Applications of gauged linear sigma models Zhuo Chen Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Eric Sharpe, Chair Lara Anderson James Gray Uwe T¨auber March 26, 2019 Blacksburg, Virginia Keywords: Heterotic compactification, Mirror symmetry, Gauged linear sigma model, Topological field theory, Superconformal field theory Copyright 2019, Zhuo Chen Applications of gauged linear sigma models Zhuo Chen (ABSTRACT) This thesis is devoted to a study of applications of gauged linear sigma models. First, by constructing (0,2) analogues of Hori-Vafa mirrors, we have given and checked proposals for (0,2) mirrors to projective spaces, toric del Pezzo and Hirzebruch surfaces with tangent bundle deformations, checking not only correlation functions but also e.g. that mirrors to del Pezzos are related by blowdowns in the fashion one would expect. Also, we applied the recent proposal for mirrors of non-Abelian (2,2) supersymmetric two-dimensional gauge theories to examples of two-dimensional A-twisted gauge theories with exceptional gauge groups G2 and E8. We explicitly computed the proposed mirror Landau-Ginzburg orbifold and derived the Coulomb ring relations (the analogue of quantum cohomology ring relations). We also studied pure gauge theories, and provided evidence (at the level of these topological- field-theory-type computations) that each pure gauge theory (with simply-connected gauge group) flows in the IR to a free theory of as many twisted chiral multiplets as the rank of the gauge group. Last, we have constructed hybrid Landau-Ginzburg models that RG flow to a new family of non-compact Calabi-Yau threefolds, constructed as fiber products of genus g curves and noncompact K¨ahlerthreefolds. We only considered curves given as branched double covers of P1. Our construction utilizes nonperturbative constructions of the genus g curves, and so provides a new set of exotic UV theories that should RG flow to sigma models on Calabi-Yau manifolds, in which the Calabi-Yau is not realized simply as the critical locus of a superpotential. Applications of gauged linear sigma models Zhuo Chen (GENERAL AUDIENCE ABSTRACT) This thesis is devoted to a study of vacua of supersymmetric string theory (superstring theory) by gauged linear sigma models. String theory is best known as the candidate to unify Einstein's general relativity and quantum field theory. We are interested in theories with a symmetry exchanging bosons and fermions, known as supersymmetry. The study of superstring vacua makes it possible to connect string theory to the real world, and describe the Standard model as a low energy effective theory. Gauged linear sigma models are one of the most successful models to study superstring vacua by, for example, providing insights into the global structure of their moduli spaces. We will use gauged linear sigma models to study mirror symmetry and its heterotic generalization \(0; 2) mirror symmetry." They are both world-sheet dualities relating different interpretations of the same (internal) superstring vacua. Mirror symmetry is a very powerful duality which exchanges classical and quantum effects. By studying mirror symmetry and (0; 2) mirror symmetry, we gain more knowledge of the properties of superstring vacua. Acknowledgments I would like to express my sincere gratitude to my advisor professor Eric Sharpe for his endless help and patient guidance. From you, I not only learned a great deal about string theory, but also the way to conduct research. I am very fortunate to have had the opportunity to work with you and I am really grateful for all that you have done for me. I would also like to acknowledge two other professors in the string theory group at Virginia Tech, professor Lara Anderson and professor James Gray, for all their help and valuable conversations. In addition, I appreciate all useful discussions with other graduate students and postdocs of string theory group at Virginia Tech. Special thanks to my wife and my parents. Without your consistent supports, I wouldn't be able to accomplish my Ph.D. study. iv Contents 1 Introduction 1 1.1 The N = 2 superconformal algebra ....................... 3 1.2 Nonlinear sigma models ............................. 5 1.3 Gauged linear sigma models ........................... 13 1.4 Mirror symmetry ................................. 19 2 Toda-like (0,2) mirrors 26 2.1 Introduction .................................... 26 2.1.1 Review of (2,2) Toda dual theories ................... 29 2.1.2 Review of (0,2) Landau-Ginzburg models ................ 31 2.2 Pn × Pm ...................................... 34 2.2.1 The A/2-twisted nonlinear sigma model ................ 34 2.2.2 The Toda-like mirror theory ....................... 35 2.3 Example: Toda-like duals to P1 × P1 ...................... 39 2.3.1 The (0,2) NLSM ............................. 39 2.3.2 The Toda-like mirror theory ....................... 41 2.3.3 Moduli ................................... 45 2.3.4 Redundancies and equivalent descriptions ................ 45 v 2.4 Del Pezzo surfaces ................................ 50 2.4.1 The first del Pezzo surface, dP1 ..................... 50 2.4.2 The second del Pezzo surface, dP2 .................... 58 2.4.3 The third del Pezzo surface, dP3 ..................... 70 2.5 Hirzebruch surfaces ................................ 78 2.5.1 Review of the (2,2) mirror ........................ 78 2.5.2 (0,2) deformations and proposed (0,2) mirrors ............. 80 2.6 Correlation functions in some examples .................... 84 2.6.1 A/2 correlation functions on P1 × P1 .................. 84 2.6.2 Toda-like dual to P1 × P1 ........................ 85 2.6.3 A/2 correlation functions on P1 × P2 .................. 86 2.7 Tangent bundle moduli .............................. 89 2.8 Quantum cohomology of dP1 .......................... 93 3 Two-dimensional supersymmetric gauge theories with exceptional gauge groups 95 3.1 Introduction .................................... 95 3.2 G2 ......................................... 98 3.2.1 Mirror Landau-Ginzburg orbifold .................... 98 3.2.2 Weyl group ................................ 101 3.2.3 Coulomb ring relations .......................... 104 3.2.4 Vacua ................................... 108 3.2.5 Pure gauge theory ............................ 111 3.2.6 Comparison with A model results .................... 114 3.2.7 Comparison to other bases for weight lattice .............. 116 vi 3.3 E8 ......................................... 117 3.3.1 Mirror Landau-Ginzburg orbifold .................... 117 3.3.2 Superpotential .............................. 119 3.3.3 Coulomb ring relations .......................... 128 3.3.4 Pure gauge theory ............................ 144 4 Landau-Ginzburg models for certain fiber products with curves 146 4.1 Introduction .................................... 146 4.2 GLSM for P2g+1[2; 2] and curves of genus g ................... 148 4.3 Hybrid Landau-Ginzburg models for fiber products .............. 149 4.3.1 Fiber products with vector bundles on P1 ................ 149 4.3.2 Fiber products with hypersurfaces in vector bundles .......... 153 4.4 Fiber products with twistor spaces ....................... 154 4.4.1 Fiber product with twistor space of R4 ................. 155 2 4.4.2 Fiber product with twistor space of C =Zk ............... 156 4.4.3 Fiber product with twistor space of S1 × R3 .............. 157 4.5 Review of pertinent mathematics ........................ 158 5 Conclusion 161 References 164 vii List of Figures 1.1 Mirror symmetry for Calabi-Yau manifolds ................... 22 1.2 Mirror symmetry for Fano spaces ........................ 22 2.1 A toric fan of P2 can be obtained by removing the edge (0; −1) from the toric fan of dP1. .................................... 56 2.2 A toric fan for dP1 can be obtained by removing the edge (−1; 0) from the toric fan for dP2. ................................. 64 2.3 A toric fan for P1 × P1 can be obtained by removing the edge (−1; −1) from the toric fan for dP2. ............................... 69 2.4 A toric fan for dP2 can be obtained by removing the edge (1; 1) from the toric fan for dP3. .................................... 76 3.1 Roots of G2. ................................... 99 3.2 Weights of 7 of G2. ............................... 100 viii List of Tables 3.1 Roots and weights for G2 and associated fields. ................ 98 3.2 Weyl group actions on the vacua of the case n = 4 ............... 110 3.3 Weyl group actions on the vacua of five fundamental matter multiplets ... 112 3.4 Roots and weights for G2 and associated fields. ................ 116 3.5 First set of roots of E8 and associated fields. ................. 118 3.6 Second set of roots of E8 and associated fields. ................ 119 3.7 Third set of roots of E8 and associated fields. ................. 120 ix Chapter 1 Introduction Ever since the beginning of superstring theory, the study of superstring vacua has been one of the most interesting problems in string theory. String theory predicts that the uni- verse has ten dimensions, nine space and one time. To get the four-dimensional world we observe, string theorists typically make the ansatz that the universe is a product of our four-dimensional universe and a compact six-dimensional space, the `internal space.' Such an ansatz is known as a compactification of string theory, and the data defining the compact- ification correspond