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Duality and Integrability in Topological String Theory

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Duality and Integrability in Topological String Theory View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by SISSA Digital Library International School of Advanced Studies Mathematical Physics Sector Academic Year 2008–09 PhD Thesis in Mathematical Physics Duality and integrability in topological string theory Candidate Andrea Brini Supervisors External referees Prof. Boris Dubrovin Prof. Marcos Marino˜ Dr. Alessandro Tanzini Dr. Tom Coates 2 Acknowledgements Over the past few years I have benefited from the interaction with many people. I would like to thank first of all my supervisors: Alessandro Tanzini, for introducing me to the subject of this thesis, for his support, and for the fun we had during countless discussions, and Boris Dubrovin, from whose insightful explanations I have learned a lot. It is moreover a privilege to have Marcos Mari˜no and Tom Coates as referees of this thesis, and I would like to thank them for their words and ac- tive support in various occasions. Special thanks are also due to Sara Pasquetti, for her help, suggestions and inexhaustible stress-energy transfer, to Giulio Bonelli for many stimulating and inspiring discussions, and to my collaborators Renzo Cava- lieri, Luca Griguolo and Domenico Seminara. I would finally like to acknowledge the many people I had discussions/correspondence with: Murad Alim, Francesco Benini, Marco Bertola, Vincent Bouchard ;-), Ugo Bruzzo, Mattia Cafasso, Hua-Lian Chang, Michele Cirafici, Tom Claeys, Andrea Collinucci, Emanuel Diaconescu, Bertrand Ey- nard, Jarah Evslin, Fabio Ferrari Ruffino, Barbara Fantechi, Davide Forcella, Antonio Grassi, Tamara Grava, Amar Henni, Kentaro Hori, Hiroshi Iritani, Igor Krichever, Ernst Krupnikov, Melissa Liu, Carlo Maccaferri, Cristina Manolache, Etienne Mann, Andrei Marshakov, Luca Mazzucato, Luca Mertens, Andrei Mironov, Alexei Mo- rozov, Fabio Nironi, Alexei Oblomkov, Nicolas Orantin, Orlando Ragnisco, Andrea Raimondo, Antonio Ricco, Paolo Rossi, Yongbin Ruan, Simon Ruijsenaars, Houman Safaai, Emanuel Scheidegger, Bal´azs Szendr˝oi, Nico Temme, Alessandro Tomasiello, Hsian-Hua Tseng, Stefan Vandoren, Marcel Vonk, Marlene Weiss, Katrin Wendland, Jian Zhou. Finally, I want to thank my family, Dario, Mariacarla and Michele, and Cristina for their love and patience and for enduring my many omissions during the last few years. This thesis is dedicated to them. 3 Contents Acknowledgements 3 1 Introduction 3 1.1 Background ................................ 4 1.2 Outlineofthethesis ........................... 7 2 Gromov-Witten theory of toric CY3 and mirror symmetry: a re- view 11 2.1 TheA-side................................. 12 2.1.1 Moduli spaces of holomorphic maps and stable compactification 12 2.1.2 Gromov-Witteninvariants . 13 2.1.3 Generating functions and relations between correlators .... 16 2.1.4 Quantum co-homology and the J-function . .. 19 2.2 TheB-side................................. 21 2.2.1 Asummaryofthecompactcase . 21 2.2.2 Closed local mirror symmetry at tree level . 25 2.2.3 Openlocalmirrorsymmetry . 31 2.2.4 TheBKMPformalism . .. .. 34 3 Toric GW theory I: mirror symmetry and wall-crossings 41 3.1 Introduction................................ 42 3.2 The genus zero B–model on resolved Y p,q singularities . 43 3.2.1 Cones over Y p,q .......................... 43 3.2.2 Hori–Vafa curves, integrable systems and five-dimensional gauge theories .............................. 45 3.2.3 Solving the P F system in the full B-model moduli space . 48 3.2.4 Examples ............................. 51 3 3.3 A case study: the C /Z4 orbifold .................... 59 3.3.1 Intermezzo: the disc and torus amplitudes . 59 3.3.2 Thechangeofbasisfromlargeradius . 62 3.3.3 Γ(2)modularforms. .. .. 63 3.3.4 BulletproofingBKMP . 66 i Contents 4 Toric GW theory II: conifold transitions and Gopakumar-Vafa du- ality 75 4.1 Overview.................................. 76 4.2 The closed string side: conifold transition for T ∗L(p, q) ........ 78 4.2.1 Geometrictransition . 78 4.2.2 Mirrorsymmetry ......................... 83 4.3 The open string side: CS theory on L(p, q)andmatrixmodels . 84 4.3.1 Large N limitoftheCSmatrixmodel . 86 5 Toric GW theory III: local curves and integrable hierarchies 93 5.1 Topological string theory and integrable systems . ....... 94 5.2 Dispersionless hierarchies from associativity equations ................................. 96 5.2.1 WDVV and the Gauss-Manin connection . 96 5.2.2 ThePrincipalHierarchy . 98 5.3 The local Gromov–Witten theory of curves . 101 Xk 5.3.1 Calculating F0,qu(u)........................ 102 Xk 5.3.2 Calculating F0,cl(u)........................ 103 5.3.3 Warming up: principal hierarchies in the equivariantly CY case 104 5.4 A case study: the resolved conifold and the Ablowitz–Ladik hierarchy 106 5.4.1 Solving the flatness condition . 106 5.4.2 Normalizing the deformed flat coordinates . 107 5.4.3 The dispersionless Ablowitz–Ladik hierarchy . 110 5.4.4 The equivariant Gromov–Witten theory of the conifold and the dispersionful Ablowitz–Ladik hierarchy: the Main Conjecture . 112 6 Conclusions 117 Bibliography 121 3 A The C /Z4 computation 133 3 A.1 Predictions for open orbifold Gromov-Witten invariants of C /Z4 . 134 A.1.1 g = 0, h =1............................ 134 A.1.2 g = 0, h =2............................ 136 A.1.3 g = 0, h =3............................ 138 A.1.4 g = 1, h =1............................ 140 A.1.5 g = 1, h =2............................ 142 A.1.6 g = 2, h =1............................ 144 B Useful formulae 147 B.1 Euler integral representations, analytic continuation and generalized hypergeometricfunctions. 148 B.2 Lauricellafunctions. .. .. 150 B.2.1 Definition ............................. 150 ii Contents B.2.2 Analytic continuation formulae for Lauricella FD ....... 151 B.3 CSMMasaunitarymatrixmodel . 152 B.4 Expansion formulae for Gauss’ hypergeometric function around integer parameters................................. 153 B.5 Hypergeometric Yukawas as algebraic functions . ..... 154 1 Contents 2 Chapter 1 Introduction 3 1.1. Background 1.1 Background The interplay between Mathematics and Physics has been for long time a most fruit- ful and conceptually rich arena of modern Science. On one hand, the mathematical formalization of physical models has proven to be remarkably successful in the quan- titative description of natural phenomena; on the other, recent times have witnessed how surprisingly effective many ideas from particle physics (e.g supersymmetry) can be in the search of new mathematical structures. The modern prototypical example of such cross–fertilization is provided by Super- string Theory. In addition to being the leading candidate for a unified theory of all forces - including a consistent quantum mechanical description of gravitational phe- nomena - and a surprisingly powerful tool for the study of gauge theories at strong coupling, the theory of superstrings has had a truly remarkable impact on sometimes distant areas of Mathematical Physics and Mathematics in general, both as a heuris- tic guiding principle and as a major unifying framework. A central role in this respect has been played by the so-called topological phase of string theory (see [43,93,103,118,144,149] for reviews). Physically, this consists of a class of two–dimensional = 2 superconformal field theories coupled to 2d–gravity N which are characterized by an exact nilpotent fermionic symmetry QBRST [145,149]. The most striking feature of these string theories is that fact the QBRST symmetry singles out a distinguished vector space of operators in the worldsheet σ-model, whose correlation functions do not dependent on the background metric on the Riemann surface. As the main consequence, these correlators are computed in a drastically simplified fashion as a finite-dimensional integral over on-shell, classical trajectories only. A natural place of appearance for this theories is in the context of twisted = (2, 2) σ-models, namely supersymmetric quantum field theories of maps from aN compact connected Riemann surface to a K¨ahler manifold X, whose energy-momentum tensor is redefined by the abelian automorphisms of the = (2, 2) superalgebra. When the target manifold X is the internal part of a vacuumN configuration for superstrings, i.e. a Ricci-flat K¨ahler manifold, such topological string theories may come in two guises: the so-called A-model1 and B-model. From the physical point of view their most attractive feature is the close relationship with the F -term sector of the parent Type IIA and IIB string theories on 1,3 X, for which the topological invariance can be exploited to full power to giveM exact× results. This has yielded a great deal of non-trivial information about the effective holomorphic dynamics of supersymmetric gauge and gravity theories [12,22,45,143], and has provided at the same time a very useful laboratory for general ideas about black–hole dynamics [132] and dualities in 1In fact, conformality is not needed for the definition of the topological A-twist, which can in principle be performed on a target manifold X with a symplectic form ω, and a not necessarily integrable ω-tamed almost complex structure [17,145]. Such level of generality will not however be needed in this thesis. 4 Chapter 1. Introduction string theory and gauge theory [83]. From the mathematical viewpoint, topological string theories have perhaps an even more striking significance, as physical arguments suggest that their correlation function should capture deep and subtle aspects of the geometry of the target space. The most remarkable instance is given by the A-model on a target manifold XA: the stationary phase reduction of the path integral in this case heuristically boils down to some sort of integration over a moduli space of holomorphic maps from the source Riemann surface to X, whose result should be invariant under deformations of the K¨ahler (or in general almost-K¨ahler) structure on X. The outcome of the integration, roughly speaking, should be related to a “count” of the number of curves in X in a fixed homology class and subject to intersect various cycles. These geometric invariants go under the name of Gromov–Witten invariants, and have very important applications in symplectic topology and enumerative algebraic geometry.
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