. Scalar Fields on the Fuzzy Sphere 4.1 4.2 4.3

Loop Expansion . . . . . . . . . . . . . The One-Loop Two-Point Function . . . Numerical Simulations . . . . . . . . . . 4.3.1 Scalar Field Theory on Fuzzy S 2 4.3.2 Gauge Theory on Fuzzy S 2 × S 2

35 . . . . .

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5. Instantons, Monopoles and Projective Modules 5.1 5.2 5.3 5.4

Free Modules, Projective Modules . . . . . . . . . . . . . Projective Modules on A = C ∞ (S 2 ) . . . . . . . . . . . . Equivalence of Projective Modules . . . . . . . . . . . . . Projective Modules on the Fuzzy Sphere . . . . . . . . . . (k) 5.4.1 Fuzzy Monopoles and Projectors PF . . . . . . . 5.4.2 The Fuzzy Module for the Tangent Bundle and the Fuzzy Complex Structure . . . . . . . . . . . . . .

6. Fuzzy Nonlinear Sigma Models 6.1 6.2 6.3 6.4 6.5

6.6

26 27 28 28 31

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . CP 1 Models and Projectors . . . . . . . . . . . . . . . . . An Action . . . . . . . . . . . . . . . . . . . . . . . . . . . CP 1 -Models and Partial Isometries . . . . . . . . . . . . . 6.4.1 Relation Between P (κ) and Pκ . . . . . . . . . . . Fuzzy CP 1 -Models . . . . . . . . . . . . . . . . . . . . . . 6.5.1 The Fuzzy Projectors for κ > 0 . . . . . . . . . . 6.5.2 The Fuzzy Projectors for κ < 0 . . . . . . . . . . 6.5.3 The Fuzzy Winding Number . . . . . . . . . . . . 6.5.4 The Generalized Fuzzy Projector : Duality or BPS States . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 The Fuzzy Bound . . . . . . . . . . . . . . . . . . CP N -Models . . . . . . . . . . . . . . . . . . . . . . . . .

37 39 43 45 46 47 47 49 51 54 54 56 59 59 60 63 65 68 69 69 70 71 71 71 73

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Contents

7. Fuzzy Gauge Theories 7.1 7.2 7.3 7.4 7.5

Limits on Gauge Groups . . . . . . . . . . . . Limits on Representations of Gauge Groups . Connection and Curvature . . . . . . . . . . . Instanton Sectors . . . . . . . . . . . . . . . . The Partition Function and the θ-parameter .

77 . . . . .

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8. The Dirac Operator and Axial Anomaly 8.1 8.2 8.3

8.4

8.5 8.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . A Review of the Ginsparg-Wilson Algebra . . . . . . . . . Fuzzy Models . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Review of the Basic Fuzzy Sphere Algebra . . . . 8.3.2 The Fuzzy Dirac Operator (No Instantons or Gauge Fields) . . . . . . . . . . . . . . . . . . . . 8.3.3 The Fuzzy Gauged Dirac Operator (No Instanton Fields) . . . . . . . . . . . . . . . . . . . . . . . . The Basic Instanton Coupling . . . . . . . . . . . . . . . . 8.4.1 Mixing of Spin and Isospin . . . . . . . . . . . . . 8.4.2 The Spectrum of the Dirac operator . . . . . . . . Gauging the Dirac Operator in Instanton Sectors . . . . Further Remarks on the Axial Anomaly . . . . . . . . . .

9. Fuzzy Supersymmetry 9.1 9.2 9.3

9.4 9.5 9.6

osp(2, 1) and osp(2, 2) Superalgebras and their Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Passage to Supergroups . . . . . . . . . . . . . . . . . . . On the Superspaces . . . . . . . . . . . . . . . . . . . . . 9.3.1 The Superspace C 2,1 and the Noncommutative CF2,1 9.3.2 The Supersphere S (3,2) and the Noncommutative S (3,2) . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 The Commutative Supersphere S (2,2) . . . . . . . (2,2) 9.3.4 The Fuzzy Supersphere SF . . . . . . . . . . . More on Coherent States . . . . . . . . . . . . . . . . . . The Action on the Supersphere S (2,2) . . . . . . . . . . . (2,2) The Action on the Fuzzy Supersphere SF . . . . . . . . 9.6.1 The Integral and Supertrace . . . . . . . . . . . . 9.6.2 OSp(2, 1) IRR’s with Cut-Off N . . . . . . . . . .

78 79 80 81 82 85 85 85 88 88 89 91 93 94 94 95 96 99 100 106 107 107 108 108 112 114 116 119 119 121

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9.6.3

9.7

9.8 9.9

The Highest Weight States and the osp(2, 2)Invariant Action . . . . . . . . . . . . . . . . . . . 9.6.4 The Spectrum of V . . . . . . . . . . . . . . . . . 9.6.5 The Fuzzy SUSY Action . . . . . . . . . . . . . . The ∗-Products . . . . . . . . . . . . . . . . . . . . . . . . (2,2) 9.7.1 The ∗-Product on SF . . . . . . . . . . . . . . . 9.7.2 The ∗-Product on Fuzzy “Sections of Bundles” . . More on the Properties of Kab . . . . . . . . . . . . . . . . The O(3) Nonlinear Sigma Model on S (2,2) . . . . . . . . 9.9.1 The Model on S (2,2) . . . . . . . . . . . . . . . . . (2,2) 9.9.2 The Model on SF . . . . . . . . . . . . . . . . . 9.9.3 Supersymmetric Extensions of Bott Projectors . . 9.9.4 The SUSY Action Revisited . . . . . . . . . . . . 9.9.5 Fuzzy Projectors and Sigma Models . . . . . . . .

10. SUSY Anomalies on the Fuzzy Supersphere 10.1

10.2 10.3 10.4 10.5

10.6

137

Overview . . . . . . . . . . . . . . . . . . . . . . 10.1.1 The Fuzzy Sphere . . . . . . . . . . . . . 10.1.2 SUSY . . . . . . . . . . . . . . . . . . . . 10.1.3 Irreducible Representations . . . . . . . . 10.1.4 Casimir Operators . . . . . . . . . . . . . 10.1.5 Tensor Products . . . . . . . . . . . . . . 10.1.6 The Supertrace and the Grade Adjoint . 10.1.7 The Free Action . . . . . . . . . . . . . . SUSY Chirality . . . . . . . . . . . . . . . . . . . Eigenvalues of V0 . . . . . . . . . . . . . . . . . . Fuzzy SUSY Instantons . . . . . . . . . . . . . . Fuzzy SUSY Zero Modes and their Index Theory 10.5.1 Spectrum of K2 . . . . . . . . . . . . . . 10.5.2 Index Theory and Zero Modes . . . . . . Final Remarks . . . . . . . . . . . . . . . . . . .

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11. Fuzzy Spaces as Hopf Algebras 11.1 11.2 11.3 11.4

Overview . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . The Group and the Convolution Algebras A Prelude to Hopf Algebras . . . . . . . .

121 122 124 125 125 127 129 131 131 132 133 134 135

137 137 138 139 140 141 141 141 143 144 145 146 147 149 149 151

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151 153 154 155

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11.5 11.6 11.7 11.8

The ∗-Homomorphism G∗ → SF2 . Hopf Algebra for the Fuzzy Spaces Interpretation . . . . . . . . . . . . The Preˇsnajder Map . . . . . . . .

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159 161 165 166

Bibliography

169

Index

179

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Chapter 1

Introduction

We can find few fundamental physical models amenable to exact treatment. Approximation methods like perturbation theory are necessary and are part of our physics culture. Among the important approximation methods for quantum field theories (qft’s) are strong coupling methods based on lattice discretization of the underlying spacetime or perhaps its time-slice. They are among the rare effective approaches for the study of confinement in QCD and for nonperturbative regularization of qft’s. They enjoyed much popularity in their early days and have retained their good reputation for addressing certain fundamental problems. One feature of naive lattice discretizations however can be criticized. They do not retain the symmetries of the exact theory except in some rough sense. A related feature is that topology and differential geometry of the underlying manifolds are treated only indirectly, by limiting the couplings to “nearest neighbors”. Thus lattice points are generally manipulated like a trivial topological set, with a point being both open and closed. The upshot is that these models have no rigorous representation of topological defects and lumps like vortices, solitons and monopoles. The complexities in the ingenious lattice representations of the QCD θ-term [12] illustrate such limitations. There do exist radical attempts to overcome these limitations using partially ordered sets [13], but their potentials are yet to be adequately studied. As mentioned in the preface, a new approach to discretization, under the name of “fuzzy physics” inspired by noncommutative geometry (NCG), is being developed for a while now. The key remark here is that when the underlying spacetime or spatial cut can be treated as a phase space and ˆ assuming the role of ~, the emergent quanquantized, with a parameter ~ 1

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tum space is fuzzy, and the number of independent states per (“classical”) unit volume becomes finite. We have known this result after Planck and Bose introduced such an ultraviolet cut-off and quantum physics later justified it. A “fuzzified” manifold is expected to be ultraviolet finite, and if the parent manifold is compact too, supports only finitely many indepenˆ → 0 limit. This dent states. The continuum limit is the semi-classical ~ unconventional discretization of classical topology is not at all equivalent to the naive one, and we shall see that it does significantly overcome the previous criticisms. There are other reasons also to pay attention to fuzzy spaces, be they spacetimes or spatial slices. There is much interest among string theorists in matrix models and in describing D-branes using matrices. Fuzzy spaces lead to matrix models too and their ability to reflect topology better than elsewhere should therefore evoke our curiosity. They let us devise new sorts of discrete models and are interesting from that perspective. In addition, as mentioned in the preface, it has now been discovered that when open strings end on D-branes which are symplectic manifolds, then the branes can become fuzzy. In this way one comes across fuzzy tori, CP N and many such spaces in string physics. The central idea behind fuzzy spaces is discretization by quantization. It does not always work. An obvious limitation is that the parent manifold has to be even dimensional. If it is not, it has no chance of being a phase space. But that is not all. Successful use of fuzzy spaces for qft’s requires good fuzzy versions of the Laplacian, Dirac equation, chirality operator and so forth, and their incorporation can make the entire enterprise complicated. The torus T 2 is compact, admits a symplectic structure and on quantization becomes a fuzzy, or a non-commutative torus. It supports a finite number of states if the symplectic form satisfies the Dirac quantization condition. But it is impossible to introduce suitable derivations without escalating the formalism to infinite dimensions. But we do find a family of classical manifolds elegantly escaping these limitations. They are the co-adjoint orbits of Lie groups. For semi-simple Lie groups, they are the same as adjoint orbits. It is a theorem that these orbits are symplectic. They can often be quantized when the symplectic forms satisfy the Dirac quantization condition. The resultant fuzzy spaces are described by linear operators on irreducible representations (IRR’s) of the group. For compact orbits, the latter are finite-dimensional. In addition, the elements of the Lie algebra define natural derivations, and that helps to find the Laplacian and the Dirac operator. We can even

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Introduction

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3

define chirality with no fermion doubling and represent monopoles and instantons. (See chapters 5, 6 and 8). These orbits therefore are altogether well-adapted for qft’s. Let us give examples of these orbits: • S 2 ' CP 1 : This is the orbit of SU (2) through the Pauli matrix σ3 or any of its multiples λ σ3 (λ 6= 0). It is the set {λ g σ3 g −1 : g ∈ SU (2)}. The symplectic form is j d (cos) θ ∧ dφ with θ, φ being the usual S 2 coordinates. Quantization gives the spin j SU (2) representations. • CP 2 : CP 2 is of particular interest being of dimension 4. It is the orbit of SU (3) through the hypercharge Y = 1/3 diag(1, 1, −2) (or its non-zero multiples): CP 2 : {g Y g −1 : g ∈ SU (3)}.

(1.1)

The associated representations are symmetric products of 3’s or ¯ 3’s. In a similar way CP N are adjoint orbits of SU (N + 1) for any N ≥ 3. They too can be quantized and give rise to fuzzy spaces. • SU (3)/[U (1) × U (1)]: This 6-dimensional manifold is the orbit of SU (3) through λ3 = diag(1, −1, 0) and its non-zero multiples. These orbits give all the IRR’s containing a zero hypercharge state. In this book, we focus on the fuzzy spaces emerging from quantizing S 2 . They are called the fuzzy spheres SF2 and depend on the integer or half integer j labelling the irreducible representations of SU (2). Physics on SF2 is treated in detail. Scalar and gauge fields, the Dirac operator, instantons, index theory, and the so-called UV-IR mixing [14–20] are all covered. Supersymmetry can be elegantly discretized in the approach of fuzzy physics by replacing the Lie algebra su(2) of SU (2) by the superalgebras osp(2, 1) and osp(2, 2). Fuzzy supersymmetry is also discussed in these lectures including its instanton and index theories. We also briefly discuss the fuzzy spaces associated with CP N (N ≥ 2). These spaces, especially CP 2 , are of physical interest. We refer to the literature [29–33] for their more exhaustive treatment. Fuzzy physics draws from many techniques and notions developed in the context of noncommutative geometry. There are excellent books and reviews on this vast subject some of which we include in the bibliography [3, 34–38].

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Chapter 2

Fuzzy Spaces

In the present chapter, we approach the problem of quantization of classical manifolds like S 2 and CP N using harmonic oscillators. The method is simple and transparent, and enjoys generality too. The point of departure in this approach is the quantization of complex planes. We focus on quantizing C2 and its associated S 2 first. We will consider other manifolds later in the chapter. 2.1

Fuzzy C2

The two-dimensional complex plane C2 has coordinates z = (z1 , z2 ) where zi ∈ C. We want to quantize C2 turning it into fuzzy C2 ≡ C2F . This is easily accomplished. After quantization, zi become harmonic oscillator annihilation operators ai and zi∗ become their adjoint. Their commutation relations are [ai aj ] = [a†i a†j ] = 0 ,

˜ ij , [ai a†j ] = ~δ

(2.1)

˜ need not be the “Planck’s constant/2π”. The classical manifold where the ~ ˜ → 0. We set the usual Planck’s constant ~ to 1 hereafter unless emerges as ~ otherwise stated. In the same way, we can quantize CN +1 for any N using an appropriate number of oscillators and that gives us fuzzy CP N as we shall later see. 2.2

Fuzzy S 3 and Fuzzy S 2

There is a well-known descent chain from C2 to the 3-sphere S 3 and thence to S 2 . Our tactic to obtain fuzzy S 2 ≡ SF2 is to quantize this chain, 5

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Lectures on Fuzzy and Fuzzy SUSY Physics

obtaining along the way fuzzy S 3 ≡ SF3 . Let us recall this chain of manifolds. Consider C2 with the origin re1 P z with |z| = moved, C2 \ {0}. As z 6= 0, |z| |zi |2 2 makes sense here. i z z = 1, it gives the 3-sphere S 3 . Thus we Since |z| is normalized to 1, |z| have the fibration z z 2 3 , z→ . (2.2) R → C \ {0} → S = ξ = |z| |z| Now S 3 is a U (1)-bundle (“Hopf fibration”) [45] over S 2 . If ξ ∈ S 3 , then ~x(ξ) = ξ † ~τ ξ (where τi , i = 1, 2, 3 are the Pauli matrices) is invariant under the U (1) action ξ → ξeiθ and is a real normalized three-vector: ~x(ξ)∗ = ~x(ξ) ,

2

~x(ξ) · ~x(ξ) = 1 .

(2.3)

So ~x(ξ) ∈ S and we have the Hopf fibration Note that ~x(ξ) =

U (1) → S 3 → S 2 ,

1 ∗ 1 τ z |z| . |z| z ~

The fuzzy S 3 is obtained by replacing is the number operator: 1 1 zi zi∗ → ai p , → p a†i , |z| |z| b b N N

ξ → ~x(ξ) .

zi |z|

(2.4)

ˆ = P a† aj by ai √1 where N j j

b= N

ˆ N

X j

b 6= 0 . a†j aj , N

(2.5)

ˆ 6= 0 means that the vacuum is omitted from the The quantum condition N Hilbert space, so that it is the orthogonal complement of the vacuum in Fock space. This omission is like the deletion of 0 from C2 . There is a problem with this omission as ai √1 and its polynomials will b N

b = n state. For this reason, and because ai √1 and its create it from any N b N

adjoint need the infinite-dimensional Fock space to act on and do not give finite-dimensional models for SF3 , we will not dwell on this space. 2.3

The Fuzzy Sphere SF2

The problems of SF3 melt away for SF2 . Quantization of S 2 gives SF2 with xi (ξ) becoming the operator xi : 1 1 1 b 6= 0 . xi (ξ) → xi = p a† ~τ a p = a† τi a , N (2.6) ˆ ˆ ˆ N N N Since

ˆ = 0, [xi , N]

(2.7)

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7

Fuzzy Spaces

ˆ = n (6= 0). we can restrict xi to the subspace Hn of the Fock space where N This space is (n+1)-dimensional and is spanned by the orthonormal vectors (a†1 )n1 (a†2 )n2 √ √ |0i ≡ |n1 n2 i , n1 ! n2 !

n1 + n 2 = n .

(2.8)

The xi act irreducibly on this space and generate the full matrix algebra M at(n + 1). The SU (2) angular momentum operators Li are given by the Schwinger construction: τi (2.9) Li = a† a , [Li , Lj ] = iijk Lk . 2 a†i transform as spin 12 spinors and (2.8) spans the n-fold symmetric product of these spinors. It has angular momentum n2 : n n + 1 1 H . Li Li H = n n 2 2

Since

2 xi H n = L i H n , n

we find

2 [xi , xj ] Hn = iijk xk Hn , n X 2 2 xi H n = 1 + . 1 n Hn i

(2.10)

(2.11)

(2.12)

1 SF2 has radius 1 + n2 2 which becomes 1 as n → ∞. We generally write the equations in (2.12) as [xi , xj ] = P 2 2 2 , omitting the indication of Hn . SF2 should i x , = 1 + x i i n ijk k n have an additional label n, but that too is usually omitted. The xi ’s are seen to commute in the naive continuum limit n → ∞ giving back the commutative algebra of functions on S 2 . The fuzzy sphere SF2 is a “quantum” object. It has wave functions which are generated by xi restricted to Hn . Its Hilbert space is M at(n + 1) with the scalar product (m1 , m2 ) =

1 T r m†1 m2 , n+1

mi ∈ M at(n + 1) .

We denote M at(n + 1) with this scalar product also as M at(n + 1).

(2.13)

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Observables of SF2

The observables of SF2 are associated with linear operators on M at(n + 1). We can associate two linear operators αL and αR to each α ∈ M at(n + 1). They have left- and right-actions on M at(n + 1); αL m = αm ,

αR m = mα ,

∀ m ∈ M at(n + 1)

(2.14)

and fulfill (αβ)L = αL β L ,

(αβ)R = β R αR .

(2.15)

Such left- and right- operators commute: [αL , β R ] = 0 ,

∀

α , β ∈ M at(n + 1) .

(2.16)

We denote the two commuting matrix algebras of left- and right- operators by M atL,R (n + 1). M at(n + 1) is generated by a†i aj with the understanding that their domain is Hn . Accordingly, M atL,R (n + 1) are generated by (a†i aj )L,R . , (a†j )L,R : We can also define operators aL,R i aL i m = ai m , † a†L j m = aj m ,

aR i m = mai † a†R j m = maj .

(2.17)

They are operators changing n: aL,R : i a†L,R j

:

Hn → Hn−1

Hn → Hn+1 .

(2.18)

Such operators are important for discussions of bundles. (See chapter 5.) With the help of these operators, we can write L (a†i aj )L = a†L i aj ,

†R (a†i aj )R = aR j ai .

(2.19)

Of particular interest are the three angular momentum operators R LL i , Li ,

R Li = L L i − Li .

(2.20)

Of these, Li annihilates 1 as does the continuum orbital angular momentum. It is the fuzzy sphere angular momentum approaching the orbital angular momentum of S 2 as n → ∞: ~ ≡ −iijk x(ξ)j Li → −i ~x(ξ) ∧ ∇

∂ ∂x(ξ)k

as n → ∞.

(2.21)

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2.5

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9

Diagonalizing L2

P P R 2 n n 2 We have i (LL i ) = i (Li ) = 2 2 + 1 so that orbital angular momentum is the sum of two angular momenta with values n2 . Hence the spectrum of L2 is ∞

h`(` + 1) : ` ∈ {0, 1, 2, ..., n}i.

(2.22)

2

A function f in C (S ) has the expansion X a`m Y`m f=

(2.23)

`,m

in terms of the spherical harmonics Y`m . The spectrum of orbital angular momentum is thus h`(` + 1) : ` ∈ {0, 1, 2, . . . , n, , . . .}i. The spectrum of L2 is thus precisely that of the continuum orbital angular momentum cut off at n. There is no distortion of eigenvalues upto n. ` The eigenstates Tm , m ∈ {−`, −` + 1, ..., `} of L2 are known as polarization tensors [46]. They are eigenstates of L3 and also orthonormal: ` ` L 2 Tm = `(` + 1)Tm ,

` ` = mTm , L 3 Tm `0 ` Tm0 , Tm = δ``0 δm0 m .

2.6

(2.24)

Scalar Fields on SF2

We will be brief here as they are treated in detail in chapter 4. A complex scalar field Φ on S 2 is a power series in the coordinate functions mi := xi , X Φ= ai1 ...in mi1 · · · min . (2.25)

~ 2 and (Note again that m ~ ·m ~ = 1) The Laplacian on S 2 is ∆ := −(−i~x ∧ ∇) a simple Euclidean action is Z dΩ ∗ Φ ∆Φ dΩ = d cos(θ)dφ . (2.26) S=− 4π We can simplify (2.26) by the expansion X Φ(~x) = Φ`m Y`m (m) ~ . (2.27) R dΩ ~ ∗ Y`m (m) ~ = Then since ∆Y`m (m) ~ = −`(` + 1)Y`m (m), ~ and 4π Y`0 m0 (m) δ``0 δmm0 , X S= `(` + 1)Φ∗`m Φ`m . (2.28) `,m

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From (2.25), we infer that the fuzzy scalar field Ψ is a power series in the matrices xi and is hence itself a matrix. The Euclidean action replacing (2.26) is S = (Li Ψ , Li Ψ) = −(Ψ , ∆Ψ) . On expanding ψ according to Ψ=

X

` ψ`m Tm ,

(2.29)

(2.30)

`≤n+1

this reduces to S=

X

`≤n

2.7

`(` + 1)|ψ`m |2 .

(2.31)

The Holstein-Primakoff Construction

There is an interesting construction of Li for fixed n using just one oscillator due to Holstein and Primakoff [47]. We outline this construction here. ˆ commutes with Li , we can eliminate a2 from Li and In brief, since N restrict Li to Hn without spoiling their commutation relations. The result is the Holstein-Primakoff construction. We now give the details. (2.5) gives the following polar decomposition of a2 : q U †U = U U † = 1 , (2.32) a2 = U N − a†1 a1 , where we choose the positive square root: q N − a†1 a1 ≥ 0 .

(2.33)

We can understand U better by examining the action of a2 on the orthonormal states (2.8) spanning Hn . We find √ a2 |n1 , n2 i = n2 |n1 , n2 − 1i q = U N − a†1 a1 |n1 , n2 i √ = n2 U |n1 , n2 i (2.34)

or U |n1 , n2 i = |n1 , n2 − 1i

(2.35)

Thus if A† = a†1 U ,

A = U † a1 ,

(2.36)

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then

√

A† |n1 , n2 i =

n1 + 1|n1 + 1 , n2 − 1i , √ A|n1 , n2 i = n1 |n1 − 1 , n2 − 1i ,

(2.37)

and [A , A† ] = 1 ,

[A† , A† ] = [A , A] = 0 ,

[A , N ] = [A† , N ] = 0 .

(2.38)

a2 vanishes on |n , 0i and U and A† are undefined on that vector. That is, A and A† can not be defined on Hn . In any case, the oscillator algebra of (2.38) has no finite-dimensional representation. But this is not the case for Li . We have p L+ = L1 + iL2 = a†1 a2 = A† N − A† A p L− = L1 − iL2 = a†2 a1 = N − A† A A L3 = a†1 a1 − a†2 a2 = A† A − N

(2.39)

On Hn , (2.39) gives the Holstein-Primakoff realization of the SU (2) Lie algebra for angular momentum n2 . CP N and Fuzzy CP N

2.8

S 2 is CP 1 as a complex manifold. The additional structure for CP 1 as compared to S 2 is only the complex structure. So we can without great harm denote S 2 and SF2 also as CP 1 and CPF1 . In section 5.4.2, we will in fact consider the complex structure and its quantization. Generalizations of CP 1 and CPF1 are CP N and CPFN . They are associated with the groups SU (N + 1). Classically CP N is the complex projective space of complex dimension N . It can described as follows. Consider the (2N + 1)-dimensional sphere D E X S 2N +1 = ξ = ξ1 , ξ2 · · · , ξN +1 : ξi ∈ C , |ξ|2 := |ξi |2 = 1 . (2.40) It admits the U (1) action CP

N

is the quotient of S

2N +1

ξ → eiθ ξ .

(2.41)

by this action giving rise to the fibration

U (1) → S 2N +1 → CP N .

(2.42)

If λi are the Gell-Mann matrices of SU (N + 1), a point of CP N is ~ X(ξ) = ξ †~λξ , ξ ∈ S 2N +1 . (2.43)

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For N = 1, these become the previously constructed structures. There is another description of S 2N +1 and CP N . SU (N + 1) acts transitively on S 2N +1 and the stability group at (1 , ~0) is 10 SU (N ) = u ∈ SU (N + 1) : u = . (2.44) 0u ˆ Hence S 2N +1 = SU (N + 1) SU (N ) .

Consider the equivalence class

h(1 , ~0)i = (eiθ , ~0) eiθ ∈ U (1)

(2.45)

(2.46)

of all elements connected to (1 , ~0) by the U (1) action (2.41). Its orbit under SU (N + 1) is CP N . The stability group of (2.46) is iθ e 0 S U (1) × U (N ) = υ ∈ SU (N + 1) : υ = . (2.47) 0 υˆ Thus

CP N = SU (N + 1) S U (1) × U (N ) .

(2.48) S U (1) × U (N ) is commonly denoted as U (N ). The two groups are isomorphic. To obtain CPFN , we think of S 2N +1 as a submanifold of CN +1 \ {0}: E D z : z = (z1 , z2 , · · · , zN +1 ) ∈ CN +1 \ {0} . (2.49) S 2N +1 = ξ = |z| Just as before, we can quantize CN +1 by replacing zi by annihilation operators ai and zi∗ by a†i : [ai , aj ] = [a†i , a†j ] = 0 ,

[ai , a†j ] = δij .

(2.50)

With b = a † ai N i

(2.51)

as the number operator, the quantized ξ is given by the correspondence ξi =

zi 1 −→ ai p , |z| b N

N 6= 0 .

(2.52)

Then as in (2.6), we get the CPFN coordinates Xi (z) −→ Xi =

1 † a λi a , b N

N 6= 0 .

(2.53)

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The rest of the discussion follows that of CP 1 with SU (N + 1) replacing SU (2). Because of (the analogue of) (2.7), Xi can be restricted to Hn , the b = n. It is spanned by the orthonormal subspace of the Fock space with N vectors N +1 Y X (a†i )ni √ |0i := |n1 n2 , · · · , N + 1i , ni = n , (2.54) ni ! i=1 and is of dimension

(N + n)! . (2.55) n!N ! The SU (N +1) angular momentum operators are given by a generalized Schwinger construction : λi [Li , Lj ] = ifijk Lk . (2.56) Li = a † a , 2 M=

N +1+n

Cn =

a†i transform by the unitary irreducible representation (UIR) of dimension (N +1) of SU (N +1) and (2.54) span the space of n fold symmetric product of these UIR’s. It carries a UIR of dimension (2.55) and has the quadratic Casimir operator X N n2 +n 1. (2.57) L2i = 2 N +1

Its remaining Casimir operators are fixed by (2.57). As before 2 Xi H n = L i H n , n 2 [Xi , Xj ] H = ifijk Xk H , n n n X 2N 2N 2 (2.58) Xi H n = 1 Hn . + N +1 n r 2N 2N The “size” of CPFN is measured by the “radius” + N +1 n . In the n → ∞ limit, the Xi ’s also commute and generate C ∞ (CP N ). The “wave functions” of CPFN are polynomials in Xi , that is they are elements of M at(M ), with a scalar product like (2.13). As before, for each α ∈ M at(M ), we have two observables αL,R and they constitute the matrix algebras M atL,R (M ). The discussions leading up to (2.18) and (2.20) can be adapted also to CPFN . As for (2.21), it generalizes to Li −→ −ifijk Xj (ξ)

∂ . ∂Xk (ξ)

(2.59)

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Diagonalization of Li involves the reduction of the product of the UIR’s of R SU (N + 1) given by LL i and its complex conjugate given by Li to their irreducible components. The corresponding polarization operators can also in principle be constructed. The scalar field action (2.28) generalizes easily to CPFN . 2.9

The CP N Holstein-Primakoff Construction

The generalization of this construction to CP N and SU (N + 1) is due to Sen [48]. Consider for specificity N = 2 and SU (3) first. SU (3) has 3 oscillators a1 , a2 , a3 . There are also the SU (2) algebras with generators 3 2 ~σ ~σ X X a†i a†i aj , aj , (2.60) 2 ij 2 ij i,j=2 i=j=1 acting on the indices 1, 2 and 2, 3 respectively, of a’s and a† ’s. Taking their commutators, we can generate the full SU (3) Lie algebra. We will eliminate a2 , a†2 from both these sets using the previous Holstein-Primakoff construction. In that way, we will obtain the SU (3) Holstein-Primakoff construction. As previously we write the polar decompositions p p a2 = U2 N2 , a†2 = N2 U2† , N2 = a†2 a2 , U2† U2 = 1 . (2.61) The oscillators act on the Fock space ⊕N HN spanned by (2.54) for N = 2. The actions of U2 and A†12 = a†1 U2 ,

A12 = U2† a1

(2.62)

follow (2.35) and (2.36). They do not affect n3 . Using (2.39), we can write the SU (2) generators acting on (12) indices as p I+ = a†1 a2 = A†12 N2 , p I− = a†2 a1 = N2 A12 , 1 1 (2.63) I3 = a†1 a1 − a†2 a2 = A†12 A12 − N2 . 2 2 We follow the I , U , V spin notation of SU (3) in particle physics [49]. They are connected by Weyl reflections. In a similar manner, the SU (3) generators acting on (23) indices are constructed from A†32 = a†3 U2 , A32 = U2† a3 ,

(2.64)

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and read

p U+ = a†3 a2 = A†32 N2 , p U− = a†2 a3 = N2 A32 , 1 1 (2.65) U3 = a†2 a2 − a†3 a3 = N2 − A†32 A32 . 2 2 In a UIR of SU (3), the total number operator N = N1 + N2 + N3 is fixed. Acting on Hn , it becomes n. Keeping this in mind, we now substitute N2 = N − N1 − N3 = N − A†12 A12 − A†32 A32

in (2.63) and (2.65) to eliminate the second oscillator. That gives p I+ = A†12 N − N1 − N3 , p I− = N − N1 − N3 A12 , N3 N I3 = N 1 + − 2 p2 U+ = A†32 N − N1 − N3 , p U− = N − N1 − N2 A32 , N N1 − U 3 = − N3 + 2 2

(2.66)

(2.67)

(2.68)

These operators and their commutators generate the full SU (3) Lie algebra when restricted to Hn . That is the SU (3) Holstein-Primakoff construction. If the restriction to Hn is not made, N is a new operator and we get instead the U (3) Lie algebra with N generating its central U (1). The Holstein-Primakoff construction for CP N is much the same. One introduces N +1 oscillators ai , a†i (i ∈ [1 , · · · N ]) with which the SU (N +1) Lie algebra can be realized using the Schwinger construction. The SU (N + 1) UIR’s we get therefrom are symmetric products of the fundamental representation of dimension (N + 1). The number operator N = a† · a has a fixed value in one such UIR. Next a2 , a†2 are eliminated from SU (N + 1) generators in favor of the N and the remaining operators to obtain the generalized Holstein-Primakoff construction. SU (N + 1) is of rank N , and we can realize its Lie algebra with N oscillators. There is a similar result in quantum field theory where with the help of the vertex operator construction, a (simply laced) rank N Lie algebra can be realized with N scalar fields on S 1 × R valued on S 1 [50]. This resemblance perhaps is not an accident.

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Chapter 3

Star Products

3.1

Introduction

The algebra of smooth functions on a manifold M under point-wise multiplication is commutative. In deformation quantization [51], this point-wise product is deformed to a non-commutative (but still associative) product called the ∗-product. It has a central role in many discussions of noncommutative geometry. It has been fruitfully used in quantum optics for a long time. The existence of such deformations was understood many years ago by Weyl, Wigner, Groenewold and Moyal [52, 40, 43]. They noted that if there is a linear injection (one-to-one map) ψ of an algebra A into smooth functions C∞ (M) on a manifold M, then the product in A can be transported to the image ψ(A) of A in C∞ (M) using the map. That is then a ∗-product. Let us explain this construction with greater completeness and generality [31]. For concreteness we can consider A to be an algebra of bounded operators on a Hilbert space closed under the Hermitian conjugation of ∗. It is then an example of a ∗-algebra. More generally, A can be a generic “∗-algebra’, that is an algebra closed under an anti-linear involution: a , b ∈ A , λ ∈ C ⇒ a∗ , b∗ ∈ A , (ab)∗ = b∗ a∗ , (λa)∗ = λ∗ a∗ .

(3.1)

A two-sided ideal A0 of A is a subalgebra of A with the property a0 ∈ A0 ⇒ αa0 and a0 α ∈ A0 , ∀α ∈ A .

(3.2)

That is AA0 , A0 A ⊆ A0 . A two-sided ∗-ideal A0 by definition is itself closed under ∗ as well. 17

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An element of the quotient A/A0 is the equivalence class {α + A0 ⊂ A} = [α + a0 ] a0 ∈ A0 .

(3.3)

If A0 is a two-sided ideal, A/A0 is itself an algebra with the sum and the product (α + A0 ) + (β + A0 ) = α + β + A0 , (α + A0 )(β + A0 ) = αβ + A0 .

(3.4)

If A0 is a two-sided ∗-ideal, then A/A0 is a ∗-algebra with the ∗-operation (α + A0 )∗ = α∗ + A0 .

(3.5)

We note that the product and ∗ are independent of the choice of the representatives α , β from the equivalence classes α+A0 and β +A0 because A0 is a two-sided ideal. So they make sense for A/A0 . Let C ∞ (M) denote the complex-valued smooth functions on a manifold M. Complex conjugation −(bar) is defined on these functions. It sends a ¯ function f to its complex conjugate f. We consider the linear maps

ψ

X

λi a i

ψ : A −→ C ∞ (M) , X = λi ψ(ai ) , ai ∈ A ,

(3.6) λi ∈ C .

(3.7)

The kernel of such a map is the set of all α ∈ A for which ψ(α) is the zero function 0. (Its value is zero at all points of M): Ker ψ = hα0 ∈ A ψ(α0 ) = 0i . (3.8)

ψ descends to a linear map, called Ψ, from A/Ker ψ = {α+Ker ψ : α ∈ A} to C ∞ (M): Ψ(α + Ker ψ) = ψ(α) .

(3.9)

ψ(α) does not depend on the choice of the representative α from α + Ker ψ because of (3.8). Clearly Ψ is an injective map from A/Ker ψ to C ∞ (M). If Ker ψ is also a two-sided ideal, Ψ is a linear map from the algebra A/Ker ψ to C ∞ (M). Using this fact, we define a product, also denoted by ∗, on Ψ(A/Ker ψ) = ψ(A) ⊆ C ∞ (M) : Ψ(α + Ker ψ) ∗ Ψ(β + Ker ψ) = Ψ (α + Ker ψ) (β + Ker ψ) . (3.10) or

ψ(α) ∗ ψ(β) = ψ(αβ) .

(3.11)

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With this product, ψ(A) is an algebra (ψ(A) , ∗) isomorphic to A/Ker ψ. (The notation means that ψ(A) is considered with product ∗ and not say point-wise product). We assume that A/Ker ψ is a ∗-algebra and that Ψ preserves the stars on A/Ker ψ and C ∞ (M), the ∗ on the latter being complex conjugation denoted by bar: Ψ (α + Ker ψ)∗ = Ψ(α + Ker ψ) , ψ(α∗ ) = ψ(α) .

(3.12)

Such ψ and Ψ are said to be ∗-morphisms from A and A/Ker ψ to (ψ(A) , ∗). The two algebras A/Ker ψ and (ψ(A) , ∗) are ∗-isomorphic. Remark: Star (∗) occurs with two meanings. (1) It refers to involution on algebras in the phrase ∗-morphism. (2) It refers to the new product on functions in (ψ(A) , ∗). These confusing notations, designed to keep the reader alert, are standard in the literature. The above is the general framework. In applications, we encounter more than one linear bijection (one-to-one, onto map) from an a algebra A to C ∞ (M) and that produces different-looking ∗’s on C ∞ (M) and algebras (C ∞ (M) , ∗), (C ∞ (M) , ∗0 ) etc. As they are ∗-isomorphic to A, they are mutually ∗-isomorphic as well. A simple example we encounter below is C ∞ (C) with Moyal- and coherent-state-induced ∗-products. These algebras are ∗-isomorphic. 3.2

Properties of Coherent States

It is useful to have the Campbell-Baker-Hausdorff (CBH) formula written down. It reads ˆ ˆ

ˆ

ˆ

1

ˆ ˆ

eA eB = eA+B e 2 [A ,B]

(3.13)

ˆ = [B ˆ , [Aˆ , B]] ˆ = 0. [Aˆ , [Aˆ , B]]

(3.14)

ˆ if for two operators Aˆ , B

For one oscillator with annihilation-creation operators a,a† , the coherent state |zi = eza

†

−¯ za

1

2

†

|0i = e− 2 |z| eza |0i ,

z∈C

(3.15)

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has the properties a|zi = z|zi ;

1

0 2

hz 0 |zi = e 2 |z−z | .

(3.16)

The coherent states are overcomplete, with the resolution of identity Z 2 d z x1 + ix2 √ 1= |zihz| , d2 z = dx1 dx2 , where z = . (3.17) π 2 R The factor π1 is easily checked: T r 1|0ih0| = 1 while d2 z|h0|zi|2 is π in view of (3.16). A central property of coherent states is the following: an operator Aˆ is determined just by its diagonal matrix elements ˆ , A(z , z¯) = hz|A|zi

(3.18)

ˆ that is by its “symbol” A, a function∗ on C with values A(z , z¯) = hz|A|zi. An easy proof uses analyticity [54]. Aˆ is certainly determined by the colˆ or equally by lection of all its matrix elements h¯ η |A|ξi 1

e 2 (|η|

2

+|ξ|2 )

ˆ = h0|eηa Ae ˆ ξa† |0i . h¯ η |A|ξi

(3.19)

The right hand side (at least for appropriate A) is seen to be a holomorphic function of η and ξ, or equally well of η+ξ η−ξ , v= . (3.20) 2 2i Holomorphic functions are globally determined by their values for real arguments. Hence the function A˜ defined by u=

˜ v) = h0|eηa† Ae ˆ ξa† |0i A(u,

(3.21)

¯ Thus hξ|A|ξi ˆ is globally determined by its values for u, v real or η = ξ. ˆ determines A as claimed. ¯ [55]. There are also explicit formulas for Aˆ in terms of hξ|A|ξi 3.3

The Coherent State or Voros ∗-Product on the Moyal Plane

As indicated above, we can map an operator Aˆ to a function A using coherent states as follows: ∗ The

Aˆ −→ A ,

ˆ A(z , z¯) = hz|A|zi.

(3.22)

z¯ argument in A(z , z¯) is redundant. It is there to emphasize that A is not necessarily a holomorphic function of the complex variable z.

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This map is linear and also bijective by the previous remarks and induces a product ∗C on functions (C indicating “coherent state”). With this product, we get an algebra (C ∞ (C) , ∗C ) of functions. Since the map Aˆ → A ¯ this map is a ∗-morphism from operators has the property Aˆ∗ → A∗ ≡ A, ∞ to (C (C) , ∗C ). Let us get familiar with this new function algebra. The image of a is the function α where α(z , z¯) = z. The image of an has the value z n at (z , z¯), so by definition, (α ∗C α . . . ∗C α)(z , z¯) = z n .

(3.23)

The image of a∗ ≡ a† is α ¯ where α ¯ (z, z¯) = z¯ and that of (a∗ )n is α ¯ ∗C α ¯ · · · ∗C α ¯ where (¯ α ∗C α ¯ · · · ∗C α ¯ )(z , z¯) = z¯n .

(3.24)

Since hz|a∗ a|zi = z¯z and hz|aa∗ |zi = z¯z + 1, we get α ¯ ∗C α = α ¯α ,

α ∗C α ¯ = αα ¯ +1,

(3.25)

where αα ¯ = αα ¯ is the pointwise product of α and α, ¯ and 1 is the constant function with value 1 for all z. For general operators fˆ, the construction proceeds as follows. Consider : eξa

†

¯ −ξa

:

(3.26)

where the normal ordering symbol : · · · : means as usual that a† ’s are to be put to the left of a’s. Thus : aa† a† a : = a† a† aa , : eξa

†

¯ −ξa

†

¯

: = eξa e−ξa .

(3.27)

Hence hz| : eξa

†

¯ −ξa

¯

: |zi = eξz¯−ξz .

Writing fˆ as a Fourier transform, Z 2 d ξ ξa† −ξa ¯ ˜ , ξ) ¯ , ˆ :e : f(ξ f= π

˜ , ξ) ¯ ∈ C, f(ξ

(3.28)

(3.29)

its symbol is seen to be

f=

Z

d2 ξ ξz¯−ξz ¯ ˜ ¯ . e f (ξ , ξ) π

˜ This map is invertible since f determines f.

(3.30)

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Consider also the second operator Z 2 d η ηa† −¯ηa gˆ = : g˜(η , η¯) , :e π and its symbol Z 2 d η ηz¯−¯ηz g= g˜(η , η¯) . e π The task is to find the symbol f ∗C g of fˆgˆ. Let us first find ¯ eξz¯−ξz ∗C eηz¯−¯ηz . We have † † † † ¯ ¯ ¯ : eξa −ξa : : eηa −¯ηa :=: eξa −ξa eηa −¯ηa : e−ξη and hence ¯ ¯ ¯ eξz¯−ξz ∗C eηz¯−¯ηz = e−ξη eξz¯−ξz eηz¯−¯ηz ¯

(3.31)

(3.32)

(3.33) (3.34)

− ← − →

(3.35) = eξz¯−ξz e ∂ z ∂ z¯ eηz¯−¯ηz . ← − → − k The bidifferential operators ∂ z ∂ z¯ , (k = 1, 2, ...) have the definition ← − → − k ∂ k α(z , z¯) ∂ k β(z , z¯) . (3.36) α ∂ z ∂ z¯ β (z , z¯) = ∂z k ∂ z¯k The exponential in (3.35) involving them can be defined using the power series. f ∗C g follows from (3.35): ← − → − (3.37) (f ∗C g) (z , z¯) = f e ∂ z ∂ z¯ g (z , z¯) . (3.37) is the coherent state ∗-product [56]. We can explicitly introduce a deformation parameter θ > 0 in the discussion by changing (3.37) to ← − → − f ∗C g (z , z¯) = f eθ ∂ z ∂ z¯ g (z , z¯) . (3.38) After rescaling z 0 = √zθ , (3.38) gives (3.37). As z 0 and z¯0 after quantization become a , a† , z and z¯ become the scaled oscillators aθ , a†θ : [aθ , aθ ] = [a†θ , a†θ ] = 0 , [aθ , a†θ ] = θ . (3.39) (3.39) is associated with the Moyal plane with Cartesian coordinate func+ix2 −ix2 tions x1 , x2 . If aθ = x1√ , a†θ = x1√ , 2 2 [xi , xj ] = iθεij , εij = −εji , ε12 = 1 . (3.40) 2 The Moyal plane is the plane R , but with its function algebra deformed in accordance with (3.40). The deformed algebra has the product (3.38) or equivalently the Moyal product derived below. Discussion of the equivalence between Moyal and Voros star products, using Stratonovich’s methods [57] and with several physical applications is given in [58, 59].

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3.4

The Moyal ∗-Product on the Groenewold-Moyal Plane

We get this by changing the map fˆ → f from operators to functions. For a given function f , the operator fˆ is thus different for the coherent state and Moyal ∗’s. The ∗-product on two functions is accordingly also different. 3.4.1

The Weyl Map and the Weyl Symbol

The Weyl map of the operator Z 2 d ξ ˜ ¯ ξa† −ξa ¯ ˆ f (ξ , ξ)e , f= π

(3.41)

to the function f is defined by

f (z , z¯) =

Z

d2 ξ ˜ ¯ ξz¯−ξz ¯ f (ξ , ξ)e . π

(3.42) makes sense since f˜ is fully determined by fˆ as follows: Z 2 d ξ ˜ ¯ − 1 ξξ¯ ξz¯−ξz ¯ ˆ f (ξ , ξ)e 2 e . hz|f|zi = π

(3.42)

(3.43)

f˜ can be calculated from here by Fourier transformation. The map is invertible since f˜ follows from f by Fourier transform of (3.42) and f˜ fixes fˆ by (3.41). ˆ f is called the Weyl symbol of f. As the Weyl map is bijective, we can find a new ∗ product, call it ∗W , between functions by setting f ∗W g = Weyl symbol of fˆgˆ. For † ¯ fˆ = eξa −ξa ,

gˆ = eηa

†

−¯ ηa

,

(3.44)

to find f ∗W g, we first rewrite fˆgˆ according to

† 1 ¯ ¯ fˆgˆ = e 2 (ξη¯−ξη) e(ξ+η)a −(ξ+¯η)a .

(3.45)

Hence ¯

1

¯

(f ∗W g) (z , z¯) = eξz¯−ξz e 2 (ξη¯−ξη) eηz¯−¯η z − → − ← − → − 1 ← = f e 2 ∂ z ∂ z¯− ∂ z¯ ∂ z g (z , z¯) .

(3.46)

˜ , ξ), ¯ g˜(η , η¯) and integrating, we get (3.46) for arbitrary Multiplying by f(ξ functions: 1 ← − − ← − → − → (3.47) (f ∗W g) (z , z¯) = f e 2 ∂ z ∂ z¯− ∂ z¯ ∂ z g (z , z¯) .

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Note that ← − → − ← − → − ← − → − ← − → − ← − → − ∂ z ∂ z¯ − ∂ z¯ ∂ z = i( ∂ 1 ∂ 2 − ∂ 2 ∂ 1 ) = iεij ∂ i ∂ j .

(3.48)

Introducing also θ, we can write the ∗W -product as ← − → − ∂j

θ

f ∗W g = f ei 2 εij ∂ i

g.

(3.49)

By (3.40), θεij = ωij fixes the Poisson brackets, or the Poisson structure on the Moyal plane. (3.49) is customarily written as ← − → − ∂j

i

f ∗W g = f e 2 ωij ∂ i

g.

(3.50)

using the Poisson structure. (But we have not cared to position the indices so as to indicate their tensor nature and to write ω ij .) 3.5

Properties of ∗-Products

A ∗-product without a subscript indicates that it can be either a ∗C or a ∗W . 3.5.1

Cyclic Invariance

The trace of operators has the fundamental property ˆ = T rB ˆ Aˆ T rAˆB

(3.51)

which leads to the general cyclic identities T r Aˆ1 . . . Aˆn = T r Aˆn Aˆ1 . . . Aˆn−1 . We now show that Z 2 d z ˆ ˆ A ∗ B (z , z¯) , T r AB = π

∗ = ∗C

or

(3.52)

∗W .

(3.53)

ˆ are fixed). (The functions on R.H.S. are different for ∗C and ∗W if Aˆ , B From this follows the analogue of (3.52): Z 2 Z 2 d z d z A1 ∗A2 ∗· · ·∗An ) (z , z¯ = An ∗A1 ∗· · ·∗An−1 ) (z , z¯ . (3.54) π π For ∗C , (3.53) follows from (3.17). † ¯ The coherent state image of eξa −ξa is the function with value ¯

1

¯

eξz¯−ξz e− 2 ξξ

(3.55)

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at z, with a similar correspondence if ξ → η. So Z 2 † † 1 d z ξz¯−ξz ¯ ¯ ¯ ¯ − 1 ξξ eηz¯−¯ηz e− 2 η¯η e−ξη T r eξa −ξa eηa −¯ηa = e e 2 π

(3.56)

The integral produces the δ-function Y ξ1 + ξ 2 η1 + η 2 2δ(ξi + ηi ) , ξi = √ , ηi = √ . (3.57) 2 2 i 1 1 ¯ ¯ 1 ¯ ¯ ξη by e 2 (ξη¯−ξη) and get (3.53) for We can hence substitute e− 2 ξξ+ 2 ηη+ Weyl ∗ for these exponentials and so for general functions by using (3.41). 3.5.2

A Special Identity for the Weyl Star

The above calculation also gives, the identity Z 2 Z 2 d z d z (A ∗W B) (z , z¯) = A(z , z¯) B (z , z¯) . π π

(3.58)

That is because Y

1

¯

δ(ξi + ηi ) e 2 (ξη¯−ξη) =

i

Y

δ(ξi + ηi ) .

(3.59)

i

In (3.54), A and B in turn can be Weyl ∗-products of other functions. Thus in integrals of Weyl ∗-products of functions, one ∗W can be replaced by the pointwise (commutative) product: Z 2 d z A1 ∗W A2 ∗W · · · AK ∗W (B1 ∗W B2 ∗W · · · BL (z , z¯) π Z 2 d z A1 ∗W A2 ∗W · · · AK (B1 ∗W B2 ∗W · · · BL (z , z¯) . (3.60) = π This identity is frequently useful.

3.5.3

Equivalence of ∗C and ∗W

For the operator † ¯ Aˆ = eξa −ξa ,

(3.61)

the coherent state function AC has the value (3.55) at z, and the Weyl symbol AW has the value ¯

AW (z , z¯) = eξz¯−ξz .

(3.62)

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As both C ∞ (R2 ) , ∗C and C ∞ (R2 ) , ∗W are isomorphic to the operator algebra, they too are isomorphic. The isomorphism is established by the maps AC ←→ AW

(3.63)

and their extension via Fourier transform to all operators Aˆ and functions AC ,W . Clearly 1

AW = e− 2 ∂z ∂z¯ AC ,

1

AC = e 2 ∂z ∂z¯ AW ,

AC ∗C BC ←→ AW ∗W BW .

(3.64)

The mutual isomorphism of these three algebras is a ∗-isomorphism ˆ † −→ B ¯C ,W ∗C ,W A¯C ,W . since (AˆB) 3.5.4

Integration and Tracial States

This is a good point to introduce the ideas of a state and a tracial state on a ∗-algebra A with unity 1. A state ω is a linear map from A to C, ω(a) ∈ C for all a ∈ A with the following properties: ω(a∗ ) = ω(a) , ω(a∗ a) ≥ 0 ,

ω(1) = 1 .

(3.65)

If A consists of operators on a Hilbert space and ρ is a density matrix, it defines a state ωρ via ωρ (a) = T r(ρa) .

(3.66)

If ρ = e−βH /T r(e−βH ) for a Hamiltonian H, it gives a Gibbs state via (3.66). Thus the concept of a state on an algebra A generalizes the notion of a density matrix. There is a remarkable construction, the Gel’fand- NaimarkSegal (GNS) construction which shows how to associate any state with a rank-1 density matrix [60]. A state is tracial if it has cyclic invariance [60]: ω(ab) = ω(ba) .

(3.67)

The Gibbs state is not tracial, but fulfills an identity generalizing (3.67). It is a Kubo-Martin-Schwinger (KMS) state [60].

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A positive map ω 0 is in general an unnormalized state: It must fulfill all the conditions that a state fulfills, but is not obliged to fulfill the condition ω 0 (1) = 1. Let us define a positive map ω 0 on (C ∞ (R2 ) , ∗) (∗ = ∗C or ∗W ) using integration: Z 2 d z ˆ ω 0 (A) = A(z , z¯) . (3.68) π

It is easy to verify that ω 0 fulfills the properties of a positive map. A tracial positive map ω 0 also has the cyclic invariance (3.67). The cyclic invariance (3.67) of ω 0 (A∗B) means that it is a tracial positive map. 3.5.5

The θ-Expansion

On introducing θ, we have (3.38) and f ∗W g(z , z¯) = f e

θ 2

The series expansion in θ is thus

Introducing the notation

g (z , z¯) .

(3.69)

∂f ∂g (z , z¯) (z , z¯) + O(θ 2 ) , ∂z ∂ z¯

(3.70)

θ ∂f ∂g ∂f ∂g − (z , z¯) + O(θ 2 ) . 2 ∂z ∂ z¯ ∂ z¯ ∂z

(3.71)

f ∗C g (z , z¯) = f g (z , z¯) + θ f ∗W g (z , z¯) = f g(z , z¯) +

← − → − ← − → − ∂ z ∂ z¯ − ∂ z¯ ∂ z

[f , g]∗ = f ∗ g − g ∗ f ,

∗ = ∗C

or ∗W ,

(3.72)

We see that ∂f ∂g − ∂z ∂ z¯ ∂f ∂g =θ − ∂z ∂ z¯

[f , g]∗C = θ [f , g]∗W We thus see that

∂f ∂g (z , z¯) + O(θ 2 ) , ∂ z¯ ∂z ∂f ∂g (z , z¯) + O(θ 2 ) , ∂ z¯ ∂z

[f , g]∗ = iθ{f , g}P.B. + O(θ2 ) ,

(3.73)

(3.74) 2

where {f , g} is the Poisson Bracket of f and g and the O(θ ) term depends on ∗C ,W . Thus the ∗-product is an associative product which to leading order in the deformation parameter (“Planck’s” constant) θ is compatible with the rules of quantization of Dirac. We can say that with the ∗-product,

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we have deformation quantization of the classical commutative algebra of functions. But it should be emphasized that even to leading order in θ, f ∗C g and ∞ 2 f ∗W g do not agree. Still the algebras C (R , ∗C ) and C ∞ (R2 , ∗W ) are ∗-isomorphic. Suppose we are given a Poisson structure on a manifold M with Poisson bracket {. , .}. Then Kontsevich ([61]) has given a ∗-product f ∗g as a formal power series in θ such that (3.74) holds. For further developments on the physical equivalence of ∗C and ∗W see, [62]. 3.6

The ∗-Products for the Fuzzy Sphere

Star products for K¨ ahler manifolds have been known for a long time. The approach we take here was initiated by Preˇsnajder [63](See also [62], it produces particularly compact expressions.). Let Pn be the orthogonal projection operator to the subspace with N = n. The fuzzy sphere algebra is then the algebra with elements Pn γ(a†i aj )Pn where γ is any polynomial in a†i aj . As any such polynomial commutes with N , if γ and δ are two of these polynomials, Pn γ(a†i aj )Pn Pn δ(a†i aj )Pn = Pn γ(a†i aj )δ(a†i aj )Pn .

(3.75)

This algebra, more precisely, is the orthogonal direct sum M at(n + 1) ⊕ 0 b = n subspace and is the fuzzy sphere. But where M at(n + 1) acts on the N the extra 0 here is entirely harmless. 3.6.1

The Coherent State ∗-Product ∗C

There are now two oscillators a1 , a2 , so the coherent states are labeled by two complex variables, being |Z1 , Z2 i = eZa

†

¯ −Za

|0i ,

Z = (Z1 , Z2 ) .

(3.76)

We use capital Z’s for unnormalized Z’s and z’s for normalized ones: z = P Z 2 |Zi |2 . |Z| , |Z| = The normalized coherent states |zin for SF2 , as one can guess, are obtained by projection from |Zi, P † n Pn |Zi i zi ai √ |zin = = |0i , (3.77) |hPn |Zi| n!

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where we have used (Zi a†i )n |0i . (3.78) n! They are called Perelomov coherent states [54] ˆ n , the coherent state symbol has the value For an operator Pn AP Pn |Zi =

|z|2n ˆ n hz|A|zi (3.79) n! at Z. By a previous result, the diagonal coherent state expectation values ˆ n |zin determines Pn AP ˆ n uniquely and there is a ∗-product for S 2 . hz|Pn AP F We call it a ∗C -product in analogy to the notation used before. We can find it explicitly as follows [63, 31, 9]. For n = 1 (spin n2 = 21 ), a basis for 2 × 2 matrices is σA : σ0 = 1 , σi (i = 1, 2, 3) = Pauli Matrices , T rσA σB = 2δAB . (3.80) Let ˆ n |Zi = hZ|Pn AP

|ii = a†i |0i ,

i = 1, 2

(3.81)

be a pair of orthonormal vectors for n = 1. A general operator is (3.82) Fˆ = fA σ ˆA , σ ˆA = a† σA a n=1 , fA ∈ C . , we mean the restriction of and σ ˆA |ii = |ji(σA )ji . In above, by a† σA a n=1

a† σA a to the subspace with n = 1. Call the coherent state symbol of σ ˆA for n = 1 as χA : χA (z) = hz|ˆ σA |zi ,

χ0 (z) = 1 ,

χi = z¯σi z ,

i = 1, 2, 3 .

(3.83)

The ∗-product for n = 1 now follows: χA ∗C χB (z) = hz|ˆ σA σ ˆB |zi .

(3.84)

σA σB = δAB + EABi σi

(3.85)

Write

to get χA ∗C χB (z) = δAB + EABi χi (z)

:= χA (z)χB (z) + KAB (z) .

(3.86)

Let us use the notation ni = χi (z) ,

n0 = 1 .

(3.87)

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~n is the coordinate on S 2 : ~n · ~n = 1. Then (3.86) is nA ∗C nB (z) = nA nB + KAB (n) ,

KAB (z) := KAB (n) .

(3.88)

This KAB has a particular significance for complex analysis. Since χ0 (z) = 1, χ0 (z) ∗ χA = χ0 χA by (3.86) and K0A = 0 .

(3.89)

The components Kij (n) of K can be calculated from (3.85), (3.86). Let θ(α) be the spin 1 angular momentum matrices: θ(α)ij = −iεαij .

(3.90)

Then {~ θ · ~n (~ θ · ~n − 1)}ij 2 θ~ · ~n := θ(α)nα .

Kij (~n) =

(3.91)

The eigenvalues of ~ θ · ~n are ±1 , 0 and Kij (~n) is the projection operator to the eigenspace θ~ · ~n = −1, K(~n)2 = K(~n) .

(3.92)

It is related to the complex structure of S 2 in the projective module picture treated in chapter 5. The vector space for angular momentum n2 is the n-fold symmetric tensor product of the spin- 12 vector spaces. The general linear operator on this space can be written as Fb = fA1 A2 ···An σ ˆ A1 ⊗ σ ˆ A2 ⊗ · · · σ ˆ An

(3.93)

where f is totally symmetric in its indices. Its symbol is thus F (~n) = fA1 A2 ···An nA1 nA2 · · · nAn ,

n0 := 1 .

(3.94)

The symbol of another operator b = gB1 B2 ···Bn σ G ˆ B1 ⊗ σ ˆ B2 ⊗ · · · σ ˆ Bn ,

(3.95)

G(~n) = gB1 B2 ···Bn nB1 nB2 · · · nBn .

(3.96)

where g is symmetric in its indices, is

Since b = fA1 A2 ···An gB1 B2 ···Bn σA1 σB1 ⊗ σA2 σB2 ⊗ · · · ⊗ σAn σBn , FbG

we have that

F ∗ G(~n) = fA1 A2 ···An gB1 B2 ···Bn

Y i

nAi nBi + K Ai Bi

(3.97)

(3.98)

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or F ∗ G(~n) = n X F G(~n) +

n! fA A ···A A nA · · · n An ···A nA m!(n − m)! 1 2 m m+1 n m+1 m+2 m=1

×KA1 B1 (~n) KA2 B2 (~n) · · · KAm Bm (~n)gB1 B2 ···Bm Bm+1 ···Bn nBm+1 ×

nBm+2 · · · nBn .

(3.99)

Now as f and g are symmetric in indices, there is the expression n! fA A ···A A nA · · · n An , ···A nA (n − m)! 1 2 m m+1 n m+1 m+2 ∂ (3.100) ≡ ∂nAAi

∂A1 ∂A2 · · · ∂Am F (~n) = ∂ Ai

for F and a similar expression for G. Hence F ∗C G(~n) =

n X (n − m)! ∂A1 ∂A2 · · · ∂Am F (~n) m!n! m=0

× KA1 B1 (~n) KA2 B2 (~n) · · · KAm Bm (~n) ∂B1 ∂B2 · · · ∂Bm G (~n)

(3.101)

which is the final answer. Here the m = 0 term is to be understood as F G(~n), the pointwise product of F and G evaluated at ~n. This formula was first given in [63]. Differentiating on nA ignoring the constraint ~n · ~n = 1 is justified in the final answer (3.101) (although not in (3.100)), since KAB (~n)∂A (~n · ~n) = KAB (~n)∂B (~n · ~n) = 0. (3.100) being only an intermediate step on the way to (3.101), this sloppiness is immaterial. For large n, (3.101) is an expansion in powers of n1 , the leading term giving the commutative product. Thus the algebra SF2 is in some sense a deformation of the commutative algebra of functions C ∞ (S 2 ). But as the maximum angular momentum in F and G is n, we get only the spherical harmonics Y`m , ` ∈ {0, 1, · · · n} in their expansion. For this reason, F and G span a finite-dimensional subspace of C ∞ (S 2 ) and SF2 is not properly a deformation of the commutative algebra C ∞ (S 2 ). 3.6.2

The Weyl ∗-Product ∗W

The Weyl ∗-products are characterized by the special identity described before. For this reason they are very convenient for use in loop expansions in quantum field theory (see chapter 4).

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A simple way to find ∗W is to find it via its connection to ∗C . For this purpose let us consider Z 1 1 n+1 ` † ˆ `0 T r(Tˆm ) T m0 = dΩ Tn (`) 2 Y `m ∗C Tn (`0 ) 2 Y`0 m0 (~x) , (3.102) 4π where 1

` hz, n|Tm |z, ni = Tn (`) 2 Y`m (ˆ n) .

(3.103)

` Tm is defined in section 2.5 while we can show (3.103) as follows. The factor 1 Tn (`) 2 is independent of m by rotational invariance. It is real as shown by complex conjugating (3.103) and using ` † ` (Tm ) = (−1)m T−m ,

Y¯`m (ˆ n) = (−1)m Y` ,−m (~n) .

(3.104)

It can be chosen to be positive as well by explicit calculation. We shall evaluate it later. ` The normalization of Tm and Y`m are Z ` † `0 (3.105) T r(Tm ) Tm0 = dΩY `m (~x)Y`0 m0 (~x) = δ``0 δmm0 .

Hence using (3.102) Z 0 0 δ`` δmm = dΩY `m (~x)Y`0 m0 (~x) Z n+1 1 1 = dΩ Tn (`) 2 Y `m (~x) ∗C Tn (`0 ) 2 Y`0 m0 (~x) (. 3.106) 4π

Equation (3.106) suggests that the fuzzy sphere algebra (SF2 , ∗W ) with the Weyl-Moyal product ∗M is obtained from the fuzzy sphere algebra (SF2 , ∗C ) with the coherent state ∗C product from the map χ : (SF2 , ∗C ) −→ (SF2 , ∗W ) ! r n+1 1 χ Tn (`) 2 Y`m = Y`m . 4π

(3.107)

The induced ∗, call it for a moment as ∗0 , on the image of χ is ! r r 1 1 n + 1 n + 1 Y`m ∗0 Y`0 m0 = χ Tn (`) 2 Y`m ∗C Tn (`0 ) 2 Y`0 m0 . (3.108) 4π 4π For the evaluation of (3.108), Y`m ∗C Y`0 m0 has to be written as a series in Y`00 m00 and χ applied to it term-by-term. We will not need its full details here. Now replace Y`m by Y `m and integrate. As χ commutes with rotaq tions, only the angular momentum 0 component of

1 n+1 2 4π Tn (`) Y `m

∗C

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q

n+1 0 12 0 0 4π Tn (` ) Y` m

contributes to the integral.

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This component is

1 . δ``0 δmm0 4π

Using (3.107) for ` = 0 and the value δ``0 δmm0 Y 00 ∗C Y00 = q 1 4π Tn (0) 2 = n+1 to be derived below, we get Z Z 0 (3.109) dΩY `m ∗ Y`0 m0 = δ``0 δmm0 = dΩY `m Y`0 m0 .

Hence ∗0 enjoys the special identity characterizing the Weyl-Moyal product for the basis of functions in our algebra and hence for all functions. ∗0 is the Weyl-Moyal product ∗W . Tn is a function Tn of `(` + 1). The latter is the eigenvalue of L2 , the square of angular momentum. The map χ can hence be defined directly on all functions α by r 1 n+1 Tn (L2 ) 2 α (3.110) χ(α) = 4π where R.H.S. can be calculated for example by expanding α in spherical harmonics. 1 The evaluation of Tn2 (`) can be done as follows. It is enough to compare the two sides of (3.103) for m = `. For m = `, p (2` + 1)! ` ` z¯2 z1 . (3.111) Y`` (~x) = `! ` The operator T` being the highest weight state commutes with L+ = a†2 a1 while [L3 , T`` ] = ` T``. Hence in terms of ai and a†j , ` T`` = N` a†` 2 a1 where the constant N` is to be fixed by the condition T r(T`` )† T`` = 1 . (a†1 )n1 (a†2 )n2 √ n1 !n2 !

(3.112) (3.113)

Evaluating L.H.S. in the basis |0i , n1 + n2 = n, we get after a choice of sign, p r 4π (n − `)!(n + 1)! (2` + 1)! (3.114) N` = n+1 n!`!(n + ` + 1)! and p r 4π (n − `)!(n + 1)! (2` + 1)! †` ` ` a2 a1 . (3.115) T` = n+1 n!`!(n + ` + 1)! Inserting (3.115) in (3.103) and using (3.111), we get, after a short calculation, r 1 4π n!(n + 1)! Tn (`) 2 = (3.116) n + 1 (n − `)!(n + ` + 1)! q 1 4π as claimed earlier. which gives Tn (0) 2 = n+1

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Chapter 4

Scalar Fields on the Fuzzy Sphere

The free Euclidean action for the fuzzy sphere for a scalar field is 2 1 1 ˆ i , φ] ˆ + µ φˆ2 S0 = Tr − [Li , φ][L n+1 2 2

(4.1)

where we now hat all operators or (n + 1) × (n + 1) matrices. As we saw in chapter 2, the scalar field can be expanded in terms of the ` polarization tensors Tˆm , X ` φˆ = φ`m Tˆm , (4.2) `,m

where φ`m are complex numbers. For concreteness, we will restrict our ˆ Since (Tˆ` )† = (−1)m Tˆ ` , this attention to Hermitian scalar fields φˆ† = φ. m m m implies that φ¯`,m = (−1) φ`,−m . In terms of φ`m ’s, the action (4.1) is S0 =

n+1 X `,m

=

n+1 X `=0

|φ`m |2 (`(` + 1) + µ2 ) 2 n+1 ` XX φ2`,0 |φ`m |2 2 (`(` + 1) + µ ) + 2 (`(` + 1) + µ2 ) . (4.3) 2 2 m=1 `=0

The generating function for correlators in this model is Z 1 ˆ TrJˆφ ˆ −S0 + n+1 ˆ = N0 Dφe Z0 (J)

(4.4)

ˆ the “external current” is an (n + 1) × (n + 1) Hermitian matrix. where J, Also Z −1 −S0 ˆ (4.5) N0 = Dφe 35

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is the usual normalization chosen so that Z0 (0) = 1

(4.6)

while Dφˆ = Let us write Jˆ =

Y dφ`0 Y dφ¯`m dφ`m √ . 2πi 2π |m|≤` `≤n/2

X

` J`m Tˆm ,

J¯`m = (−1)m J`m .

(4.7)

(4.8)

`,m

Then TrJˆφˆ =

X

J¯`m φ`m =

`,m

and

n+1 X

J`0 φ`0 +

`=0

` XX `

(J¯`m φ`m + J`m φ¯`m )

(4.9)

m≥1

! −φ2`,0 2 ˆ = N0 dφˆ exp (`(` + 1) + µ ) + J`0 φ`0 + Z0 (J) 2 ` # n+1 ` XX 2 2 −|φ`m | (`(` + 1) + µ ) + J¯`m φ`m + J`m φ¯`m . (4.10) "

Z

X

`=0 m=1

It is a product of Gaussians. Substituting φ`m = χ`m +

J`m `(` + 1) + µ2

(4.11)

and fixing N0 by the condition Z(0) = 1, we get Y 1 ˆ† 1 J¯`m J`m ˆ . (4.12) = exp Tr Z0 (Jˆ) = exp J J 2[`(` + 1) + µ2 ] 2 (−∆ + µ2 ) `m

Using (4.10) and (4.12) we can compute all correlators (Schwinger functions) of φ’s. For example, Z 2 ˆ) ∂ Z ( J δ ` 0 ` δ m0 m 0 −S = . hφ¯`0 m0 φ`m i := N0 Dφˆφ¯`0 m0 φ`m e = `(` + 1)µ2 ∂J`0 m0 ∂ J¯`m J=0 (4.13) All the correlators of φˆ follow from (4.13). For instance hφˆ2 i =

X

`,m,`0 ,m0

0

` † ˆ` ¯ Tˆm 0 Tm hφ`0 m0 φ`m i =

X `,m

` ˆ †` Tˆm Tm . `(` + 1) + µ2

(4.14)

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From this follow the correlators under the coherent state or Weyl maps. The latter (or working with matrices) is more convenient for current purposes. We have not given ∗W explicitly earlier for SF2 . But we will give the needed details here. The image φW under the Weyl map of φˆ has been defined earlier using ` ˆ φc (z) being hz|φ|zi. ˆ the coherent state symbol φc of φ, Since Tˆm becomes ` ` m ` Ym under the Weyl map, we get, using Y¯m = (−1) Ym , and dropping the subscript W , hφ(~x)φ(x~0 )i ≡ Gn (~x, x~0 ) = =

n ` X X Ym` (~x)Y¯m` (x~0 ) `(` + 1) + µ2 `=0 m=−`

` n X X

(−1)m

`=0 m=−`

` Ym` (~x)Y−m (x~0 ) . `(` + 1) + µ2

(4.15)

So as (−1)m = (−1)−m ,

(4.16)

Gn (~x, x~0 ) = Gn (x~0 , ~x) .

(4.17)

The symmetry of Gn is important for calculations. 4.1

Loop Expansion

There is a standard method to develop the loop expansion in the presence of interactions. Suppose the partition function is Z 1 ˆ TrJˆφ ˆ −S+ n+1 ˆ , (4.18) Z(J) = N Dφe 1 λ ˆ4 Trφ := S0 + SI , S = S0 + n + 1 4! Z ˆ −S ⇒ Z(0) = 1. N = Dφe

Let

λ > 0,

`1 ˆ `2 ˆ `3 ˆ `4 V (`1 m1 ; `2 m2 ; `3 m3 ; `4 m4 ) = Tr Tˆm T T T . 1 m2 m3 m4

(4.19) (4.20)

(4.21)

We can further abbreviate L.H.S. as follows:

V (`1 m1 ; `2 m2 ; `3 m3 ; `4 m4 ) := V (1234) .

(4.22)

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Now since

1 λ ˆ `1 ˆ `2 ˆ `3 ˆ `4 Tr Tm1 Tm2 Tm3 Tm4 φ`1 m1 φ`2 m2 φ`3 m3 φ`4 m4 n + 1 4! λ ≡ V (l1 , m1 ; l2 , m2 ; l3 , m3 ; l4 , m4 ; j)φ`1 m1 φ`2 m2 φ`3 m3 φ`4 m4 4! λ ≡ V (1234)φ`1 m1 φ`2 m2 φ`3 m3 φ`4 m4 , (4.23) 4! we can write, using (4.9), λ ∂ ∂ ∂ ∂ ˆ × Z(J) = N exp − V (1234) ¯ 4! ∂ J`1 m1 ∂ J¯`2 m2 ∂ J¯`3 m3 ∂ J¯`4 m4 Z 1 ˆ TrJˆφ ˆ −S0 + n+1 Dφe SI =

=

λ ∂ N ∂ ∂ ∂ exp − V (1234) ¯ N0 4! ∂ J`1m1 ∂ J¯`2 m2 ∂ J¯`3 m3 ∂ J¯`4 m4   X 1 1 J¯`m exp  J`m  , 2 −∆` + µ2 `,m

(−∆` + µ2 )−1 =

1 . `(` + 1) + µ2

(4.24)

Even before proceeding to calculate the one-loop two-point function, one can see that the interaction V (1234) in (4.23) has invariance only under cyclic permutation of its factors `i , mi and is not invariant under transpositions of adjacent factors. This means that we have to take care to distinguish between “planar” and “non-planar” graphs while doing perturbation theory as we shall see later below. The function V (1234) may be conveniently written as V (1234) = (n + 1)

4 Y

i=1

(2`i + 1)1/2 ×

l=n X `3 `4 l `1 `2 l l,m

n n n 2 2 2

n n n 2 2 2

`1 `2 ` `3 `4 ` (−1)m Cm Cm3 m4 −m . 1 m2 m

(4.25)

l1 l2 l The Cm are the Clebsch-Gordan (C-G) coefficients and the objects 1 m2 m with 6 entries within brace brackets are the 6j symbols. Although less obvious, the R.H.S of (4.25) has cyclic symmetry, as can be verified using properties of 6j symbols and C-G coefficients. The loop expansion of Z(J) is its power series expansion in λ. By differentiating it with respect to the currents followed by setting them zero,

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we can generate the loop expansion of the correlators. The K-loop term is the λK -th term, the zero loop being referred to as the tree term. We can write ∞ X λK zK (J) (4.26) Z(J) = 0

where λK zK (J) is the K-loop term. The factor N /N0 contributes multiplicative vacuum fluctuation diagrams to the correlation functions. It is a common factor to all correlators, and is a phase in Minkowski (real time) regime. 4.2

The One-Loop Two-Point Function

Of particular interest is the one-loop two-point function where one can see a “non-planar” graph unique to noncommutative theories. ˆ Expanding its numerator and denominator to O(λ), we get for Z(J), ∂ ∂ ∂ ˆ ≈ 1 − λ V (1234) ∂ × Z(J) ¯ ¯ ¯ ¯ 4! ∂ J` 1 m 1 ∂ J` 2 m 2 ∂ J` 3 m 3 ∂ J` 4 m 4   1X ¯ 1 exp  J`m J`m  × 2 −∆` + µ2 `,m

1−

λ V (1234)hφ`1 m1 φ`2 m2 φ`3 m3 φ`4 m4 i 4!

−1

.

(4.27)

Here, the argument i ∈ (1, 2, 3, 4) in V (1234) is to be interpreted as `i mi and `i , mi are to be summed over. Also the denominator comes from expanding N as power series in λ: ∞ X N = N (λ) := λ K NK . (4.28) K=0

This contributes disconnected diagrams, two of which are planar and one is non-planar. The disconnected diagrams are precisely canceled by other terms of (4.24) as we shall see. The O(λ) term of (4.24) or (4.27) is λz1 (J) where N1 ∂ ∂ ∂ 1 ∂ z1 (J) = × − V (1234) ¯ N0 4! ∂ J`1m1 ∂ J¯`2 m2 ∂ J¯`3 m3 ∂ J¯`4 m4   1X ¯ 1 J`m  , (4.29) exp  J`m 2 −∆` + µ2 `,m

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The two-point function follows by differentiation as in (4.13). Expanding the exact two-point function hφ`m φ¯`0 m0 i in powers of λ, hφ`m φ¯`0 m0 i = hφ`m φ¯`0 m0 i0 + λhφ`m φ¯`0 m0 i1 + . . .

(4.30)

we get

∂ z (J) 1 ∂ J¯`m ∂J`0 m0 J=0 ∂ N1 ∂ ∂ ∂ ∂ ∂ ¯ 0 0 × hφ`m φ` m i0 − = N0 ∂ J¯`m ∂J`0 m0 ∂J`1 m1 ∂J`2 m2 ∂ J¯`3 m3 ∂ J¯`4 m4    X 1  λ V (1234) exp  1 (4.31) J`m  J¯`m 4! 2 −∆` + µ2

hφ`m φ¯`0 m0 i1 =

`,m

J=0

(4.31) has both disconnected and connected diagrams. We briefly examine them. i. Disconnected Diagrams:

They come when the differentiations ∂J ∂0 0 , ∂ J¯∂`m both hit the same fac` m tor in the product of (4.31) to produce the free propagator. There are three such terms, two of which are planar diagrams and a non-planar diagram. 2 −1 1 These add up to − N δ``0 δmm0 : N0 [−∆` + µ ] N1 hφ`m φ¯`0 m0 iD [∆` + µ2 ]−1 δ``0 δmm0 1 =− N0 thus canceling the first term of (4.31).

(4.32)

ii. Connected Diagrams: They arise when the differentiation on external currents is applied to different factors in the product. There are 4 × 3 = 12 such terms, giving λ hφ`m φ¯`0 m0 iC 1 =− × 4! δ``4 δm+m4 ,0 (−1)m4 δ`0 `3 δm+m3 ,0 (−1)m3 δ`1 `2 δm1 +m2 ,0 (−1)m2 8 V (1234)+ −∆` + µ2 −∆`0 + µ2 −∆`1 + µ2 δ`` δm+m2 ,0 (−1)m2 δ`0 `4 δm+m4 ,0 (−1)m4 δ`1 `3 δm1 +m3 ,0 (−1)m3 4 2 V (1234) −∆` + µ2 −∆`0 + µ2 −∆`1 + µ2 (4.33) where, keeping in mind the symmetries of the trace, we have decomposed (4.33) into planar and nonplanar contributions. In the planar case, the

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indices of adjacent Tˆ’s get contracted. There are 8 such terms. In the non-planar case, it is the indices of the alternate Tˆ’s that get contracted, and there are 4 such terms. The planar term can be further simplified, by observing that `1 ˆ `1 `1 ˆ `1 † Tˆm T (−1)m1 = Tˆm T is rotationally invariant, and thus proportional 1 −m1 1 m1 to 1, the constant of proportionality being 1/(n + 1) (as seen by taking the trace). Incising the external legs, the one-loop planar contribution is thus (−∆`0 + µ2 )−1 = (−∆` + µ2 )−1 hφ`m φ¯`0 m0 iC,planar 1 n X 1 2` + 1 . (4.34) − δ``0 δm+m0 ,0 (−1)m 3 `(` + 1) + µ2 `=0 In the non-planar case, the indices of nonadjacent Tˆ’s get contracted. To evaluate the non-planar term, we need to make explicit use of the form (4.25). There are four such terms giving (−∆` + µ2 )−1 hφ`m φ¯`0 m0 iC,nonplanar (−∆`0 + µ2 )−1 = 1 4 l=n X Y X 1 `3 `4 l `1 `2 l 1/2 (2`i + 1) × − (n + 1) n n n n n n 6 2 2 2 2 2 2 i=1 `1 ,m1 ,`3 ,m3

l,m

m3 `1 `2 ` `3 `4 ` δ`1 `3 δm1 +m3 ,0 (−1) , ×(−1)m Cm Cm3 m4 −m 1 m2 m 2 `1 (`1 + 1) + µ X p 1 `1 `4 l ` ` l = − (n + 1) (2`2 + 1)(2`4 + 1) × (2` + 1) n1 n2 n n n n 6 2 2 2 2 2 2 `,m,`1 ,m1

m−m1

`1 `4 ` `1 `2 ` Cm C−m 1 m2 m 1 m4 −m

×(−1) We first perform the X sum `1 `4 ` `1 `2 ` (−1)m−m1 Cm C−m 1 m2 m 1 m4 −m

(4.35)

(4.36)

m,m1

for which we need the identities `1 `2 ` Cm 1 m2 m

X

`1 `4 ` C−m 1 m4 −m

r

2` + 1 `1 ` `2 , (4.37) C 2`2 + 1 m1 −m−m2 r 2` + 1 `1 ` `4 = (−1)`−`4 +m1 C , (4.38) 2`2 + 1 m1 −mm4

= (−1)

`1 −m1

Cm`11 m`22 m`33 Cm`11 m`22 m`44 = δ`3 `4 δm3 m4 .

(4.39)

m1 ,m2

This simplifies the non-planar contribution to 1 (−∆` +µ2 )−1 hφ`m φ¯`0 m0 iC,nonplanar (−∆`0 +µ2 )−1 = − (n+1)δ`2 `4 δm2 +m4 × 1 6 X `1 `2 l `1 `4 l `1 +` (2` + 1)(2`1 + 1) m2 −`2 (−1) . (4.40) (−1) n n n n n n `1 (`1 + 1) + µ2 2 2 2 2 2 2 `,`1

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This can be simplified even further, using the following identity involving the 6j symbols: n n X `1 `4 l `1 2 2 ` ` l = . (4.41) (−1)n+` (2` + 1) n1 n2 n n n n `4 n2 n2 2 2 2 2 2 2 `

We finally get

(−∆` + µ2 )−1 hφ`m φ¯`0 m0 iC,nonplanar (−∆`0 + µ2 )−1 = 1 1 − (n + 1)δ`2 `4 δm2 +m4 (−1)m2 (−1)`4 +n × 6 X (n + 1)(2`1 + 1) `1 n2 n2 (4.42) (−1)`1 `1 (`1 + 1) + µ2 `4 n2 n2 `1

The surprising fact is that this nonplanar contribution to the one-loop two-point function does not vanish even in the limit of n → ∞ [16]. In particular the difference between planar and non-planar contributions remains finite. To see this, we can use the Racah formula [46] n n (−1)`1 +`4 +n 2`4 2 `1 2 2 ' (4.43) P` 1 1 − 2 `4 n2 n2 n n where P` are the usual Legendre polynomials. Recall that the planar contribution from each Feynman diagram is proportional to X 2` + 1 (4.44) `(` + 1) + µ2 `=0

which is logarithmically divergent. The difference, proportional to n n X X 2`1 + 1 `1 2 2 `1 (n + 1)(2`1 + 1) (−1) δ≡ (4.45) − `1 (`1 + 1) + µ2 `1 (`1 + 1) + µ2 `4 n2 n2 `1 =0

`1

between planar and nonplanar terms then simplifies to n X 2` + 1 2`2 δ= 1 − P `1 1 − 2 . `(` + 1) + µ2 n

(4.46)

`=0

Using y = `/n, this sum is easily approximated by an integral for large n. Finally, substituting 1 − y 2 /2 = x gives us Z 1 `1 X 1 1 − P`1 (x) =2 (4.47) dx δ' 1−x k −1 k=1

This incorporates the UV-IR mixing [16–18]: δ depends on the external momentum `1 is non-vanishing for even small values of `1 . Thus integrating

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out high energy (or UV) modes in the loop produces non-trivial effects even at low (or IR) external momenta. This mixing has the potential to pose a serious challenge to any lattice program that uses matrix models on SF2 to discretize continuum models on the sphere. It is therefore important to ask if its effect can effectively be restricted to a class of n-point functions. To this end, one can calculate the four-point function at one-loop. Interestingly in this case, careful analysis shows that the difference between planar and the non-planar diagrams vanishes in the limit of large n [18]. Since only the quadratic term is affected by UV-IR mixing (albeit by a complicated momentum dependence), it suggests that appropriately “normal-ordered” vertices may completely eliminate this problem. That this is indeed the case was shown by Dolan, O’Connor and Presnajder [18]. Working with a modified action 2 1 1 ˆ i , φ] ˆ + µ φˆ2 + λ : φˆ4 : Tr − [Li , φ][L (4.48) S0 = n+1 2 2 4! where

 X [φ, X φˆTˆ † Tˆ`m φˆ ˆ Tˆ`m ]† [φ, ˆ Tˆ`m ] `m , +2 Tr : φˆ4 := Tr φˆ4 − 12 `(` + 1) + µ2 `(` + 1) + µ2 

`,m

`,m

(4.49) they showed that one gets the standard action on the sphere in the continuum limit n → ∞. One may ask if normal-ordering can help cure the UV-IR mixing problem in higher dimensions, say, on SF2 × SF2 . Here the problem is much more severe, and unfortunately persists after normal-ordering [19, 20]. 4.3

Numerical Simulations

One of the principal motivations of studying the fuzzy sphere is to approximate the ordinary sphere without losing any of the symmetries. We will briefly indicate the issues involved here, drawing from [21]. Functions are approximated by (n + 1) × (n + 1) matrices, and we are interested in a quantum partition function constructed from these matrices. For example, while studying the action (4.19), one may be interested in ˆ expectation values of some operator h(φ): Z ˆ ˆ −S[φ] ˆ . ˆ = 1 Dφe h(φ) (4.50) hh(φ)i Z

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ˆ the above integral involves (n + 1)2 integraFor Hermitian matrices φ, tions. Were one to use a typical integration routine which requires p steps, 2 calculation of the partition function would require p(n+1) steps. Since this diverges exponentially with matrix size, we require another strategy. Notice that in the exponential in the definition of the partition function, only a small number of configurations of φˆ makes a significant contribution to the integral. If there were a way to focus on these regions of the configuration space, the computation could be speeded up considerably. To this end, if the matrix entries of φˆ could be generated randomly with the ˆ ˆ = e−S[φ] probability distribution p(φ) /Z, then the expectation value of ˆ can be calculated simply is a simple average of a random number of h(φ) ˆ The nice thing about this strategy is that since φˆ is “draws” of the field φ. drawn according to the probability distribution, it would typically be drawn from regions where the probability is large, and hence make a significant contribution to the integral. ˆ The fields (or matrices) φˆ are randomly chosen (with distribution p(φ)) using a one-step Markov process. Starting from an initial configuration, a sequence of matrices is drawn inductively, using the Monte-Carlo Metropolis method. For example, at the (k + 1)-th step, a matrix ψ is drawn randomly from a uniform distribution near the random matrix ψk obtained at step k. One then calculates difference between the actions δS ≡ S[ψ] − S[ψk ]. The matrix ψ is accepted as the next one in the sequence with probability min(e−δS , 1), otherwise one retains ψk as ψk+1 . After a sufficiently large number of such steps, one is reasonably sure that the matrices generated in this way are sufficiently random. Having thus created a large bank of such random matrices, one can then explicitly calculate the statistical average of any operator of interest as X ˆ = 1 h(φˆi ) (4.51) hh(φ)i N N

ˆ we can investigate various By judiciously choosing the operator h(φ), physical phenomena. For example, the finite volume susceptibility given by ˆ 2, χ ≡ hTr φˆ2 i − h|Tr φ|i

(4.52)

which diverges in the continuum theory at a continuous phase transition, is one such indicator: sharp peaks in susceptibility will separate regions of different phases. We will briefly discuss the results for 3 situations: 1. Scalar field theory with quartic interaction on fuzzy S 2 .

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2. Scalar field theory with quartic interaction on fuzzy S 2 and discrete Euclidean time. 3. Gauge theory on fuzzy S 2 × S 2 . 4.3.1

Scalar Field Theory on Fuzzy S 2

Following [22], the action of interest is S[Φ] = Tr [aΦ† [Li , [Li , Φ]] + bΦ2 + cΦ4 ] .

(4.53)

Three phases of this model have been identified: 1. Disordered: For small negative values of b, the fluctuations fluctuate in the neighborhood of Φ = 0: hTr Φi = 0 = hTr Φ2 i. 2. Non-uniform ordered: As b becomes more negative, there develop symmetrical peaks for the probability distribution of hTr Φi. In particular, there is evidence of a spontaneous breakdown of rotational invariance. 3. Uniform ordered: For large negative values of b, the p probability distribution of hTr Φi are located approximately at ±(n+1) −b/c. In addition, p hTr Φ2 i ' hTr Φi/(n + 1), indicating that Φ ' −b/c1, so that rotational symmetry is again restored. 3 N=2 N=3 N=4 N=6 N=8 N=10 -2 -3/2 2 cN =(bN ) /4

2.5

Disorder phase

Non-Uniform Order phase

c/N

2

2

1.5

1

Uniform Order phase

0.5

Triple point (0.80 ± 0.08,0.15 ± 0.05) 0 0

2

4

6

8

10

12

14

-b/N3/2

Fig. 4.1 Phase diagram obtained from Monte–Carlo simulations of the model (4.53), from Garcia Flores et. al. [22]

In the limit of a → 0 or equivalently, if |b|, c |a|, the model reduces to the usual random matrix model, which has a third order phase transition

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between the disordered and non-uniform ordered phase at c = b2 /4N [24]. Again, the numerical evidence confirms this behavior. When (discretised) Euclidean time is included, numerical results again confirm the existence of the abovementioned three phases [23]. One important advantage of this model is that the radius R of the fuzzy sphere is now an independent parameter, and novel and interesting scaling limits are now accessible. For example, keeping R fixed, n → ∞ gives the fuzzy sphere, taking both R and n to infinity but with R2 /n → 0 gives the commutative plane, while keeping R2 /n finite and non-zero when R and n → ∞ gives the noncommutative plane. 4.3.2

Gauge Theory on Fuzzy S 2 × S 2

Another class of models closely related to the fuzzy sphere comes from the action 2i 1 2 (4.54) [Aµ , Aν ] + αfµνρ Aµ Aν Aρ S[A] = nTr 4 3 where Aµ are n × n traceless Hermitian matrices, and fµνρ is a totally antisymmetric tensor. The equation of motion is [Aν , [Aµ , Aν ]] − iαfµνρ [Aν , Aρ ] = 0.

(4.55)

Theoretical interest in these models in two-fold. Firstly, they arise as limits of certain nonperturbative formulations of string theory (see for eg [141] and references therein). Secondly, these models provide a potential regularization of Yang-Mills theories, and are thus amenable to numerical study. For fµνρ = µνρ , it is easy to see that the fuzzy sphere Aµ = αLµ is a solution, apart from the trivial solution Aµ = 0. Similarly, fµνρ = µνρ for µ, ν, ρ ∈ {1, 2, 3} or {4, 5, 6} and zero otherwise, one obtains the fuzzy S 2 × S 2 as a solution. These models hold out the possibility of studying topology changes as phase transitions [25–28]. Their results indicate that for α smaller than a critical value, the true ground state of the theory is Aµ equal to diagonal matrices. For α above the critical value, it is the fuzzy S 2 that is the ground state, while fuzzy S 2 × S 2 can exist as a metastable state. We refer to the original papers for further details.

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Chapter 5

Instantons, Monopoles and Projective Modules

The two-sphere S 2 admits many nontrivial field configurations. One such configuration is the instanton. It occurs when S 2 is Euclidean space-time. It is of particular importance as a configuration which tunnels between distinct “classical vacua” of a U (1) gauge theory. An instanton can be regarded as the curvature of a connection for a U (1)-bundle on S 2 . As there are an infinite number of U (1)-bundles on S 2 characterized by an integer k (Chern number), there are accordingly an infinite number of instantons as well. We can also think of S 2 as the spatial slice of space-time S 2 ×R. In that case, the instantons become monopoles. (The monopoles can be visualized as sitting at the center of the sphere embedded in R3 . If a charged particle moves in its field, k is the product of its electric charge and monopole charge [45, 66].). In algebraic language, what substitutes for bundles are “projective modules” [3]. Here we describe what they mean and find them for monopoles and instantons.

5.1

Free Modules, Projective Modules

Consider M at(N + 1) = M at(2L + 1). It carries the left- and right-regular representations of the fuzzy algebra. Thus for each a ∈ M at(2L + 1) there are two operators aL and aR acting on M at(2L + 1) (thought of as a vector space) defined by aL b = ab,

aR b = ba,

with aL bL = (ab)L and aR bR = (ba)R . 47

b ∈ M at(N + 1)

(5.1)

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Definition: A module V for an algebra A is a vector space which carries a representation of A. Thus V = M at(N + 1) is an A- (= M at(N + 1)−) module. As this V carries two actions of A, it is a bimodule. (But note that aR bR = (ba)R .) For an A-module, linear combinations of vectors in V can be taken with coefficients in A. Thus if vi ∈ V and ai ∈ A, ai vi ∈ V . A vector space over complex numbers in this language is a C-module. We consider only A-modules V whose elements are finite-dimensional vectors vi = (vi1 , · · · viK ) with vij ∈ A. The action of a ∈ A on V is then vi → avi = (avi1 , · · · , avik ). Consider the identity 1 belonging to V = M at(N + 1). Then all its elements can be got by (left- or right-) A-action. As an A-module, it is one-dimensional. It is also “generated” by 1 as an A-module. It is a “free” module as it has a basis. Generally, an A-module V is said to be free if it has a basis {ei }, ei ∈ V . P That means that any x ∈ V can be uniquely written as ai ei , ai ∈ A. P Uniqueness implies linear independence: ai ei = 0 ⇔ all ai = 0. The phrase “free” merits comment. It just means that there is no (additional) condition of the form bi ei = 0, bi ∈ A, with at least one bj 6= 0. In other words, {ei } is a basis. A class of free M at(N + 1)-bimodules we can construct from V = M at(N +1) are V ⊗CK ≡ V K . Elements of V K are v := (v1 , . . . vK ), vi ∈ V . The left- and right- actions of a ∈ A on V K are the natural ones: aL v = (av1 , . . . , avK ), aR v = (v1 a, . . . , vK a). V K is a free module as it has the basis h{ei } : ei = (0, . . . , 0, |{z} 1 , 0, . . . , 0)i. ith entry

A projector P on the A-module V K is a K × K matrix P = (Pij ) with entries Pij ∈ A, fulfilling P † = P, P 2 = P where Pij† = Pji∗ . Consider P V K . (We can also apply P on the right: ξ ∈ V K P ⇒ ξi = ξj Pji ). On P V K , we can generally act only on the right with A, so it is only a right-A-module and not a left one. Any vector in P V K is a linear combination of P ei with coefficients in A P (acting on the right): ξ ∈ P V K ⇒ ξ = i (P ei )ai , a ∈ A. But {P ei = fi } cannot be regarded as a basis as fi are not linearly independent. There P P exist ai ∈ A, not all equal to zero, such that i P ei ai = 0, that is, ei a i P is in the kernel of P , without ei ai being 0. P V K is an example of a projective module. A module, projective or otherwise, is said to be trivial if it is a free

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module. Note that P V K is a summand in the decomposition V K = P V K ⊕ (1 − P )V K of the trivial module V K . These ideas are valid (with possible technical qualifications) for any algebra A and an A-module V . In particular, they are valid if A is the commutative algebra C ∞ (M ) of smooth functions on a manifold with point-wise multiplication. We now show that the elements of A-modules are sections of bundles on M , picking M = S 2 for concreteness. In this picture, sections of twisted bundles on S 2 , such as twisted U (1)-bundles, are elements of nontrivial projective modules. Such sections have a natural interpretation as charge-monopole wave functions. It is a theorem of Serre and Swan [36] that all such sections can be obtained from projective modules using preceding algebraic constructions.

5.2

Projective Modules on A = C ∞ (S 2 )

Consider the free module A2 = A ⊗ C2 . If xˆ is the coordinate function, (ˆ xi a)(x) = xi a(x), a ∈ A, we can define the projector 1 + ~τ · x ˆ (5.2) P (1) = 2 where τi are the Pauli matrices. P (1) A2 is an example of a projective module. P (1) A2 carries an A-action, left- and right- actions being the same. The projector P (1) occurs routinely when discussing the chargemonopole system [70, 71] or the Berry phase [67]. We will now establish that P (1) A2 is a nontrivial projective module. Its elements are known to be the wave functions for Chern number k (= product of electric and magnetic τ ·ˆ x charges) = 1. For k = −1, we can use the projector P (−1) = 1−~ 2 . (1) (1) 2 At each x, P (x) is of rank 1. If P A has a basis e, then e(x) is an eigenstate of P (1) (x), P (1) (x)e(x) = e(x), and smooth in x. But there is no such e. For suppose that is not so. Let us normalize e(x) : e† (x)e(x) = 1. Let fa = ab eb (εab = −εba , ε12 = +1). Then f is a smooth normalized vector perpendicular to e and annihilated by P (1) : P (1) f = 0. The operator e1 f1 U= . (5.3) e2 f2 is unitary at each x (U † (x)U (x) = 1) and 1 + τ3 U † P (1) U = 2

(5.4)

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So we have rotated the hedgehog (winding number 1) map xˆ : x → xˆ(x) to the constant map x → (0, 0, 1). As that is impossible [45], e does not exist. For higher k, we can proceed as follows. Take k copies of C2 and consider k C2 = C2 ⊗ · · · ⊗ C2 . Let ~τ (i) be the Pauli matrices acting on the ith slot k in C2 . That is ~τ (i) = 1 ⊗ · · · ⊗ ~τ ⊗ · · · ⊗ 1. Then the projector for k is P (k) =

k Y 1 + ~τ (i) · xˆ 2 i=1

(5.5)

and the projective module is k

k

P (k) [A ⊗ C2 ] := P (k) A2 .

(5.6)

For k = −|k|, the projector in (5.5) gets replaced by P (−|k|) =

|k| Y 1 − ~τ (i) · x ˆ . 2 i=1

(5.7)

We can also construct the modules in another way. Let k > 0. Consider P z = (z1 , z2 ) with i |zi |2 = 1. These are the z’s of Chapter 2. For k > 0, let k X 1 z1 vk (z) = √ , Zk = |zi |2k . (5.8) k Zk z 2 i

It is legitimate to put Zk in the denominator: it cannot vanish without both zi being 0, and that is not possible. vk (z) is normalized: vk† (z)vk (z) = 1.

(5.9)

So vk (z) ⊗ vk† (z) is a projector. Under zi → zi eiθ , vk (z) → vk (z)eikθ and the projector is invariant, so it depends only on x = z †~τ z ∈ S 2 . In this way, we get the projector P 0(k) P0

(k)

(x) = vk (z) ⊗ vk† (¯ z)

(5.10)

For k = −|k| < 0, such a projector is P0

(−|k|)

† (x) = v¯|k| (¯ z ) ⊗ v¯|k| (z)

(5.11)

The projectors (5.10 , 5.11) are sometimes refered to as “Bott” projectors.

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5.3

Equivalence of Projective Modules (k)

We briefly explain the sense in which the projectors P (k) , P 0 and the k (k) modules P (k) A2 and P 0 A2 are equivalent. Two modules are said to be equivalent if the corresponding projectors are equivalent. But there are several definitions of equivalence of projectors [68]. We pick one which appears best for physics. k k (k) The 22 × 22 matrix P (k) or the 2 × 2 matrix P 0 can be embedded in the space of linear operators on an infinite-dimensional Hilbert space H. The elements of H consist = (a1 , a2 , . . .), ai ∈ C ∞ (S 2 ). The scalar R of aP product for H is (b, a) = S 2 dΩ l b∗l (x)al (x). H is clearly an A-module. (k) The embedding is accomplished by putting P (k) and P 0 in the top left- corner of an “∞ × ∞” matrix. The result is ! (k) (k) P 0 P0 0 0(k) (k) . (5.12) , P = P = 0 0 0 0 A matrix U acting on H has “coefficients” in A : Uij ∈ C ∞ (S 2 ). It is said to be unitary if U † U = 1 where each diagonal entry in 1 is the constant function on S 2 with value 1 ∈ C. The projectors P (k) and P 0(k) are said to be equivalent if there exists a unitary U such that U P (k) U † = P 0(k) . If there is such a U , then U P (k) a = P 0(k) U a , a ∈ H. That means that wave functions given by P (k) H and k (k) P 0 H are unitarily related. It is then reasonable to regard P (k) A2 and (k) P 0 A2 as equivalent. Illustration: (k)

We now illustrate this notion of equivalence using P (k) and P 0 . Since (±1) (±1) , k = ±2 is the first nontrivial example. P = P0 Let zi be as above. Then the matrix with components zi z¯j is a projector. It is invariant under zi → zi eiθ and is a function of x. In fact P (1) (x)ij = zi z¯j .

(5.13)

P (−1) (x)ij = z¯i zj .

(5.14)

Similarly,

Inspection shows that z and ¯ z = (ij z¯j ) are eigenvectors of P (1) (x) with eigenvalues 1 and 0, whereas z¯ and z are those of P (−1) (x) with the same eigenvalues.

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Previous remarks on the impossibility of diagonalizing P (k) (x) using a unitary U (x) for all x do not contradict the existence of these eigenvectors: their domain is not S 2 , but S 3 . (k) Just as P (±1) , P 0 has eigenvectors vk , ¯ vk for k > 0, and v¯|k| , v−|k| for k < 0. (k) As P (k) is 2|k| × 2|k| , let us embed P 0 inside a 2|k| × 2|k| matrix P 0(k) in the manner described above. Let us first assume that k > 0. Let ξ (k) (j)be orthonormal eigenvectors of P (k) constructed as follows: For ξ (k) (1), we set ξ (k) (1) =

z ⊗ z ··· ⊗ z 1 2 k

(5.15)

The integers 1, 2, · · · , k below z’s label the vector space C2 which contains the z above it: the z above j belongs to the C2 of the j-th slot in the tensor k product C2 ⊗ C2 ⊗ · · · ⊗ C2 = C2 . The next set of vectors ξ (k) (j) (j = 2, · · · , k+1) is obtained by replacing z above j by ¯ z and not touching the remaining z’s. We say we have “flipped” one z at a time to get these vectors. Next we flip 2 z’s at a time: there are k C2 of these. We proceed in this manner, flipping 3,4, etc z’s. When all are flipped, we get the vector ξ (k) (2k ) = ¯ z ⊗ ¯ z ⊗ · · · ⊗ ¯ z.

(5.16)

The following is important: a basis vector after j flips has the property ξ (k) (l) → ei(k−2j)θ ξ (k) (l),

when z → eiθ z.

(5.17)

Our task is to find an orthonormal basis η (k) (l) where η (k) (1) is the following eigenvector of P 0(k) (x) with eigenvalue 1, η (k) (1) = (vk , ~0), P 0(k) (x)η (k) (1) = η (k) (1).

(5.18)

Then the rest are in the null space of P 0(k) (x): P 0(k) (x)η (k) (j) = 0,

j 6= 1.

(5.19)

We require in addition that η (k) (l) transforms in exactly the same manner as ξ (k) (l): η (k) (l) → ei(k−2j)θ η (k) (l),

when z → eiθ z.

(5.20)

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Then the operator ˆ (z) = U

X l

ξ (k) (l) ⊗ η¯(k) (l)

(5.21)

is unitary, ˆ (z)† U(z) ˆ U = 1,

(5.22)

and invariant under z → zeiθ : ˆ iθ ) = U(z). ˆ U(ze

(5.23)

ˆ (z) = U (x) U

(5.24)

Hence we can write

and U provides the equivalence between P 0

(k)

and P (k) :

U P 0(k) U † = P (k)

(5.25)

There are indeed such orthonormal vectors. η (k) (1) clearly has the required property. As for the rest, we show how to find them for k = 2 and 3. The general construction is similar. If k = −|k| < 0, the same considerations apply after changing z to ¯ z (k) in P (k) and vk to v¯|k| in P 0 . k=2 k

In this case, C2 = C4 . The basis is    0 v2 (2) (2)    η (1) = 0 , 0 , η (2) = v2 0     0 ¯ v2 η (2) (3) =  0  , η (2) (4) =  0  . ¯ v2 0 

k=3

(5.26)

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Now C2 = C8 . The basis is       v3 0 0 0 0 0       0 v  0    3         (3) (3) (3) η (1) =  0  , η (2) =  0  , η (3) =  0  ,       0 0  v3        0 0 0 0 0 0       0 0 0  0   0  0        0    ¯ 0   v3           η (3) (4) =  0  , η (3) (5) =  0  , η (3) (6) =  0  ,        ¯  0  0 v   3      0   0  0 0 0 v3     0 ¯ v3  0   0       0   0          η (3) (7) =  0  , η (3) (8) =  0  .      0   0       0   0  ¯ v3 0

(5.27)

In this manner, we can always construct η (k) (j).

5.4

Projective Modules on the Fuzzy Sphere

We want to construct the analogues of P (k) and P 0 (k) for the fuzzy sphere. They give us the monopoles and instantons of SF2 . Let us consider P (k) (k) first, and denote the corresponding projectors as PF . 5.4.1

(k)

Fuzzy Monopoles and Projectors PF

We begin by illustrating the ideas for k = 1. On C2 , the spin 12 representation of SU (2) acts with generators τ2i . On (1) SF2 , the spin ` representation of SU (2) acts with generators LL i . Let PF 1 1 be the projector coupling ` and 2 to ` + 2 . Consider the projective module

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PF (SF2 ⊗ C2 ). On this module, ~L + (L

1 3 ~τ 2 ) = (` + )(` + ), 2 2 2

(5.28)

or ~L L · ~τ = 1. `

(5.29) (1)

Passing to the limit ` → ∞, this becomes x ˆ · ~τ = 1, so PF ` → ∞. (1) We can find PF explicitly. (1)

−2PF − 1 ≡ ΓL =

~L + 1 ~τ · L 2 . ` + 12

→ P (1) as

(5.30)

ΓL is an involution, (ΓL )2 = 1

(5.31)

and will turn up in the theory of fuzzy Dirac operators and the GinspargWilson system (see chapter 8). It is the chirality operator of the Watamuras’ [69]. (1) An important feature of PF (SF2 ⊗ C2 ) is that it is still an SU (2)R bimodule. On the right, Li act as before. On the left, LL i do not, but (1) τi L Li + 2 do as they commute with PF . ~ L stands here for the phenomenon of “mixing of This addition of ~τ2 to L spin and isospin” in the ”t Hooft-Polyakov monopole theory [70]. (1) But PF (SF2 ⊗ C2 ) is not a free SF2 -module as it does not have a basis {ei = (ei1 , ei2 ) : ei,j ∈ SF2 }. That is because if α = (α1 , α2 ) ∈ SF2 ⊗C2 , αi ∈ (1) SF2 the projector PF mixes up the rows of αi . (−1) For k = −1, the projector PF couples ` and 1/2 to ` − 1/2. It is just (1) 1 − PF . k The construction for any k is similar. For k = |k|, we consider C2 = C2 ⊗ C2 · · · ⊗ C2 . On this, the SU (2) acts on each C2 , the generators for (j) the jth slot being τi /2 ≡ 1 ⊗ · · · ⊗ τ2i ⊗ · · ·⊗ 1, the τ2i being in the jth slot. (k) Let PF be the projector coupling ` and all the spin 21 ’s to the maximum k (k) value ` + k2 . The projective module is PF (SF2 ⊗ C2 ). (k) For k = −|k|, PF couples ` and the spins to the least value ` − |k| 2 . (j)

L

We can show that (τ `·L ) tends to +1 for k > 0 and -1 for k < 0 on these modules, so that the τ (j) · xˆ have the correct values in the limit. Thus consider for example k > 0. As all angular momenta are coupled to the

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maximum possible value, every pair must also be so coupled. So on this ~ L + ~τ (j) )2 = (` + 1 )(` + 3 ) and the result follows as for k = 1. module (L 2 2 2 Similar considerations apply for k < 0. For higher k, we can also proceed in a different manner. If k = |k|, SU (2) acts on Ck+1 by angular momentum k2 representation. Hence there is the projector P 0(k) coupling the left ` and k2 to ` + k2 . The projective module is then P 0(k) (SF2 ⊗ Ck+1 ). |k| For k < 0 we can couple ` and |k| to ` − |k| 2 instead (we assume ` > 2 ). P 0(k) and P (k) are equivalent in the sense discussed earlier. We can in fact exhibit the two modules so that they look the same: diagonalize the ~ L + P ~τ (j) )2 and its third component on P (k) (S 2 ⊗ angular momentum (L F F j 2 k

C2 ). Their right angular momenta being both `, their equivalence (in any sense!) is clear. For reasons indicated above, none of these SF2 -modules are free.

5.4.2

The Fuzzy Module for the Tangent Bundle and the Fuzzy Complex Structure

The projectors for k = 2 are of particular interest as they can be interpreted as fuzzy sections of the tangent bundle. To see this, let us begin with the commutative algebra A = C ∞ (S 2 ) and the module A3 = C ∞ (S 2 ) ⊗ C3 . In this case, SU (2) acts on C3 with the spin 1 generators θ(α) where θ(α)ij = −iαij . (5.32) Consider θ(α)ˆ xα ≡ θ · x ˆ. (5.33) (T ) Its eigenvalues at each x are ±1, 0. Let P be the projector to the subspace (θ · x ˆ)2 = 1: P (T ) = (θ · x ˆ)2 . (5.34) Any vector in the module P (T ) A3 can be written as ξ + +ξ − where θ · xˆξ ± = ±ξ ± , that is −iαij xα ξj± (x) = ±ξi± (x). It follows from the antisymmetry of αij that xi ξi± (x) = 0 or that ξ ± (x) are tangent to S 2 at x. The ξ ± give sections of the (complexified) tangent bundle T S 2 . 2 A smooth split for all x of T S 2 (x) into two subspaces T S± (x) gives a 2 2 complex structure J on T S . J(x) is ±i1 on T S± (x). Thus a complex structure on T S 2 is defined by the decomposition 2 2 T S 2 = T S+ ⊕ T S− , 2 = ±i1. J|T S±

(5.35)

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Now P (T ) is the sum of projectors which give eigenspaces of θ · xˆ for eigenvalues ±1: (T )

(T )

P (T ) = P+ + P− , θ·x ˆ(θ · x ˆ ± 1) (T ) . P± = 2

(5.36)

With (T )

(T )

= ±iP±

JP±

we get the required decomposition of P (T )

(1)

(5.37)

3

A for a complex structure: (T )

P (T ) A3 = P+ A3 ⊕ P− A3 .

(5.38)

Fuzzification of these structures is easy and elegant. Instead of working with SF2 ⊗ C2 we work with SF2 ⊗ C3 . The projector (T ) PF we thereby obtain is the fuzzy version of P T . We can show this as follows. (T,±) Let PF be the projectors coupling LL α and θ(α) to the values ` ± 1. Then (T )

PF (T,+)

On the module PF

(T,+)

= PF

(T,−)

+ PF

.

(5.39)

(SF2 ⊗ C3 ),

2 [LL α + θ(α)] = (` + 1)(` + 2)

(5.40)

LL α θ(α) = 1. `

(5.41)

or

(T,−)

On the module PF

(SF2 ⊗ C3 ),

2 (LL α + θ(α)) = −1 −

Thus as ` → ∞

1 `

(5.42)

LL (T,±) α θ(α) → ±1 on PF (SF2 ⊗ C3 ) . `

(5.43) (T )

As the left hand side tends to θ(α)ˆ xα as ` → ∞, we have that PF (SF2 ⊗C3 ) (T,+) defines the fuzzy tangent bundle and its decomposition PF (SF2 ⊗ C3 ) ⊕ (T,−) PF (SF2 ⊗ C3 ) defines the fuzzy complex structure. The corresponding (T,±) J, call it JF , is ±i on PF (SF2 ⊗ C3 ).

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Chapter 6

Fuzzy Nonlinear Sigma Models

6.1

Introduction

In space-time dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are massless Nambu-Goldstone modes with values in the coset space G/H. Being massless, they dominate low energy physics as is the case with pions in strong interactions and phonons in crystals. Their theoretical description contains new concepts because G/H is not a vector space. Such G/H models have been studied extensively in 2 − d physics, even though in that case there is no spontaneous breaking of continuous symmetries. A reason is that they are often tractable nonperturbatively in the two-dimensional context, and so can be used to test ideas suspected to be true in higher dimensions. A certain amount of numerical work has also been done on such 2 − d models to control conjectures and develop ideas, their discrete versions having been formulated for this purpose. This chapter develops discrete fuzzy approximations to G/H models. We focus on two-dimensional Euclidean quantum field theories with target space G/H = SU (N + 1)/U (N ) = CP N . The novelty of this approach is that it is based on fuzzy physics [3] and non-commutative geometry [34–38]. Although fuzzy physics has striking elegance because it preserves the symmetries of the continuum and because techniques of non-commutative geometry give us powerful tools to describe continuum topological features, still its numerical efficiency has not been fully tested. This chapter approaches σ-models with this in mind, the idea being to write fuzzy G/H models in a form adapted to numerical work. This is not the only approach on fuzzy G/H. In [81], a particular description based on projectors and their orbits was discretized. We shall

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refine that work considerably in this paper. Also in the continuum there is another way to approach G/H, namely as gauge theories with gauge invariance under H and global symmetry under G [72]. This approach is extended here to fuzzy physics. Such a fuzzy gauge theory involves the decomposition of projectors in terms of partial isometries [68] and brings new ideas into this field. It is also very pretty. It is equivalent to the projector method as we shall also see. Related work on fuzzy G/H model and their solitons is due to Govindarajan and Harikumar [73]. A different treatment, based on the HolsteinPrimakoff realization of the SU (2) algebra, has been given in [74]. A more general approach to these models on noncommutative spaces was proposed in [75]. The first two sections describe the standard CP 1 -models on S 2 . In section 2, we discuss it using projectors, while in section 3, we reformulate the discussion in such a manner that transition to fuzzy spaces is simple. Sections 4 and 5 adapt the previous sections to fuzzy spaces. Long ago, general G/H-models on S 2 were written as gauge theories [72]. Unfortunately their fuzzification for generic G and H eludes us. Generalization of the considerations here to the case where S 2 ' CP 1 is replaced with CP N , or more generally Grassmannians and flag manifolds associated with (N + 1) × (N + 1) projectors of rank ≤ (N + 1)/2, is easy as we briefly show in the concluding section 6. But extension to higher ranks remains a problem. 6.2

CP 1 Models and Projectors

Let the unit vector x = (x1 , x2 , x3 ) ∈ R3 describe a point of S 2 . The field n in the CP 1 -model is a map from S 2 to S 2 : X na (x)2 = 1 . (6.1) n = (n1 , n2 , n3 ) : x → n(x) ∈ R3 , n(x)·n(x) := a

These maps n are classified by their winding number κ ∈ Z: Z 1 abc na (x) dnb (x) dnc (x) . (6.2) κ= 8π S 2 That κ is the winding of the map can be seen taking spherical coordinates (Θ, Φ) on the target sphere (n2 = 1) and using the identity sin ΘdΘ dΦ = 1 2 abc na dnb dnc . We omit wedge symbols in products of forms. We can think of n as the field at a fixed time t on a (2+1)-dimensional manifold where the spatial slice is S 2 . In that case, it can describe a field

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of spins, and the fields with κ 6= 0 describe solitonic sectors. We can also think of it as a field on Euclidean space-time S 2 . In that case, the fields with κ 6= 0 describe instantonic sectors. Let τa be the Pauli matrices. Then each n(x) is associated with the projector 1 (6.3) P (x) = (1 + ~τ · ~n(x)) . 2 Conversely, given a 2 × 2 projector P (x) of rank 1, we can write

1 (α0 (x) + ~τ · α ~ (x)) . 2 Using Tr P (x) = 1, P (x)2 = P (x) and P (x)† = P (x), we get P (x) =

α0 (x) = 1, 1

2

α ~ (x) · α ~ (x) = 1,

α∗a (x) = αa (x) .

(6.4)

(6.5)

Thus CP -fields on S can be described either by P or by na = Tr(τa P ) [76]. In terms of P , κ is Z 1 κ= Tr P (dP ) (dP ) . (6.6) 2πi S 2

There is a family of projectors, called Bott projectors [77, 78], which play a central role in our approach. Let z = (z1 , z2 ),

|z|2 := |z1 |2 + |z2 |2 = 1 .

(6.7)

The z’s are points on S 3 . We can write x ∈ S 2 in terms of z: xi (z) = z † τi z .

The Bott projectors are Pκ (x) =

vκ (z)vκ† (z),

1 z1κ vκ (z) = κ √ z2 Zκ

(6.8)

if κ ≥ 0 ,

Zk ≡ |z1 |2|κ| + |z2 |2|κ| , " # ∗|κ| 1 z1 if κ < 0 . vκ (z) = ∗|κ| √ Zκ z2

(6.9)

The field n(κ) associated with Pκ is given by † n(κ) a (x) = Tr τa Pκ (x) = vκ (z)τa vκ (z) .

(6.10)

Under the phase change z → zeiθ , vκ (z) changes, vκ (z) → vκ (z)eiκθ , whereas x is invariant. As this phase cancels in vκ (z)vκ† (z), Pκ is a function of x as written.

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The κ that appears in eqs.(6.9)(6.10) is the winding number as the explicit calculation of section 3 will show. But there is also the following argument. In the map z → vκ (z), for κ = 0, all of S 3 and S 2 get mapped to a point, giving zero winding number. So, consider κ > 0. Then the points 2π 2π (6.11) z1 ei κ (l+m) , z2 ei κ m , l, m ∈ {0, 1, .., κ − 1} 2π

have the same image. But the overall phase ei κ m of z cancels out in x. Thus, generically κ points of S 2 (labeled by l) have the same projector Pκ (x), giving winding number κ. As for κ < 0, we get |κ| points of S 2 mapped to the same Pκ (x). But because of the complex conjugation in eq.(6.9), there is an orientation-reversal in the map giving −|κ| = κ as winding numbers. One way to see this is to use P−|κ| (x) = P|κ| (x)T

(6.12)

Substituting this in (6.6), we can see that P±|κ| have opposite winding numbers. The general projector Pκ (x) is the gauge transform of Pκ (x): Pκ (x) = U (x)Pκ (x)U (x)† ,

(6.13)

where U (x) is a unitary 2 × 2 matrix. Its n(κ) is also given by (6.10), with Pκ replaced by Pκ . The winding number is unaffected by the gauge transformation. That is because U is a map from S 2 to U (2) and all such maps can be deformed to identity since π2 (U (2)) = {identity e}. The identity Pκ (dPκ ) = (dPκ )(1l − Pκ )

(6.14)

which follows from Pκ2 = Pκ , is valuable when working with projectors. The soliton described by Pκ has the action (below) peaked at the north 1 +ix2 = 0 and a fixed width and shape. The solitons with pole x3 = 1 or x1+x 3 1 +ix2 = η and variable width and shape are energy density peaked at x1+x 3 given by the projectors Pκ (x , η , λ) = vκ (z, η, λ)vκ (z, η, λ)† , 1 λz1κ vκ (z, η, λ) = . z2κ − ηz1κ (|λz1 |2κ + |z2κ − ηz1κ |2 ) 21

(6.15)

In order to find the explicit form of the unitary transformation in (6.13) corresponding to the choice (6.15) we proceed as follows. There is a unit

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vector wκ (.) perpendicular to vκ (.) (The dot represents z or z, η, λ). An explicit realization of wκ is given by ∗ ∗ wκ,α = i(τ2 )αβ vκ,β := αβ vκ,β

from which it is easy to see that fulfills

wκ† (.)vκ (.)

wκ (.)wκ† (.) = 1l − Pκ (.) ,

=0=

(6.16) vκ† (.)wκ (.).

wκ† (.)wκ (.) = 1 .

wκ (.) also (6.17)

Let us consider V (x) = vκ (z, η, λ)vκ† (z) ,

W (x) = wκ (z, η, λ)wκ† (z) .

(6.18)

From these definitions it is easy to verify that U (x) = V (x) + W (x)

(6.19)

does the required job for κ > 0. We call the field associated with Pκ (., η, λ) as n(κ) (., η, λ): ~n(κ) (x, λ, η) = vκ (z, η, λ)† ~τ vκ (z, η, λ) .

(6.20)

We can use vκ (z, η, λ) = v|κ| (¯ z , η, λ) and wκ (z, η, λ) = w|κ| (¯ z , η, λ) to write the solitons for κ < 0. 6.3

An Action

Let Li = −i(x∧∇)i be the angular momentum operator. Then a Euclidean action in the κ-th topological sector (or a static Hamiltonian in the (2 + 1) picture) is Z c (κ) (κ) dΩ (Li nb )(Li nb ) , c = a positive constant, (6.21) Sκ = − 2 S2

where dΩ is the S 2 volume form d cos θ dϕ. We can also write Z dΩ Tr (Li Pκ )(Li Pκ ) . Sκ = −c

(6.22)

S2

The following identities, based on (6.14), are also useful:

Tr Pκ (Li Pκ )2 = Tr (Li Pκ )(1l − Pκ )(Li Pκ ) = 1 Tr(1l − Pκ )(Li Pκ )2 = Tr(Li Pκ )2 . 2 Hence Sκ = −2c

Z

S2

dΩ Tr Pκ Li Pκ Li Pκ

(6.23)

(6.24)

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The Euclidean functional integral for the actions Sκ is Z X iκψ Z(ψ) = D P κ e−Sκ e

(6.25)

κ

where the angle ψ is induced by the instanton sectors as in QCD. Using the identity dP = −ijk dxi xj iLk P , we can rewrite the definition (6.2) or (6.6) of the winding number as Z 1 (κ) (κ) κ= dΩ ijk xi abc n(κ) (6.26) a iLj nb iLk nc 8π S 2 Z 1 dΩ Tr Pκ ijk xi iLj Pκ iLk Pκ . (6.27) = 2πi S 2 The Belavin-Polyakov bound [79] Sκ ≥ 4π c |κ|

(6.28)

follows from (6.26) on integration of

(κ)

(κ) 2 (iLi n(κ) a ± ijk xj abc nb iLk nc ) ≥ 0 ,

or from (6.27) on integration of

Pκ (iLi Pκ )±iij 0 k0 xj 0 Pκ (iLk0 Pκ ) ≥ 0 . (6.30) From this last form it is easy to rederive the bound in a way better adapted to fuzzification. Using Pauli matrices {σi } we first rewrite (6.24) and (6.27) as Z Sκ = c dΩ Tr Pκ (iσ · L Pκ )(iσ · L Pκ ) , S2 Z −1 κ= dΩ Tr σ · x Pk (iσ · L Pk )(iσ · L Pk ) . (6.31) 4π S 2 The trace is now over C2 × C2 = C4 , where τa acts on the first C2 and σi on the second C2 (so they are really τa ⊗ 1l and 1l ⊗ σi ) Then, with 1 , 2 = ±1, 1 + 2 τ · n(κ) σi (iLi Pκ ) + 1 iijk xj (iLk Pκ ) = 2 1 + 2 τ · n(κ) (1 + 1 σ · x) (iσ · LPκ ) , (6.32) 2 since x · L = 0. The inequality (6.30) is equivalent to † 1 + 1 σ · x 1 + 2 τ · n(κ) Tr (iσ · LPκ ) × 2 2 1 + 1 σ · x 1 + 2 τ · n(κ) (iσ · LPκ ) ≥0, (6.33) 2 2 from which (6.28) follows by integration. Tr Pκ (iLi Pκ )±iijk xj Pκ (iLk Pκ )

†

(6.29)

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6.4

CP 1 -Models and Partial Isometries

If P(x) is a rank 1 projector at each x, we can find its normalized eigenvector u(z): P(x)u(z) = u(z) ,

u† (z)u(z) = 1 .

(6.34)

Then P(x) = u(z)u† (z) .

(6.35)

If P = Pκ , an example of u is vκ . u can be a function of z, changing by a phase under z → zeiθ . Still, P depends only on x. We can regard u(z)† (or a slight generalization of it) as an example of a partial isometry [68] in the algebra A = C ∞ (S 3 ) ⊗C M at2×2 (C) of 2 × 2 matrices with coefficients in C ∞ (S 3 ). A partial isometry in a ∗−algebra A is an element U † ∈ A such that U U † is a projector. U U † is the support projector of U † . It is an isometry if U † U = 1l. With u1 0 U= ∈ A, (6.36) u2 0 we have P = U U†

(6.37)

Aκ = vκ† dvκ .

(6.38)

so that U † is a partial isometry. We will be free with language and also call u† as a partial isometry. The partial isometry for Pκ is vκ† . Now consider the one-form

Under zi → zi eiθ(x) , Aκ transforms like a connection: Aκ → Aκ + iκ dθ

(6.39)

(Aκ are connections for U (1) bundles on S 2 for Chern numbers κ, see later.) Therefore Dκ = d + A κ is a covariant differential, transforming under z → ze

(6.40) iθ

as

Dκ → eiκθ Dκ e−iκθ

(6.41)

Dκ2 = dAκ

(6.42)

and

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is its curvature. Recall that w(z) perpendicular to v(z) for all z was defined in section 6.2. An explicit realization of w(z) is given in (6.16). We have that Bκ = wκ† dvκ and Bκ∗ = (dvκ† )wκ = −vκ† dwκ

(6.43)

are gauge covariant, Bκ (z) → eiθ(x) Bκ eiθ(x) ,

Bκ (z)∗ → e−iθ(x) Bκ∗ e−iθ(x)

(6.44)

under z → zeiθ . We can account for U (x) by considering Vκ = U v κ ,

Aκ = Vκ† dVκ ,

Dκ = d + A κ ,

Wκ = (τ2 U ∗ τ2 )wκ ,

Dκ2 = dAκ ,

Bκ = Wκ† dVκ .

(6.45)

Aκ is still a connection, and the properties (6.44) are not affected by U . Pκ is the support projector of Vκ† , and Wκ Wκ† = 1l − Pκ ,

(1l − Pκ )Vκ = 0 .

(6.46)

2

Gauge invariant quantities being functions on S , we can contemplate a formulation of the CP 1 -model as a gauge theory. Let Ji be the lift of Li to angular momentum generators appropriate for functions of z, (eiθi Ji f )(z) = f (e−iθi τi /2 z) ,

(6.47)

Bκ,i = Wκ† Ji Vκ .

(6.48)

and let

Now, Wκ Bκ,i Vκ† is gauge invariant, and should have an expression in terms of Pκ . Indeed it is, in view of (6.46), Wκ Bκ,i Vκ† = Wκ Wκ† (Ji Vκ )Vκ† =

(1l − Pκ )Ji (Vκ Vκ† ) = (1l − Pκ )(Li Pκ ) = (Li Pκ )Pκ .

(6.49)

Therefore we can write the action (6.22, 6.24) in terms of the Bκ,i : Z Sκ = −2c dΩ Tr Pκ (Li Pκ )(Li Pκ ) 2 Z S = 2c dΩ Tr ((Li Pκ )Pκ )† ((Li Pκ )Pκ ) S2 Z = 2c dΩ Tr(Wκ Bκ,i Vκ† )† (Wκ Bκ,i Vκ† ) S2 Z ∗ = 2c dΩ Bκ,i Bκ,i . (6.50) S2

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It is instructive also to write the gauge invariant (dAκ ) in terms of Pκ and relate its integral to the winding number (6.6). The matrix of forms Vκ (d + Aκ )Vκ†

(6.51)

dVκ† = (dVκ† ) + Vκ† d

(6.52)

is gauge invariant. Here

where d in the first term differentiates only Vκ† . Now Vκ (d + Vκ† (dVκ ))Vκ†

(6.53)

and Pκ dPκ = Vκ Vκ† d (Vκ Vκ† ) = Vκ Vκ† (dVκ )Vκ† + Vκ (dVκ† ) + Vκ Vκ† d

(6.54)

are equal. Hence, squaring Vκ (d + Aκ )2 Vκ† = Vκ (dAκ )Vκ† = Pκ (dPκ ) (dPκ ) on using d2 = 0, eq.(6.54) and Pκ (dPκ )Pκ = 0 . Thus Z Z Z (dAκ ) = Tr Vκ (dAκ )Vκ† = Tr Pκ (dPκ ) (dPκ ) . S2

S2

(6.55)

(6.56)

S2

We can integrate the LHS. For this we write (taking a section of the bundle U (1) → S 3 → S 2 over S 2 \{north pole(0, 0, 1)}), z(x) = e

−iτ3 ϕ/2 −iτ2 θ/2 −iτ3 ϕ/2

e

e

−iϕ 1 e cos θ2 = . 0 sin θ2

(6.57)

Taking into account the fact that U (~x) is independent of ϕ at θ = 0, we get Z Z (dAκ ) = − eiκϕ de−iκϕ = 2πiκ . (6.58) S2

This and eq.(6.56) reproduce eq.(6.6). The Belavin-Polyakov bound [79] for Sκ can now be got from the inequality † Tr Cκ,i Cκ,i ≥ 0 ,

Cκ,i = Wκ Bκ,i Vκ† ± Wκ (ijl xj Bκ,l )Vκ† .

(6.59)

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Relation Between P (κ) and Pκ

In the treatment in [81], for κ > 0, the fuzzy σ-model was based on the continuum projector κ Y 1 P (κ) (x) = P1 (x) ⊗ ... ⊗ P1 (x) = (1 + τ (i) · x) (6.60) 2 i=1 and its unitary transform P (κ) (x) = U (κ) (x)P (κ) (x)U (κ) (x)−1 , U (κ) (x) = U (x)⊗. . . U (x) (κ factors). (6.61) P At each x, the stability group of P (κ) (x) is U (1) with generator 21 κi=1 τ (i) · x, and we get a sphere S 2 as U (x) is varied. Thus U (κ) (x) gives a section of a sphere bundle over a sphere, leading us to identify P (κ) with a CP 1 -field. Furthermore, the R.H.S. of eq.(6.56) (with P (κ) replacing Pκ ) gives κ as the invariant associated with P (κ) , suggesting a correspondence between κ and winding number. We can write P (κ) = V (κ) V (κ)† , V (κ) = V1 ⊗ ... ⊗ V1 (κfactors), (6.62) its connection A(κ) and an action as previously. A computation similar to the one leading to eq.(6.56) shows Z that i dA(κ) = κ . (6.63) − 2π So κ is the Chern invariant of the projective module associated with P (κ) . For κ < 0, we must change x to −x in (6.60), and accordingly change other expressions. We note that κ cannot be identified with the winding number of the map x → P (κ) (x). To see this, say for κ > 0, we show that there is a winding number κ map from P (κ) to Pκ (x). As that is also the winding number of the map x → Pκ (x), the map x → P (κ) (x) must have winding number 1. The map P (κ) → Pκ (x) is induced from the ! map (κ)

V11...1 (6.64) (κ) V22...2 and their expressions in terms of V (κ) and Vκ . In (6.64) all the points V (κ) (z1 e2πi j/κ , z2 e2πi l/κ ), j, l ∈ {0, 1, ..., κ − 1}, have the same image, but in the passage to P (κ) and Pκ , the overall phase of z is immaterial. However, the projectors for V (κ) (z1 e2πij/κ , z2 ) and Vκ† (z1 , z2 e2πij/κ ) are distinct and map to the same Pκ , giving winding number κ. We have not understood the relation between the models based on P (κ) and Pκ . V (κ) → Vκ =

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Fuzzy CP 1 -Models

The advantage of the preceding formulation using {zα } is that the passage to fuzzy models is relatively transparent. Thus let ξ = (ξ1 , ξ2 ) ∈ C2 \{0}. We can then identify z and x as p ξ (6.65) , |ξ| = |ξ1 |2 + |ξ2 |2 xi = z † τi z . z= |ξ| Quantization of the ξ’s and ξ ∗ ’s consists in replacing ξα by annihilation operators aα and ξα∗ by a†α . |ξ| is then the square root of the number operator: ˆ =N ˆ1 + N ˆ2 , N ˆ 1 = a † a 1 , N2 = a † a 2 , N 1

1 1 , = p a†α = a†α p ˆ +1 ˆ N N 1 x ˆ i = p a † τi a . ˆ N

zˆα†

2

zˆα = p

1 1 aα = a α p , ˆ +1 ˆ N N

(6.66)

(We have used hats on some symbols to distinguish them as fuzzy operators). We will apply these operators only on the subspace of the Fock space ˆ , where √1 is well-defined. This restriction is with eigenvalue n ≥ 1 of N ˆ N

natural and reflects the fact that ξ cannot be zero. 6.5.1

The Fuzzy Projectors for κ > 0

On referring to (6.9), we see that if κ > 0, for the quantized versions vˆκ , vˆκ† of vκ , vκ∗ , we have κ 1 1 h † κ † κi a , vˆκ† = p vˆκ† vˆκ = 1l , vˆκ = 1κ p (a1 ) (a2 ) , a2 Zˆκ Zˆκ ˆ α (N ˆα − 1)...(N ˆα − κ + 1) . Zˆκ = Zˆ (1) + Zˆ (2) , Zˆ (α) = N (6.67) κ

κ

κ

ˆ whose entries U ˆij The fuzzy analogue of U is a 2 × 2 unitary matrix U are polynomials in a†a ab . As for Vˆκ , the quantized version of Vκ , it is just ˆ vˆκ Vˆκ = U (6.68)

and fulfills Vˆκ† Vˆκ = 1l ,

(6.69)

Vˆκ† being the quantized version of Vκ† . We thus have the fuzzy projectors Pˆκ = Vˆκ Vˆκ† . (6.70) Pˆκ = vˆκˆvκ† ,

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Unlike vˆκ , Vˆκ and their adjoints, Pˆκ and Pˆκ commute with the number ˆ . So we can formulate a finite-dimensional matrix model for operator N these projectors as follows. Let Fn be the subspace of the Fock space where ˆ = n. It is of dimension n + 1, and carries the SU (2) representation with N angular momentum n/2, the SU (2) generators being Li =

1 † a τi a . 2

(6.71)

Its standard orthonormal basis is | n2 , m > , m = − n2 , − n2 + 1, ..., n2 . Now (2) consider Fn ⊗C C2 := Fn , with elements f = (f1 , f2 ), fa ∈ Fn . Then (2) Pˆκ , Pˆκ act on Fn in the natural way. For example f → Pˆκ f,

† (Pˆκ f )a = (Pˆκ )ab fb = (Vˆκ,a Vˆκ,b )fb .

(6.72)

We can now write explicit matrices for Pˆκ and Pˆκ . We have: Pˆκ = aκ1

aκ1 Zˆ1 a†1 κ aκ1 Zˆ1 a†2 κ κ

κ

κ

κ

aκ2 Zˆ1 a†1 κ aκ2 Zˆ1 a†2 κ

!

,

(6.73)

1 1 = aκ , ˆ1 + κ)...(N ˆ1 + 1) + Zˆκ(2) 1 Zˆκ (N

ˆ1 + κ)...(N ˆ1 + 1) , aκ1 a†1 κ = (N

ˆ = n can be obtained. from which its matrix Pˆκ (n) for N ˆ ˆ ˆ Pˆκ (n)U ˆ † where U ˆ The matrix Pκ (n) of Pκ is the unitary transform U ˆ is a 2 × 2 matrix and Uab is itself an (n + 1) × (n + 1) matrix. As for the fuzzy analogue of Li , we define it by Li Pˆκ (n) = [Li , Pˆκ (n)] .

(6.74)

The fuzzy action SF,κ (n) =

c Tr ˆ (Li Pˆκ (n))† (Li Pˆκ (n)) 2(n + 1) N =n

c = constant , (6.75)

(2)

follows, the trace being over the space Fn . 6.5.2

The Fuzzy Projectors for κ < 0

For κ < 0, following an early indication, we must suitably exchange the roles of aa and a†a .

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The Fuzzy Winding Number

In the literature [80], there are suggestions on how to extend (6.6) to the fuzzy case. They do not lead to an integer value for this number except in the limit n → ∞. There is also an approach to topological invariants using the Dirac operator and cyclic cohomology. Elsewhere this approach was applied to the fuzzy case [81, 82] and gave integer values, and even a fuzzy analogue of the Belavin-Polyakov bound. However they were not for the action SF,κ , but for an action which approaches it as n → ∞. In the subsection below, we present an alternative approach to this bound which works for SF,κ . It looks like (6.28), except that κ becomes an integer only in the limit n → ∞. There is also a very simple way to associate an integer to Vˆκ [80, 89, 82]. It is equivalent to the Dirac operator approach. We can assume that the ˆ . Then after applying Vˆκ , domain of Vˆκ are vectors with a fixed value n of N n becomes n − κ if κ > 0 and n + |κ| is κ < 0.Thus κ is just the difference ˆ , or equivalently twice the difference in the value of the in the value of N angular momentum, between its domain and its range. We conclude this section by deriving the bound for SF,κ (n). 6.5.4

The Generalized Fuzzy Projector : Duality or BPS States

We introduced the projectors Pκ (·, η, λ) and their fields n(κ) (·, η, λ) earlier. 1 +ix2 = η and a [See (6.15), (6.20)] They describe solitons localized at x1+x 3 shape and width controlled by λ. As inspection shows, they are very easy to quantize by replacing ξi by ai and ξ¯j by a†j . The fields n(κ) (·, η, λ) and their projectors Pκ (·, η, λ) have a particular significance. P|κ| (·, η, λ) saturates the bounds (6.30) with the plus sign, P−κ (·, η, λ) saturates it with the minus sign. This result is due to their holomorphicity (anti-holomorphicity) properties as has been explained elsewhere [66]. It is very natural to identify their fuzzy versions as fuzzy BPS states. But as we note below, they do not saturate the bound on the fuzzy action. 6.5.5

The Fuzzy Bound

A proper generalization of the Belavin-Polyakov bound to its fuzzy version involves a slightly more elaborate approach. This is because the straightforward fuzzification of ~σ · ~x and ~τ · ~n(κ) and their corresponding projectors

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do not commute, and the product of such fuzzy projectors is not a projector. We use this elaborated approach only in this section. It is not needed elsewhere. In any case, what is there in other sections is trivially adapted to this formalism. ˆ = n generate The operators a†α aβ acting on the vector space with N the algebra M at(n + 1) of (n + 1) × (n + 1) matrices. The extra structure comes from regarding them not as observables, but as a Hilbert space of † 1 TrCn+1 m0 m, with matrices m, m0 , ... with scalar product (m0 , m) = n+1 the observables acting thereon. To each α ∈ M at(n + 1), we can associate two linear operators αL,R on M at(n + 1) according to αL m = αm ,

αR m = mα ,

m ∈ M at(n + 1) .

(6.76)

αL − αR has a smooth commutative limit for operators of interest. It actually vanishes, and αL,R → 0 if α remains bounded during this limit. Consider the angular momentum operators Li ∈ M at(n + 1). The associated ‘left’ and ‘right’ angular momenta LL,R fulfil i 2 R 2 (LL i ) = (Li ) =

n n ( + 1) . 2 2

(6.77)

†L ˆ L We now regard aα , a†α of section 6.5.1 as left operators aL α and aα . Pκ thus becomes a 2 × 2 matrix with each entry being a left multiplication operator. It is the linear operator PˆκL on M at(n + 1) ⊗ C2 . We tensor this vector space with another C2 as before to get H = M at(n + 1) ⊗ C2 ⊗ C2 , with σi acting on the last C2 , and σ ·LPˆκL denoting the operator σi (Li Pˆκ )L . We can repeat the previous steps if there are fuzzy analogues γ and Γ of continuum ‘world volume’ and ‘target space’ chiralities ~σ · ~x and ~τ · ~n(κ) which mutually commute. Then 21 (1±γ), 12 (1±Γ) are commuting projectors and the expressions derived at the end of Section 3 generalize, as we shall see. There is such a γ, due to Watamuras[69], and discussed further by [81]. Following [81], we take

γ ≡ γL =

2σ · LL + 1 . n+1

(6.78)

The index L has been put to emphasize its left action on M at(n + 1). As for Γ, we can do the following. Pˆκ acts on the left on M at(n + 1), let us call it PˆκL . It has a PˆκR acting on the right and an associated ˆR Γ ≡ ΓR κ = 2 Pκ − 1 ,

2 (ΓR κ) = 1 .

(6.79)

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As it acts on the right and involves τ ’s while γ acts on the left and involves σ’s, R L γ L ΓR (6.80) κ = Γκ γ . The bound for (6.75) now follows from † 1 + 1 γ L 1 + 2 ΓR 1 + 1 γ L 1 + 2 ΓR κ κ L L ˆ ˆ TrH σ · LPκ σ · LPκ ≥ 0 2 2 2 2 (6.81) (1 , 2 = ±1), and reads c TrH (σ · LPˆκL )† (σ · LPˆκL ) SF,κ = 4(n + 1) c ˆL ˆL TrH (1 γ L + 2 ΓR ≥ κ )(σ · LPκ )(σ · LPκ ) 4(n + 1) c (6.82) TrH 1 2 γ L ΓR (σ · LPˆκL )(σ · LPˆκL ) . + 4(n + 1) The analogue of the first term on the R.H.S. is zero in the continuum, being absent in (6.28), but not so now. As n → ∞, (6.82) reproduces (6.28) to leading order n, but has corrections which vanish in the large n limit. A minor clarification: if τ ’s are substituted by σ’s in 2Pˆ1L − 1, then it is γ L . The different projectors are thus being constructed using the same principles.

6.6

CP N -Models

We need a generalization of the Bott projectors to adapt the previous approach to all CP N . Fortunately this can be easily done. The space CP N is the space of (N + 1) × (N + 1) rank 1 projectors. The important point is the rank. So we can write CP N = hU (N +1) P0 U (N +1)† : P0 = diag. (0, ...., 0, 1) U (N +1) ∈ U (N + 1)i . | {z } N +1 entries

(6.83) As before, let z = (z1 , z2 ), |z1 |2 + |z2 |2 = 1, and xi = z † τi z. Then we define     z1κ z1∗κ z κ  z ∗κ   2  2    1   1 0  0  (N ) (N ) vκ (z) =   √ , κ > 0 ; vκ (z) =   √ , κ < 0 . (6.84)  .  Zκ  .  Zκ      .  . 0 0

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Since vκ(N ) (z)† vκ(N ) (z) = 1 , Pκ(N ) (x) = vκ(N ) (z)vκ(N ) (z)† ∈ CP N .

(6.85) (N )

We can now easily generalize the previous discussion, using Pκ for Pκ and U (N +1) for U , and subsequently quantizing zα , zα∗ . In that way we get fuzzy CP N -models. CP N -models can be generalized by replacing the target space by a general Grassmannian or a flag manifold. They can also be elegantly formulated as gauge theories [72]. But we are able to formulate only a limited class of such manifolds in such a way that they can be made fuzzy. The natural idea would be to look for several vectors (N )(i)

v ki

(z) ,

i = 1, .., N

(6.86)

in (N + 1)-dimensions which are normalized and orthogonal, (N )(i)†

v ki

(N )(j)

(z)vkj

(z) = δij ,

(6.87)

and have the equivariance property (N )(i)

(N )(i)

(6.88) (zeiθ ) = vki (z)ei ki θ . PM (N )(i) (N )(i)† The orbit of the projector (z)vki (z) under U (N +1) will i=1 vki then be a Grassmannian for each M ≤ N , while the orbit of P (N )(i) (N )(i) (z)vki (z)† with possibly unequal λi under U (N +1) will be i λi v ki a flag manifold. (N )(i) But we can find such vki only for i = 1, 2, ..., M ≤ N2+1 . For instance in an (N + 1) = 2L-dimensional vector space, for integer L, we can form the vectors   0  k   0  z1 1    z k1   z k2   2  1  1  1     (N )(1) (N )(2) v k1 (z) =  0  p , v k2 (z) = z2k2  p ,··· ,   Z k1   Z k2  0   ·     ·  0 0   0  ·    1   (N )(L) (z) =  0  p v kL (6.89)  kL  Z kL z 1  z2kL v ki

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for ki > 0. For those ki which are negative, we replace vki (N )(i)

v|ki |

(z) here by

(z)∗ : (N )(i)

(N )(i)

(z)∗ , ki < 0 . (6.90) P (N )(i) These vki are orthonormal for all z with α |zα |2 = 1, so that we can handle Grassmannians and flag manifolds involving projectors up to rank L. If N instead is 2L, we can write   0  k   0  z1 1   z k2   z k1   1   2  1 1     (N )(2) (N )(1) , v k2 (z) = z2k2  p ,··· , v k1 (z) =  0  p     Z k1 Z k 2  0  . . .   . . . 0 0   0  ·     0  1   (N )(L) (6.91) v kL (z) =  kL  p  z 1  Z kL  kL  z 2  0 v ki

(z) = v|ki |

for ki > 0, and use (6.90) for ki < 0. (N )(L+1) But we can find no vector vkL+1 (z) fulfilling (N )(i)

v ki

(N )(L+1)

(z)† vkL+1

(N )(L+1)

vkL+1

(z) = δi,L+1 ,

i = 1, 2, .., L + 1 ,

(N )(L+1)

(zeiθ ) = vkL+1

(z)eikL+1 θ .

(6.92)

The quantization or fuzzification of these models can be done as before. (i) But lacking suitable vki for i > L, the method fails if the target flag manifold involves projectors of rank > N2+1 . Note that we cannot consider vectors like   0 ·   0   1   0 v (z) = zik  , k > 0 , i = 1 or 2 (6.93)   |zi |k 0   · 0

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and v 0 (z)∗ . That is because zi can vanish compatibly with the constraint |z1 |2 + |z2 |2 = 1, and v 0 (z), v 0 (z)∗ are ill-defined when zi = 0. The flag manifolds are coset spaces M = SU (K)/S[U (k1 ) ⊗ U (k2 ) ⊗ P .. ⊗ U (kσ )] , ki = K. Since π2 (M) = Z ⊕ ... ⊕ Z, a soliton on M is now | {z } σ−1 terms

characterized by σ − 1 winding numbers, with each number allowed to take either sign. [We get (σ − 1) and not σ winding numbers. That is because, Pσ det(u1 ⊗ u2 ⊗ · · · ⊗ u( kσ )) = 1 if ui ∈ U (ki ), and hence i=1 ni = 0 if ni is the winding number associated with U (ki ).] The two possible signs for ki (i) in vki reflect this freedom.

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Chapter 7

Fuzzy Gauge Theories

Gauge transformations on commutative spaces are based on transformations which depend on space-time points P . Thus if G is a conventional global group, the associated gauge group is the group of maps G from spacetime to G, the group multiplication being point-wise multiplication. For each irreducible representation (IRR) σ of G, there is an IRR Σ of G given by Σ(g ∈ G)(p) = σ(g(p)). The construction works for any connected Lie group G. There is no problem in composing representations of G either: if Σi are representations of G associated with representations of σi of G, then we can ˆ 2 which has the same relation to σ1 ⊗ σ2 define the representations Σ1 ⊗Σ ˆ 2 (g)(p) = [σ1 ⊗ σ2 ](g(p)) = σ1 (g(p)) ⊗ σ2 (g(p)). that Σi have to σi : Σ1 ⊗Σ Thus such products of Σ are defined using those of G at each p. Existence of these products is essential to describe gauge theories of particles and fields transforming by different representations of G. An additional point of significance is that there is no condition on G, except that it is a compact connected Lie group. For general noncommutative manifolds, several of these essential features of G are absent. Thus in particular • Noncommutative manifolds require G to be a U (N ) group, • Only a very limited and quite inadequate number of representations of the gauge group can be defined. We shall illustrate these points below for the fuzzy gauge groups GF based on SF2 , but one can see the generalities of the considerations. There is an important map, the Seiberg-Witten(SW), map for a noncommutative deformation of RN . In that case the deformed algebra RN θ depends continuously on a parameter θ, becoming the commutative algebra for θ = 0. If a certain gauge group on RN θ is Gθ , it becomes a standard 77

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N gauge group G0 on RN 0 = R . The SW map is based on a homomorphism N N from Rθ to R0 and connects gauge theories for different θ. The aforementioned problems can be more or less overcome on RN θ using this map. But fuzzy spheres have no continuous parameter like θ. What plays the role of θ is L1 where 2L is the cut-off angular momentum, and L1 assumes discrete values. Fuzzy spheres have no SW map as originally conceived, and we can not circumvent its gauge-theoretic problems along the lines for RN θ . There is however a complementary positive feature of fuzzy spaces. While SF2 for example presents problems in describing particles of charge 13 and 23 at the same time (because we cannot “tensor” representations of the fuzzy U (1) gauge group GF (U (1))), we can describe particles with differing magnetic charges. The projective modules for all magnetic charges were already explained in Chapter 5 and 6. There is no symmetry (“duality”) here between electric and magnetic charges.

7.1

Limits on Gauge Groups

The conditions on gauge groups on the fuzzy sphere arise algebraically. They can be understood at the Lie algebraic level. If {λa } is a basis for the Lie algebra of G in a representation σ, the Lie algebra of GF , the fuzzy gauge group of G, are generated by λa ξ a

(7.1)

where ξa are (2L + 1) × (2L + 1) matrices. ξa become functions on S 2 in the large L-limit. Now consider the commutator [λa ξa , λb ηb ] ,

ηb = (2L + 1) × (2L + 1) matrix

(7.2)

of two such Lie algebra elements. We get c [λa , λb ]ξa ηb + λa λb [ξa , ηb ] = iCab ξa ηb λc + λa λb [ξa , ηb ] , c Cab = structure constants of the Lie algebra of G .

(7.3)

c Since Cab ξa ηb ∈ SF2 , the first term is of the appropriate form for a fuzzy gauge group of G. But the last term is not, it involves λa λb which is a product of two generators of G. By taking repeated commutators, we will generate products of all orders of λa ’s and their commutators. If σ is irreducible and of dimension d, we will get all the d × d hermitian matrices

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in this way and not just the λa . This means that the fuzzy gauge group is that of U (d). In the commutative limit, [ξa , ηb ] is zero and this problem does not occur. This escalation of the gauge group to U (d) is difficult to control. No convincing proposal to minimize its effects exists. [See in this connection [84–86]]. In any case, U (d) gauge theories without matter fields can be consistently formulated on fuzzy spheres. For applications, there is one mitigating circumstance: In the standard model, if we gauge just SU (3)C and U (1)EM , namely the SU (3) of colour and U (1) of electromagnetism, the group is actually U (3) [88]. Likewise, the weak group is not SU (2) × U (1), but U (2). Thus gauge fields without matter in these sectors can be studied on fuzzy spheres. Unfortunately, this does not mean that these gauge theories can be formulated satisfactorily on SF2 or (for a four-dimensional continuum limit) on SF2 × SF2 say, when quarks and leptons are included. For example with different flavours, different charges like 2/3 and −1/3 occur, and there is no good way to treat arbitrary representations of gauge groups in noncommutative geometry [86]. We explain this problem now.

7.2

Limits on Representations of Gauge Groups

For the fuzzy U (d) gauge group on the fuzzy sphere SF2 (2L+1), we consider SF2 (2L + 1) ⊗ Cd . The fuzzy U (d) gauge group U (d)F consists of d × d matrices U with coefficients in SF2 (2L + 1) : Uij ∈ SF2 (2L + 1). The U (d)F can act in three different ways on SF2 (2L + 1) ⊗ Cd : on left, right and both: i. Left action : U → U L where U L X = U X for X ∈ SF2 (2L + 1) ⊗ Cd , ii. Right action : U → (U † )R : (U † )R X = XU † , iii. Adjoint action : U → AdU : AdU X = U XU † . If i. gives representation Λ, then ii. is its complex conjugate Λ∗ and iii. is its adjoint representation Ad Λ. We are guaranteed that these representations can always be constructed. But can we construct other representations such as the one correspondb 2 ? The answer appears to be no. ing to Σ1 ⊗Σ b is not the tensor product ⊗. In Σ ⊗ Σ, we get The reason is as follows ⊗ functions of two variables p and q: (Σ(g) ⊗ Σ(g))(p, q) = σ(g(p)) ⊗ σ(g(q)).

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b We must restrict (Σ(g) ⊗ Σ(g)) to the diagonal points (p, p) to get ⊗. In noncommutative geometry, the tensor product Λ1 ⊗Λ2 exists of course since Λ1 (U ) ⊗ Λ2 (U ) is defined, and gives a representation of U (d)F . But noncommutative geometry has no sharp points. That obstructs the construction of an analogue of diagonal points, or the restriction of ⊗ to an b analogue of ⊗. There exist proposals [86] to get around this problem using Higgs fields. There is also the work of K¨ urk¸cu ¨oˇ glu and S¨ amann [87] on the use of “twisted” coproducts for defining the action of diffeomorphisms and gauge groups on the fuzzy sphere. It has the potentiality to overcome this problem. 7.3

Connection and Curvature

As a convention we choose the gauge potential to act on the left of SF2 (2L + 1) ⊗ Cd . So the components of the gauge potentials are L a AL i = (Ai ) λa ,

a 2 (AL i ) ∈ SF (2l + 1)

(7.4)

where λa , (a = 1 , · · · , d2 ) are the d × d basis matrices for the Lie algebra of U (d). They can be the Gell-Mann matrices. The covariant derivative ∇ is then the usual one: ∇i = L i + A L i

(7.5)

The curvature is Fij = [∇i , ∇j ] − iεijk ∇k

L L L L = [Li , Lj ] + Li AL j − Lj Ai + [Ai , Aj ] − iεijk (Lk + Ak )

L L L L = L i AL j − Lj Ai + [Ai , Aj ] − iεijk Ak .

(7.6)

The subtraction of iεijk ∇k is needed to cancel the [Li , Lj ] term in [∇i , ∇j ]. There is one important condition on ∇i . On S 2 , AL becomes a commutative gauge field a and its components ai have to be tangent to S 2 : xi a i = 0 .

(7.7)

We need a condition on ∇i which becomes this condition for large L. A simple condition of such a nature is due to Nair and Polychronakos [90] and reads L 2 (LL i + Ai ) = L(L + 1) .

(7.8)

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This is compatible with gauge invariance. Its expansion is L L L L L LL i A i + A i Li + A i A i = 0 .

(7.9)

AL i

We have that L → 0 as L → ∞. Dividing (7.9) by L and passing to the limit, we thus get (7.7). The fuzzy Yang-Mills action is 1 (7.10) SF = 2 T rFij2 + λ(∇2i − L(L + 1)) , λ ≥ 0 , 4e where the second term is a Lagrange multiplier: it enforces the constraint (7.8) as λ → ∞. There is an approach to fuzzy gauge theories which has connections to string-theoretical matrix models such as those of Ishibashi, Kawai, Kitazawa and Tsuchiya [91] and Alekseev, Recknagel and Schomerus [136]. This approach developed by Watumaras [92], and Steinacker [93] is conceptually important and can also have practical advantages. We refer to the literature for details.

7.4

Instanton Sectors

The above action is good in the sector with no instantons. But U (d) gauge theories on S 2 have instantons, or equivalently, twisted U (1)-bundles on S 2 . We outline how to incorporate instantons on the fuzzy sphere, taking d = 1 for simplicity. The projective modules for instanton sectors were constructed previously. We review it briefly constructing the modules in a different (but Morita equivalent [36]) manner. The instanton sectors on S 2 correspond to U (1) bundles thereon. To build the corresponding projective module for Chern number 2T ∈ Z+ , introduce C2T +1 carrying the angular momentum T representation of SU (2). Let Ti be the angular momentum operators in this representation with standard commutation relations. Let M at(2L + 1) ⊗ C(2T +1) ≡ M at(2L + 1)(2T +1) . We let P L+T be the projector coupling left angular momentum operators LL and T to produce maximum angular momentum L + T . Then the projective module P L+T M at(2L + 1)(2T +1) is a fuzzy analogue of sections of U (1) bundles on S 2 with Chern number 2T > 0 [81]. If instead we couple LL and T to produce the least angular momentum L − T using the projector P L−T , then the projective module P L−T M at(2L + 1)(2T +1) corresponds to Chern number −2T . (We assume that L ≥ T ).

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The derivation Li does not commute with P L±T and has no action on these modules. But, Ji = Li + Ti does commute with P L±T . Thus Li must be replaced with Ji in further considerations. Ji is to be considered the total angular momentum . The addition of Ti to Li here is the algebraic analogue of “mixing of spin and isospin”. [71, 70]. It is interesting that the mixing of “spin and isospin” occurs already in our finite-dimensional matrix model and does not need noncompact spatial slices and spontaneous symmetry breaking. We must next gauge Ji . In the zero instanton sector, the fuzzy gauge L L fields AL i were functions of Li . But that is not possible now since Ai does ~L ~ not commute with P L±T . Instead we require AL i to be a function of L + T and write for the covariant derivative ∇i = J i + A L i .

(7.11)

L 2 L 2 (LL i + Ti + Ai ) = (Li + Ti )

(7.12)

2 (LL i + Ti ) = (L ± T )(L ± T + 1)

(7.13)

When L → ∞, T~ can be ignored, and then AL i becomes a function of x, just as we want. The transversality condition must be modified. It is now

where

on P L±T M at(2L + 1)(2T +1) . The curvature Fij and the action SF are as in (7.6) and (7.10). 7.5

The Partition Function and the θ-parameter

Existence of instanton bundles on a commutative manifold brings in a new parameter, generally called θ, as in QCD. The partition function Zθ depends on θ. Let us denote the action in the instanton number K ∈ Z sector by SFK . Then XZ K −SF +iKθ Zθ = DAL . (7.14) i e k

We thus have a matrix model for U (d) gluons. In the continuum, K can be written as the integral of trF (where trace tr (with lower case t) is over the internal indices). In four dimensions it is the integral of trF ∧ F . But on SF2 , T rεij Fij (where T r is over both

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internal indices and SF2 ) is not an integer. A similar difficulty arises for SF2 × SF2 or CP 2 . In continuum gauge theory, F and F ∧ F play a role in discussions of chiral symmetry breaking. They arise as the local anomaly term in the continuity equation for chiral current. Therefore although Zθ defines the theory, it is still helpful to have fuzzy analogues of the topological densities T rF and T rF ∧ F . It is possible to construct fuzzy topological densities using cyclic cohomology [34]. We will not review cyclic cohomology here.

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Chapter 8

The Dirac Operator and Axial Anomaly

8.1

Introduction

The Dirac operator is central for fundamental physics. It is also central in noncommutative geometry. In Connes’ approach [34], it is possible to formulate metrical, differential geometric and bundle-theoretic ideas using the Dirac operator in a form generalisable to noncommutative manifolds. In this chapter, we explain the theory of the fuzzy Dirac operator basing it on the Ginsparg-Wilson (GW) algebra [94]. This algebra appeared first in the context of lattice gauge theories as a device to write the Dirac operator overcoming the well-known fermion-doubling problem. The same algebra appears naturally for the fuzzy sphere. The theory of the fuzzy Dirac operator can be based on this algebra. It has no fermion doubling and correctly and elegantly reproduces the integrated U (1)A -(axial) anomaly. Incidentally the association of the GW-algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. Below we review the GW-algebra in its generality. We then adapt it to SF2 . Our discussion here closely follows [95]. Other references on the subject include [105–107]. 8.2

A Review of the Ginsparg-Wilson Algebra

In its generality, the Ginsparg-Wilson algebra A can be defined as the unital ∗-algebra over C generated by two ∗-invariant involutions Γ and Γ0 :

2 ∗ A = Γ, Γ0 : Γ2 = Γ0 = 1l, Γ∗ = Γ, Γ0 = Γ0 , (8.1)

(∗ generally denoting the adjoint in any representation). The unity of A has been indicated by 1l. 85

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In any such algebra, we can define a Dirac operator 1 D0 = Γ(Γ + Γ0 ) , a where a is the “lattice spacing”. It fulfills ∗

D0 = Γ D0 Γ,

{Γ, D0 } = a D0 Γ D0 .

(8.2)

(8.3)

(8.2) and (8.3) give the original formulation [94]. But they are equivalent to (8.1), since (8.2) and (8.3) imply that Γ0 = Γ(aD0 ) − Γ

(8.4)

is a ∗-invariant involution [98] [96]. Each representation of (8.1) is a particular realization of the GinspargWilson algebra. Representations of physical interest are reducible. Here we choose 1 (8.5) D = (Γ + Γ0 ) , a instead of D0 as our Dirac operator, as it is self-adjoint and has the desired continuum limit. From Γ and Γ0 , we can construct the following elements of A: 1 (8.6) Γ0 = {Γ, Γ0 } , 2 1 Γ1 = (Γ + Γ0 ) , (8.7) 2 1 Γ2 = (Γ − Γ0 ) , (8.8) 2 1 Γ3 = [Γ, Γ0 ] . (8.9) 2i Let us first look at the centre C(A) of A in terms of these operators. It is generated by Γ0 which commutes with Γ and Γ0 and hence with every element of A . Γ2i , i = 1, 2, 3 also commute with every element of A, but they are not independent of Γ0 . Rather, 1 (8.10) Γ21 = (1l + Γ0 ) , 2 1 Γ22 = (1l − Γ0 ) , (8.11) 2 → Γ21 + Γ22 = 1l , (8.12) Γ20 + Γ23 = 1l .

(8.13)

Notice also that {Γi , Γj } = 0 , i, j = 1, 2, 3, i 6= j .

(8.14)

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From now on by A we will mean a representation of A. The relations (8.10)-(8.13) contain spectral information. From (8.13) we see that −1 ≤ Γ0 ≤ 1 ,

(8.15)

where the inequalities mean that the eigenvalues of Γ0 are accordingly bounded. By (8.10), this implies that the eigenvalues of Γ1 are similarly bounded. We now discuss three cases associated with (8.15). Case 1 : Γ0 = 1l. Call the subspace where Γ0 = 1l as V+1 . On V+1 , Γ21 = 1l and Γ2 = Γ3 = 0 by (8.10-8.13). This is subspace of the top modes of the operator |D|. Case 2 : Γ0 = −1l. Call the subspace where Γ0 = −1l as V−1 . On V−1 , Γ22 = 1l and Γ1 = Γ3 = 0 by (8.10-8.13). This is the subspace of zero modes of the Dirac operator D. Case 3 : Γ20 6= 1l. Call the subspace where Γ20 6= 1l as V . On this subspace, Γ2i 6= 0 for i = 1, 2, 3 by (8.9-8.12), and therefore Γi , |Γi | = positive square root of Γ2i (8.16) sign Γi = |Γi |

are well defined and by (8.14) generate a Clifford algebra on V : {sign Γi , sign Γj } = 2δij .

(8.17)

Consider Γ2 . It anticommutes with Γ1 and D. Also Tr Γ2 = (TrV + TrV+1 + TrV−1 )Γ2 ,

(8.18)

where the subscripts refer to the subspaces over which the trace is taken. These traces can be calculated: TrV Γ2 = TrV (sign Γi )Γ2 (sign Γi ) (i fixed, 6= 2) = − TrV Γ2

by(8.17)

= 0,

TrV+1 Γ2 = 0,

as Γ2 = 0 on V+1 .

(8.19) (8.20)

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So 1 + Γ2 1 − Γ2 − ) = index of Γ1 . (8.21) 2 2 Following Fujikawa [96], we can use Γ2 as the generator of chiral transformations. It is not involutive on V ⊕ V+1 Tr Γ2 = TrV−1 Γ2 = TrV−1 (

1l + Γ0 . (8.22) 2 But this is not a problem for fuzzy physics. In the fuzzy model below, in the continuum limit, Γ0 → −1l on all states with |D| ≤ a fixed ‘energy’ E0 independent of a (and is −1l on V−1 where D = 0). We can see this as follows. Γ1 = aD, so that if |D| ≤ E0 , Γ1 → 0 as a → 0. Hence by (8.10,8.12), Γ0 → −1l and Γ22 → 1l on these levels. There are of course states, such as those of V+1 , on which Γ22 does not go to 1l as a → 0. But their (Euclidean) energy diverges and their contribution to functional integrals vanishes in the continuum limit. We can interpret (8.22) as follows. The chiral charge of levels with D 6= 0 gets renormalized in fuzzy physics. For levels with |D| ≤ E0 , this renormalization vanishes in the naive continuum limit. We note that the last feature is positive: it resolves a problem faced in [99] for fuzzy spheres, where all the top modes of the Dirac operator had to be projected out because of the insistence that chirality squares to 1l on V+1 (see below). For Dirac operators of maximum symmetry on SF2 , Γ0 is a function of the conserved total angular momentum J~ as we shall show. It increases with J~2 so that V+1 consists of states of maximum J~2 . As the calculations below show, on SF2 = M at(2` + 1), where a = O L1 , this maximum value diverges as a → 0. Γ22 = 1l −

8.3 8.3.1

Fuzzy Models Review of the Basic Fuzzy Sphere Algebra

Let us briefly recollect the basic algebraic details. The algebra for the fuzzy sphere characterized by cut-off 2L is the full matrix algebra M at(2L + 1) ≡ M2L+1 of (2L + 1) × (2L + 1) matrices. On M2L+1 , the SU (2) Lie algebra acts either on the left or on the right. Call R the operators for left action as LL i and for right action as Li . We have R LL i a = Li a , Li a = aLi , a ∈ M at(2L + 1) ,

(8.23)

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R R [LR i , Lj ] = −iijk Lk ,

2 R 2 (LL i ) = (Li ) = L(L+1)1l , (8.24) where Li is the standard matrix for the i-th component of the angular momentum in the the (2L + 1)-dimensional irreducible representation (IRR). ~ i as L → ∞ is The orbital angular momentum which becomes −i(~r ∧ ∇)

R Li = L L Li a = [Li , a] . (8.25) i − Li , L R ~ ~ As L → ∞, both L /L and L /L approach the unit vector xˆ with commuting components: ~ L,R L −→ x ˆ, x ˆ·x ˆ = 1 , [ˆ xi , x ˆj ] = 0 . (8.26) L L→∞ x ˆ labels a point on the sphere S 2 in the continuum limit.

8.3.2

The Fuzzy Dirac Operator (No Instantons or Gauge Fields)

Consider M2L+1 ⊗ C2 . C2 is the carrier of the spin 1/2 representation of SU (2) with generators 21 σi , σi = Pauli matrices. We can couple its spin 1/2 and the angular momentum L of LL i to the value L + 1/2. If (1 + Γ)/2 is the corresponding projector, then [99] [69] [81] ~ L + 1/2 ~σ · L . (8.27) Γ= L + 1/2 Γ is a self-adjoint involution, Γ∗ = Γ ,

Γ2 = 1l .

(8.28)

There is likewise the projector (1l + Γ0 )/2 coupling the spin 1/2 of C2 and the right angular momentum −LR i to L + 1/2, where Γ0 =

~ R + 1/2 −~σ · L ∗ = Γ0 L + 1/2

2

Γ0 = 1l .

(8.29)

The algebra A is generated by Γ and Γ0 . The fuzzy Dirac operator of Grosse et al.[7] is 2 1 1 ~L − L ~ R) + 1 , a = . (8.30) D = (Γ + Γ0 ) = Γ1 = ~σ · (L a a L + 1/2 Thus the Dirac operator is in this case an element of the Ginsparg-Wilson algebra A. ~ + ~σ /2: We can calculate Γ0 in terms of J~ = L Γ0 =

a2 ~2 1 [J − 2L(L + 1) − ] . 2 4

(8.31)

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Thus the eigenvalues of Γ0 increase monotonically with the eigenvalues j(j+1) of J~2 starting with a minimum for j = 1/2 and attaining a maximum of 1 for j = 2L + 1/2. Γ2 is chirality. It anticommutes with D. For fixed j, as L → ∞, Γ0 → −1l and Γ22 = 1l as expected. In fact, Γ2 in the naive continuum limit is the standard chirality for fixed j. As L → ∞, Γ2 → σ · x ˆ. As mentioned earlier, use of Γ2 as chirality resolves a difficulty addressed elsewhere [99], where sign (Γ2 ) was used as chirality. That necessitates projecting out V+1 and creates a very inelegant situation. Finally we note that there is a simple reconstruction of Γ and Γ0 from ·~ x σ · ~x| ≡ their continuum limits [104]. If ~x is not normalized, ~σ · x ˆ = |~~σσ ·~ x| , |~ 1/2 ~ L or L ~ R in fuzzy physics, | (~σ · ~x)2 |. As ~x can be represented by L natural choices for Γ and Γ0 are sign (~σ · LL ) and −sign (~σ · LR ). The first ~ L > 0 and −1 if instead ~σ · L ~ L < 0. operator is +1 on vectors having ~σ · L L 2 L ~ + ~σ /2) = (L + 1/2)(L + 3/2), then ~σ · L ~ = L > 0, while if But if (L ~ L + ~σ /2)2 = (L − 1/2)(L + 1/2), ~σ · L ~ L = −(L + 1) < 0. Γ is +1 on (L

former states and −1 on latter states. Thus

~ L) = Γ , sign (~σ · L

(8.32)

~ R ) = −Γ0 . sign (~σ · L

(8.33)

and similarly

It is easy to calculate the spectrum of D. We can write ~2 − 3 + 1 aD = J~ 2 − L 4

(8.34)

~2 ] = 0. The spectrum of L ~2 is We observe that [J~ 2 , L

~ 2 = {`(` + 1) : ` = 0, 1, · · · , 2L} , spec L

whereas that of J~ 2 is

spec J~ 2 =

1 1 3 j(j + 1) : j = , , · · · , 2L + 2 2 2

(8.35)

.

(8.36)

Here each j can come from ` = j± 12 by adding spin, except j = 2L+ 21 which comes only from` = 2L. It follows that the eigenvalue of D for ` = j − 21 is j + 12 = ` + 1 , ` ≤ 2L and for ` = j + 21 is −(j + 21 ) = −` , ` ≤ 2L. The spectrum found here agrees exactly with what is found in the continuum for j ≤ 2L − 23 . For j = 2L + 12 we get the positive eigenvalue correctly, but the negative one is missing. That is an edge effect caused by cutting off the angular momentum at 2L.

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8.3.3

The Fuzzy Gauged Dirac Operator (No Instanton Fields)

We adopt the convention that gauge fields are built from operators on M at(2L + 1) which act by left multiplication. For U (k) gauge theory, we start from M at(2L + 1) ⊗ Ck . The fuzzy gauge fields AL i are k × k matrices L [(Ai )mn ] where each entry is the operator of left-multiplication by (Ai )mn ∈ M at(2L + 1) on M at(2L + 1). AL i thus acts on ξ = (ξ1 , . . . , ξk ), ξi ∈ M at(2L + 1) according to (AL i ξ)m = (Ai )mn ξn .

(8.37)

The gauge-covariant derivative is then L R L ∇i (AL ) = Li + AL i = Li − Li + Ai .

(8.38)

Note how only the left angular momentum is augmented by a gauge field. The hermiticity condition on AL i is ∗ L (AL i ) = Ai ,

(8.39)

∗ ∗ ((AL i ) ξ)m = (Ai )mn ξn ,

(8.40)

where

(A∗i )nm being Hermitian conjugate of (Ai )nm . The corresponding field strength Fij is defined by L L [(L + A)L i , (L + A)j ] = iijk (L + A)k + iFij .

(8.41)

There is a further point to attend to. We need a gauge-invariant condition which in the continuum limit eliminates the component of Ai normal to S 2 . There are different such conditions, the following simple one was disccussed in chapter 7, (cf. 7.8): L 2 L 2 (LL i + Ai ) = (Li ) = L(L + 1) .

(8.42)

The Ginsparg-Wilson system can be introduced as follows. As Γ squares ~ L +1/2. By continuity, to 1l, there are no zero modes for Γ and hence for ~σ · L L L L ~ ~ ~ for generic A , its gauged version ~σ ·(L + A )+1/2 also has no zero modes. Hence we can set ~L + A ~ L ) + 1/2 ~σ · (L , Γ(AL )∗ = Γ(AL ) , Γ(AL )2 = 1l . Γ(AL ) = ~L + A ~ L ) + 1/2| |~σ · (L (8.43) ~L. It is the gauged involution which reduces to Γ = Γ(0) for zero A

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As for the second involution Γ0 (AL ), we can set Γ0 (AL ) = Γ0 (0) ≡ Γ0 .

(8.44)

On following (8.6-8.9), these idempotents generate the Ginsparg-Wilson algebra with operators Γλ (AL ), where Γλ (0) = Γλ . ~ L,R do not individually have continuum limits as their The operators L ~ and A ~ L do have consquares L(L + 1) diverge as L → ∞. In contrast L ~ just tinuum limits. This was remarked earlier on for the latter, while L becomes orbital angular momentum. To see more precisely how D(AL ), the Dirac operator for gauge field L A , (D(0) being the D of (8.30)), and Γ2 (AL ), behave in the continuum limit, we note that from (8.41),(8.42), ~L + A ~ L ) + 1 2 = (L + 1 )2 − 1 ijk σi Fij . (8.45) ~σ · (L 2 2 2 Therefore we have the expansions Z ∞ 2 2 1 2 1 ~ L ~L √ = ds e−s (~σ·(L +A )+ 2 ) ~L + A ~L) + 1 | π 0 |~σ · (L 2

1 = L+

1 2

1 + ijk σi Fjk + . . . , 4(L + 12 )3

(8.46)

D(AL ) = (2L + 1)Γ1 (AL ) = 1 ~ L ~L ~L − L ~R + A ~ L ) + 1 + ~σ · (L + A ) + 2 ijk σk Fij + . . . ~σ · (L 1 2 4(L + 2 )

Γ2 (AL ) = ~L + A ~L) + ~σ · (L

~R + 1 ~L + A ~L) + 1 −~σ · L ~σ · (L 2 2 + ijk σk Fij + . . . . 1 2(L + 2(L + 2 ) 8(L + 12 )3 (8.47) L ~ ~ So in the continuum limit, D(A ) → ~σ · (L + A) + 1 , and Γ2 (A) → ~σ · xˆ, exactly as we want. It is remarkable that even in the presence of gauge field, there is the operator ~ L ), Γ0 (A ~ L )]+ ~ L ) = 1 [Γ(A (8.48) Γ0 (A 2 which is in the centre of A. It assumes the role of J~2 in the presence of ~ L . In the continuum limit, it has the following meaning. With D(AL ) A denoting the Dirac operator for gauge field AL , (D(0) being the D of (8.30)), sign (D(AL )) and Γ2 (AL ) generate a Clifford algebra in that limit and the Hilbert space splits into a direct sum of subspaces, each carrying its IRR. Γ0 (AL ) is a label for these subspaces. 1 2)

1 2

−

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The Basic Instanton Coupling

The instanton sectors on S 2 correspond to U (1) bundles thereon. The connection on these bundles is not unique. Those with maximum symmetry have a particular simplicity and are therefore important for analysis. In a similar way, on SF2 , there are projective modules which in the algebraic approach substitute for sections of bundles [34] [78] [81](see chapter 5 and 6). There are particular connections on these modules with maximum symmetry and simplicity. In this section we build the Ginsparg-Wilson system for such connections. The Dirac operator then is also simple. It has zero modes which are responsible for the axial anomaly. Their presence will also be shown by simple reasoning. To build the projective module for Chern number 2T , T > 0, we follow chapters 6 and 7 and introduce C2T +1 carrying the angular momentum T representation of SU (2). Let Tα , α = 1, 2, 3 be the angular momentum operators in this representation with standard commutation relations. Let M at(2L + 1)2T +1 ≡ M at(2L + 1) ⊗ C2T +1. We let P (L+T ) be the projector ~ L with T~ to produce maximum coupling left angular momentum operators L angular momentum L + T . Then the projective module P (L+T ) M at(2L + 1)2T +1 is the fuzzy analogue of sections of U (1) bundles on S 2 with Chern ~ L and T~ to produce the least number 2T > 0 [81]. If instead we couple L angular momentum (L − T ) using the projector P (L−T ) , P (L−T ) M at(2L + 1)2T +1 corresponds to Chern number −2T . (We assume that L ≥ T ). We go about as follows to set up the Ginsparg-Wilson system. For Γ, we now choose Γ± =

~ L + T~ ) + 1/2 ~σ · (L . L ± T + 1/2

(8.49)

The domain of Γ± is P (L±T ) M at(2L + 1)2T +1 ⊗ C2 with σ acting on C2 . ~ L + T~ )2 = (L ± T )(L ± T + 1) and (Γ± )2 = 1l. On this module, (L 0 As for Γ , we choose it to be the same as in eq.(8.29). Γ± and Γ0 generate the new Ginsparg-Wilson system. The operators Γλ are defined as before as also the new Dirac operator D (L±T ) = a2 Γ1 . With T > 0 it is convenient to choose

for the two cases.

a= q

1 (L + 21 )(L ± T + 21 )

(8.50)

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8.4.1

Mixing of Spin and Isospin

The total angular momentum J~ which commutes with P (L±T ) and hence ~ L −L ~ R +~σ /2, but L ~ L + T~ − L ~ R +~σ /2. acts on P (L±T ) M at(2L+1)⊗C2 is not L ~ The addition of T here is the algebraic analogue of the ‘mixing of spin and isospin’ [70] as remarked in chapter 7. Such a term is essential in J~ ~L − L ~ R + ~σ /2, not commuting with P (L±T ) , would not preserve the since L modules. 8.4.2

The Spectrum of the Dirac operator

The spectrum of Γ1 and D(L±T ) can be derived simply by angular momentum addition, confirming the results of section 2. On the P (L±T ) M at(2L + ~ L + T~ )2 has the fixed values (L ± T )(L ± T + 1), and 1)2T +1 modules, (L 1 ~ L + T~ − L ~ R + 1 ~σ )2 + 1 − T 2 , (8.51) (L (Γ1 )2 = (2(L ± T ) + 1)(2L + 1) 2 4

~ L + T~ + 1 ~σ )2 − (L ± T )(L ± T + 1) − 1 (L 2 4 , (8.52) (L ± T ) + 21 ~ R + 1 ~σ )2 − L(L + 1) − 1 (−L 2 4 . (8.53) Γ0 = L + 12 Comparing (8.51) with (8.10) we see that the ‘total angular momentum’ ~ 2 = (L ~ L + T~ − L ~ R + 1 ~σ )2 is linearly related to Γ0 = 1 [Γ± , Γ0 ]+ . The (J) 2 2 ~ 2 , call them j(j+1). eigenvalues (γ1 )2 of (Γ1 )2 are determined by those of (J) For j = jmax = L ± T + L + 21 we have (Γ1 )2 = 1, so this is V+1 , and the degeneracy is 2jmax + 1 = 2(2L ± T + 1). The maximum value of j can be achieved only if ~ L + T~ + 1 ~σ )2 = (L±T + 1 )(L±T + 3 ) , (−L ~ R + 1 ~σ )2 = (L+ 1 )(L+ 3 ) . (L 2 2 2 2 2 2 (8.54) Replacing these values in (8.52,8.53) we see that on V+1 we have γ1 = 1, and Γ2 = 0. The case T = 0 has been treated before [7][81][99].So we here assume that T > 0. In that case, for either module jmin = T − 12 , which gives an eigenvalue (γ1 )2 = 0 with degeneracy 2T ; we are in V−1 , the space of the zero modes. To realize this minimum value of j we must have ~ L + T~ + 1 ~σ )2 = (L ± T ∓ 1 )(L ± T ∓ 1 + 1) , (8.55) (L 2 2 2 ~ R + 1 ~σ )2 = (L ± 1 )(L ± 1 + 1) . (−L (8.56) 2 2 2 Γ± =

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Replacing these values in (8.52, 8.53) we find that on the corresponding eigenstates Γ2 = ∓1: they are all either chiral left or chiral right. These are the results needed by continuum index theory and axial anomaly. For jmin < j < jmax , that is on V , we have 0 < (γ1 )2 < 1, and by (8.12), Γ2 6= 0. Since [Γ1 , Γ2 ]+ = 0, to each state ψ such that Γ1 ψ = γ1 ψ corresponds a state ψ 0 = Γ2 ψ such that Γ1 ψ 0 = −γ1 ψ 0 . For any value of j we can write j = n + T − 21 with n = 0, 1, ..., 2L + 1 when the projector is P (L+T ) , and n = 0, 1, ..., 2(L − T ) + 1 when the projector is P (L−T ) , while correspondingly, (γ1 )2 =

n(n + 2T ) . (2(L ± T ) + 1)(2L + 1)

(8.57)

With the choice (8.50) for a this gives for the squared Dirac operator the eigenvalues ρ2 = n(n + 2T ). This spectrum agrees exactly with what one finds in the continuum [100], except at the top value of n. Such a result is true also for T = 0 [99][81]. For the top value of n, Γ2 = 0, and we get only the eigenvalue γ1 = 1, whereas in the continuum, Γ2 6= 0 and both eigenvalues γ1 = ±1 occur. This result [99][81], valid also for T = 0, has been known for a long time. Finally, we can check that summing the degeneracies of the eigenvalues we have found, we get exactly the dimension of the corresponding module. In fact: 2T + 2

2L X

n=1

1 2(n + T − ) + 1 + 2(2L + T + 1) 2

= 2(2L + 1)(2(L + T ) + 1) , 2(L−T )

2T + 2

X

n=1

1 2(n + T − ) + 1 + 2(2L − T + 1) 2

= 2(2L + 1)(2(L − T ) + 1) .

(8.58)

We show below that the axial anomaly on SF2 is stable against perturbations compatible with the chiral properties of the Dirac operator, and is hence a ‘topological’ invariant. 8.5

Gauging the Dirac Operator in Instanton Sectors

~ + T~ commutes with P (L±T ) and hence preserves the proThe operator L jective modules. It is important to preserve this feature on gauging as well.

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~ L is taken to be a function of L ~ L + T~ (which remains So the gauge field A bounded as L → ∞). For L → ∞, it becomes a function of x. The limiting ~ L can be guaranteed by imposing the condition transversality of T~ + A ~ L + T~ + A ~ L )2 = ( L ~ L + T~ )2 = (L ± T )(L ± T + 1) , (L

(8.59)

which generalizes (8.42). We can now construct the Ginsparg-Wilson system using Γ(AL ) =

~ L + T~ + A ~ L ) + 1/2 σ · (L ~ L + T~ + A ~ L ) + 1/2| |σ · (L

(8.60)

~ L + T~ ) + 1/2 has no zero and the Γ0 of (8.29), Γ(0) being Γ of (8.49). σ · (L ~ L . We can now use modes, and therefore (8.60) is well-defined for generic A section 2 to construct the Dirac theory. We have a continuous number of Ginsparg-Wilson algebras labeled by ~ L . For each, (8.21) holds: A Tr Γ2 (AL ) = n(AL ) .

(8.61)

Here as n(AL ) ∈ Z, it is in fact a constant by continuity. The index of the Dirac operator and the global U (1)A axial anomaly implied by (8.61) are ~ L as previously indicated. [See Fujikawa [96] and [97] thus independent of A for the connection of (8.61) to the global axial anomaly.] The expansions (8.45-8.47) are easily extended to the instanton sectors, ~ L ) and chirality Γ2 (A ~ L ): and imply the desired continuum limit of D (L±T ) (A ~ L ) → ~σ · (L ~ + T~ + A) ~ +1 , D(L±T ) (A Γ2 (AL ) → ~σ · x ˆ.

(8.62)

Chirality is thus independent of the gauge field in the limiting case, but not otherwise. 8.6

Further Remarks on the Axial Anomaly

The local form of U (1)A -anomaly has not been treated in the present approach. (See however [63][100][101].) As for gauge anomalies, the central and familiar problem is that noncommutative algebras allow gauging only by the particular groups U (N ), and that too by their particular representations (see chapter 7). This is so in a naive approach. There are clever methods to overcome this problem on the Moyal planes [102] using the Seiberg-Witten map [103], but they fail for the fuzzy spaces. Thus gauge

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anomalies can be studied for fuzzy spaces only in a very limited manner, but even this is yet to be done. More elaborate issues like anomaly cancellation in a fuzzy version of the standard model have to wait till the above mentioned problems are solved.

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Chapter 9

Fuzzy Supersymmetry

Another important feature we encounter in studying fuzzy discretizations is their ability to preserve supersymmetry (SUSY) exactly: They allow the formulation of regularized and exactly supersymmetric field theories. It is very difficult to formulate models with exact SUSY in conventional lattice discretizations. At least for this reason, fuzzy supersymmetric spaces merit careful study. The original idea of a fuzzy supersphere is due to Grosse et al.[7, 8]. A slightly different approach for its construction, which is closer to ours, is given in [122]. We start this chapter describing the supersphere S (2,2) and its fuzzy (2,2) version SF . Although the mathematical structure underlying the formulation of the supersphere is a generalization of that of the 2-sphere, it is not widely known. Therefore, we here collect the necessary information on representation theory and basic properties of Lie superalgebras osp(2, 1) and osp(2, 2) and their corresponding supergroups OSp(2, 1) and OSp(2, 2) from sections 9.1 to 9.3: they underlie the construction of S (2,2) and con(2,2) sequently that of SF as discussed in these sections. In section 9.4, the construction of generalized coherent states is extended to the supergroup OSp(2, 1). In section 9.5, we outline the SUSY action of Grosse et al. [7] on S (2,2) . It is a quadratic action in scalar and spinor fields. It is the simplest SUSY action one can formulate and is closest to the quadratic scalar field action on S 2 . We then discuss its fuzzy version. The latter has exact SUSY. Following three sections discuss the construction and differential geo(2,2) metric properties of an associative ∗-product of functions on SF and on (2,2) “sections of bundles” on SF . We conclude the chapter by a brief discussion on construction of non99

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linear sigma models on SF . Our discussion in this chapter follows and expands upon [9]. 9.1

osp(2, 1) and osp(2, 2) Superalgebras and their Representations

Here we review some of the basic features regarding the Lie superalgebras osp(2, 1) and osp(2, 2). For detailed discussions, the reader is refered to the references [108–112]. The Lie superalgebras osp(2, 1) and osp(2, 2) can be defined in terms of 3 × 3 matrices acting on C3 . The vector space C3 is graded: it is to be regarded as C2 ⊕ C1 where C2 is the even- and C1 is the odd-subspace. As C3 is so graded, it is denoted by C(2, 1) while linear operators on C(2, 1) are denoted by M at(2, 1). (C(2, 1) is to be distinguished from the superspace C (2,1) which will appear in section (9.3). By convention, the above C2 and C1 are embedded in C3 as follows: C2 = {(ξ1 , ξ2 , 0) : ξi ∈ C} ⊂ C(2,1) ,

C1 = {(0, 0, η) : η ∈ C} ⊂ C(2,1) .

(9.1)

The grade of C2 is 0 (mod 2) and that of C1 is 1 (mod 2). The grading of C(2, 1) induces a grading of M at(2, 1). A linear operator L ∈ M at(2, 1) has grade |L| = 0 (mod 2) or is “even” if it does not change the grade of underlying vectors of definite grade. Such an L is block-diagonal:   `1 `2 0 (9.2) L =  `3 `4 0  `i , ` ∈ C if |L| = 0 (mod 2) . 0 0 `

If L instead changes the grade of an underlying vector of definite grade by 1 (mod 2) unit, its grade is |L| = 1 (mod 2) or it is “odd”. Such an L is off-diagonal:   0 0 s1 (9.3) L =  0 0 s2  si , ti ∈ C if |L| = 1 (mod 2) . t1 t2 0

A generic element of C(2, 1) and M at(2, 1) will be a sum of elements of both grades and will have no definite grade. If M, N ∈ M at(2, 1) have definite grades |M |, |N | their graded Lie bracket [M, N } is defined by [M , N } = M N − (−1)|M ||N | N M .

(9.4)

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1 2

The even part of osp(2, 1) is the Lie algebra su(2) for which C2 has spin and C1 has spin 0. su(2) has the usual basis 1 σi 0 ( 21 ) , σi = Pauli matrices . (9.5) Λi = 2 0 0

The superscript 12 here denotes this representation: irreducible representations of osp(2, 1) are labelled by the highest angular momentum. (1)

osp(2, 1) has two more generators Λα2 (α = 4, 5) in its basis:     0 0 −1 00 0 1 1 ( 12 ) ( 21 ) 0 0 0  , Λ5 =  0 0 −1  . Λ4 = 2 2 0 −1 0 10 0

(9.6)

The full osp(2, 1) superalgebra is defined by the graded commutators (1)

(1)

(1)

(1)

(1)

[Λi 2 , Λj 2 ] = iijk Λk2 , [Λi 2 , Λα2 ] =

1 (1) (σi )βα Λβ2 , 2

1 (1) (Cσi )αβ Λi 2 , (9.7) 2 where Cαβ = −Cβα is the Levi-Civita symbol with C45 = 1. (Here the rows and columns of σi and C are being labeled by 4, 5). The abstract osp(2, 1) Lie superalgebra has basis Λi , Λα (i = 1, 2, 3 , α = 4, 5) with graded commutators obtained from (9.7) by dropping the superscript 12 : (1)

(1)

{Λα2 , Λβ2 } =

[Λi , Λj ] = iijk Λk ,

[Λi , Λα ] =

1 (σi )βα Λβ , 2

{Λα , Λβ } =

1 (Cσi )αβ Λi . 2 (9.8)

Thus Λα transforms like an su(2) spinor. The Lie algebra su(2) is isomorphic to the Lie algebra osp(2) of the ortho-symplectic group OSp(2). The above graded Lie algebra has in addition one spinor in its basis. For this reason, it is denoted by osp(2, 1). In customary Lie algebra theory, compactness of the underlying group is reflected in the adjointness properties of its Lie algebra elements. Thus these Lie algebras allow a star ∗ or adjoint operation † and their elements are invariant under † (in the convention of physicists) if the underlying group is compact. As † complex conjugates complex numbers, the Lie algebras of compact Lie groups are real as vector spaces: they are real Lie algebras. In graded Lie algebras, the operation † is replaced by the grade adjoint (or grade star) operation ‡. Its relation to the properties of the underlying supergroup will be indicated later. The properties and definition of ‡ are as follows.

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First, we note that the grade adjoint of an even (odd) element is even (odd). Next, one has (A‡ )‡ = (−1)|A| A for an even or odd (that is homogeneous) element A of degree |A| (mod 2), or equally well, integer (mod 2). (So, depending on |A|, |A| itself can be taken to be 0 or 1.) Thus, it is the usual † on the even part, while on an odd element A, it squares to −1. ‡ Further (AB)‡ = (−1)|A||B| B ‡ A‡ so that, [A, B} = (−1)|A||B| [B ‡ , A‡ } for homogeneous elements A, B. Henceforth, we will denote the degree of a (which may be a Lie superalgebra element, a linear operator or an index) by |a|(mod 2), |a| denoting any integer in its equivalence class h|a| + 2n : n ∈ Zi. The basis elements of the osp(2, 1) (and osp(2, 2), see later) graded Lie algebras are taken to fulfill certain “reality” properties implemented by ‡. For the generators of osp(2, 1), these are given by X Λ‡i = Λ†i = Λi , Λ‡α = − Cαβ Λβ α = 4, 5 . (9.9) β=4,5

Let V be a graded vector space V so that V = V0 ⊕ V1 where V0 and V1 are even and odd subspaces [112]. In a (grade star) representation of a graded Lie algebra on V , V0 and V1 are invariant under the even elements of the graded Lie algebra while its odd elements map one to the other. This representation becomes a grade-∗ representation if the following is also true. Let us assume that V is endowed with the inner product hu|vi for all u, v ∈ V . Now if L is a linear operator acting on V , then the grade adjoint of L is defined by hL‡ u|vi = (−1)|u| |L| hu|L vi

(9.10)

for homogeneous elements u, L. In a basis adapted to the above decomposition of V , a generic L has the matrix representation 0 α2 α1 α2 α1 0 , M1 = ML = = M 0 + M 1 , M0 = 0 α4 α3 0 α3 α4 (9.11) where M0 and M1 are the even and odd parts of ML . The formula for ‡ is then ! ML‡ =

α†1 −α†3 α†2 α†4

,

(9.12)

α†i being matrix adjoint of αi . Then in a grade-∗ representation, the image of L‡ is ML‡ . We note that the supertrace str of ML is by definition strML = T rα1 − T rα4 .

(9.13)

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The irreducible representations of osp(2, 1) are characterized by an integer or half-integer non-negative quantum number Josp(2,1) called superspin. From the point of view of the irreducible representations of su(2), the superspin Josp(2,1) representation has the decomposition 1 , (9.14) Josp(2,1) = Jsu(2) ⊕ J − 2 su(2) where Jsu(2) is the su(2) representation for angular momentum J. All these are grade-∗ representations : the relations (9.9) are preserved in the representation. The fundamental and adjoint representations of osp(2, 1) correspond to Josp(2,1) = 12 and Josp(2,1) = 1 respectively, being 3 and 5 dimensional. The quadratic Casimir operator is osp(2,1)

K2

= Λi Λi + Cαβ Λα Λβ .

(9.15)

1 2 ).

It has eigenvalues J(J + It is also worthwhile to make the following technical remark. The superspin multiplets in Josp(2,1) representation may be denoted by |Josp(2,1) , Jsu(2) , J3 i, and |Josp(2,1) , J − 21 su(2) , J3 i. One of the multiplets generates the even and the other generates the odd subspace of the representation space. Although, this grade can be arbitrarily assigned, the choice consistent with the reality conditions we have chosen in (9.9) and the definition of grade adjoint operation in (9.10) fixes themultiplet |Josp(2,1) , Jsu(2) , J3 i to be of even degree while |Josp(2,1) , J − 21 su(2) , J3 i is odd. The osp(2, 2) superalgebra can be defined by introducing an even generator Λ8 commuting with the Λi and odd generators Λα with α = 6, 7 in addition to the already existing ones for osp(2, 1). The graded commutation relations for osp(2, 2) are then 1 [Λi , Λj ] = iijk Λk , [Λi , Λα ] = (˜ σi )βα Λβ , [Λi , Λ8 ] = 0 , 2 1 1 ˜ [Λ8 , Λα ] = ε˜αβ Λβ , {Λα , Λβ } = (C˜ σ˜i )αβ Λi + (˜ εC)αβ Λ8 , (9.16) 2 4 where i, j = 1, 2, 3 and α, β = 4, 5, 6, 7. In above we have used the matrices 0 I2×2 C 0 σi 0 . (9.17) , ε˜ = , C˜ = σ ˜i = I2×2 0 0 −C 0 σi

Their matrix elements are indexed by 4, . . . , 7. In addition to (9.9), the new generators satisfy the “reality” conditions X Λ‡α = − C˜αβ Λβ , α = 6, 7 , Λ‡8 = Λ†8 = Λ8 . (9.18) β=6,7

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So we can write the osp(2, 2) reality conditions for all α as Λ‡α = −C˜αβ Λβ . Irreducible representations of osp(2, 2) fall into two categories, namely the typical and non-typical ones. Both are grade ∗-representations which preserve the reality conditions (9.9) and (9.18). Typical ones are reducible with respect to the osp(2, 1) superalgebra (except for the trivial representation) whereas non-typical ones are irreducible. Typical representations are labeled by an integer or half integer non-negative number Josp(2,2) , called osp(2, 2) superspin and the maximum eigenvalue k of Λ8 in that IRR. They can be denoted by (Josp(2,2) , k). Independently of k, these have the osp(2, 1) content Josp(2,1) ⊕ (J − 12 )osp(2,1) for Josp(2,2) ≥ 12 while (0)osp(2,2) contains just (0)osp(2,1) . Hence on restriction to su(2), Josp(2,2) , k → ( Jsu(2) ⊕ J − 21 su(2) ⊕ J − 21 su(2) ⊕ (J − 1)su(2) , Josp(2,2) ≥ 1 ; ( 12 )su(2) + (0)su(2) + (0)su(2) , Josp(2,2) = 21 . (9.19) osp(2, 2) has the quadratic Casimir operator osp(2,2)

K2

= Λi Λi + C˜αβ Λα Λβ −  X osp(2,1) = K2 −

α,β=6,7

1 2 Λ 4 8



1 −C˜αβ Λα Λβ + Λ28  . 4

(9.20)

It has also a cubic Casimir operator [108, 113]. We do not show it here , as we will not use it. osp(2,2) Note that since all the generators of osp(2, 1) commute with K2 osp(2,1) and K2 , they also commute with X 1 osp(2,1) osp(2,2) K2 − K2 (9.21) =− C˜αβ Λα Λβ + Λ28 . 4 α,β=6,7

osp(2,2)

The osp(2, 2) Casimir K2

vanishes on non-typical representations: osp(2,2) K2 = 0. (9.22) nontypical

The substitutions Λi → Λ i ,

Λα → Λ α ,

Λ8 → −Λ8 (9.23) define an automorphism of osp(2, 2). This automorphism changes the irreducible representation (Josp(2,2) , k) into an inequivalent one (Josp(2,2) , −k) α = 4, 5;

Λα → −Λα ,

α = 6, 7;

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(except for the trivial representation with J = 0), while preserving the reality conditions given in (9.9) and (9.18) [109]. In the nontypical case, we discriminate between these two representations associated with Josp(2,1) as follows: For J > 0, Josp(2,2)+ will denote the representation in which the eigenvalue of the representative of Λ8 on vectors with angular momentum J is positive and Josp(2,2)− will denote its partner where this eigenvalue is negative. (This eigenvalue is zero only in the trivial representation with J = 0.) Here while considering nontypical IRR’s we concentrate on Josp(2,2)+ . The results for Josp(2,2)− are similar and will be occasionally indicated. Another important result in this regard is that every non-typical representation Josp(2,2)± of osp(2, 2), is at the same time an irreducible representation of osp(2, 1) with superspin Josp(2,1) . For this reason the osp(2, 2) generators Λ6,7,8 can be nonlinearly realized in terms of the osp(2, 1) generators. Repercussions of this result will be seen later on. Below we list some of the well-known results and standard notations that are used throughout the text. The fundamental representation of osp(2, 2) is non-typical and we concentrate on the one given by Josp(2,2)+ = (1)

( 21 )osp(2,2)+ . It is generated by the (3 × 3) supertraceless matrices Λa2 satisfying the “reality” conditions of (9.9) and (9.18): 1 σi 0 1 1 0 ξ 0 η (1) (1) ( 12 ) Λi 2 = , Λ4 2 = , Λ , = 5 2 0 0 2 ηT 0 2 −ξ T 0 1 1 I2×2 0 0 −ξ 0 −η ( 21 ) ( 12 ) ( 12 ) , Λ6 = , Λ7 = , Λ8 = 0 2 2 ηT 0 2 −ξ T 0 (9.24) where ξ=

−1 0

and η =

0 −1

.

(9.25)

These generators satisfy (1)

(1)

Λa2 Λb 2 = Sab 1 +

(1) 1 dabc + ifabc Λc 2 2

(a, b, c = 1, 2, . . . 8) .

It is possible to write 1 (1) (1) (1) ( ) (1) Sab = str Λa2 Λb 2 , fabc = str − i[Λa2 , Λb 2 }Λc 2 , (1) (1) (1) dabc = str {Λa2 , Λb 2 ]Λc 2 .

(9.26)

(9.27)

Here a = i = 1, 2, 3, and a = 8 label the even generators whereas a = α = 4, 5, 6, 7 label the odd generators.

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In above [A, B}, {A, B] denote the graded commutator and the graded anticommutator respectively. The former is already defined, while the latter is given by {A, B] = AB + (−1)|A||B| BA for homogeneous elements A and B. Sab defines the invariant metric of the Lie superalgebra osp(2, 2). In their block diagonal form, S and its inverse read   1  2I 2 I   , S −1 =  S= . (9.28) 2 C˜ − 21 C˜ −2 8×8 − 21 8×8

The explicit values of the structure constants fabc can be read from (9.17), since [Λa , Λb } = ifabc Λc . Those of dabc are as follows∗ : 3 1 dij8 = − δij , dαβ8 = C˜αβ , dα8β = 3δαβ , di8j = 2δij , 2 4 1 ˜ 1 dαβi = − (˜ εC σ ˜i )αβ , diαβ = − (˜ εσ ˜i )βα , d888 = 6 . (9.29) 2 2 We close this subsection with a final remark. Discussion in the subsequent sections will involve the use of linear operators acting on the adjoint b acting on Λa acrepresentation of osp(2, 2). These are linear operators Q b a = Λb Qba , Q being the matrix representation of Q. b They cording to QΛ are graded because Λa ’s are, and hence the linear operators on the adjoint representation are graded. The degree (or grade) of a matrix Q with only the nonzero entry Qab is (|Λa | + |Λb |) (mod 2) ≡ (|a| + |b|) (mod 2). The b now follows from the sesquilinear form grade star operation on Q −1 α = α a ΛA , β = β b Λb = α ¯a Sab βb , α a , βb ∈ C (9.30) and is given by

b

b‡ α, β) = (−1)|α| |Q| (α, Qβ) b . (Q 9.2

(9.31)

Passage to Supergroups

We recollect here the passage from these superalgebras to their corresponding supergroups [112, 114]. Let ξ ≡ (ξ1 , · · · , ξ8 ) be the elements of the superspace R(4,4) . Here ξa for a = i = 1, 2, 3 and a = 8 label the even and for a = α = 4, 5, 6, 7 label the odd elements of a real Grassmann algebra G. ξa ’s satisfy the graded commutation relations mutually and with the algebra elements: ∗ The

[ξa , ξb } = 0 ,

[ξa , Λb } = 0 .

(9.32)

tensor dabc given explicitly in (9.29) for Josp(2,2)+ becomes −dabc for Josp(2,2)− .

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We assume that ξi‡ = ξi , ξ8‡ = ξ8 and ξα‡ = −C˜αβ ξβ . Then ξa Λa is grade-∗ even: (ξa Λa )‡ = ξa Λa .

(9.33)

An element of OSp(2, 2) is given by g = eiξa Λa , while for a restricted to a ≤ 5, g gives an element of OSp(2, 1). (9.33) corresponds to the usual hermiticity property of Lie algebras which yields unitary representations of the group. 9.3 9.3.1

On the Superspaces The Superspace C 2,1 and the Noncommutative CF2,1

C 2,1 is the (2 , 1)-dimensional superspace specified by two even and one odd element of a complex Grassmann algebra G. Let G0 and G1 denote the even and odd subspaces of G. We write C 2,1 ≡ {ψ ≡ (z1 , z2 , θ)} , where z1 , z2 ∈ G0 and θ ∈ G1 satisfy

¯ ≡ θθ¯ + θθ ¯ = 0, {θ , θ}

We note that under ‡ operation

zi‡ = zi† = z¯i ,

θθ = θ¯θ¯ = 0 .

θ‡ = θ¯ ,

θ¯‡ = −θ .

(9.34)

(9.35)

(9.36)

The noncommutative C 2,1 , denoted by CF2,1 hereafter, is obtained by replacing ψ ∈ C 2,1 , by Ψ ≡ (a1 , a2 , b), where the operators ai and b obey the commutation and anticommutation relations [ai aj ] = [a†i a†j ] = 0 ,

[ai a†j ] = δij ,

{b , b} = {b† , b† } = 0 , a‡i

= Under † they fulfill Using the notation

ai , (a†i )‡

[ai , b] = [ai , b† ] = 0

{b , b† } = 1 .

(9.37)

= ai , b‡ = b† , (b† )‡ = −b.

(Ψ1 , Ψ2 , Ψ0 ) ≡ (a1 , a2 , b) ,

(9.38)

the commutation relations can be more compactly expressed as [Ψµ , Ψν } = [Ψ†µ , Ψ†ν } = 0 ,

[Ψµ , Ψ†ν } = δµν ,

(9.39)

where µ = 1, 2, 0. Ψµ , Ψ†µ and the identity operator 1 span the graded Heisenberg-Weyl algebra, with 1 generating its center.

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9.3.2

The Supersphere S (3,2) and the Noncommutative S (3,2)

¯ we define ψ 0 = Dividing ψ by its modulus |ψ| ≡ |z1 |2 + |z2 |2 + θθ, 0

2,1

C \ {0} with |ψ | = 1. The (3, 2) dimensional supersphere S be defined as

ψ S (3,2) ≡ ψ 0 = ∈ C2,1 \ {0} . |ψ|

(3,2)

b= N

†

+ b b is the number operator. We have ψµ0

−→

ψµ0†

−→

1 1 Ψµ , Sµ := Ψµ p = p b b N +1 N 1 1 Sµ† := p Ψ†µ = Ψ†µ p , b b N +1 N

∈

can then

(9.40)

Obviously S (3,2) has the 3-sphere S 3 as its even part. The noncommutative S (3,2) is obtained by replacing ψ 0 by Ψ √1 a†i ai

ψ |ψ|

b N

where

(9.41)

b 6= 0. Furthermore, we have that [Sµ , Sν } = [Sµ† , Sν† } = 0, while where N after a small calculation, we get 1 (9.42) δµν − (−1)|Sµ ||Sν | Sν† Sµ . [Sµ , Sν† } = b +1 N b approaches to infinity, we recover S (3,2) We note that as the eigenvalue of N back. Noncommutative S (3,2) suffers from the same problem as noncommutative S 3 does: Sµ an Sµ† act on an infinite-dimensional Hilbert space so that we do not obtain finite-dimensional models for noncommutative S (3,2) either. Nevertheless, the structure of the non-commutative S (3,2) described (2,2) above is quite useful in the construction of SF as well as for obtaining (2,2) ∗-products on the “sections of bundles” over SF as we will discuss later in this chapter. 9.3.3

The Commutative Supersphere S (2,2)

There is a supersymmetric generalization of the Hopf fibration. In this subsection we construct this (super)-Hopf fibration through studying the actions of OSp(2, 1) and OSp(2, 2) on S (3,2) . We also establish that the supersphere S (2,2) is the adjoint orbit of OSp(2, 1), while it is a closely related (but not the adjoint) orbit of OSp(2, 2). We elaborate on the subtle

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features of the latter, which are important for future developments in this chapter. We first note that the group manifold of OSp(2, 1) is nothing but S (3,2) . Also note that |ψ|2 is preserved under the group action ψ −→ gψ for g ∈ OSp(2, 1). Let us then consider the following map Π from the functions on the (3, 2)-dimensional supersphere S (3,2) to functions on S (2,2) : Π

:

ψ 0 −→

1 1 ¯ (a2 ) ψ . ¯ := ψ¯0 Λa( 2 ) ψ 0 = 2 ψΛ wa (ψ , ψ) 2 |ψ|

(9.43)

The fibres in this map are U (1) as the overall phase in ψ → ψeiγ cancels out while no other degree of freedom is lost on r.h.s. Quotienting S (3,2) ≡ OSp(2, 1) by the U (1), fibres, we get the (2, 2) dimensional base space † n o S (2,2) := S (3,2) / U (1) ≡ w(ψ) = w1 (ψ), · · · , w5 (ψ) . (9.44)

Π is thus the projection map of the “super-Hopf fibration” over S (2,2) [117, 118, 78], and S (2,2) can be thought as the supersphere generalizing S2. We now characterize S (2,2) as an adjoint orbit of OSp(2, 1). First observe that w(ψ) is a (super)-vector in the adjoint representation of OSp(2, 1). Under the action w → gw,

(gw)(ψ) = w(g −1 ψ),

g ∈ OSp(2, 1) ,

(9.45)

it transforms by the adjoint representation g → Ad g : wa (g −1 ψ) = wb (ψ) (Ad g)ba .

(9.46)

The generators of osp(2, 1) in the adjoint representation are adΛa where (ad Λa )cb = ifabc .

(9.47)

From this and the infinitesimal variations δw(ψ) = εa ad Λa w(ψ) of w(ψ) under the adjoint action, where εi ’s are even and εα ’s are odd Grassmann variables, we can verify that δ(wi (ψ)2 + Cαβ wα (ψ)wβ (ψ)) = 0 .

(9.48)

Hence, S (2,2) is an OSp(2, 1) orbit with the invariant 1 (wa (S −1 )ab wb ) = wi (ψ)2 + Cαβ wα (ψ)wβ (ψ) . 2

(9.49)

† In what follows we do not show the ψ ¯ dependence of wa to abbreviate the notation a little bit.

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The value of the invariant can of course be changed by scaling. Now the even components of wa (ψ) are real while its odd entries depend on both θ ¯ and θ: 1 1 ¯ , w5 (ψ) = 1 (−¯ ¯ . z¯σi z , w4 (ψ) = − 2 (¯ z1 θ +z2 θ) z2 θ +z1 θ) 2 |ψ| |ψ| |ψ|2 (9.50) From (9.36) and (9.50), one deduces the reality conditions

wi (ψ) =

wi (ψ)‡ = wi (ψ)

wα (ψ)‡ = −Cαβ wβ (ψ) .

(9.51)

The OSp(2, 1) orbit is preserved under this operation as can be checked directly using (9.51) in (9.49). The reality condition (9.51) reduces the degrees of freedom in wα (ψ) to two. The (3, 2) number of variables wa (ψ) are further reduced to (2, 2) on fixing the value of the invariant (9.49). As (2, 2) is the dimension of S (2,2) , there remains no further invariant in this orbit. Thus (2,2) SD = E (−) (+) η ∈ R(3,2) ηi2 + Cαβ ηα(−) ηβ = 1 , (ηi )‡ = ηi , (ηα(−) )‡ = −Cαβ ηβ ,

(9.52)

1 4

for the value of the invariant. It is important where we have chosen to note that the superspace R(3,2) in (9.52) is defined as the algebra of (−) polynomials in generators ηi and ηα satisfying the reality conditions ηi‡ = (+) ηi , η (−)‡ = −Cαβ ηβ . Thus S (2,2) is embedded in R(3,2) as described by (9.52). As OSp(2, 2) acts on ψ, that is on S (3,2) , preserving the U (1) fibres in the map S (3,2) → S (2,2) , it has an action on the latter. It is not the adjoint action, but closely related to it, as we now explain. The nature of the OSp(2, 2) action on S (2,2) has elements of subtlety. If g ∈ OSp(2, 2) and ψ ∈ S (3,2) then g ψ ∈ S (3,2) and hence w(g ψ) ∈ S (2,2) : wi (g ψ)2 + Cαβ wα (g ψ) wβ (g ψ) = 1 , wi (g ψ)‡ = wi (g ψ) ,

wα‡ (g ψ) = −Cαβ wβ (g ψ) .

(9.53)

But the expansion of wα (g ψ) for infinitesimal g contains not only the (−) (+) (+) odd Majorana spinors ηα , but also the even ones ηα , where (ηα )‡ = P (+) − β=6,7 C˜αβ ηβ (α = 6, 7). We cannot thus think of the OSp(2, 2) action as the adjoint action on the adjoint space of OSp(2, 1). The reason of course is that the Lie superalgebra osp(2, 1) is not invariant under graded commutation with the generators Λ6,7,8 of osp(2, 2).

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Now consider the generalization of the map (9.43) to the osp(2, 2) Lie algebra, 2 ¯ ( 12 ) (1) ψ 0 −→ Wa (ψ) := ψ¯0 Λa2 ψ 0 = ψΛa ψ , a = (1, . . . , 8) , (9.54) |ψ|2 where the ψ¯ dependence of Wa has been suppressed for notational brevity. Just as for OSp(2, 1), we find, Wa (g −1 ψ) = Wb (ψ)(Ad g)ba , a, b = 1, . . . , 8 , g ∈ OSp(2, 2) . (9.55) Thus this extended vector W(ψ) = (W1 (ψ) , W2 (ψ), . . . , W8 (ψ)) transforms as an adjoint (super)-vector of osp(2, 2) under OSp(2, 2) action. The formula given in (9.50) extends to this case when index a there also takes the values (6, 7, 8). Explicitly we have 1 ¯ , W7 (ψ) = 1 (¯ ¯ , (¯ z1 θ − z2 θ) z2 θ + z1 θ) W6 (ψ) = |ψ|2 |ψ|2 1 ¯ = 2 2 − 1 z¯i zi . zi zi + 2θθ) (9.56) W8 (ψ) = 2 2 (¯ 2 |ψ| |ψ| The reality conditions for W6 (ψ) , W7 (ψ) , W8 (ψ) are X W8 (ψ)‡ = W8 (ψ) , Wα (ψ)‡ = − C˜αβ Wβ (ψ) , α = 6, 7 , (9.57) β=6,7

showing that the new spinor Wα (ψ), (α = 6, 7) is an even Majorana spinor as previous remarks suggested. As W(ψ) transforms as an adjoint vector under OSp(2, 2), the OSp(2, 2) Casimir function is a constant on this orbit: 1 1 (Wa (S −1 )ab Wb ) = Wi2 (ψ) + C˜αβ Wα (ψ)Wβ (ψ) − W82 (ψ) = constant . 2 4 (9.58) But we saw that the sum of the first term, and the second term with α , β = 4, 5 only, is invariant under OSp(2, 1). Hence so are the remaining terms: X 1 C˜αβ Wα (ψ)Wβ (ψ) − W8 (ψ)2 = constant . (9.59) 4 α,β=6,7

Its value is −1 as can be calculated by setting ψ = (1, 0, 0). In fact, since the OSp(2, 1) orbit has the dimension of S (3,2) /U (1) and Wa (ψ) = Wa (ψ eiγ ) are functions of this orbit, we can completely express the latter in terms of w(ψ). We find‡ σ · w(ψ) Wα (ψ) = −wβ r β ,α−2 W8 (ψ) = ‡W

6,7,8

2 2 (r + Cαβ wα wβ ) , r

become −W6,7,8 for Josp(2,2)− .

r 2 = w i wi .

(9.60)

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9.3.4

(2,2)

The Fuzzy Supersphere SF

(2,2)

We are now ready to construct the fuzzy supersphere SF replacing the coordinates wa of S (2,2) by w ˆa :

. We do so by

1 1 1 (1) (1) (1) wa −→ w ˆa = S † Λa2 S = p Ψ† Λa2 Ψ p = Ψ† Λa2 Ψ . b b b N N N

b: Obviously, we have w ˆa commuting with the number operator N b = 0. [w ˆa , N]

(9.61)

(9.62)

˜ n of the Fock space of Consequently, we can confine w ˆa to the subspace H dimension (2n + 1) spanned by the kets (a† )n1 (a†2 )n2 † n3 √ (b ) |0i , |n1 , n2 , n3 i ≡ √1 n1 ! n2 !

n 1 + n2 + n3 = n ,

(9.63)

˜ n splits into where n3 takes on the values 0 and 1 only. The Hilbert space H e o ˜ ˜ the even subspace Hn and the odd subspace Hn of dimensions n + 1 and n, respectively. ˜ n generate the algebra of Linear operators, and hence wa , acting on H supermatrices M at(n + 1, n) of dimension (2n + 1)2 which is customarily (2,2) identified with the fuzzy supersphere. Similar to the fuzzy sphere, SF is also finite-dimensional: M at(n + 1, n) is an inner product space with the inner product (m1 , m2 ) = Str m‡1 m2 ,

mi ∈ M at(n + 1, n) ,

(9.64)

where the identity matrix is already normalized to have the unit norm in this form. In order to be more explicit, we first note that the osp(2, 1) (and hence osp(2, 2)) Lie superalgebras can be realized as a supersymmetric generalization of the Schwinger construction by (1)

λa = Ψ† (Λa2 )Ψ ,

[λa , λb } = ifabc λc .

(9.65)

The vector states in (9.63) for n = 1 give the superspin J = 12 representation of osp(2, 1), while for generic n they correspond to the n-fold graded symmetric tensor product of J = 21 superspins that span the superspin ˜ n , we J = n2 representation of osp(2, 1). Therefore, on the Hilbert space H have ˜n = n n + 1 H ˜n . λi λi + Cαβ λα λβ H (9.66) 2 2 2

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Using the relation ˜n , ˜ n = 2 λa H w ˆa H n

(9.67)

we obtain 2 ˜n ifabc w ˆc H (9.68) n ˜n . ˜n = 1 + 1 H (9.69) w ˆi w ˆi + Cαβ w ˆα w ˆβ H n r (2,2) 1 + n1 of SF goes to 1 as n tends to infinity. The graded The radius ˜n = [w ˆa , w ˆb }H

commutative limit is recovered when J → ∞: in that limit, [w ˆa , w ˆb } → 0. The Schwinger construction above naturally extends to the generators of osp(2, 2) as well. In general we can write ca := 2 λa , a = (1 , · · · , 8) . (9.70) W n (9.70) generate the osp(2, 2) algebra where ca → Wa as n → ∞. W (9.71) c6,7,8 can be realized in terms of the osp(2, 1) generators. The generators W (2,2) This fact becomes important for field theories on both S (2,2) and SF ; Even though, these field theories have OSp(2, 1) invariance, the osp(2, 2) structure is needed to treat them as we will see later in the chapter. (2,2) The observables of SF are defined as the linear operators α ∈ M at(n+ 1, n) acting on M at(n + 1, n). They have the graded right- and left- action on the Hilbert space M at(n + 1, n) given by αL m = αm ,

αR m = (−1)|α||m|mα ,

They satisfy

∀ m ∈ M at(n + 1, n) .

(9.72)

(αβ)R = (−1)|α||β| βα ,

(9.73)

∀ α , β ∈ M at(n + 1, n) .

(9.74)

(αβ)L = αL β L , and commute in the graded sense: [αL , β R } = 0 ,

In particular osp(2, 1) and osp(2, 2) act on M at(n+1, n) by the (super)adjoint action: R (9.75) ad Λa m = ΛL a − Λa m = [Λa , m} , which is a graded derivation on the algebra M at(n + 1, n). Before closing this section we note that left- and right-action of Ψµ and † Ψµ can also be defined on M at(n + 1, n). They shift the dimension of the Hilbert space by 1 and will naturally arise in discussions of “fuzzy sections of bundles” in section 9.7.

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More on Coherent States

In this section we construct the OSp(2, 1) supercoherent states (SCS) by projecting them from the coherent states associated to C2,1 [9]. In the literature the construction of OSp(2, 1) coherent states has been discussed [114, 115]. Here we explicitly show that our SCS is equivalent to the one obtained using the Perelomov’s construction of the generalized coherent states, considered in chapter 3. We start our discussion by introducing the coherent states including the bosonic and fermionic degrees of freedom [54, 53]: 2

†

|ψi ≡ |z, θi = e−1/2 |ψ| eaα zα +b

†

θ

|0i .

(9.76)

We can see from section 9.3 that the labels ψ of the states |ψi are in one to one correspondence with points of the superspace C (2,1) . We recall that ¯ Hence |ψi’s are normalized to 1 as written. |ψ|2 ≡ |z1 |2 + |z2 |2 + θθ. ˜ n of the Fock space can be The projection operator to the subspace H written as X 1 (a†1 )n1 (a†2 )n2 (b† )n3 |0ih0|(b)n3 (a2 )n2 (a1 )n1 , Pn = n ! n ! 1 2 n=n +n +n 1

2

3

(9.77) where n3 = 0 or 1. Clearly Pn2 = Pn , Pn† = Pn . Projecting |ψi with Pn and renormalizing the result by the factor (hψ|Pn |ψi)−1/2 , we get (Ψ†µ ψµ0 )n 1 (a†α zα + b† θ)n √ |0i = |0i . |ψ 0 , ni = √ (|ψ|)n n! n!

(9.78)

This is the supercoherent state associated to OSp(2, 1). It is normalized to unity : hψ 0 , n|ψ 0 , ni = 1 .

(9.79)

We first establish the relation of (9.78) to Perelomov’s construction of coherent states. To this end consider the following highest weight state in b = 1: the Josp(2,1) = 21 representation of osp(2, 1) for which N 1 1 1 |Josp(2,1) Jsu(2) , J3 i = | , , i . 2 2 2

(9.80)

This is also the highest weight state in the associated non-typical representation Josp(2,2)+ = ( 21 )osp(2,2)+ of osp(2, 2). Consider now the action of the OSp(2, 1) on (9.80). This can be realized by taking g ∈ OSp(2, 1) and

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U(g) as the corresponding element in the 3 × 3 fundamental representation. Thus let 1 1 1 |gi = U(g)| , , i , (9.81) 2 2 2 where |gi is the super-analogue of the Perelomov coherent state [54]. We can write 1 1 1 (9.82) | , , i = Ψ†1 |0i 2 2 2 where Ψ† = Ψ†1 , Ψ†2 , Ψ†0 ≡ a†1 , a†2 , b† as given in (9.38). In the basis

spanned by the {Ψ†µ |0i}, (µ = 1, 2, 0) the matrix of U(g) can be expressed as [114]  0  z1 −¯ z20 −θ0 X |zi0 |2 + θ¯0 θ0 = 1 . (9.83) D(g) =  z20 z¯10 −θ¯0  , i χ −χ ¯ λ Then |gi = D(g) 1µ Ψ†µ |0i = a†α zα0 + b† θ0 |0i = Ψ†µ ψµ0 |0i . (9.84) Clearly (9.84) is exactly equal to |ψ 0 , 1i in (9.78). For the case of general n, we start from the highest weight state | n2 , n2 , n2 i in the n-fold graded symmetric tensor product ⊗nG of the Josp(2,1) = 21 n representation and the corresponding representative U ⊗G (g) of g: 1 1 1 1 1 1 n n n | , , i := | , , i ⊗G · · · · · · ⊗G | , , i , 2 2 2 2 2 2 2 2 2 n U ⊗G (g) := U (g) ⊗G · · · · · · ⊗G U (g) . (9.85) Note that, since U (g) is an element of OSp(2, 1), it is even. The corresponding coherent state is n n n n n 1 1 1 1 1 1 |g; i = U ⊗G | , , i = U (g) | , , i ⊗G · · · · · · ⊗G U (g) | , , i . 2 2 2 2 2 2 2 2 2 2 (9.86) Upon using (9.84), this becomes equal to (9.78) as we intended to show. The coherent state in (9.76) can be written as a sum of its even and odd components by expanding it in powers of b† : 2 † |ψi ≡ |z, θi = e−1/2 |ψ| eaα zα |0 , 0i − θ |0 , 1i = |z , 0i − θ |z , 1i . (9.87) We proved in chapter 2 that the diagonal matrix elements of an operator K in the coherent states |zi completely determine K. That proof can be adapted to |ψi as can be infered from (9.87). It can next be adapted to |ψ 0 , ni for operators leaving the the subspace N = n invariant. The line of reasoning is similar to the one used for SU (2) coherent states in chapter 2.

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The Action on the Supersphere S (2,2)

The simplest Osp(2, 1)-invariant Lagrangian density L can be written as Φ‡ V Φ, where Φ is the scalar superfield and V an appropriate differential operator. We focus on L in what follows. The superfield Φ is a function on S (2,2) , that is , it is a function of wa , (a = 1, 2, · · · , 5) fulfilling the constraint in (9.51). For functional integrals, what is important is not L, but the action S. Thus we need a method to integrate L over S (2,2) maintaining SUSY. We also need a choice of V to find S. The appropriate choice is not obvious, and was discovered by Fronsdal [119]. It was adapted to Osp(2, 1) by Grosse et al. [7]. We now describe these two aspects of S and indicate also the calculation of S. i. Integration on S (2,2) Let K be a scalar superfield on S (2,2) . It is a function of wi and wα . We can write it as K = k0 + Cαβ kα wβ + k1 Cαβ wα wβ

(9.88)

0

where k0 and k1 are even, kα (α = 4, 5) are odd and k s do not depend on wα ’s, but can depend on wi ’s. There is no need to include w6,7 in (9.88) as they are nonlinearly related to w4,5 . The integral of K over S (2,2) (of radius R) can be defined as Z I(K) = dΩ r2 dr dw4 dw5 δ(r2 + Cαβ wα wβ − R2 ) K (9.89)

where R > 0 and dΩ = d cos(θ)dψ is the volume form on S 2 . In the volume form in the integrand of I(K), we do not constrain wi , wα to fulfil wi2 + Cαβ wα wβ = R2 . This constraint is taken care of by the δfunction. The grade-adjoint representation of osp(2, 1) is 5-dimensional. It acts on R3,2 := R3 ⊕ R2 with an even subspace R3 (spanned by wi ) and an odd subspace R2 (spanned by wα ). Integration in (9.89) uses the OSp(2, 1)invariant volume form on R3,2 and the OSp(2, 1)-invariant δ-function to restrict the integral to S (2,2) . Thus I(K) is invariant under the action of SUSY on K. I(K) is in fact OSp(2, 2) invariant. That is because OSp(2, 2) leaves the argument of the δ-function invariant as we already saw. The volume

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form as well is invariant because of the nonlinear realization of w6,7,8 as is easily checked. We can write d δ(r2 + Cαβ wα wβ − R2 ) = δ(r2 − R2 ) + 2w4 w5 2 δ(r2 − R2 ) dr 1 d 1 δ(r − R) + w4 w5 δ(r − R) ,(9.90) = 2R 2Rr dr d δ(r + R) as they where we have dropped terms involving δ(r + R) and dr do not contribute to the dr-integral from 0 to ∞. Thus using also Z dw4 dw5 w4 w5 = −1 , (9.91) we get

I(K) = This is a basic formula.

Z

dΩ

d (rk0 ) − Rk1 dr

.

(9.92)

r=R

ii. The OSp(2, 1)-invariant operator V The first guess would be the Casimir K2 of OSp(2, 1), written in terms of differential and superdifferential operators [119, 7]. But this choice is not satisfactory. The simplest OSp(2, 1)-invariant action is that of the WessZumino model [120] and contains just the standard quadratic (“kinetic energy”) terms of the scalar and spinor fields. But K2 gives a different action with nonstandard spinor field terms [119, 7]. But any OSp(2, 1) representation is also a nontypical representation of OSp(2, 2) and its OSp(2, 2) Casimir K20 is certainly OSp(2, 1) invariant. Thus so is 1 (9.93) V := K20 − K2 = Λ6 Λ7 − Λ7 Λ6 + Λ28 . 4 It happens that this V correctly reproduces the needed simple action. iii. How to calculate : A sketch SUSY calculations are typically a bit tedious. For that reason, we just sketch the details and give the final answer. We first expand the superfield Φ in the standard manner: Φ(wi , wα ) = ϕ0 (wi ) + Cαβ ψα wβ + χ(wi )Cαβ wα wβ .

(9.94)

Here (α, β = 4, 5), ϕ0 and χ are even fields (commuting with wα ) and ψα are odd fields (anticommuting with wα ).

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The aim is to calculate S = I(Φ‡ V Φ) .

(9.95)

For V we take (9.93) where Λ6,7,8 represent the OSp(2, 2) generators acting on wi , wα . Thus we need to know how they act on the constituents of Φ in (9.94). The action of Λα on wβ follows from (9.16) since wβ transform like osp(2, 2) generators: 1 1 Λα wi = wβ (˜ σi )βα , Λα wβ−2 = Cαβ w8 , α , β = 6, 7 . (9.96) 2 2 We now write w6,7 in terms of w4,5 using the relation (9.60) to find 1 Λα wi = − wγ−2 (σ · w) ˆ γβ (˜ σi )βα , 2 1 2 Λα wβ−2 = Cαβ (r2 + 2w4 w5 ) , α, β, γ = 6, 7 . (9.97) 2 r The action of of Λα on the fields of (9.94) follows from the chain rule. For example, ∂ ϕ0 (wi ) . (9.98) Λα ϕ0 (wi ) = (Λα wi ) ∂wi The ingredients for working out the action are now at hand. The calculation can be conveniently done for a real superfield: Φ‡ = Φ . Φ can be decomposed in component fields as follows: 1 Φ = ψ0 + Cαβ ψα θβ + χCαβ θαβ 2 Then with θα an odd Majorana spinor, we find that so is ψ:

(9.99)

(9.100)

θα‡ = −Cαβ θβ ,

(9.101)

ψα‡ = −Cαβ ψβ .

(9.102)

We give the answer for the action Z S(Φ) = dΩ r2 dr δ(r2 + Cαβ wα wβ − 1)ΦV Φ .

(9.103)

We have set R = 1 whereas in previous sections we had R = 12 . We have Z 1 1 1 2 0 2 S(Φ) = dΩ − (Lϕ0 ) + (χ − ϕ0 ) − (Cψ)α (Dψ)α 4 4 4 1 d ~ i. ϕ00 = ψ0 , D = −˜ σ · L + 1 , Li = i(~r × ∇) (9.104) r dr

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The Dirac operator D here is unitarily equivalent to the Dirac operator in chapter 8. (χ0 − ϕ00 ) is the auxiliary field F . Having no kinetic energy term, it can be eliminated. SUSY transformations mix all the fields. A complex superfield Φ can be decomposed into two real superfields: Φ = Φ(1) + iΦ(2) Φ=

Φ+Φ 2

‡

, Φ(2) =

Φ−Φ 2i

(9.105) ‡

(9.106)

The action for Φ is the sum of the actions for Φ(i) . We can use (9.104) to write it. No separate calculation is needed. 9.6

(2,2)

The Action on the Fuzzy Supersphere SF (2,2)

Finding the action on SF is the crucial step for regularizing supersymmetric field theories using finite-dimensional matrix models, preserving OSp(2, 1)-invariance. We have seen that S 2 and SF2 allow instanton sectors. They affect chiral symmetry and are important for physics. There are SUSY generalizations of these instantons. They are discussed in Chapter 10. 9.6.1

The Integral and Supertrace

In fuzzy physics with no SUSY, trace substitutes for SU (2)-invariant integration. The trace trM of an (n + 1) × (n + 1) matrix M is invariant under the SU (2) action M → U (g)M U (g)−1 by its angular momentum n2 representation SU (2) : g → U (g). It becomes the invariant integration in the large n -limit. In fuzzy SUSY physics, the corresponding OSp(2, 2) invariant trace is supertrace str. But (9.104) gives invariant integration in the (graded) commutative limit. We now establish that str goes over to the invariant integration as the cut-off n → ∞. A simple way to establish this is to use the supercoherent states. We have already defined them in (9.78). Here we drop the 0 on ψ and write |ψ, ni =

(a†α zα + b† θ)n √ |0i . n!

(9.107)

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This ψ is to be distinguished from those in (9.100). ˆ commuting with N = a† ai + b† b, Then as we saw, to every operator K i we can define its symbol K, a function of w 0 s, by ˆ K(w) = hψ, N |K|ψ, Ni . (9.108) ˆ can then be defined as An invariant “integral” Iˆ on K ˆ K) ˆ = I(K) . I( (9.109) With the normalization Z dcosθ ∧ dφ dΩ = 1 or dΩ = , (9.110) 4π we can show that ˆ K) ˆ = 1 strK . I( (9.111) 2 It is then clear that str becomes 2I as n → ∞. The proof is easy. First note that for the non-SUSY coherent state (a† · zˆn ) |z , ni = √ |0i , zˆ · zˆ = 1 . (9.112) n! Z ˆ z , ni = 1 T rAˆ (9.113) dΩhˆ z , n|A|ˆ n+1 if Aˆ is an operator on the subspace spanned by |ˆ z , ni for fixed n. Terms linear in b and b† have zero str. Hence we can assume that ˆ = M 0 + M 1 b† b K (9.114) where Mj are polynomials in a†i aj . ˆ It can be easily checked Nthat str K is OSp(2, 2)-invariant as well. In the OSp(2, 1) IRR 2 osp(2,1) , the even subspace of its carrier space

has angular momentum N2 and the odd subspace has angular momentum N −1 2 . Hence ˆ = trN +1 M0 − trN M0 − trN M1 strK (9.115) where trm indicates trace over an m-dimensional space. ˆ K), ˆ we note that As for I( √ |ψ, N i = |z, N i + Nb† θ|z, N − 1i . (9.116) Hence ¯ , N − 1|M0 |z, N − 1i K(w) = hz , N |M0 |z , N i + N θθhz ¯ , N − 1|M1 |z , N − 1i . + N θθhz (9.117) ¯ = w4 w5 . So on using (9.92), we get But by (9.50), θθ Z n 1 I(K) = − dΩ N hz , N − 1|M0 |z , N − 1i + N hz , N − 1|M1 |z , N − 1i 2 o 1 ˆ (9.118) −(N + 1) iz , N |M0 |z , N = strK 2 as claimed.

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9.6.2

OSp(2, 1) IRR’s with Cut-Off N

The Clebsch-Gordan series for OSp(2, 1) is [J]osp(2,1) ⊗ [K]osp(2,1) = 1 ⊕ · · · ⊕ [|J − K|]osp(2,1)(9.119) . [J + K]osp(2,1) ⊕ J + K − 2 osp(2,1) The series on R.H.S thus descends in steps of 12 (and not in steps of 1 as for su(2)) from J + K to |J − K|. Under the (graded) adjoint action of osp(2, 1), the linear operators in the representation space of N2+1 osp(2,1) transform as N2+1 osp(2,1) ⊗ N +1 . Hence the osp(2, 1) content of the fuzzy supersphere is 2 osp(2,1) N +1 2

N +1 = 2 osp(2,1) osp(2,1) 1 N +1 [N + 1]osp(2,1) ⊕ ⊕ N+ ⊕ · · · ⊕ [0]osp(2,1) . 2 2 osp(2,1) osp(2,1) (9.120)

⊗

We now discuss • The highest weight angular momentum states in each of these IRR’s and the realization of osp(2, 2) on these osp(2, 1) multiplets, and • The spectrum of V and the free supersymmetric scalar field action on the fuzzy supersphere. 9.6.3

The Highest Weight Invariant Action

States

and

the

osp(2, 2)-

The graded Lie algebra osp(2, 1) is of rank 1. We can diagonalize (a multiple of) one operator in osp(2, 1) in each IRR. We choose it to be Λ3 , the third component of angular momentum. Λ4 is a raising operator for Λ3 , raising its eigenvalues by 12 . The vector state annihilated by Λ4 in an IRR of osp(2, 1) is it highest weight state. Λ+ = Λ1 + iΛ2 is also a raising operator for Λ3 , raising its eigenvalue by +1. Vector states annihilated by Λ4 are the highest weight states for the su(2) IRR’s contained in an osp(2, 1) IRR. A vector state in an IRR annihilated by Λ4 is also annihilated by Λ+ .

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The matrices of the fuzzy supersphere are polynomials in a†i aj , a†i b, b† ai restricted to the subspace with N = a†i ai + b† b fixed. Supersymmetry acts on them by adjoint action. The expression for Λ4 is given in (9.65) while Λ+ = a†1 a2 .

(9.121)

It follows that for J integral, The highest weight state for [J]osp(2,1) = (a†1 a2 )J , 1 the highest weight state for [J − ]osp(2,1) = (a†1 a2 )J−1 Λ6 . 2

(9.122)

The fact that (a†1 a2 )J−1 Λ6 anticommutes with Λ4 follows from {Λ4 , Λ6 } = 0. The states with angular momentum J − 12 in [J]osp(2,1) and J − 1 in [J − 12 ]osp(2,1) which are su(2)-highest weight states can be got acting with adΛ5 on highest weight states in (9.122). Thus [J]osp(2,1) :

(a†1 a2 )J

adΛ5 −→

(a†1 a2 )J−1 Λ4

adΛ7 ↓

. adΛ8

adΛ7 ↓

adΛ5 −→

X

[J − 21 ]osp(2,1) : (a†1 a2 )J−1 Λ6

(9.123)

where 1 + N − J † J−1 2J − 1 † J−1 † (a1 a2 ) + (a1 a2 ) b b. (9.124) 4 4 As usual, ad denotes graded adjoint action as in 9.75. The vectors are not normalized. The arrows indicate the adjoint actions of Λ5,7,8 . They establish that osp(2, 2) acts irreducibly on [J]osp(2,1) ⊕ [ J−1 2 ]osp(2,1) . We also see that 1 N +1 J = 0, , · · · , . (9.125) 2 2 X=

9.6.4

The Spectrum of V

We show that for J integer V [J]

V [J− 1 ]

osp(2,1)

2 osp(2,1)

J 1, 2 J = − 1. 2

=

(9.126a) (9.126b)

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Proof of (9.126a) It is enough to evaluate V on the highest weight state (a†1 a2 )J . Since adΛ8 (a†1 a2 )J = adΛ6 (a†1 a2 )J = 0 ,

(9.127)

we have V (a†1 a2 )J = (adΛ6 adΛ7 − adΛ7 adΛ6 )(a†1 a2 )J = (adΛ6 adΛ7 + adΛ7 adΛ6 )(a†1 a2 )J

= ad{Λ6 , Λ7 } (a†1 a2 )J 1 = − (εσi )67 adΛi (a†1 a2 )J 2 1 = adΛ3 (a†1 a2 )J 2 J = (a†1 a2 )J . 2

(9.128)

Proof of (9.126b) We evaluate V on (a†1 a2 )J−1 Λ6 . We have adΛ8 (a†1 a2 )J−1 Λ6 = (a†1 a2 )J−1 Λ4 .

(9.129)

Thus 1 1 (adΛ8 )2 (a†1 a2 )J−1 Λ6 = (a†1 a2 )J−1 Λ6 . (9.130) 4 4 Now the osp(2, 1) Casimir K2 has value J(J + 12 )1 in the IRR [J]osp(2,1) while 1 1 (9.131) (adΛi )2 (a†1 a2 )J−1 Λ4 = (J − )(J + ) (a†1 a2 )J−1 Λ4 . 2 2 Hence with α , β ∈ [4, 5], 2J + 1 † J−1 (εαβ adΛα adΛβ ) (a†1 a2 )J−1 Λ4 = (a1 a2 ) Λ4 . (9.132) 4 But π

π

ei 2 Λ8 Λ4,5 e−i 2 Λ8 = i Λ6,7 .

(9.133)

Hence π

π

ei 2 Λ8 (εαβ adΛα adΛβ (a†1 a2 )J−1 Λ4 ) e−i 2 Λ8 = 2J + 1 −(εαβ adΛα0 ad Λβ 0 ) (i(a†1 a2 )J−1 Λ6 ) = i (a†1 a2 )J−1 Λ6 (. 9.134) 4 (9.126b) follows upon using this and (9.131).

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9.6.5

The Fuzzy SUSY Action

Let J be integral. We can write the highest weight component in angular momentum J of the superfield in the IRR [J]osp(2,1) as ΦJ = cj (a†1 a2 )J + (a†1 a2 )J−1 ξJ− 21 Λ4

(9.135)

where cj is a (commuting) complex number and ξJ− 12 is a Grassmann number. The osp(2, 2) transformations map [J]osp(2,1) to [J − 21 ]osp(2,1) . The highest weight component in the latter can be written as ΦJ− 12 = ηJ− 21 (a†1 a2 )J−1 Λ5 + dJ−1 X ,

(9.136)

where ηJ− 12 is a Grassmann and dJ−1 a complex number. The fuzzy action for the highest weight state in [J]osp(2,1) is J strΦ‡J ΦJ 2 Jh ‡ = ξ 1 ξ 1 str[Λ‡4 (a†2 a1 )J−1 (a†1 a2 )J−1 ] 2 J− 2 J− 2 i

SFJ (M = J) =

+ |cJ |2 str[(a†2 a1 )J (a†1 a2 )J ] ,

Λ‡4 = −Λ5 ,

(9.137)

since the two terms in ΦJ are str-orthogonal. For [J − 12 ]osp(2,1) , instead, J− 12

SF

1 J (M = J − ) = strΦ‡J− 1 ΦJ− 12 2 2 2 Jh ‡ =− η 1 η 1 str[Λ‡5 (a†2 a1 )J−1 (a†1 a2 )J−1 Λ5 ] 2 J− 2 J− 2 + |dJ−1 |2 strX ‡ X , Λ‡5 = Λ4 , (9.138)

since the two terms in ΦJ− 12 are also str-orthogonal. The second term here is the integral spin term. It is positive since str X ‡ X < 0

(9.139)

as can be verified. Str-orthogonality extends also to ΦJ and ΦJ− 12 : strΦ‡ ΦJ− 12 = 0 .

(9.140)

Hence for the highest weight states of [J]osp(2,2) = [J]osp(2,1) ⊕ [J − the actions add up:

1 2 ]osp(2,1) ,

J⊕(J− 12 )

SF

for highest weight states

J− 21

= SFJ (J) + SF

1 (J − ) . (9.141) 2

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The superfield Φ is a superposition of such terms. We must first include all angular momentum descendants of ΦJ and ΦJ− 12 . We must also sum on J from 0 to N in steps of 12 . For the fuzzy sphere SF2 , such calculations are best performed using spherical tensors TbLM (N ) and their properties. Similarly, perhaps such calculations are best performed on the fuzzy supersphere using supersymmetric spherical tensors. But as yet only certain basic results about these tensors are available [8]. Reality conditions like Φ‡ = Φ constrain the Fourier coefficients cj , ξJ− 12 , ηJ− 21 , dJ−1 . 9.7 9.7.1

The ∗-Products (2,2)

The ∗-Product on SF

The diagonal matrix elements of operators in the supercoherent state |ψ 0 , ni (2,2) (2,2) define functions on SF . The ∗-product of functions on SF is induced by this map of operators to functions. To determine this map explicitly, it ca . Generalis sufficient to compute the matrix elements of the operators W ization to arbitrary operators can then be made easily as we will see. ca ’s are The diagonal coherent state matrix element for W 1 1 ¯ (a2 ) ψ = ψ¯0 Λa( 2 ) ψ 0 . ca |ψ 0 , ni = 2 ψΛ Wa (ψ 0 , ψ¯0 , n) = hψ 0 , n|W |ψ|2

(9.142)

This defines the map

ca −→ Wa W

(9.143)

ca W cb |ψ 0 , ni Wa ∗ Wb (ψ 0 , ψ¯0 , n) = hψ 0 , n|W

(9.144)

ca to functions Wa . Wa is a superfunction on S (2,2) since of the operator W F it is invariant under the U (1) phase ψ 0 → ψ 0 eiγ . We are now ready to define and compute the ∗-product of two functions of the form Wa and Wb . It depends on n, and to emphasise this we include it in the argument of the product. It is given by

which becomes, after a little manipulation, 1 ( 1 ) ( 1 ) n − 1 ¯0 ( 21 ) 0 ¯0 ( 21 ) 0 ψ Λa ψ Wa ∗n Wb ψ 0 , ψ¯0 , n = ψ¯0 Λa2 Λb 2 ψ 0 + ψ Λb ψ . n n (9.145)

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( ) ( ) Furthermore, since ψ 0 Λa2 Λb 2 ψ 0 is Wa ∗ Wb (ψ 0 , ψ¯0 , 1), (9.145) can be rewritten as n−1 1 Wa (ψ 0 , ψ¯0 ) Wb (ψ 0 , ψ¯0 ) . Wa ∗n Wb ψ 0 , ψ¯0 , n = Wa ∗1 Wb (ψ 0 , ψ¯0 , 1)+ n n (9.146) Introducing the matrix K with

Kab := Wa ∗1 Wb − Wa Wb ,

(9.147)

we can express (9.146) as 1 Kab + Wa Wb . (9.148) n In this form it is apparent that in the graded commutative limit n → ∞, we recover the graded commutative product of functions Wa and Wb . (2,2) The ∗-product of arbitrary functions on SF can be obtained via a 2 similar procedure used to derive that on SF . In this case, one also needs to pay attention to the graded structure of the operators. Thus we can start from the generic operators F and G in the representation ( n2 )osp(2,2)+ expressed as Wa ∗n Wb =

ca1 ⊗G · · · ⊗G W can , Fb = F a1 a2 ···an W cbn , b = Gb1 b2 ···bn W cb1 ⊗G · · · ⊗G W G

(9.149)

where for example F a1 ···ai aj ···an = (−1)|ai ||aj | F a1 ···aj ai ···an , |ai | (mod 2) being the degree of the index ai . After a long but a straightforward calculation, the following finite-series formula is obtained (details can be found in [9]): Fn ∗n Gn (W) = Fn Gn (W)+ n X . . ← → ← → (n − m)! Fn (W) .. ∂ K ∂ · · · ∂ K ∂ ) .. Gn (W . n! m! | {z } m=1

(9.150)

m f actors

. . ← Here we have introduced the ordering .. · · · .., in which ∂Wa ←

→

→

i

∂ Wb

i

are

moved to the left (right) extreme and ∂Wa ’s (∂Wb )’s act on everything i i to their left (right). In doing so one always has to remember to include the overall factor coming from graded commutations. Thus for example, .. ← → ← → .. ← ← → → . ∂ K ∂ ∂ K ∂ ) . = (−1)|a||c|+|b|(|c|+|d|)∂ ∂ K K ∂ ∂ . From Wa Wc

ab

cd Wb Wd

(9.150) it is apparent that, in the graded commutative limit (n → ∞), we get back the ordinary point-wise multiplication Fn Gn (W). This formula was first derived in [9].

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A consequence of (9.146) is the graded commutator of the ∗-product

i fabc Wc (9.151) n which generalizes a familiar result for the usual ∗-products. A special case of our result for the ∗-product follows if we restrict our(2,2) selves to the even subspace SF2 of SF , namely the fuzzy sphere. In this case, Fn (W) and Gn (W) become Fn (~x) and Gn (~x) and we get from (9.150): [Wa , Wb }∗n =

Fn ∗n Gn (~x) = Fn Gn (~x) +

n X (n − m)! ∂i1 · · · ∂im Fn (~x) n! m! m=1

× Ki1 j1 · · · Kim jm ∂j1 · · · ∂jm Gn (~x) , (9.152)

which is the formula given in 3.101. 9.7.2

The ∗-Product on Fuzzy “Sections of Bundles”

Let us first remark that the left- and right-actions of ΨL,R and (Ψ†µ )L,R on µ M at(n + 1, n) are defined and change n by an increment of 1: ΨL,R µ M at(n + 1, n) : † (ΨL,R µ ) M at(n + 1, n) :

On |ψ 0 , ni, we find Sµ |ψ 0 , ni = ψµ0 |ψ 0 , n − 1i ,

˜n → H ˜ n−1 , H ˜n → H ˜ n+1 . H

(9.153)

0

hψ 0 , n|Sµ† = hψ 0 , n − 1|ψ¯µ .

(9.154)

0 hψ 0 , n|Sµ† |ψ 0 , n − 1i = ψ¯µ .

(9.155)

Thus we get the matrix elements hψ 0 , n − 1|Sµ |ψ 0 , ni = ψµ0 ,

We observe that the r.h.s. of the equations in (9.155) define functions on S (3,2) . Thus these matrix elements correspond to fuzzy sections of bundles (2,2) on SF . It is possible to obtain the ∗-product for these fuzzy sections of bundles. The results below also provide an alternative way to compute the ∗-products in (9.146) and (9.150). For the ∗-product of ψ 0 with ψ¯0 we find from hψ 0 , n|Sµ Sν† |ψ †0 , ni = hψ 0 , n|(−1)|Sµ ||Sν |

n 1 Sν† Sµ + δµν |ψ 0 , ni , n+1 n+1 (9.156)

that ψµ0 ∗ ψ¯ν0 =

1 n ψ 0 ψ¯0 + δµν . n+1 µ ν n+1

(9.157)

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[There is an abuse of notation here. ∗-products are for coordinate functions while (9.157) involves their values.] Here we have used (9.42) and the fact that ψµ0 ψ¯ν0 = (−1)|Sµ ||Sν | ψ¯ν0 ψµ0 to get rid of (−1)|Sµ ||Sν | . Rearranging the last result we can write 1 ψµ0 ∗ ψ¯ν0 = Ωµν + ψµ0 ψ¯ν0 , n+1 Ωµν ≡ δµν − ψµ0 ψ¯ν0 . (9.158)

The significance of Ωµν will be be discussed shortly. Before that, as a check of our results of the previous section, we can compute Wa ∗n Wb , using the method above. First note that (1) (1) (9.159) Wa (ψ 0 , ψ¯0 , n) = ψ¯0 Λa2 ψ 0 = hψ 0 , n|S † Λa2 S|ψ 0 , ni . Hence

Wa ∗n Wb (ψ 0 , ψ¯0 , n) (1)

(1)

= hψ 0 , n|Sµ† (Λa2 )µν Sν Sα† (Λb 2 )αβ Sβ |ψ 0 , ni 1 ( 21 ) (1) 0 0 ¯0 ¯ = ψµ (Λa )µν Ωνα + ψν ψα (Λb 2 )αβ ψβ0 n 1 1 n − 1 0 ¯0 (1) (2) 0 ¯ (9.160) δνα + ψν ψα (Λb 2 )αβ ψβ0 = ψµ (Λa )µν n n n−1 1 Wa (ψ 0 , ψ¯0 , n) Wb (ψ 0 , ψ¯0 , n) = Wa ∗1 Wb (ψ 0 , ψ¯0 , n) + n n which is (9.146). Comparing the second line of the last equation with (9.148) we get the important result ← Kab = (Wa ∂ ) Ωµν (∂~¯ν Wb ) µ

≡ Wa ∂ Ω ∂~¯ Wb , ←

(9.161)

where ∂ Ω ∂~¯ ≡ ∂µ Ωµν ∂~¯ and ∂µ = ∂ ∂ψ0 . µ We would like to note that this result can be used to write (9.150) in ← ~¯ To this end we write terms of ∂ Ω ∂. P Y ← b1 b2 ···bn ~¯ Fn ∗n Gn (W) = (−1) j>i |aj ||bi | F a1 a2 ···an (Wai (1+∂ Ω ∂)W . bi ) G ←

←

i

(9.162) Carrying out a similar calculation that lead to (9.150), one finally finds Fn ∗n Gn (W) = Fn Gn (W)+ n X . ← ~ ← (n − m)! ~¯ ... G (W) , ¯ · · · (∂ Fn (W) .. (∂ Ω ∂) Ω ∂) n | {z } n!m! m=1

mf actors

(9.163)

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. . ← where now .. · · · .. takes ∂ and ∂~¯ to the left and right extreme respectively. ← ~¯ are moved in this fashion, the phases coming from the (When ∂ ’s and ∂’s graded commutators should be included just as for (9.150)). It can be explicitly shown that Ω = (Ωµν ) is a projector, i.e., Ω2 = Ω and Ω‡ = Ω . Due to (9.161), the last equation implies similar properties for

(9.164) §

Kab ≡ (K S −1 )ab .

(9.165)

which we discuss next. 9.8

More on the Properties of Kab

A closer look at the properties of Kab ≡ (K S −1 )ab , where Kab (ψ) = Wa ∗1 Wb (ψ) − Wa (ψ) Wb (ψ)

cb |ψ 0 , 1i − hψ 0 , 1|W ca |ψ 0 , 1ihψ 0 , 1|W cb |ψ 0 , 1i , ca W = hψ 0 , 1|W

(9.166)

will give us more insight on the structure of the ∗-product found in the ¯ We denote previous section. First note that Kab depends on both ψ and ψ. ¯ this dependence by Kab (ψ) for short, omitting to write the ψ dependence. Now we would like to show that the matrix K(ψ) = (Kab (ψ)) is a projector. We first recall that the ( 21)osp(2,2)+ , representation of osp(2, 2) is at the same time the Josp(2,1) = 21 osp(2,1) irreducible representation of osp(2, 1). Their highest and lowest weight states are given by 1 1 1 | 2 , 2 , 2 i ≡ |highest weight state, |Josp(2,1) , Jsu(2) , J3 i = (9.167) | 12 , 12 , − 21 i ≡ |lowest weight state We note that, starting from the lowest weight state |1/2, 1/2, −1/2i = Ψ†2 |0i, one can construct another supercoherent state, expressed by a formula similar to (9.84). Now consider Wa± (ψ 0 ) at ψ = ψ 0 = (1, 0, 0) obca in the supercoherent states induced from the tained from computing W states given in (9.167): § We

1 W ± (ψ 0 ) = (W1 (ψ 0 ) · · · W8 (ψ 0 )) = 0, 0, ± , 0, 0, 0, 0, 1 . 2

(9.168)

consider all the indices down through out this chapter. In the following section the relevant object under investigation is Kab corresponding to Ka b in a notation where indices are raised and lowered by the metric.

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In (9.168) +(−) corresponds to upper(lower) entries in (9.167) and the calculation is done using (9.50) and (9.60). Although not essential in what follows, we remark that W − (ψ = (1, 0, 0)) = W + (ψ = (0, 1, 0)), that is, (1) 2

Wa− (ψ 0 ) = Wb+ (ψ 0 ) (Ad eiπΛ2 )ba . (2,2) SF

Note that all other points in the adjoint action of the group, i.e.,

(9.169)

can be obtained from W ± (ψ 0 ) by

Wa± (ψ) = Wb± (ψ 0 )(Ad g −1 )ba

(9.170)

0

where ψ = gψ . We define K± (ψ 0 ) using W ± (ψ 0 ) for W, and the equations (9.165), (9.166). The matrices K± (ψ 0 ) when computed at the fiducial points (using for instance (9.24), (9.145), (9.147)) have the block diagonal forms

with

± K± (ψ 0 ) = (Kab (ψ 0 )) =   ± ± i 1 0 0 0 0 2 δij ± 2 ij3 − 2 Wi (ψ ) (Wj (ψ )) 3×3    (9.171) 0 Σ± 0 αβ  4×4  0 0 0

Σ± = (Σ± αβ ) =

1 4

1 ± σ3 −(1 ± σ3 ) −(1 ± σ3 ) 1 ± σ3

(9.172)

where the upper (lower) sign stands for the upper (lower) sign in W ± (ψ 0 ). The supermatrices K± (ψ 0 ) are even and consequently do not mix the 1, 2, 3, 8 and 4, 5, 6, 7 entries of a (super)vector. Its grade adjoint is its ordinary adjoint †. Now from (9.171), it is straightforward to check that the relations (K± (ψ 0 ))2 = K± (ψ 0 ) , (K± (ψ 0 ))‡ = K± (ψ 0 ) , K+ (ψ 0 ) K− (ψ 0 ) = 0

(9.173)

are fulfilled. (9.173) establishes that K± (ψ 0 ) are orthogonal projectors. By the adjoint action of the group, we have ± ± 0 T Kab (ψ) = ((Ad g)T )−1 ad Kde (ψ ) (Ad g)eb ,

(9.174)

with T denoting the transpose. (9.174) implies that K ± (ψ) are projectors for all g ∈ OSp(2, 2).

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We further observe that a super-analogue J of the complex structure can be defined over the supersphere. To show this, we first observe that the projective module for “sections of the supertangent bundle” T S (2,2) over S (2,2) is PA8 , where A is the algebra of superfunctions over S (2,2) , A8 = A ⊗C C8 and P(ψ) = K+ (ψ) + K− (ψ)

(9.175)

is a projector. The super-complex structure is the operator with eigenvalues (2,2) (2,2) ±i on the subspaces T S± of T S (2,2) with K± A8 ≡ T S (2,2) = T S+ ⊕ (2,2) T S− . It is given by the matrix J with elements Jab (ψ) = −i (K+ − K− )ab (ψ) ,

and acts on PA8 . Since

J 2 (ψ)

PA8

= −P(ψ)

PA8

= −1

PA8

(9.176)

(9.177)

(δ denoting the restriction of δ to ε), it indeed defines a super complex ε

structure. Furthermore, due to the relation J ± 8 = ∓i ± K A

K A8

,

(9.178)

K± A8 give the “holomorphic” and “anti-holomorphic” parts of PA8 . Finally, we can also write K± (ψ) = 9.9

1 (−J 2 ± iJ )(ψ) . 2

(9.179)

The O(3) Nonlinear Sigma Model on S (2,2)

As a final topic in this chapter, we describe the “O(3) nonlinear SUSY (2,2) sigma model” on S (2,2) and SF . We follow the discussion in [123]. 9.9.1

The Model on S (2,2)

On S (2,2) , it is defined by the action Z 1 1 S SU SY = − dµ Cαβ dα Φa dβ Φa + γΦa γΦa , 4π 4

(9.180)

Φa Φa = 1 ,

(9.181)

where Φa = Φa (xi , θα ), (a = 1, 2, 3) is a real triplet superfield fulfilling the constraint (a = 1, 2, 3) .

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Obviously, the world sheet for this theory is S (2,2) while the target manifold is a 2-sphere. A closely related model is the one formulated on the standard (2, 1)dimensional superspace C (2,1) , first studied by Witten, and by Di Vecchia et al.[124, 125]. The triplet superfield Φa can be expanded in powers of θα as 1 Φa (xi , θα ) = na (xi ) + Cαβ θβ ψαa (xi ) + F a (xi )Cαβ θα θβ (9.182) 2 where ψ a (xi ) are two component Majorana spinors : ψαa‡ = Cαβ ψβa , and F a (xi ) are auxiliary scalar fields. In terms of the component fields, the constraint equation (9.181) splits into na na = 1 , 1 na F a = ψ a‡ ψ a , 2 na ψαa = 0 .

(9.183a) (9.183b) (9.183c)

(9.183a) is the usual constraint of O(3) non-linear sigma model defined earlier in chapter 6 by the action [66] Z 1 dΩ(Li na )(Li na ) . (9.184) S=− 8π S 2 Thus, we see that bosonic sector of the S SU SY coincides with the CP 1 sigma model. The other two constraints are additional. We note that (9.183b) can be used along with the equations of motion for F a to eliminate F a ’s from the action. The techniques for performing such calculations can be found for instance in [125]. 9.9.2

(2,2)

The Model on SF

The fuzzy action approaching (9.180) for large n is [123] ˆ a ] [Γ , Φ ˆ a] , ˆ a } [Dβ , Φ ˆ a } + 1 [Γ , Φ S SU SY = str Cαβ [Dα , Φ 4 where ˆ aΦ ˆ a = 12n+1 , 12n+1 ∈ M at(n + 1, n) . Φ

(9.186) can be expressed in terms of the ∗-product on Φa ∗ Φa (ψ 0 , ψ¯0 , n) = 1 .

(2,2) SF

(9.185) (9.186)

as

(9.187)

This expression involves the product of derivatives of Φa up to nth order, and is not easy to work with. Alternatively we can construct supersymmetric extensions of “Bott Projectors” introduced in chapter 6 to study this model, as we indicate below.

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Supersymmetric Extensions of Bott Projectors

A possible supersymmetric extension of the projector Pκ (x) can be obtained in the following manner. Let U(xi , θα ) be a graded unitary operator : UU ‡ = U ‡ U = 1 .

(9.188)

U(xi , θα ) can be thought as a 2 × 2 supermatrix whose entries are functions on S (2,2) . U(xi , θα ) acts on Pκ by conjugation and generates a set of supersymmetric projectors Qκ (xi , θα ): Qκ (xi , θα ) = U ‡ Pκ (x) U .

(9.189)

It is easy to see that Qκ (xi , θα ) satisfies Q2κ (xi , θα ) = Qκ (xi , θα ) ,

and Q‡κ (xi , θα ) = Qκ (xi , θα ) .

(9.190)

Thus Qκ (xi , θα ) is a (super-)projector. The real superfields on S (2,2) associated to Qκ (xi , θα ) are given by Φ0a (xi , θα ) = T r τa Qκ .

(9.191)

In order to check that Qκ (xi , θα ) reproduces the superfields on S (2,2) subject to Φ0a Φ0a = 1 ,

(9.192)

we proceed as follows. First we expand U(xi , θα ) in powers of Grassmann variables as 1 U(xi , θα ) = U0 (xi ) + Cαβ θβ Uα (xi ) + U2 (xi )Cαβ θα θβ (9.193) 2 where U0 , Uα (α = ±) and U2 are all 2 × 2 graded unitary matrices. The requirement of graded unitarity for U(xi , θα ) implies the following for the component matrices: i. U0 (xi ) is unitary, ii. Uα (xi ) are uniquely determined by Uα (xi ) = Hα (xi )U0 (xi ) ,

(9.194)

where Hα are 2 × 2 odd supermatrices satisfying the reality condition Hα‡ = −Cαβ Hβ , iii. U2 is of the form U2 = AU0 with A being an 2 × 2 even supermatrix, whose symmetric part satisfies A + A† = −Cαβ Hα Hβ .

(9.195)

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Using (9.193) in (9.189) and the conditions listed above, we can extract the component fields of the superfield Φ0a (xi , θα ). We find † nκ0 a := T r τa U0 Pκ U0 ,

T r τa U0† [Hα

(9.196)

~ α0 )a , , Pκ ]U0 = −2i(~n × H

(9.197)

Fa0 := T r τa U0† (Pκ A + A† Pκ − Cαβ Hβ Pκ Hα )U0 ~0 ·H ~ 0 )nκ0 − 2H ~ a0 (~nκ0 · H ~0 ) = 4(H

(9.198)

ψαa0

:=

κ0

and, after using (9.195),

+ κ0

−

a

+

−

0 a0 ~+ ~− ~0 − A ~ †0 ))a , −(~n · H )2H + i(~nκ0 × (A

~ α0 = Hα10 τ 1 + Hα20 τ 2 and A ~ 0 = A30 τ 3 . By direct computation from where H above it follows that 1 ‡0 0 κ0 0 a0 nκ0 nκ0 nκ0 (9.199) a na = 1 , a Fa = ψa ψa , a ψ± = 0 . 2 Comparing (9.199) with (9.183) we observe that they are identical. Therefore, we conclude that the superfield associated to the super-projector Qκ is the same as the superfield of the supersymmetric nonlinear sigma model discussed previously. 9.9.4

The SUSY Action Revisited

We now extend (9.129) by including winding number sectors. Equipped with the supersymmetric projector Qκ we can write, in close analogy with the CP 1 model, the action for the supersymmetric nonlinear O(3) sigma model for winding number κ as Z h i 1 1 dµ T r Cαβ (dα Qκ )(dβ Qκ ) + (γQκ )(γQκ ) . (9.200) SκSU SY = − 2π 4 The even part of this action, as well as the one given in (9.180) are nothing but the action Sκ of the CP 1 theory given in (6.22) and (9.184), respectively. In other words, the action SκSU SY is the supersymmetric extension of Sκ on S 2 to S (2,2) . Consequently, in the supersymmetric theory, it is possible to interpret the index κ carried by the action as the winding number of the corresponding CP 1 theory. For κ = 0 we get back (9.129). We recall that dα and γ are both graded derivations in the superalgebra osp(2, 2). Therefore, they obey a graded Leibniz rule. From Q2κ = Qκ , we find Qκ dα Qκ = dα Qκ (1 − Qκ ) .

(9.201)

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This enables us to write 1 T r(dα Qκ )2 . (9.202) 2 Equations (9.201) and (9.202) continue to hold when dα is replaced by γ as well. The action can also be written as Z h i 1 1 SU SY dµ T r Cαβ Qκ (dα Qκ )(dβ Qκ ) + Qκ (γQκ )(γQκ ) . Sκ =− π 4 (9.203) T rdα Qκ (1 − Qκ )dα Qκ = T r(1 − Qκ )(dα Qκ )2 =

9.9.5

Fuzzy Projectors and Sigma Models

In much the same way that the supersymmetric projectors Qκ have been constructed from Pκ in the previous section, we can construct the superbκ by the graded unitary transformation symmetric extensions of P bκ = Ub‡ P bκ Ub Q

(9.204)

bκ , N b} = 0 , [Q

(9.205)

where now Ub is a 2 × 2 supermatrix whose entries are polynomials in not ˜n. only a†α aβ , but also in b† b. The domain of Uij is H 2 bκ acts on the finite-dimensional space H ˜n = H ˜ n ⊗ C2 . We can check Q that b = a† aα + b† b is the number operator on H ˜ n . In close analogy with where N α 1 the fuzzy CP model, it is now possible to write down a finite-dimensional bκ . (super-)matrix model for the (super-)projectors Q The action for the fuzzy supersymmetric model becomes 1 bκ } [Dβ , Q bκ } + 1 [Γ , Q bκ ] [Γ , Q bκ ] . StrNb =n Cαβ [Dα , Q SFSU,κSY = 2π 4 (9.206) 2 ˜ b =n Str in the above expression is the supertrace over Hn . In the large N limit, (9.206) approximates the action given in (9.200). (2,2) This concludes our discussion of the nonlinear sigma model on SF .

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Chapter 10

SUSY Anomalies on the Fuzzy Supersphere

10.1

Overview

In this section, we give an overview of fuzzy SUSY for convenience. Much of it is in chapter 9 also. The notation for irreducible representations is different from that of chapter 9. 10.1.1

The Fuzzy Sphere

We recall that the fuzzy sphere SF2 (n) is the (n + 1) × (n + 1) matrix algebra M at(n + 1). It can be realized as linear operators on H n+1 with the orthonormal basis vectors (a†1 )n1 (a†2 )n2 √ √ |0i , n1 + n2 = n , n1 ! n2 !

(10.1)

where ai , a†i are bosonic oscillators. The vectors (10.1) span a subspace of the Fock space with fixed particle number n: X † N := ai ai , N |Hn+1 = n . (10.2) i

In this representation, the elements of SF2 (n) are the linear operators X † m m cm , cm (10.3) i,j (ai ) (aj ) i,j ∈ C , i,j

restricted to the subspace Hn+1 . The group SU (2) acts on Hn+1 and hence on SF2 (n) by its spin n2 unitary irreducible representation. The angular momentum generators are σi (10.4) Li = a† a , σi are Pauli matrices. 2 137

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10.1.2

SUSY

The N = 1 SUSY version of SU (2) is OSp(2, 1). It has the graded Lie algebra osp(2, 1). Its generators (basis) can be written using oscillators if we introduce one additional fermionic oscillator b and its adjoint b† . They commute with ai , a†j . Then the osp(2, 1) generators are Λi = a †

σi 1 a , Λ4 = − (a†1 b + b† a2 ) , 2 2

(10.5)

1 (−a†2 b + b† a1 ) , σi = Pauli matrices. 2 The N = 2 SUSY version of SU (2) is OSp(2, 2). It has the graded Lie algebra osp(2, 2). Its basis consists of the osp(2, 1) generators and three additional generators 1 1 Λ40 ≡ Λ6 = (a†1 b − b† a2 ) , Λ50 ≡ Λ7 = (a†2 b + b† a1 ) , (10.6) 2 2 Λ8 = a† a + 2b† b . Λ5 =

If {·, ·} denotes the anticommutator, osp(2, 2) has the defining relations

1 1 Λβ (σi )βα , {Λα , Λβ } = (εσi )αβ Λi , 2 2 1 [Λi , Λ8 ] = 0 , [Λα , Λ8 ] = −Λα0 , {Λα , Λα0 } = εαβ Λ8 , 4 1 0 1 0 0 0 . (10.7) {Λα , Λβ } = − (εσi )αβ Λi , [Λα , Λ8 ] = −Λα , ε = −1 0 2 [Λi , Λj ] = iεijk Λk , [Λi , Λα ] =

These relations show in particular that the additional three generators form a triplet under osp(2, 1). Conventional Lie algebras like that of su(2) have a ∗ or an adjoint operation † defined on them. For Λi , it is just Λ†i = Λi . This follows from the fact that a†i is the adjoint of ai . For osp(2, 1) and osp(2, 2), † is replaced by the grade adjoint ‡ . On the oscillators, ‡ is defined by a‡i = a†i , (a†i )‡ = (a†i )† = ai , b‡ = b† , (b‡ )‡ = −b .

(10.8)

Hence ‡ = † on bosonic oscillators. On products of operators, ‡ is defined as follows. We assign the grade 0 to ai , a†j and their products and 1 to b and b† . The grades are additive (mod 2). The grade of an operator L with definite grade is denoted by |L|. Then if L, M have definite grades, (LM )‡ ≡ (−1)|L||M |M ‡ L‡ . Hence (b† b)‡ = b† b and Λ‡i = Λi , Λ‡α = −εαβ Λβ , Λ‡α0 = εαβ Λβ 0 Λ‡8 = Λ8 .

(10.9)

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Irreducible Representations

Let osp(2, 0) denote su(2), the Lie algebra of SU (2). Its IRR’s are Γ0J , J ∈ N/2. (Here N = {0, 1, 2, ...}.) J has the meaning of angular momentum. The osp(2, 1) algebra is of rank 1 just as osp(2, 0). We can take Λ3 to be the generator of its Cartan subalgebra. Since 1 Λ4 , [Λ3 , Λ+ = Λ1 + iΛ2 ] = Λ+ , 2 are its raising operators. They commute: [Λ3 , Λ4 ] =

Λ4 , Λ+

[Λ4 , Λ+ ] = 0 .

(10.10)

(10.11)

In an IRR, both vanish on the highest weight vector. The eigenvalue J ∈ N/2 of Λ3 on the highest weight vector can be used to label its IRR’s. They are denoted by Γ1J in this chapter. When restricted to its subalgebra osp(2, 0), Γ1J splits as follows: 1 (10.12) Γ1J osp(2,0) = Γ0J ⊕ Γ0J− 1 , J ≥ . 2 2 Γ10 is the trivial IRR. The dimension of Γ1J is 4J + 1. The graded Lie algebra osp(2, 2) is of rank 2. A basis for its Cartan subalgebra is {Λ3 , Λ8 }. Since

1 (10.13) (Λ4 + Λ40 ) , [Λ8 , Λ4 + Λ40 ] = Λ4 + Λ40 , 2 Λ4 + Λ40 serves as the raising operator for both Λ3 and Λ8 . We also have that Λ1 + iΛ2 = Λ+ is the raising operator for Λ3 alone: [Λ3 , Λ4 + Λ40 ] =

[Λ3 , Λ+ ] = Λ+ , [Λ8 , Λ+ ] = 0 .

(10.14)

The raising operators Λ4 + Λ40 and Λ+ commute: [Λ4 + Λ40 , Λ+ ] = 0 .

(10.15)

Both vanish on the highest weight vector in an IRR while the eigenvalues J ∈ N/2 and k ∈ Z of Λ3 and Λ8 on the highest weight vector can be used as labels of the IRR. They are denoted in this paper by Γ2J (k). The osp(2, 2) IRR’s fall into classes, the typical and atypical (or short) IRR’s. In the typical IRR’s, 2|k| 6= J or k = J = 0, while in the atypical IRR’s, 2|k| = J 6= 0. The typical IRR with 2|k| 6= J restricted to osp(2, 1) splits as follows: 1 Γ2J (k) osp(2,1) = Γ1J ⊕ Γ1J− 1 , J ≥ . (10.16) 2 2

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Γ20 (0) is the trivial representation. The atypical IRR’s Γ2J (±2J) (J ≥ 1/2) remain irreducible on restriction to osp(2, 1): Γ2J (±2J) osp(2,1) = Γ1J . (10.17)

Γ2J (±J/2) can also be abbreviated to Γ2J± :

Γ2J (±2J) ≡ Γ2J± , J ≥ 1/2 .

(10.18)

osp(2, 2) admits the automorphism τ : Λi → Λi , Λα → Λα , Λα0 → −Λα0 , Λ8 → −Λ8

(10.19)

which interchanges Γ2J (±k): τ : Γ2J (k) → Γ2J (−k) . 10.1.4

(10.20)

Casimir Operators

The osp(2, 0) := su(2) Casimir operator K0 is well-known: K0 = Λ2i .

(10.21)

The osp(2, 1) Casimir operator is K1 = Λ2i + εαβ Λα Λβ .

(10.22)

1 K1 |Γ1 = J(J + )1l . J 2

(10.23)

We have that

The osp(2, 2) quadratic Casimir operator is 1 K2 = K1 − εαβ Λα0 Λβ 0 − Λ28 := K1 − V0 . 4

(10.24)

It has the property K2 |Γ2 (k) = J 2 − J

K 2 |Γ 2

J±

=0.

k2 , 4 (10.25)

As already mentioned, the IRR’s Γ2J± can be distinguished by the sign of Λ8 on the highest weight vector. osp(2, 2) also has a cubic Casimir operator [126], but we will not have occasion to use it.

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Tensor Products

The basic Clebsch-Gordan series we need to know is as follows: Γ1J ⊗ Γ1K = Γ1J+K ⊕ Γ1J+K−1/2 ⊕ · · · ⊕ Γ1|J−K| . 10.1.6

(10.26)

The Supertrace and the Grade Adjoint

Because of the decomposition (10.12), the vector space C4J+1 on which Γ1J acts can be written as C2J+1 ⊕ C2J where the first term has angular momentum J and the second term has angular momentum J − 1/2. By definition, the first term is the even subspace and the second term is the odd subspace. The supertrace str of a matrix P(2J+1)×(2J+1) Q(2J+1)×2J M= (10.27) R2J×(2J+1) S2J×2J is accordingly strM = trP − trS .

(10.28)

The grade adjoint M ‡ can be calculated using the rules of graded vector spaces (cf. chapter 9). The result is † P −R† ‡ (10.29) M = Q† S † This formula is coherent with (10.9). If Q, R = 0, we say that M is even, while if P, S = 0, we say that M is odd. We assign a number |M | = 0, 1 (mod 2) to even and odd matrices M respectively. 10.1.7

The Free Action

The space with N = n has maximum angular momentum J = n/2. It carries the osp(2, 1) IRR Γ1n/2 which splits under su(2) into Γ0n/2 ⊕Γ0(n−1)/2 . It carries either of the short osp(2, 2) IRR’s as well. The dimension of the Hilbert space with N = n is 2n + 1. We denote it by H2n+1 . It is the direct sum Hn+1 ⊕ Hn where Hn+1 is the even subspace carrying the IRR Γ0n/2 and Hn is the odd subspace carrying the representation Γ0(n−1)/2 . A basis for H2n+1 is X (a†1 )n1 (a†2 )n2 † n3 √ √ (b ) |0i , ni = n , n3 ∈ (0, 1) , (b† )0 := 1l . n1 ! n2 !

(10.30)

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The fuzzy SUSY SF2,2 (in the zero instanton sector) is the matrix algebra M at(4J +1) = M at(2n+1). Just as SF2 , it can be realized using oscillators. In terms of oscillators, a typical element is X X † m m−1 † m−1 m−1 m ∈C. di,j (ai ) (aj )m−1 b† b , cm cm + i,j , di,j i,j (ai ) (aj ) i,j

i,j

(10.31)

It is to be restricted to the space H2n+1 . The left- and right-actions R |Λρ ||M | ΛL M Λρ ρ M = Λρ M , Λρ M = (−1)

(10.32)

of osp(2, N ) on M at(2n + 1) give two commuting IRR’s of osp(2, N ). Here, Λρ ∈ osp(2, N ) , N = 1, 2, M ∈ M at(2n + 1) and both Λρ and M are of definite grade |Λρ |, |M | (mod 2). Combining the left- and right- representations, we get the grade adjoint representation R 0 gad : Λρ → gadΛρ = ΛL ρ − Λρ , ρ ∈ (i, α, α , 8)

(10.33)

of osp(2, N ). With regard to gad, M at(4J + 1) transforms as Γ1J ⊗ Γ1J = Γ12J ⊕ Γ12J−1/2 ⊕ Γ12J−1 ⊕ · · · ⊕ Γ10 .

(10.34)

osp(2, 2) acts on M at(4J + 1) by L, R and gad representations as well. The L and R are the short representations Γ2J± so that under gad, M at(4J + 1) transforms as Γ2J+ ⊗Γ2J− . Its reduction can be inferred from (10.34) once we know that Γ2j (0)|osp(2,1) = Γ1j ⊕ Γ1j−1/2 . We will see this later. Hence Γ2J+ ⊗ Γ2J− = Γ22J (0) ⊕ Γ22J−1 (0) ⊕ · · · ⊕ Γ20 (0) .

(10.35)

The fuzzy field Φ is an element of fuzzy SUSY. The free action for Φ is f2 str Φ‡ V0 Φ , (10.36) 2 where f is a real constant and V0 is an osp(2, 1)-invariant operator. When restricted to the odd subspace, it should become the Dirac operator of chapter 8. The limit of this operator for “J = ∞” (that is on S 2,2 ) was found by Fronsdal [127] and later used effectively by Grosse et al [7]. For J = ∞, it is the difference K1 − K2 of the Casimir operators K1 and K2 written as graded differential operators. This operator, for finite J, becomes 1 V0 = εαβ (Λα0 )(Λβ 0 ) + (Λ8 )2 . (10.37) 4 S0 =

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It is evident that V0 is osp(2, 1)-invariant. But it is less obvious that gad Λα0 , gad Λ8 anti-commute with V0 : {gadΛα0 , V0 } = {gadΛ8 , V0 } = 0 .

(10.38)

This means that these generators are realized as chiral symmetries. Of these, gad Λ8 , restricted to the odd sector, is just standard chirality. Thus, these generators, associated with osp(2, 2)/osp(2, 1) are SUSY generalizations of conventional chirality. We now show these results.

10.2

SUSY Chirality

Let us first exhibit the highest weight vectors of the su(2) IRR’s which occur in Γ2j (0). Here j is an integer. Referring to (10.12), we have that Γ1j su(2) = Γ0j ⊕ Γ0j−1/2 for j ≥ 1. The highest weight vector of Γ0j is (a†1 a2 )j as it commutes with Λ4 and carries the eigenvalue j of gad Λ3 . gad Λ5 maps it to −j(a†1 a2 )j−1 Λ4 , the highest weight vector of Γ0j−1/2 ⊂ Γ1j . Thus ⊕ Γ0j−1/2 Γ1j su(2) = Γ0j (10.39) gadΛ5 Highest weight −j(a†1 a2 )j−1 Λ4 , j ≥ 1 . (a†1 a2 )j −→ vectors The equation also indicates the operator mapping one highest weight vector of su(2) to the other. Next consider Γ1j−1/2 ⊃ Γ0j−1/2 ⊕ Γ0j−1 for j ≥ 1. To distinguish the su(2) IRR’s here from those in Γ1j , we put a prime on them: 0 0 = Γ0j−1/2 ⊕ Γ0j−1 . (10.40) Γ1j−1/2 su(2)

The highest weight state of Γ1j−1/2 , commuting with Λ4 and with eigenvalue

j − 1/2 for gad Λ3 is −j(a†1 a2 )j−1 Λ6 . And Λ5 maps it to the highest weight 0 vector Xj−1 of Γ0j−1 . We show Xj−1 below. Thus 0 0 = Γ0j−1/2 ⊕ Γ0j−1 , Γ1j−1/2 su(2) (10.41) gadΛ5 Highest weight −j(a†1 a2 )j−1 Λ6 −→ Xj−1 , vectors j − 2J − 1 † j−1 1 − 2j † j−1 † Xj−1 = (a1 a2 ) + (a1 a2 ) b b , j ≥ 1 . 4 4 In calculating Xj−1 , we use 1 Λ4 Λ6 = − (a†1 a2 )(2b† b − 1) , a† a + b† b = 2J . (10.42) 4

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Now gad Λ7 , gad Λ8 map the vectors in (10.39) The full table is gadΛ5 Γ1j 3 (a†1 a2 )j −→ 0 % ∈ Γj Γ2j (0) gadΛ7 ↓ . gadΛ8 & gadΛ5 Γ1j−1/2 3 −j(a†1 a2 )j−1 Λ6 −→ 00 ∈ Γj−1/2

to the vectors in (10.41). −j(a†1 a2 )j−1 Λ4 ∈ Γ0j−1/2 ↓ gadΛ7 Xj−1 , 0 ∈ Γ0j−1

j≥1.

(10.43) For j = 0, we get the trivial IRR of osp(2, N )’s. Eq. (10.43) shows that gad Λα0 , gad Λ8 map the vectors of Γ1j to those of Γ1j−1/2 (j ≥ 1) and vice versa. So if V0 has opposite eigenvalues in the representations in (10.43), then we can conclude that∗ {gadΛα0 , V0 } = {gadΛ8 , V0 } = 0

(10.44)

identically, since V0 |Γ10 = 0. That means that these operators associated with osp(2, 2)/osp(2, 1) are chirally realized symmetries. 10.3

Eigenvalues of V0

As V0 is an osp(2, 1) scalar, it is enough to compute its eigenvalue on the highest weight state of Γ1j to find V0 |Γ1j . As Λ40 = Λ6 commutes with (a†1 a2 )j , we have that

εαβ gadΛα0 gadΛβ 0 (a†1 a2 )j = ( gadΛ40 gadΛ50 + gadΛ50 gadΛ40 ) (a†1 a2 )j (10.45) where the sign of the second term has been switched as it is zero anyway. Thus the left-hand side of the previous formula can be written as gad{Λ40 , Λ50 }(a†1 a2 )j =

1 j gadΛ3 (a†1 a2 )j = (a†1 a2 )j . 2 2

(10.46)

Also gadΛ8 (a†1 a2 )j = 0 .

(10.47)

Hence V0 (a†1 a2 )j = ∗ To

j † j (a a2 ) . 2 1

(10.48)

show that {gadΛ6 , V0 } = 0 we use the fact that gadΛ6 = −[gadΛ4 , gadΛ8 ]. The result follows from the graded Jacobi identity.

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One quick way to evaluate V0 |Γ1j−1/2 is as follows. Since K1 |Γ1j = j(j + 1/2), we have K2 |Γ1j = (K1 − V0 )|Γ1j = j 2 .

(10.49)

But K2 is osp(2, 2)-invariant. Hence K2 |Γ1j−1/2 = j 2 .

(10.50)

Since also K1 |Γ1j−1/2 = j(j − 1/2), we have j V0 |Γ1j−1/2 = (K1 − K2 )|Γ1j−1/2 = − 1l . 2 1 1 Thus V0 has opposite eigenvalues on Γj and Γj−1/2 . It is important to notice that K2 = (2V0 )2 .

(10.51)

(10.52)

That is, 2V0 is a square root of K2 , a little like in the way that the Dirac operator is the square root of the Laplacian. 10.4

Fuzzy SUSY Instantons

The manifold S 2 admits twisted U (1) bundles labelled by a topological index or Chern number k ∈ Z. In the algebraic language, sections of vector bundles associated with these U (1) bundles are described by elements of projective modules (see chapter 5,6). When S 2 becomes the graded supersphere S 2,2 , we expect these modules to persist, and become in some sense supersymmetric projective modules. That is in fact the case. We shall see that explicitly after first studying their fuzzy analogues. The projective modules on S 2 and SF2 are associated with SU (2) ' S 3 via Hopf fibration and Lens spaces. In the same way, the supersymmetric projective modules on S 2,2 and SF2,2 get associated with osp(2, 1) and osp(2, 2). The fuzzy algebra SF2,2 of previous sections is to be assigned k = 0. The elements of this algebra are square matrices mapping the space with N = 2J to the same space N = 2J. We emphasize the value of k by writing SF2,2 (0) for SF2,2 . SF2,2 (0) is a bimodule for osp(2, 2) as the latter can act on the left or right of SF2,2 (0) by the IRR’s Γ2J± (0). For k 6= 0, SF2,2 (k) is not an algebra. It can be described using projectors or equally well as maps of the vector space with N = 2J to the one with

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N = 2J + k. (We take J + k2 ≥ 0. If k < 0, this means J ≥ |k| 2 .) If a basis is chosen for the domain and range of SF2,2 (k), their elements become rectangular matrices with 2J + k rows and 2J columns. SF2,2 (k) as well is a bimodule for osp(2, 2). The latter acts by Γ2(J+ k )+ on the left of SF2,2 (k) 2

and by Γ2J− on the right of SF2,2 (k). The invariant associated with SF2,2 (k) is just k. The meaning of k is k = Dimension of range of SF2,2 (k) − Dimension of domain of SF2,2 (k) . (10.53) 2,2 Scalar fields Φ are now elements of SF (k) while V0 is replaced by a new operator Vk which incorporates the appropriate connection and “topological” data. We now argue, using index theory and other considerations, that the osp(2, 1)-invariant Vk is fixed by the requirement Vk2 = K2 (10.54) where K2 is the Casimir invariant for Γ2(J+ k )+ ⊗ Γ2J− . 2

10.5

Fuzzy SUSY Zero Modes and their Index Theory

We begin by analyzing the osp(2, 1) and osp(2, 2) representation content of SF2,2 (k). As regards the gad representation of osp(2, 1), it transforms according to Γ1J+ k ⊗ Γ1J = Γ12J+ k ⊕ Γ12J+ k − 1 2 2 2 2 1 1 ⊕ Γ2J+ k −1 ⊕ Γ2J+ k − 3 ⊕ · · · ⊕ Γ1|k| +1 ⊕ Γ1|k| + 1 ⊕ Γ1|k| . 2

2

2

2

2

2

2

The analogue of (10.43) is:

2J + Γ1j Γ2j (k)

% &

|k| 2 , 2

3

|k| k ≥j≥ +1 , 2 2 Γ0j

⊕ Γ0j−1/2 (10.55)

Γ1j−1/2 3 Γ0j−1/2 ⊕ Γ0j−1

.

we get the atypical representation of osp(2, 2): Here |k| ≥ 1. For j = Γ |k| (k) → Γ1|k| = Γ0|k| ⊕ Γ0|k| −1 . (10.56) 2

2

2

2

All this becomes explicit during the following calculation of the eigenvalues of K2 .

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Spectrum of K2

For k > 0, the highest weight vector with angular momentum j =m+

|k| , m = 0, 1, ... 2

(10.57)

is (a†1 )|k| (a†1 a2 )m .

(10.58)

Since gadΛ8 (a†1 )|k| (a†1 a2 )m = |k|(a†1 )|k| (a†1 a2 )m , of Γ2j (|k|). SF2,2 (|k|).

(10.59)

Γ2j (|k|)

it is the highest weight vector Thus occurs in the reduction of the osp(2, 2) action on General theory [126] tells us the branching rules of Γ2j (|k|) as in (10.55). This equation is thus established for k > 0. We can check as before that |k| 1 m+ (a†1 )|k| (a†1 a2 )m (10.60) εαβ gadΛα0 gadΛβ 0 (a†1 )|k| (a†1 a2 )m = 2 2 while

1 k2 (gadΛ8 )2 (a†1 )|k| (a†1 a2 )m = 4 4

(10.61)

K1 |Γ1j = j(j + 1)1l .

(10.62)

and

We thus have [126] K2 |Γ2j (|k|) = For k < 0,

j2 −

k2 4

1l .

(a2 )|k| (a†1 a2 )m

(10.63)

(10.64)

is the highest weight vector for angular momentum j =m+

|k| . 2

(10.65)

Since gadΛ8 (a2 )|k| (a†1 a2 )m = −|k|(a2 )|k| (a†1 a2 )m ,

(10.66)

weight vector of Γ2j (−|k|). Hence Γ2j (−|k|) occurs in the osp(2, 2) action on SF2,2 (−|k|). We thus establish (10.55)

it is the highest reduction of the for k < 0 as well.

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The eigenvalues of Vk , when restricted to Γ1j and Γ1j−1/2 and for q 2 + 1, are ± j ≥ |k| j 2 − k4 . These eigenvalues are not zero. Hence the 2 osp(2, 2) operators which intertwine these representations, mapping vectors of one representation to the other, anticommute with Vk : they are chiral symmetries for these representations. For j = |k| 2 , Vk vanishes while the representation space carries the atypical representation Γ2|k| (k) of osp(2, 2). 2

Thus we can say that the above chiral operators all anticommute with Vk |J=0 . Hence these operators anticommute with Vk (for any j, on all vectors of SF2.2 (k)) just as standard chirality anticommutes with the massless Dirac operator. For k = 0, these operators were Λα0 , Λ8 . But they change with k. They can be worked out. They do not occur in subsequent discussion and hence we do not show them here. We now establish that Vk is the correct choice for the fuzzy SUSY action Sk : Sk = const str Φ‡ Vk Φ .

(10.67)

This formula is valid also for k = 0 as we saw earlier. We here focus on k 6= 0. The Dirac operators D for fuzzy spheres of instanton number k are known (see chapter 8). We first show that Vk coincides with this operator on the Dirac sector. It is enough to focus on typical osp(2, 2) IRR’s since both the Dirac operator and Vk vanish on the grade-odd sector of Γ1|k| . Thus consider 2

Γ2j (k) for j ≥ 21 |k| + 1. Angular momentum J in the Dirac sector of Γ2j (k) is j − 1/2. Hence |k| − 1 |k| + 1 2 Vk Γ2 (k) Dirac sector = J − J+ 1l . (10.68) j 2 2 Substituting J = n +

|k|−1 2

Vk2 Γ2 (k) j

Hence Vk2 Γ2 (k) j

and identifying |k| = 2T , we get the answer:

Dirac sector

Dirac sector

= n(n + 2T )1l .

(10.69)

is the correct Dirac operator .

This result and the osp(2, 1)-invariance of Vk are compelling reasons to identify it as the SUSY generalization of the Dirac and Laplacian operators for k 6= 0.

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10.5.2

Index Theory and Zero Modes

There is also further evidence supporting the correctness of Vk : It gives the SUSY generalization of index theory. Thus one knows from chapter 8 that 1) the Dirac operator has |k| zero modes for instanton number k on S 2 and on the fuzzy sphere SF2 (k), and that 2) they are left- (right-) chiral if k > 0 (k < 0), 3) charge conjugation interchanges these chiralities. More precisely if nL,R are the number of left- and right-chiral zero modes, nL − n R = k .

(10.70)

This number is “topologically stable”. The meaning of this statement in the fuzzy case can be found in chapter 8. If the Dirac operator is SU (2)-invariant, these zero modes organize themselves into SU (2) multiplets with angular momentum |k| 2 . Now Vk has zero modes which form the atypical multiplet Γ2|k| (k) of 2

osp(2, 2). The number of zero modes is 2|k| + 1. Of these, |k| correspond to the grade odd sector and can be identified with the zero modes of S 2 and SF2 (k) Dirac operators. The remaining grade even (|k| + 1) zero modes are their SUSY-partners. The zero modes transform by inequivalent IRR’s of osp(2, 2) for the two signs of k. These two atypical osp(2, 2) representations are SUSY generalizations of left- and right- chiralities. Identifying charge conjugation with the automorphism (10.19), we see that it exchanges these two IRR’s just as it exchanges chiralities in the Dirac sector.

10.6

Final Remarks

In this chapter, we have extended the developments of chapter 9 on the fuzzy SUSY model on S 2 to the instanton sector. A SUSY generalization of chapter 8 on index theory and zero modes of the Dirac operator has also been established. We can try introducing interactions involving just Φ. For k 6= 0, Φ can be thought of as a rectangular matrix. So Φ‡ Φ and ΦΦ‡ are square matrices of different sizes acting on osp(2, 2) representations with N = n and N = n + k. A typical interaction may then be λstr(Φ‡ Φ)2

(10.71)

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where str is over the space with N = n, the domain of Φ‡ Φ. But note that (10.71) and the use of str in interactions require further study. Fuzzy SUSY gauge theories remain to be formulated. The investigation of the graded commutative limit n → ∞ with k fixed has also not been done for k 6= 0.

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Chapter 11

Fuzzy Spaces as Hopf Algebras

11.1

Overview

In this chapter we will explore yet another intriguing aspect of fuzzy spaces, namely their potential use as quantum symmetry algebras. To be more precise, we will establish, through studying the fuzzy sphere as an example, that fuzzy spaces possess a Hopf algebra structure. It is a fact that for an algebra A, it is not always possible to compose two of its representations ρ and σ to obtain a third one. For groups we can do so and obtain the tensor product ρ ⊗ σ. Such a composition of representations is also possible for coalgebras C [128]. A coalgebra C has a coproduct ∆ which is a homomorphism from C to C ⊗C and the composition of its representations ρ and σ is the map (ρ ⊗ σ)∆. If C has a more refined structure and is a Hopf algebra, then it closely resembles a group, in fact sufficiently so that it can be used as a “quantum symmetry group” [129]. We follow reference [130] in this chapter. In order to make our discussion self contained, we review some of the basic definitions about coalgebras, bialgebras and Hopf algebras in terms of the language of commutative diagrams and set our notations and conventions, which are the standard ones in the literature. A well known example of a Hopf algebra is the group algebra G∗ associated to a group G. Our interest mainly lies on compact Lie groups G, as they are the ones whose adjoint orbits once quantized yield fuzzy spaces. The group algebra G∗ of such G consists of elements R G dµ(g)α(g)g where α(g) is a smooth complex function and dµ(g) is the G-invariant measure. It is isomorphic to the convolution algebra of functions on G. Basic definitions and properties related to G∗ will be given in section 11.3. In section 11.5 and 11.6, we establish that fuzzy spaces are irreducible

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representations ρ of G∗ and inherit its Hopf algebra structure. For fixed G, their direct sum is homomorphic to G∗ . For example both SF2 (J) and ⊕J SF2 (J) ' SU (2)∗ are Hopf algebras. This means that we can define a coproduct on SF2 (J) and ⊕J SF2 (J) and compose two fuzzy spheres preserving algebraic properties intact. A group algebra G∗ and a fuzzy space from a group G carry several actions of G. G acts on G and G∗ by left and right multiplications and by conjugation. Also for example, the fuzzy space SF2 (J) consists of (2J + 1) × (2J + 1) matrices and the spin J representation of SU (2) acts on these matrices by left and right multiplication and by conjugation. The map ρ of G∗ to a fuzzy space and the coproduct ∆ are compatible with all these actions: they are G-equivariant. Elements m of fuzzy spaces being matrices, we can take their hermitian conjugates. They are ∗-algebras if ∗ is hermitian conjugation. G∗ also is a ∗-algebra. ρ and ∆ are ∗-homomorphisms as well: ρ(α∗ ) = ρ(α)† , ∆(m∗ ) = ∆(m)∗ . The last two properties of ∆ on fuzzy spaces also derive from the same properties of ∆ for G∗ . All this means that fuzzy spaces can be used as symmetry algebras. In that context however, G-invariance implies G∗ - invariance and we can substitute the familiar group invariance for fuzzy space invariance. The remarkable significance of the Hopf structure seems to lie elsewhere. Fuzzy spaces approximate space-time algebras. SF2 (J) is an approximation to the Euclidean version of (causal) de Sitter space homeomorphic to S 1 ×R, or for large radii of S 1 , of Minkowski space [131]. The Hopf structure then gives orderly rules for splitting and joining fuzzy spaces. The decomposition of (ρ⊗σ)∆ into irreducible ∗-representations (IRR’s) τ gives fusion rules for states in ρ and σ combining to become τ , while ∆ on an IRR such as τ gives amplitudes for τ becoming ρ and σ. In other words, ∆ gives Clebsch-Gordan coefficients for space-times joining and splitting. Equivariance means that these processes occur compatibly with G-invariance: G gives selection rules for these processes in the ordinary sense. The Hopf structure has a further remarkable consequence: An observable on a state in τ can be split into observables on its decay products in ρ and σ. There are similar results for field theories on τ , ρ and σ, indicating the possibility of many orderly calculations. These mathematical results are very suggestive, but their physical consequences are yet to be explored. The coproduct ∆ on the matrix algebra M at(N + 1) is not unique. Its

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choice depends on the group actions we care to preserve, that of SU (2) for SF2 , that of SU (N + 1) for the fuzzy CP N algebra CPFN and so forth. It is thus the particular equivariance that determines the choice of ∆. We focus attention on the fuzzy sphere for specificity in what follows, but one can see that the arguments are valid for any fuzzy space. Proofs for the fuzzy sphere are thus often assumed to be valid for any fuzzy space without comment. Fuzzy algebras such as CPFN can be further “q-deformed” into certain quantum group algebras relevant for the study of D-branes. This theory has been developed in detail by Pawelczyk and Steinacker [132]. 11.2

Basics

Here we collect some of the basic formulae related to the group SU (2) and its representations which will be used later in the chapter. The canonical angular momentum generators of SU (2) are Ji (i = 1, 2, 3). The unitary irreducible representations (UIRR’s) of SU (2) act for any half-integer or integer J on Hilbert spaces H J of dimension 2J + 1. They have orthonormal basis |J, M i, with J3 |J, M i = M |J, M i and obeying conventional phase conventions. The unitary matrix D J (g) of g ∈ SU (2) acting on HJ has matrix elements hJ, M |D J (g)|J, N i = DJ (g)M N in this basis. Let Z V = dµ(g) (11.1) SU (2)

be the volume of SU (2) with respect to the Haar measure dµ. It is then well-known that [133] Z V δJK δil δjk , (11.2a) dµ(g)DJ (g)ij DK (g)†kl = 2J +1 SU (2) 2J + 1 X J J ¯ ij Dij (g) D (g 0 ) = δg (g 0 ) , (11.2b) V J ,ij

where bar stands for complex conjugation and δg is the δ-function on SU (2) supported at g: Z dµ(g 0 ) δg (g 0 )α(g 0 ) = α(g) (11.3) SU (2)

for smooth functions α on G.

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We have also the Clebsch-Gordan series X DµL2 m2 = C(K , L , J ; µ1 , µ2 ) C(K , L , J ; m1 , m2 ) DµJ1 +µ2 ,m1 +m2

DµK1 m1

J

(11.4)

where C’s are the Clebsch-Gordan coefficients. 11.3

The Group and the Convolution Algebras

The group algebra consists of the linear combinations Z dµ(g) α(g) g , dµ(g) = Haar measure on G

(11.5)

G

of elements g of G, α being any smooth C-valued function on G. The algebra product is induced from the group product: Z Z Z Z dµ(g) α(g) g dµ(g 0 ) β(g 0 ) g 0 := dµ(g) dµ(g 0 )α(g) β(g 0 )(gg 0 ) . G

G

G

G

We will henceforth omit the symbol G under integrals. The right hand side of (11.6) is Z dµ(s) (α ∗c β)(s) s

where ∗c is the convolution product: Z (α ∗c β)(s) = dµ(g)α(g) β(g −1 s) .

(11.6)

(11.7)

(11.8)

The convolution algebra consists of smooth functions α on G with ∗c as their product. Under the map Z dµ(g)α(g)g → α , (11.9) (11.6) goes over to α ∗c β so that the group algebra and convolution algebra are isomorphic. We call either as G∗ . Using invariance properties of dµ, (11.9) shows that under the action Z Z dµ(g) α(g) g → h1 dµ(g)α(g)g h−1 2 Z = dµ(g)α(g)h1 gh−1 hi ∈ G , (11.10) 2 , α → α0 where

α0 (g) = α(h−1 1 gh2 ).

(11.11)

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Thus the map (11.9) is compatible with left- and right- G-actions. The group algebra is a ∗-algebra [128], the ∗-operation being Z ∗ Z dµ(g) α(g) g = dµ(g) α ¯ (g)g −1 . (11.12)

The ∗-operation in G∗ is

∗ : α → α∗ ,

α∗ (g) = α(g ¯ −1 ) .

(11.13)

Under the map (11.9), Z

dµ(g)α(g)g

∗

→ α∗

(11.14)

since dµ(g) = i T r(g −1 dg) ∧ g −1 dg ∧ g −1 dg = −dµ(g −1 ) .

(11.15)

The minus sign Rin (11.15) is compensated by flips in “limits of integration”, R thus dµ(g) = dµ(g −1 ) = V . Hence the map (11.9) is a ∗-morphism, that is, it preserves “hermitian conjugation”. 11.4

A Prelude to Hopf Algebras

This section reviews the basic ingredients that go into the definition of Hopf algebras. It also sets some notations and conventions which are standard in the literature. Our approach here will be illustrative and will closely follow the exposition of [135]. Unless stated otherwise, we always work over the complex number field C, but the definitions given below extend to any number field k without the need for further remarks. In the language of commutative diagrams, an algebra A is defined as the triple A ≡ (A , M , u) where A is a vector space, M : A ⊗ A → A and u : C → A are morphisms (linear maps) of vector spaces such that the following diagrams are commutative. A⊗A⊗A

id ⊗ M

M ⊗ id

? A⊗A

- A⊗A M

M

? - A

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- A⊗A

d

u⊗i

C⊗A

M

∼

- ? A

id ⊗

u

A⊗C

∼

In this definition M is called the product and u is called the unit. The commutativity of the first diagram simply implies the associativity of the product M , whereas for the latter it expresses the fact that u is the unit of the algebra. The arrows labeled by ∼ are the canonical isomorphisms of the algebra onto itself. Also in above and what follows id denotes the identity map. A coalgebra C is the triple C ≡ (C , ∆ , ε), where C is a vector space, ∆ : C → C ⊗ C and ε : C → C are morphisms of vector spaces such that the following diagrams are commutative. ∆

C

- C⊗C id ⊗ ∆

∆ ? C⊗C

∆ ⊗ id ∼

C⊗C

? - C⊗C⊗C

C 6

∼

- C⊗C

∆ ε ⊗ id

C⊗C

id ⊗

ε

In this definition ∆ is called the coproduct and ε is called the counit. The commutativity of the first diagram implies the coassociativity of the coproduct ∆, whereas for the latter it expresses the fact that ε is the counit of the coalgebra. An immediate example of a coalgebra is the vector space of n × n matrices M at(n), with the coproduct and the counit X eip ⊗ epj , ε(eij ) = δ ij , ∆(eij ) = 1≤p≤n

(e )αβ = δαi δβj . ij

(11.16)

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As eij , 1 ≤ i , j ≤ n is a basis for M at(n)∗ . ∆ can be extended by linearity to all of C ∗ . In what follows we adopt the “sigma” or “Sweedler” notation which is standard in literature and write for c ∈ C X ∆(c) = c1 ⊗ c 2 , (11.17)

which with the usual summation convention should have been n X ∆(c) = ci1 ⊗ ci2 .

(11.18)

i=1

One by one we are exhausting the steps leading to the definition of a Hopf algebra. The next step is to define the bialgebra structure. A bialgebra is a vector space H endowed with both an algebra and a coalgebra structure such that the following diagrams are commutative. M - H H ⊗H ∆⊗∆

? H ⊗H ⊗H ⊗H

∆

id ⊗ τ ⊗ id

? M ⊗M H ⊗H ⊗H ⊗H H ⊗H

M

? - H ⊗H - H

ε⊗ε

? C⊗C

ε

φ ? C

id

? - C

2 , since elements of S 2 might be tempted to call (11.16) as the coproduct of SF F are described by matrices in M at(n + 1). But, (11.16) is not equivariant under SU (2) 2 actions and therefore has no chance of being the appropriate coproduct for S F . ∗ One

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- H

φ−1

id

∆ ? - H ⊗H

u⊗u

- H

-

? C⊗C

u

C

ε

u

C

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C

In the above, τ : H ⊗ H → H ⊗ H is the twist map defined by τ (h1 ⊗ h2 ) = h2 ⊗h1 , ∀ h1,2 ∈ H. In terms of the sigma notation, the above four diagrams read X ∆(hg) = h1 g1 ⊗ h2 g2 , ε(hg) = ε(h)ε(g) ∆(1) = 1 ⊗ 1 ,

ε(1) = 1 .

(11.19)

Now, let S be a map from a bialgebra H onto itself. Then S is called an antipode if the following diagram is commutative.

H

ε

u

- C

∆ ? H ⊗H

- H 6 M

id ⊗ S , S ⊗ id

In terms of the sigma notation, this means X X S(h1 )h2 = h1 S(h2 ) = ε(h)1 ,

- H ⊗H 1∈H.

(11.20)

By definition a Hopf algebra is a bialgebra with an antipode. Perhaps, the simplest example for a Hopf algebra is the group algebra, and it also happens to be the one of our interest. The group algebra G∗ can be made into a Hopf algebra by defining the coproduct ∆, the counit ε and antipode S as follows: ∆(g) = g ⊗ g ,

ε(g) = 1 ∈ C ,

S(g) = g

−1

.

(11.21a) (11.21b) (11.21c)

Here ε is the one-dimensional trivial representation of G and S maps g to its inverse. ∆, ε and S fulfill all the consistency conditions implied by the commutativity of the diagrams defining the Hopf algebra structure as can easily be verified. For instance we have S(g)g = g −1 g = 1 = ε(g)1 , and similarly gS(g) = ε(g)1 for any g ∈ G.

(11.22)

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The ∗-Homomorphism G∗ → SF2

As mentioned earlier henceforth, we identify the group and convolution algebras and denote either by G∗ . We specialize to SU (2) for simplicity. We work with group algebra and group elements, but one may prefer the convolution algebra instead for reasons of rigor. (The image of g is the Dirac distribution δg and not a smooth function.) The fuzzy sphere algebra is not unique, but depends on the angular momentum J as shown by the notation SF2 (J), which is M at(2J + 1). Let SF2 = ⊕J SF2 (J) = ⊕J M at(2J + 1) .

(11.23)

Let ρ(J) be the unitary irreducible representation of angular momentum J for SU (2): ρ(J) :

g → hρ(J), gi := D J (g) .

(11.24)

We have hρ(J), gi hρ(J), hi = hρ(J), ghi .

(11.25)

J

Choosing the ∗-operation on D (g) as hermitian conjugation, ρ(J) extends by linearity to a ∗-homomorphism on G∗ : Z D E Z ρ(J), dµ(g)α(g)g = dµ(g)α(g)DJ (g) D Z ∗ E Z ρ(J), dµ(g)α(g)g = dµ(g)¯ α(g)D J (g)† . (11.26)

ρ(J) is also compatible with group actions on G∗ (that is, it is equivariant with respect to these actions): Z D E ρ(J), dµ(g)α(g)h1 gh−1 = 2 Z dµ(g)α(g)DJ (h1 )DJ (g)DJ (h−1 hi ∈ SU (2) . (11.27) 2 ) , As by (11.2a), Z E D 2K + 1 K † dµ(g)(Dij ) (g)g = eji (J)δKJ , ρ(J), V ji e (J)rs = δjr δis , i, j, r, s ∈ [−J , · · · 0 , · · · , J] ,

(11.28)

we see by (11.25) and (11.26) that ρ(J) is a ∗-homomorphism from G∗ to SF2 (J) ⊕ {0}, where {0} denotes the zero elements of ⊕K6=J SF2 (K), the ∗operation on SF2 (J) being hermitian conjugation. Identifying SF2 (J) ⊕ {0} with SF2 (J), we thus get a ∗-homomorphism ρ(J) : G∗ → SF2 (J). It is also

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seen to be equivariant with respect to SU (2) actions, they are given on the basis eji (J) by DJ (h1 )eji (J)DJ (h2 )−1 . We can think of (11.26) as giving a map ρ:g

→

hρ(.) , gi := g(.)

(11.29)

to a matrix valued function g(.) on the space of UIRR’s of SU (2) where g(J) = hρ(J) , gi .

(11.30)

The homomorphism property (11.26) is expressed as the product g(.)h(.) of these functions where g(.)h(.)(J) = g(J)h(J)

(11.31)

is the point-wise product of matrices. This point of view is helpful for later discussions. As emphasized earlier, this discussion works for any group G, its UIRR’s, and its fuzzy spaces barring technical problems. Thus G∗ is ∗isomorphic to the ∗-algebra of functions g(.) on the space of its UIRR’s τ , with g(τ ) = Dτ (g), the linear operator of g in the UIRR τ and g ∗ (τ ) = Dτ (g)† . A fuzzy space is obtained by quantizing an adjoint orbit G/H, H ⊂ G and approximates G/H. It is a full matrix algebra associated with a particular UIRR τ of G. There is thus a G-equivariant ∗-homomorphism from G∗ to the fuzzy space. At this point we encounter a difference with SF2 (J). For a given G/H we generally get only a subset of UIRR’s τ . For example CP 2 = SU (3)/U (2) is associated with just the symmetric products of just 3’s (or just 3∗ ’s) of SU (3). Thus the direct sum of matrix algebras from a given G/H is only homomorphic to G∗ . ˆ For a compact Henceforth we call the space of UIRR’s of G as G. ˆ group, G can be identified with the set of discrete parameters specifying all UIRR’s. The properties of a group G are captured by the algebra of matrixˆ with point-wise multiplication, this algebra being valued functions g(.) on G ∗ isomorphic to G . In terms of g(.), (11.21) translate to ∆ g(.) = g(.) ⊗ g(.) , (11.32a) ε(g(.)) = 1 ∈ C , S g(.) = g −1 (.) .

ˆ ⊗ G. ˆ Note that g(.) ⊗ g(.) is a function on G

(11.32b)

(11.32c)

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11.6

book

Hopf Algebra for the Fuzzy Spaces

Any fuzzy space has a Hopf algebra structure, we show it here for the fuzzy sphere. \ Let δJ be the δ-function on SU (2): δJ (K) := δJK .

(11.33)

(Since the sets of J and K are discrete, we have Kronecker delta and not a delta function). Then Z 2J + 1 ji J e (J) δJ = dµ(g)Dij (g)† g(.) (11.34) V Hence ∆(eji (J)δJ ) =

2J + 1 V

Z

J dµ(g)Dij (g)† g(.) ⊗ g(.) .

\ \ At (K, L) ∈ SU (2) ⊗ SU (2), this is Z 2J + 1 J ji (g)† DK (g) ⊗ DL (g) . ∆ e (J) (K, L) = dµ(g)Dij V

(11.35)

(11.36)

As δJ2 = δJ and δJ eji (J) = eji (J)δJ , we can identify eji (J)δJ with e (J): ji

eji (J)δJ ' eji (J) .

(11.37)

Then (11.35) or (11.36) show that there are many coproducts ∆ = ∆KL we can define and they are controlled by the choice of K and L: ∆ eji (J)δJ (K, L) := ∆KL eji (J) . (11.38)

From section (11.4), we know that technically a coproduct ∆ is a homomorphism from C to C ⊗ C so that only ∆JJ is a coproduct. But, we will be free of language and call all ∆KL as coproducts. Indeed, it is the very fact that K 6= L in general in (11.38) that gives SF2 its “generalized” Hopf alegbra structure. Let us now simplify the r.h.s. of (11.36). Using (11.4), (11.36) can be written as Z X 2J + 1 J dµ(g)Dij (g)† C(K, L, J 0 ; µ1 , µ2 ) × ∆ eji (J)δJ µ1 µ2 ,m1 m2 = V 0 J

0

0

C(K, L, J ; m1 , m2 ) DµJ1 +µ2 ,m1 +m2 , (11.39)

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with µ1 , µ2 and m1 , m2 being row and column indices. The r.h.s of (11.39) is C(K, L, J; µ1 , µ2 ) C(K, L, J; m1 , m2 )δj ,µ1 +µ2 δi,m1 +m2 = X C(K, L, J; µ01 , µ02 ) C(K, L, J; m01 , m02 ) × µ01 +µ02 =j m01 +m02 =i

0 0 0 0 eµ1 m1 (K) µ1 m1 ⊗ eµ2 m2 (L) µ2 m2 . (11.40) Hence we have the coproduct ∆KL eji (J) = X C(K, L, J; µ1 , µ2 ) C(K, L, J; m1 , m2 ) eµ1 m1 (K) ⊗ eµ2 m2 (L) . µ1 +µ2 =j m1 +m2 =i

(11.41) Writing C(K, L, J; µ1 , µ2 , j) = C(K, L, J; µ1 , µ2 )δµ1 +µ2 ,j for the first Clebsch-Gordan coefficient, we can delete the constraint j = µ1 + µ2 in summation. C(K, L, J; µ1 , µ2 , j) is an invariant tensor when µ1 , µ2 and j are transformed appropriately by SU (2). Hence (11.41) is preserved by the SU (2) action on j, µ1 , µ2 . The same is the case for the SU (2) action on i, m1 , m2 . In other words, the coproduct in (11.41) is equivariant with respect to both SU (2) actions. P Since any M ∈ M at(2J + 1) is i,j Mji eji (J), (11.41) gives X ∆KL (M ) = C(K, L, J; µ1 , µ2 ) C(K, L, J; m1 , m2 ) µ1 ,µ2 m1 ,m2

× Mµ1 +µ2 ,m1 +m2 eµ1 m1 (K) ⊗ eµ2 m2 (L) . (11.42) This is the basic formula. It preserves conjugation ∗ (induced by hermitian conjugation of matrices): ∆(M † ) = ∆(M )† . (11.43) It is instructive to check directly that ∆KL is a homomorphism, that is that ∆KL (M N ) = ∆KL (M )∆KL (N ). Starting from (11.41) we have 0 0 ∆KL eji (J) ∆KL ej i (J) = X X C(K, L, J; µ1 , µ2 , j) C(K, L, J; m1 , m2 , i) 0 0 µ1 ,µ2 m1 ,m2 µ01 µ2 0 m1 ,m2

× C(K, L, J; µ01 , µ02 , j 0 ) C(K, L, J; m01 , m02 , i0 ) eµ1 m1 (K) ⊗ eµ2 m2 (L) 0 0 0 0 × eµ1 m1 (K) ⊗ eµ2 m2 (L) . (11.44)

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Using (A ⊗ B)(C ⊗ D) = AC ⊗ BD, we have 0 0 0 0 eµ1 m1 (K) ⊗ eµ2 m2 (L) eµ1 m1 (K) ⊗ eµ2 m2 (L) = 0

0

0

0

eµ1 m1 (K)eµ1 m1 (K) ⊗ eµ2 m2 (L)eµ2 m2 (L) 0

0

= δm1 µ01 δm2 µ02 eµ1 m1 (K) ⊗ eµ2 m2 (L) .

(11.45)

To get the second line in (11.45) we have made use of 0 0 0 0 0 eµ1 m1 (K)eµ1 m1 (K) = eµ1 m1 (K)αγ eµ1 m1 (K)γβ = δm1 µ01 eµ1 m1 (K)αβ . αβ

(11.46)

Inserting (11.45) in (11.44) we get 0 0 ∆KL eji (J) ∆KL ej i (J) = X X C(K, L, J; µ1 , µ2 , j) C(K, L, J; m1 , m2 , i)× 0 0 µ1 ,µ2 m1 ,m2 µ01 µ2 0 m1 ,m2

0

0

C(K, L, J; µ01 , µ02 , j 0 ) C(K, L, J; m01 , m02 , i0 )δm1 µ01 δm2 µ02 eµ1 m1 (K)⊗eµ2 m2 (L) X X = C(K, L, J; µ1 , µ2 , j)C(K, L, J; m01 , m02 , i0 )×

X |

m1 ,m2

µ1 ,µ2 m01 ,m02

0 0 C(K, L, J; m1 , m2 , i)C(K, L, J; m1 , m2 , j 0 ) eµ1 m1 (K)⊗eµ2 m2 (L) {z

}

=δij 0

(11.47)

where the orthogonality of Clebsch-Gordan coefficients is used to obtain δij 0 for the factor with the under-brace. Thus, 0 0 ∆KL eji (J) ∆KL ej i (J) = X 0 0 C(K, L, J; µ1 , µ2 , j)C(K, L, J; m01 , m02 , i0 )δij 0 eµ1 m1 (K) ⊗ eµ2 m2 (L) µ1 ,µ2 m1 ,m2

0

= δij 0 ∆KL (eji ) . (11.48) Upon multiplying both sides of (11.48) by the coefficients Mji Nj 0 i0 , we finally get X X 0 0 ∆KL Mji eji (J) ∆KL Nj 0 i0 ej i (J) = ∆KL (M )∆KL (N ) ji

j 0 i0

0

= (M N )ji0 ∆KL (eji ) = ∆KL (M N ) , (11.49)

as we intended to demonstrate.

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It remains to record the fuzzy analogues of counit ε and antipode S. For the counit we have Z Z 2J + 1 2J + 1 J J ε eji (J)δJ = dµ(g)Dij (g)† ε g(.) = dµ(g)Dij (g)† 1 V V Z 2J + 1 J = dµ(g)Dij (g)† D0 (g) . (11.50) V Using equation (11.2a) and the fact that D 0 (g) is a unit matrix with only one entry which we denote by 00, we have \ ε eji (J)δJ (K) = δ0J δj0 δi0 , ∀ K ∈ SU (2) . (11.51) 00

For the antipode, we have Z 2J + 1 J dµ(g)Dij (g)† S g(.) S eji (J)δJ = V Z 2J + 1 J dµ(g)Dij (g)† g −1 (.) (11.52) = V or Z 2J + 1 J S eji (J)δJ (K) = dµ(g)Dij (g)† DK (g −1 ) . (11.53) V In an UIRR K we have C = e−iπJ2 as the charge conjugation matrix. ¯ K (g). Then since DK (g −1 ) = DK (g)† , It fulfills CDK (g)C −1 = D DK (g −1 ) = CDK (g)T C −1 ,

(11.54)

where T denotes transposition. We insert this in (11.53) and use (11.2a) to find Z 2J + 1 −1 J S eji (J)δJ k` (K) = dµ(g)Dij (g)† Cku DK (g)Tuυ Cυ` V Z 2J + 1 −1 J = dµ(g)Dij (g)† Cku DK (g)υu Cυ` V −1 = δJK Cku δui δυj Cυ` −1 = δJK Cki Cj` .

This can be simplified further. Since in the UIRR K, e−iπJ2 ki = δ−ki (−1)K+k = δ−ki (−1)K−i ,

and C

−1

(11.56)

T

= C , we find S eji (J)δJ

Thus

(11.55)

k`

(K) = δJK δ−ki δ−`j (−1)2K−i−j

= δJK (−1)2J−i−j e−i ,−j (J)k` .

S eji (J)δJ (K) = δJK (−1)2J−i−j e−i ,−j (J) .

(11.57) (11.58)

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11.7

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We recall from chapter 2 that the matrix M ∈ M at(2J + 1) can be interpreted as the wave function of a particle on the spatial slice SF2 (J). The Hilbert space for these wave functions is M at(2J + 1) with the scalar product given by (M, N ) = T rM † N , M, N ∈ SF2 (J). We can also regard M as a fuzzy two-dimensional Euclidean scalar field as we did earlier or even as a field on a spatial slice SF2 (J) of a threedimensional spacetime SF2 (J) × R. Let us look at the particle interpretation. Then (11.42) gives the amplitude, up to an overall factor, for M ∈ SF2 (J) splitting into a superposition of wave functions on SF2 (K) ⊗ SF2 (L). It models the process where a fuzzy sphere splits into two others [134]. The overall factor is the reduced matrix element much like the reduced matrix elements in angular momentum selection rules. It is unaffected by algebraic operations on SF2 (J), SF2 (K) or SF2 (L) and is determined by dynamics. Now (11.42) preserves trace and scalar product: T r∆KL (M ) = T rM , ∆KL (M ), ∆KL (N ) = (M, N ) .

(11.59)

phase × T r(P ⊗ Q)† ∆KL (M ) .

(11.60)

So (11.42) is a unitary branching process. This means that the overall factor is a phase. ∆KL (SF2 (J)) has all the properties of SF2 (J). So (11.42) is also a precise rule on how SF2 (J) sits in SF2 (K) ⊗ SF2 (L). We can understand “how ∆KL (M ) sits” as follows. A basis for SF2 (K) ⊗ SF2 (L) is eµ1 m1 (K) ⊗ eµ2 m2 (L). We can choose another basis where left- and rightangular momenta are separately diagonal by coupling µ1 and µ2 to give angular momentum σ ∈ [0, 21 , 1, . . . , K + L], and m1 and m2 to give angular momentum τ ∈ [0, 12 , 1, . . . , K + L]. In this basis, ∆KL (M ) is zero except in the block with σ = τ = J. So the probability amplitude for M ∈ SF2 (J) splitting into P ⊗ Q ∈ 2 SF (K) ⊗ SF2 (L) for normalized wave functions is Branching rules for different choices of M, P and Q are independent of the constant phase and can be determined. Written in full, (11.60) is seen to be just the coupling conserving left- and right- angular momenta of P † , Q† and M . That alone determines (11.60). An observable A is a self-adjoint operator on a wave function M ∈ P L R SF2 (J). Any linear operator on SF2 (J) can be written as Bα Cα where

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Bα , Cα ∈ SF2 (J) and BαL and CαR act by left- and right- multiplication: BαL M = Bα M , CαR M = M Cα . Any observable on SF2 (J) has an action on its branched image ∆KL (SF2 (J)): ∆KL (A)∆KL (M ) := ∆KL (AM ) . (11.61) By construction, (11.61) preserves algebraic properties of operators. ∆KL (A) can actually act on all of SF2 (K) ⊗ SF2 (L), but in the basis described above, it is zero on vectors with σ 6= J and/or τ 6= J. This equation is helpful to address several physical questions. For example if M is a wave function with a definite eigenvalue for A, then ∆KL (M ) is a wave function with the same eigenvalue for ∆KL (A). This follows from ∆KL (BM ) = ∆KL (B)∆KL (M ) and ∆KL (M B) = ∆KL (M )∆KL (B). Combining this with (11.59) and the other observations, we see that the mean value of ∆KL (A) in ∆KL (M ) and of A in M are equal. In summary, all this means that every operator on SF2 (J) is a constant of motion for the branching process (11.42). Now suppose R ∈ SF2 (K) ⊗ SF2 (L) is a wave function which is not necessarily of the form P ⊗ Q. Then we can also give a formula for the probability amplitude for finding R in the state described by M . Note that R and M live in different fuzzy spaces. The answer is constant × T rR† ∆KL (M ) . (11.62) If M, P, Q are fields with SF2 (I) (I = J, K, L) a spatial slice or spacetime, (11.60) is an interaction of fields on different fuzzy manifolds. It can give dynamics to the branching process of fuzzy topologies discussed above.

11.8

The Preˇ snajder Map

This section is somewhat disconnected from the material in the rest of the chapter. We recall that SF2 (J) can be realized as an algebra generated by the spherical harmonics Ylm (l ≤ 2J) which are functions on the two-sphere S 2 . Their product can be the coherent state ∗C Moyal ∗W product. But we saw that SF2 (J) is isomorphic to the convolution algebra of 3 J functions DM N on SU (2) ' S . It is reasonable to wonder how functions on S 2 and S 3 get related preserving the respective algebraic properties. The map connecting these spaces is described by a function on SU (2) × S 2 ≈ S 3 × S 2 and was first introduced by Preˇsnajder [65, 63]. We give its definition and introduce its properties here. It generalizes to any group G.

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Let ai , a†j (i = 1, 2) be Schwinger oscillators for SU (2) and let us also recall that for J = n2 |z , 2Ji =

(zi a†i )2J √ |0i , 2J!

X

|zi |2 = 1

(11.63)

are the normalized Perelomov coherent states. If U (g) is the unitary operator implementing g ∈ SU (2) in the spin J UIRR, the Preˇsnajder function [65, 63] PJ is given by J PJ (g, ~n) = hz , 2J|U (g)|z , 2Ji = DJJ (h−1 gh) ,

~n = z †~τ z , ~n · ~n = 1 , z1 −¯ z2 . h= z2 z¯1

(11.64)

Now ~n ∈ S 2 . As the phase change zi → zi eiθ does not effect PJ , besides g, it depends only on ~n. It is a function on SU (2) ' S 3 × SU (2)/U (1) ' S 3 × S 2 . (11.65)

J A basis of SU (2) functions for spin J is Dij . A basis of S 2 functions for ij spin J is E (J , .) where

E ij (J , ~n) = hz , 2J|eij (J)|z , 2Ji = DJ (h−1 )Ji DJ (h)jJ ,

no sum on J . (11.66)

J The transform of Dij to E ij (J, .) is given by Z (2J + 1) J ij dµ(g)P¯J (g, ~n)Dij (g) . E (J , ~n) = V

(11.67)

This can be inverted by constructing a function QJ on SU (2) × S 2 such that Z X d cos θdϕ J J ¯ ij dΩ(~n)QJ (g 0 , ~n)P¯J (g , ~n) = Dij (g 0 )D (g) , dΩ(~n) = , 4π S2 ij

(11.68) θ and ϕ being the polar and azimuthal angles on S 2 . Then using (2), we get Z J Dij (g 0 ) = dΩ(~n)QJ (g 0 , ~n)E ij (J , ~n) . (11.69) S2

Consider first J =

1 2.

In that case

1 + ~σ · ~n P¯ 12 (g , ~n) = g¯kl z¯k zl = g¯kl 2 lk

(11.70)

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where g is a 2 × 2 SU (2) matrix and σi are Pauli matrices. Since Z 1 (11.71) dΩ(~n)ni nj = δij , 3 2 S

we find

Q 12 (g 0 , ~n) = T r˜ g 0 (1 + 3~σ · ~n) ,

(11.72)

g 0 = 2 × 2 SU (2) matrix , g˜0 = transpose of

g0 .

For J = n2 , DJ (g) acts on the symmetric product of n C2 ’s and can be written as g ⊗ g ⊗ · · · ⊗ g and (11.70) gets replaced by {z } | N f actors

h 1 + ~σ · ~n iN P¯J (g , ~n) = T r¯ g . (11.73) 2 Then QJ (g 0 , ~n) is defined by (11.68). It exists, but we have not found a neat formula for it. As the relation between E ij and Ylm can be worked out, it is possible to suitably substitute Ylm for E ij in these formulae. These equations establish an isomorphism (with all the nice properties like preserving ∗ and SU (2)-actions) between the convolution algebra ρ(J) (G∗ ) at spin J and the ∗-product algebra of SF2 (J). That is because we saw that ρ(J)(G∗ ) and SF2 (J) ' M at(2J + 1) are isomorphic, while it is known that M at(2J + 1) and the ∗-product algebra of S 2 at level J are isomorphic. There are evident generalizations of PJ to other groups and their orbits.

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[1] A. P. Balachandran, video conference course on “Fuzzy Physics”, at http://www.phy.syr.edu/courses/Fuzzy Physics and http://bach.if.usp.br/FUZZY/. [2] J. Madore, The Fuzzy Sphere, Class. Quant. Grav. 9, 69 (1992). [3] J. Madore, An Introduction to Non-commutative Differential Geometry and its Physical Applications, Cambridge University Press, Cambridge (1995); [4] J. Hoppe, Quantum Theory of a massless relativistic surface and a two-dimensional bound state problem, PhD Thesis, MIT, 1982, http://www.aei.mpg.de/ hoppe/. [5] A.A. Kirillov, Encyclopedia of Mathematical Sciences,vol 4, p.230; B. Kostant, Lecture Notes in Mathematics, vol.170, Springer-Verlag(1970), p.87. [6] F. A. Berezin, General Concept of Quantization, Commun. Math. Phys. 40, 153 (1975). [7] H. Grosse, C. Klimˇcik, P. Preˇsnajder, Field Theory on a Supersymmetric Lattice, Commun. Math. Phys., 185 (1997) 155-175 and hep-th/9507074; H. Grosse, C. Klimˇcik, P. Preˇsnajder, N=2 Superalgebra and NonCommutative Geometry, hep-th/9603071; [8] H. Grosse, G. Reiter, The Fuzzy Supersphere, J. Geom. and Phys., 28 (1998) 349-383 and math-ph/9804013. [9] A. P. Balachandran, S. K¨ urk¸cu ¨oˇ glu and E. Rojas, The Star Product on the Fuzzy Supersphere, JHEP 0207, 056 (2002) [arXiv:hep-th/0204170]; [10] S. K¨ urk¸cu ¨oˇ glu, Ph.D. Thesis, Syracuse University, Syracuse NY, 2004. [11] J. Ambjorn and S. Catterall, Stripes from (Noncommutative) Stars, Phys. Lett. B 549, 253 (2002) [arXiv:hep-lat/0209106]. [12] See for example, P. H. Frampton Gauge Field Theories, The Benjamin Cummings Publishing Company, Menlo Park CA, (1987). [13] R.D. Sorkin Int.J. Theory. Phys. 30 (1991) 923; A. P. Balachandran, G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Finite Quantum Physics and Noncommutative Geometry Nucl. Phys. Proc. Suppl. 37C, 20 (1995) [arXiv:hep-th/9403067]. [14] S. Minwalla, M. Van Raamsdonk and N. Seiberg, Noncommutative Pertur-

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F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Quantum Mechanics As A Deformation Of Classical Mechanics, Lett. Math. Phys. 1, 521 (1977). H. Weyl Gruppentheorie und Quantenmechanik The theory of groups and quantum mechanics, New York, Dover Publications, 1950; H. Weyl, Quantum Mechanics And Group Theory, Z. Phys. 46, 1 (1927). J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics, World Scientific (1985). A. M. Perelomov Generalized Coherent States and their Applications, Springer-Verlag (1986). G. Alexanian, A. Pinzul and A. Stern, Generalized Coherent State Approach to Star Products and Applications to the Fuzzy Sphere, Nucl. Phys. B 600, 531 (2001), [arXiv:hep-th/0010187]. A. Voros, Wentzel-Kramers-Brillouin method in the Bargmann representation Phys. Rev. A40, 6814 (1989). R.L. Stratonovich, On distributions in representation space, Sov. Phys. JETP 4 (1957), 891-898 . J. C. V´ arilly and J. M. Gracia-Bond´ıa, The Moyal representation for spin, Ann. Phys. (NY) 190 (1989), 107–148. Jos´e F. Cari˜ nena, J. M. Gracia-Bond´ıa and J. C. V´ arilly, Relativistic quantum mechanics in the Moyal representation, J. Phys. A 23 (1990), 901-933. R. Haag, Local quantum physics : fields, particles, algebras. Berlin, SpringerVerlag (1996). M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66, 157 (2003) [arXiv:q-alg/9709040]. A. B. Hammou, M. Lagraa and M. M. Sheikh-Jabbari, Coherent state induced star-product on R(lambda)**3 and the fuzzy sphere, Phys. Rev. D 66, 025025 (2002) [arXiv:hep-th/0110291]. P. Preˇsnajder, The origin of chiral anomaly and the non-commutative geometry, J. Math. Phys. 41, 2789 (2000) [arXiv:hep-th/9912050]; H. Grosse, C. Klimˇcik and P. Preˇsnajder, Towards finite quantum field theory in non-commutative geometry, Int. J. Theor. Phys. 35, 231 (1996) [arXiv:hep-th/9505175]. H. Grosse and P. Preˇsnajder, The Construction on non-commutative manifolds using coherent states, Lett. Math. Phys. 28, 239 (1993); A. P. Balachandran and G. Immirzi, Fuzzy Nambu-Goldstone physics, Int. J. Mod. Phys. A 18 (2003) 5981,arXiv:hep-th/0212133. M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proc. Roy. Soc. Lond. A 392, 45 (1984). N.E. Wegge Olsen, K-theory and C ∗ -Algebras-a Friendly Approach, Oxford University Press, Oxford, 1993. U. Carow-Watamura and S. Watamura, Chirality and Dirac operator on noncommutative sphere,, Commun. Math. Phys. 183, 365 (1997) [arXiv:hepth/9605003]. R. Jackiw and C. Rebbi, Spin From Isospin In A Gauge Theory, Phys. Rev. Lett. 36, 1116 (1976).

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Phys. Rev. Lett. 42, 1195 (1979); K. Fujikawa, Path Integral For Gauge Theories With Fermions, Phys. Rev. D 21, 2848 (1980) [Erratum-ibid. D 22, 1499 (1980)]. [97] A. P. Balachandran, G. Marmo, V. P. Nair and C. G. Trahern, A Nonperturbative Proof Of The Nonabelian Anomalies, Phys. Rev. D 25, 2713 (1982). [98] S. Randjbar-Daemi and J. A. Strathdee, On the overlap formulation of chiral gauge theory, Phys. Lett. B 348, 543 (1995) [arXiv:hep-th/9412165]; S. Randjbar-Daemi and J. A. Strathdee, Gravitational Lorentz anomaly from the overlap formula in two-dimensions, Phys. Rev. D 51, 6617 (1995) [arXiv:hep-th/9501012]; S. Randjbar-Daemi and J. A. Strathdee, Chiral fermions on the lattice, Nucl. Phys. B 443, 386 (1995) [arXiv:hep-lat/9501027]; S. Randjbar-Daemi and J. A. Strathdee, Consistent and covariant anomalies in the overlap formulation of chiral gauge theories, Phys. Lett. B 402, 134 (1997) [arXiv:hep-th/9703092]; M. Luscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys. Lett. B 428, 342 (1998) [arXiv:hep-lat/9802011]; M. Luscher, Weyl fermions on the lattice and the non-abelian gauge anomaly, Nucl. Phys. B 568, 162 (2000) [arXiv:hep-lat/9904009]; H. Neuberger, Chiral symmetry outside perturbation theory, arXiv:heplat/9912013; W. Kerler, Dirac operator normality and chiral fermions, Chin. J. Phys. 38, 623 (2000) [arXiv:hep-lat/9912022]; J. Nishimura and M. A. Vazquez-Mozo, Noncommutative chiral gauge theories on the lattice with manifest star-gauge invariance, JHEP 0108, 033 (2001) [arXiv:hep-th/0107110]. [99] A. P. Balachandran, T. R. Govindarajan and B. Ydri, The fermion doubling problem and noncommutative geometry, Mod. Phys. Lett. A 15, 1279 (2000) [arXiv:hep-th/9911087]. A. P. Balachandran, T. R. Govindarajan and B. Ydri, The fermion doubling problem and noncommutative geometry. II, arXiv:hep-th/000621. [100] A. Bassetto and L. Griguolo, Journ. of Math. Phys. 32 (1991) 3195. [101] H. Aoki, S. Iso and K. Nagao, Chiral anomaly on fuzzy 2-sphere, Phys. Rev. D 67, 065018 (2003) [arXiv:hep-th/0209137]. [102] J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C 16, 161 (2000) [arXiv:hep-th/0001203]; X. Calmet, B. Jurco, P. Schupp, J. Wess and M. Wohlgenannt, The standard model on non-commutative space-time, Eur. Phys. J. C 23, 363 (2002) [arXiv:hep-ph/0111115]. [103] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 9909, 032 (1999) [arXiv:hep-th/9908142]. [104] B. Ydri, Ph.D. Thesis, Syracuse University, NY,2001, arXiv:hepth/0110006; B. Ydri, Noncommutative chiral anomaly and the Dirac-Ginsparg-Wilson operator, JHEP 0308, 046 (2003) [arXiv:hep-th/0211209].

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[105] H. Aoki, S. Iso and K. Nagao, Ginsparg-Wilson relation, topological invariants and finite noncommutative geometry, Phys. Rev. D 67, 085005 (2003) [arXiv:hep-th/0209223]; [106] H. Aoki, S. Iso and K. Nagao, Ginsparg-Wilson relation and ’t HooftPolyakov monopole on fuzzy 2-sphere, Nucl. Phys. B 684, 162 (2004) [arXiv:hep-th/0312199]; [107] H. Aoki, S. Iso, T. Maeda and K. Nagao, Dynamical generation of a nontrivial index on the fuzzy 2-sphere, Phys. Rev. D 71, 045017 (2005) [Erratumibid. D 71, 069905 (2005)] [arXiv:hep-th/0412052]. [108] M. Scheunert, W. Nahm and V. Rittenberg, Graded Lie Algebras: Generalization Of Hermitian Representations, J. Math. Phys. 18, 146 (1977). [109] M. Scheunert, W. Nahm and V. Rittenberg, Irreducible Representations Of The Osp(2,1) And Spl(2,1) Graded Lie Algebras, J. Math. Phys. 18, 155 (1977). [110] A. Pais and V. Rittenberg, Semisimple Graded Lie Algebras, J. Math. Phys. 16, 2062 (1975) [Erratum-ibid. 17, 598 (1976)]. [111] B. Dewitt, Supermanifolds, Cambridge University Press, Cambridge (1985); M. Scheunert, The Theory of Lie Superalgebras, Springer-Verlag, Berlin (1979). [112] J. F. Cornwell, Group Theory in Physics Vol. III, Academic Press, San Diego (1989). [113] L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras, hepth/9607161. [114] M. Chaichian, D. Ellinas and P. Preˇsnajder, Path Integrals And Supercoherent States, J. Math. Phys. 32, 3381 (1991). [115] A. El Gradechi and L. M. Nieto, Supercoherent states, superKahler geometry and geometric quantization, Commun. Math. Phys.,175 (1996) 521, and hep-th/9403109; A. M. El Gradechi, On the supersymplectic homogeneous superspace underlying the OSp(1/2) coherent states,” J. Math. Phys. 34, 5951 (1993). [116] M. Bordemann, M. Brischle, C. Emmrich and S. Waldmann, subalgebras with convering star products in deformation quantization: An algebraic construction for CP n , J. Math. Phys., 37 (1996) 6311; q-alg/9512019; M. Bordemann, M. Brischle, C. Emmrich and S. Waldmann, Lett. Math. Phys., 36 (1996) 357; S. Waldmann, Lett. Math. Phys., 44 (1998) 331. [117] F. A. Berezin and V. N. Tolstoi, The Group With Grassmann Structure Uosp(1,2), Commun. Math. Phys. 78, 409 (1981). F. A. Berezin, Introduction to Superanalysis, D.Reidel Publishing Company, Dordrecht, Holland (1987). [118] A. P. Balachandran, G.Marmo, B. S. Skagerstam and A. Stern, Supersymmetric Point Particles And Monopoles With No Strings, Nucl. Phys. B164 (1980) 427; G. Landi and G. Marmo, Phy. Lett. B193 (1987) 61-66. Extensions Of Lie Superalgebras And Supersymmetric Abelian Gauge Fields, [119] C. Fronsdal Essays on Supersymmetry. Mathematical Physics Studies Vol-

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ume 8, Editor: Fronsdal, C. , Dordrecht, Reidel Pub.Co. 1986. [120] J. Wess, J. Bagger Princeton series in physics: supersymmetry and supergravity, Princeton, Princeton University Press, 1983. [121] A. P. Balachandran, A. Pinzul and B. Qureshi, SUSY anomalies break N = 2 to N = 1: The supersphere and the fuzzy supersphere, arXiv:hepth/0506037. [122] C. Klimˇcik, A nonperturbative regularization of the supersymmetric Schwinger model, Commun. Math. Phys. 206, 567 (1999) [arXiv:hepth/9903112]; C. Klimˇcik, An extended fuzzy supersphere and twisted chiral superfields, Commun. Math. Phys. 206, 587 (1999) [arXiv:hep-th/9903202]. [123] S. K¨ urk¸cu ¨oˇ glu, Non-linear sigma models on the fuzzy supersphere, JHEP 0403, 062 (2004) [arXiv:hep-th/0311031]. [124] E. Witten, A Supersymmetric Form Of The Nonlinear Sigma Model In Two-Dimensions, Phys. Rev. D 16, 2991 (1977). [125] P. Di Vecchia and S. Ferrara, Classical Solutions In Two-Dimensional Supersymmetric Field Theories, Nucl. Phys. B 130, 93 (1977); [126] L. Frappat, P. Sorba and A. Sciarrino, “Dictionary on Lie superalgebras,” arXiv:hep-th/9607161. [127] C. Fronsdal, in “Essays On Supersymmetry”, ed. by C. Fronsdal, D. Reidel Publishing Company, Dordrecht, 1986. [128] M. E. Sweedler Hopf Algebras, W. A. Benjamin, New York, 1969. A. A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, Berlin, 1976. [129] G. Mack and V. Schomerus, QuasiHopf quantum symmetry in quantum theory, Nucl. Phys. B 370, 185 (1992); G. Mack and V. Schomerus in New symmetry principles in Quantum Field Theory, Edited by J.Frohlich et al., Plenum Press, New York, 1992; G. Mack and V. Schomerus, Quantum symmetry for pedestrians, preprint, DESY-92-053. [130] A. P. Balachandran and S. K¨ urk¸cu ¨oˇ glu, Topology change for fuzzy physics: Fuzzy spaces as Hopf algebras, arXiv:hep-th/0310026. [131] R. Figari, R. H¨ oegh-Krohn and C. R. Nappi, Interacting relativistic boson fields in the de Sitter universe with two space-time dimensions, Commun. Math. Phys. 44, 265 (1975). [132] J. Pawelczyk and H. Steinacker, A quantum algebraic description of Dbranes on group manifolds, Nucl. Phys. B 638, 433 (2002) [arXiv:hepth/0203110]. [133] A. P. Balachandran and C. G. Trahern Lectures on Group Theory for Physicists, Monographs and Textbooks in Physical Science, Bibliopolis, Napoli, 1984. [134] A. P. Balachandran, E. Batista, I. P. Costa e Silva and P. TeotonioSobrinho, Quantum topology change in (2+1)d, Int. J. Mod. Phys. A 15, 1629 (2000) [arXiv:hep-th/9905136]; A. P. Balachandran, E. Batista, I. P. Costa e Silva and P. Teotonio-Sobrinho, The spin-statistics connection in quantum gravity, Nucl. Phys. B 566, 441

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(2000) [arXiv:hep-th/9906174]; A. P. Balachandran, E. Batista, I. P. Costa e Silva and P. Teotonio-Sobrinho, A novel spin-statistics theorem in (2+1)d Chern-Simons gravity, Mod. Phys. Lett. A 16, 1335 (2001) [arXiv:hep-th/0005286]; [135] S. Dascalescu, C. Nastasescu, S. Raianu Hopf algebras : an introduction, New York, Marcel Dekker, 2001 . Relation of Fuzzy Physics to Brane physics have been investigated. Some articles on this subject are: [136] A. Y. Alekseev, A. Recknagel and V. Schomerus, Non-commutative worldvolume geometries: Branes on SU(2) and fuzzy spheres, JHEP 9909, 023 (1999) [arXiv:hep-th/9908040]. A. Y. Alekseev, A. Recknagel and V. Schomerus, Open strings and noncommutative geometry of branes on group manifolds, Mod. Phys. Lett. A 16, 325 (2001) [arXiv:hep-th/0104054]. [137] C. Klimˇcik and P. Severa, Open strings and D-branes in WZNW models, Nucl. Phys. B 488, 653 (1997) [arXiv:hep-th/9609112]. [138] A. Y. Alekseev and V. Schomerus, D-branes in the WZW model, Phys. Rev. D 60, 061901 (1999) [arXiv:hep-th/9812193]. [139] K. Gawedzki, Conformal field theory: A case study, arXiv:hep-th/9904145. [140] H. Garcia-Compean and J. F. Plebanski, D-branes on group manifolds and deformation quantization, Nucl. Phys. B 618, 81 (2001) [arXiv:hepth/9907183]. [141] R. C. Myers, Dielectric-branes, JHEP 9912, 022 (1999) [arXiv:hepth/9910053]. [142] S. P. Trivedi and S. Vaidya, Fuzzy cosets and their gravity duals, JHEP 0009, 041 (2000) [arXiv:hep-th/0007011]. [143] S. R. Das, S. P. Trivedi and S. Vaidya, Magnetic moments of branes and giant gravitons, JHEP 0010, 037 (2000) [arXiv:hep-th/0008203].

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Index

Action scalar field, 9, 10, 35, 43, 45 sigma models, 63, 70 SUSY, 124, 131, 134, 135, 148 Yang-Mills, 81, 82 Angular momentum, 7–9, 11, 13, 81, 82, 88–90, 92–94, 101, 137, 165 left, 8, 88, 91 matrices, 30 orbital, 8, 9, 63, 92 right, 8, 88 Antipode, 158, 164 Axial anomaly, 85, 93, 95, 96

Coherent states, 19, 20, 28, 114 resolution of identity, 20 Commutative diagram, 151, 155 Complex Structure, 11, 30, 56, 57, 131 Connected diagrams, 39, 40 Connection, 32, 47, 65, 66, 68, 79–81, 93, 96 Continuum limit, 43 Convolution, 151, 154, 159 Coproduct, 156 for SF2 , 162 Correlation functions, 39 Counit, 156, 158, 164 CP 1 model, 134, 135 Curvature, 47, 66, 80, 82

Belavin-Polyakov bound, 64, 67, 71 Berezin, F.A., vii Bialgebra, 151, 158 Bott projector, 61, 73 BPS states, 71

Dirac operator, 85–89, 92–96, 119, 142, 145 index of,, 96 index of, 87, 93 Disconnected diagrams, 39, 40

C2 , 5, 93 C (2,1) , 100, 132 Casimir operator, 13, 103, 104, 140, 142 CBH (Campbell-Baker-Hausdorff) formula, 19 Chern number, 47, 49, 65, 81, 93, 145 Chirality, 55, 88, 90, 96, 143 Clebsch-Gordan coefficients, 38, 152, 154, 163 CN +1 , 5 Coalgebra, 151, 156

Euclidean action, 9, 10, 35, 63, 88 for a scalar field, 35 for fuzzy sphere, 88 Fock space, 6, 7, 13, 14, 69, 112 Fourier transform, 21 Functional integral, 64, 88, 116 Fuzzy S 3 , 5, 6 Fuzzy Dirac operator, 89 179

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gauged, 91 spectrum, 94 Fuzzy space, 97, 151, 152 Fuzzy sphere, 5–8, 28, 32, 35, 54, 88, 119, 165 Fuzzy supersphere, 99, 112 Gauge groups, 77–79 Gauge theories, 91 Gauge theory, 47, 60, 66, 83, 91 Gell-Mann matrices, 11, 80 Gibbs state, 26 Ginsparg-Wilson algebra, 85, 86, 89, 92, 96 Grade adjoint, 101–103, 130 Grassmann algebra, 107 Groenewold, H.J., 17 Groenewold-Moyal plane, See also Moyal plane, 23 Grosse H., 99 Group algebra, 151–155, 158, 159 Haar measure, 153 Highest weight state, 33, 114, 115, 121–124 Holstein-Primakoff construction, 10, 14, 15 Hopf algebra, 151, 155, 158 Hoppe, J., vii Ideal two-sided, 17 Index theory, 95, 96, 146, 149 Instanton, 47, 54, 61, 64, 81, 82, 89, 93, 95, 96, 145 Invariance, cyclic, 24, 27 Kirillov, A.A., vii KMS (Kubo-Martin-Schwinger) state, 26 Kostant, B., vii Laplacian, fuzzy, 9 Loop expansion, 31, 37 Madore, J., vii

Manifold flag, 60 Grassmanian, 60 M at(n + 1), 7, 8, 28, 48, 72, 157 Module, 30, 47–50, 54, 81, 82, 93–95, 131, 145, 146 free module, 47–49 projective module, 30, 47, 49, 50, 54, 68, 81, 93, 95, 131, 145 Monopole, 47, 54, 81, 89, 93, 94 Monte-Carlo Metropolis, 44 Moyal Plane, 20, 22, 23, 96 Moyal, J.E., 17 Noncommutative geometry, 85 Nonlinear sigma model, 59, 131, 135 Normal ordering, 21, 43 Number operator, 6, 12, 15, 69, 70, 108, 112, 135 Numerical simulations, 43 Observables, 8 Operator bidifferential, 22 polar decomposition of, 10 symbol of, 25, 29 OSp(2, 1), 106 osp(2, 1), 100 irreducible representations of, 103 OSp(2, 2), 106 osp(2, 2), 100 irreducible representations of, 104 overcomplete, 20 Partial isometry, 60, 65 Partition function, 37, 43, 44, 82 Perelomov coherent state, See also Coherent states, 29 Pointwise product, 31 Polar decomposition, 10, 14 Polarization tensors, 9, 35 Positive map, 27 Preˇsnajder map, 166 Preˇsnajder, P., 166 Propagator, 40

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181

Index

Scalar field, 9, 10, 165 Scalar product, 7, 13, 72, 165 Schwinger construction, 7, 13, 15, 112, 113 Sections of bundles, 49, 56 (2,2) on SF , 127 Six-J symbol, 38, 42 Spherical harmonics, 9, 31 Spinor SU (2), 101 Majorana, 110, 111 Star product, 17, 19–24, 28, 31, 32, 108, 125–127 Voros, 22, 28 Weyl-Moyal product, 32, 33 ∗-algebra, 17, 18, 26, 85, 152, 155, 160 SU (2), 7 SU (N + 1), 11–15, 153 Supergroup, 99, 106 Supersphere, 99, 108, 109, 112, 145 Supersymmetry, 99 Supertrace, 102, 119, 135

book

Tensor product, 30, 115 graded symmetric, 115 symmetric, 30 Tracial state, 26 UV-IR mixing, 42, 43 Wess-Zumino model, 117 Weyl map, 23, 37 Weyl symbol, 23, 25 Weyl, H., 17 Wigner, E.P., 17 Winding number, 60, 62, 64, 68, 71, 76, 134 Zero modes, 87, 91, 93, 96, 149 of Dirac operator, 87, 93