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Mathematical Physics Studies Mathematical Physics Studies Series Editors Giuseppe Dito, Institut de Mathematiques de Bourgogne, Universite de Bourgogne, Dijon, France Edward Frenkel, Department of Mathematics, University of California at Berkley, Berkeley, CA, USA Sergei Gukov, California Institute of Technology, Pasadena, CA, USA Yasuyuki Kawahigashi, Department of Mathematical Sciences, The University of Tokyo, Tokyo, Japan Maxim Kontsevich, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France Nicolaas P. Landsman, Chair of Mathematical Physics, Radboud Universiteit Nijmegen, Nijmegen, Gelderland, The Netherlands Bruno Nachtergaele, Department of Mathematics, University of California, Davis, CA, USA Hal Tasaki, Department of Physics, Gakushuin University, Tokyo, Japan The series publishes original research monographs on contemporary mathematical physics. The focus is on important recent developments at the interface of Mathe- matics, and Mathematical and Theoretical Physics. These will include, but are not restricted to: application of algebraic geometry, D-modules and symplectic geom- etry, category theory, number theory, low-dimensional topology, mirror symmetry, string theory, quantum field theory, noncommutative geometry, operator algebras, functional analysis, spectral theory, and probability theory. More information about this series at http://www.springer.com/series/6316 Taro Kimura Instanton Counting, Quantum Geometry and Algebra Taro Kimura Institut de Mathématiques de Bourgogne Université Bourgogne Franche-Comté Dijon, France ISSN 0921-3767 ISSN 2352-3905 (electronic) Mathematical Physics Studies ISBN 978-3-030-76189-9 ISBN 978-3-030-76190-5 (eBook) https://doi.org/10.1007/978-3-030-76190-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Kana, Taïchi, Shota, Sayako, and MJR Preface Gauge Theory in Physics and Mathematics Since Yang–Mills’ proposal to extend gauge symmetry to non-Abelian symmetry [81], gauge theory has been playing a crucial role in theoretical physics as a ubiq- uitous framework to describe fundamental interactions: electroweak interaction [29, 75], quantum chromodynamics (QCD) [23, 30–32, 35, 53, 54], and gravity [74]. In addition to the significant role in theoretical physics, the influence of gauge theory is not restricted to physics, but also extended to wide-ranging fields of mathematics: The study of self-duality equations in four dimensions [6, 7], which leads to the so-called Atiyah–Drinfeld–Hitchin–Manin (ADHM) construction of the instantons [5]; Morse theory [1, 76] in the relation to algebraic geometry; Donaldson invariants of four-manifolds [20]; Topological invariants of knots, known as Jones polynomial [36], from Chern–Simons gauge theory [78]; Seiberg–Witten invariant [79] moti- vated by Seiberg–Witten theory of N = 2 supersymmetric gauge theory [68, 69]. In fact, these developments have been motivating various interplay between physics and mathematics up to now. The aim of this monograph is to present new mathematical concepts emerging from such intersections of physics and mathematics. Universality of QFT In general, Quantum Field Theory (QFT) is a universal methodology to describe many-body interacting systems, which involves quite broad applications to particle physics, nuclear physics, astrophysics and cosmology, condensed-matter physics, and more. In order to discuss the origin of its universality, one cannot say anything without mentioning the role of symmetry on the low energy behavior in the vicinity of the vacuum/ground state of the system, e.g., spacetime/internal symmetry, global/local symmetry, and non-local symmetry. One may obtain several constraints on the spectrum, and also the conservation law from the symmetry argument, which provide useful information to discuss the vii viii Preface effective description of the low energy behavior. However, it is not straightforward to understand the vacuum structure of QFT, since it would be strongly coupled in many cases in the low energy regime, due to the so-called asymptotic freedom [34, 64, 65], and one cannot apply the systematic approach based on the perturbation theory with respect to a small coupling constant as in the weakly coupled regime. In order to overcome this difficulty, it would be plausible to incorporate additional symmetry, i.e., supersymmetry, which provides further analytic framework for the study of QFT. In fact, supersymmetric extension of gauge theory, which we mainly explore in this monograph, shows a lot of geometric and algebraic properties in the low energy regime. N = 2 Supersymmetry In this monograph, we mainly focus on N = 2 supersymmetric gauge theory in four dimensions and explore the associated geometric and algebraic structure emerging from the moduli space of the supersymmetric vacua. N = 2 theory has two sets of supersymmetries, which provide powerful tools to study its dynamics rather than non-supersymmetric and N = 1 theories. At the same time, it still shows various dynamical behaviors, e.g., the asymptotic freedom and the dynamical mass generation. Actually, the instanton plays a crucial role to explore the vacuum structure of N = 2 theory as well. Since the instanton provides a solution to the classical equation of motion in the Yang–Mills theory, one may consider the perturbative expansion around the instanton configuration [73]. Although it is still hard to control this expansion, we can apply the so-called topological twist to localize the path integral on the instanton configuration, if there exists N = 2 supersymmetry [77] (Sect. 1.3). This drastically simplifies the analysis of gauge theory path integral, and one can deal with the gauge theory path integral as a statistical model of the instantons. What remains is to evaluate the configuration space of the instantons, a.k.a., the instanton moduli space. Instanton Counting From this point of view, we will provide the instanton counting argument with detailed study of the instanton moduli space. We are in particular interested in the volume of the instanton moduli space, which gives rise to important contributions to the partition function based on the path integral formalism. Since the naively defined moduli space is non-compact and singular, we should instead define a regularized version of the moduli space, and then apply the equivariant localization scheme to evaluate the volume of the moduli space. The gauge theory partition function obtained by the equivariant integral over the instanton moduli space is called the instanton Preface ix partition function [43–45] and also the Nekrasov partition function [47, 55], which will be one of the main objects in this monograph (Sect. 1.8). The instanton partition function provides a lot of suggestive insights in the rela- tion to various branches of mathematics: Combinatorics of (2d and also higher dimensional) partitions; Geometric representation theory; τ -function and integrable systems; Vertex operator algebra and conformal field theory, and more. The latter part of this monograph is devoted to the study of quantum geometric and algebraic aspects of N = 2 gauge theory based on such interesting connections between the instanton partition function and various illuminating notions in mathematical physics. Seiberg–Witten Theory A striking application of the instanton counting is the Seiberg–Witten theory for N = 2 gauge theory in four dimensions [68, 69], which provides an algebraic geometric description for the low energy effective theory of N = 2 theory (Sect. 4.2). A remarkable property of N = 2 theory is the one-to-one corre- spondence between the Lagrangian and the holomorphic function, known as the prepotential [67]. Seiberg–Witten theory provides a geometric characterization of the low energy effective prepotential based on the auxiliary algebraic curve, called the Seiberg–Witten curve. The instanton partition function depends on the equivariant parameters associated 2 with the spacetime rotation symmetry denoted by (1,2) ∈ C (also called the -background/deformation parameters). The partition function diverges if we naively take the limit 1,2
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