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Topological Strings and Quantum Curves TOPOLOGICAL STRINGS AND QUANTUM CURVES arXiv:0911.3413v1 [hep-th] 17 Nov 2009 This work has been accomplished at the Institute for Theoretical Physics (ITFA) of the University of Amsterdam (UvA) and is financially supported by a Spinoza grant of the Netherlands Organisation for Scientific Research (NWO). Cover illustration: Lotte Hollands Cover design: The DocWorkers Lay-out: Lotte Hollands, typeset using LATEX ISBN 978 90 8555 020 4 NUR 924 c L. Hollands / Pallas Publications — Amsterdam University Press, 2009 All rights reserved. Without limiting the rights under copyright reserved above, no part of this book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the written permission of both the copyright owner and the author of the book. TOPOLOGICAL STRINGS AND QUANTUM CURVES ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op donderdag 3 september, te 14.00 uur door LOTTE HOLLANDS geboren te Maasbree PROMOTIECOMMISSIE Promotor prof. dr. R.H. Dijkgraaf Overige leden prof. dr. J. de Boer prof. dr. A.O. Klemm prof. dr. E.M. Opdam dr. H.B. Posthuma dr. S.J.G. Vandoren prof. dr. E.P. Verlinde Faculteit der Natuurwetenschappen, Wiskunde en Informatica PUBLICATIONS This thesis is based on the following publications: R. Dijkgraaf, L. Hollands, P. Sułkowski and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, arXiv/0709.4446 [hep-th], JHEP 02 (2008) 106. L. Hollands, J. Marsano, K. Papadodimas and M. Shigemori, Nonsupersymmetric Flux Vacua and Perturbed N=2 Systems, arXiv/0804.4006 [hep-th], JHEP 10 (2008) 102. R. Dijkgraaf, L. Hollands and P. Sułkowski, Quantum Curves and D-Modules, arXiv/0810.4157 [hep-th], JHEP 11 (2009) 047. M. Cheng and L. Hollands, A Geometric Derivation of the Dyon Wall-Crossing Group, arXiv/0901.1758 [hep-th], JHEP 04 (2009) 067. Contents 1 Introduction 1 1.1 Fermions on Riemann surfaces . .1 1.2 Outline of this thesis . .9 2 Calabi-Yau Geometry 15 2.1 The Strominger-Yau-Zaslow conjecture . 16 2.2 The Fermat quintic . 21 2.3 Local Calabi-Yau threefolds . 27 3 I-brane Perspective on Vafa-Witten Theory and WZW Models 31 3.1 Instantons and branes . 32 3.2 Vafa-Witten twist on ALE spaces . 42 3.3 Free fermion realization . 51 3.4 Nakajima-Vafa-Witten correspondence . 58 4 Topological Strings, Free Fermions and Gauge Theory 69 4.1 Curves in = 2 gauge theories . 70 N 4.2 Geometric engineering . 81 4.3 Topological invariants of Calabi-Yau threefolds . 89 4.4 BPS states and free fermions . 101 5 Quantum Integrable Hierarchies 113 5.1 Semi-classical integrable hierarchies . 114 5.2 -modules and quantum curves . 119 D 5.3 Fermionic states and quantum invariants . 134 6 Quantum Curves in Matrix Models and Gauge Theory 147 6.1 Matrix model geometries . 149 6.2 Conifold and c = 1 string . 163 6.3 Seiberg-Witten geometries . 169 6.4 Discussion . 185 7 Dyons and Wall-Crossing 189 7.1 Compact curves of genus 1 and 2 . 190 7.2 Quarter BPS dyons . 199 7.3 Wall-crossing in = 4 theory . 215 N 8 Fluxes and Metastability 235 8.1 Ooguri-Ookouchi-Park formalism . 237 8.2 Geometrically engineering the OOP formalism . 238 8.3 Embedding in a larger Calabi-Yau . 256 8.4 Concluding remarks . 269 A Level-rank duality 273 Bibliography 277 Samenvatting 293 Acknowledgments 299 Chapter 1 Introduction The twentieth century has seen the birth of two influential theories of physics. On very small scales quantum mechanics is very successful, whereas general relativity rules our universe on large scales. Unfortunately, in regimes at small scale where gravity is nevertheless non-negligible—such as black holes or the big bang—neither of these two descriptions suffice. String theory is the best candidate for a unified theoretical description of nature to date. However, strings live at even smaller scales than those governed by quantum mechanics. Hence string theory necessarily involves levels of energy that are so high that they cannot be simulated in a laboratory. Therefore, we need another way to arrive at valid predictions. Although string theory thus rests on physical arguments, it turns out that this framework carries rich mathematical structure. This means that a broad array of mathematical techniques can be deployed to explore string theory. Vice versa, physics can also benefit mathematics. For ex- ample, different physical perspectives can relate previously unconnected topics in mathematics. In this thesis we are motivated by this fertile interaction. We both use string the- ory to find new directions in mathematics, and employ mathematics to discover novel structures in string theory. Let us illustrate this with an example. 1.1 Fermions on Riemann surfaces This section briefly introduces the main ingredients of this thesis. We will meet all in much greater detail in the following chapters. So-called Riemann surfaces play a prominent role in many of the fruitful in- teractions between mathematics and theoretical physics that have developed in 2 Chapter 1. Introduction the second half of the twentieth century. Riemann surfaces are smooth two- dimensional curved surfaces that have a number of holes, which is called their genus g. Fig. 1.1 shows an example. Figure 1.1: A compact Riemann surface with just one hole in it is called a 2-torus T 2. We refer to its two 1-cycles as the A and the B-cycle. Riemann surfaces additionally come equipped with a complex structure, so that any small region of the surface resembles the complex plane C. An illustrative class of Riemann surfaces is defined by equations of the form Σ: F (x; y) = 0; where x; y C: 2 Here, F can for instance be a polynomial in the complex variables x and y.A simple example is a (hyper)elliptic curve defined by y2 = p(x); where p(x) is a polynomial. The curve is elliptic if p has degree 3 or 4, in which case its genus is g = 1. For higher degrees of p it is hyperelliptic and has genus g > 1. Figure 1.2: A simple example of a non-compact Riemann surface is defined by the equation 2 2 2 x + y = 1 in the complex plane C . The complex structure of a compact Riemann surface can be conveniently char- acterized by a period matrix. This is a symmetric square matrix τij of complex numbers, whose rank equals the genus of the surface and whose imaginary part is strictly positive. The complex structure of a 2-torus, illustrated in Fig. 1.1, is 1.1. Fermions on Riemann surfaces 3 for example determined by one complex number τ that takes values in the upper half plane. The period matrix encodes the contour integrals of the g independent holomor- phic 1-forms over the 1-cycles on the surface. To this end one picks a canon- ical basis of A-cycles and B-cycles, where the only non-trivial intersection is A Bj = δij. The 1-forms !i may then be normalized by integrating them 1 \ over the A-cycles, so that their integrals over the B-cycles determine the period matrix: Z Z !j = δij; !j = τij: Ai Bi Conformal field theory Riemann surfaces play a dominant role in the study of conformal field theories (CFT’s). Quantum field theories can be defined on a space-time background M, which is usually a Riemannian manifold with a metric g. The quantum field theory is called conformal when it is invariant under arbitrary rescaling of the metric g. A CFT therefore depends only on the conformal class of the metric g. The simplest quantum field theories study a bosonic scalar field φ on the back- ground M. The contribution to the action for a massless scalar is Z mn Sboson = pg g @mφ @nφ, M m yielding the Klein-Gordon equation @m@ φ = 0 (for zero mass) as equation of motion. Note that, on the level of the classical action, this theory is clearly conformal in 2 dimensions; this still holds for the full quantum theory. Since a 2-dimensional conformal structure uniquely determines a complex structure, the free boson defines a CFT on any Riemann surface Σ. Let us compactify the scalar field φ on a circle S1, so that it has winding modes along the 1-cycles of the Riemann surface. The classical part of the CFT partition function is determined by the solutions to the equation of motion. The holomor- phic contribution to the classical partition function is well-known to be encoded in a Riemann theta function X 2πi 1 ptτp+ptν θ (τ; ν) = e ( 2 ); g p2Z where the integers p represent the momenta of φ that flow through the A-cycles of the 2-dimensional geometry. Another basic example of a CFT is generated by a fermionic field on a Rie- mann surface. Let us consider a chiral fermion (z). Mathematically, this field 4 Chapter 1. Introduction transforms on the Riemann surface as a (1=2; 0)-form, whence (z)pdz = (z0)pdz0; where z and z0 are two complex coordinates. Such a chiral fermion contributes to the 2-dimensional action as Z Sfermion = (z) @A (z); Σ where @A = @ + A, and A is a connection 1-form on Σ. This action determines the Dirac equation @A (z) = 0 as its equation of motion.
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