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Topics in Two-Dimensional Heterotic and Minimal Supersymmetric Sigma Models

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Topics in Two-Dimensional Heterotic and Minimal Supersymmetric Sigma Models Topics in Two-dimensional Heterotic and Minimal Supersymmetric Sigma Models A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jin Chen IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Prof. Arkady Vainshtein May, 2016 c Jin Chen 2016 ALL RIGHTS RESERVED Acknowledgements I am deeply grateful to my advisor, Arkady Vainshtein, for his step by step guidance, constant encouragement and generous support throughout these years at University of Minnesota. Arkady has been a knowledgeable, passionate, and also patient advisor, mentor and friend to me. He teaches me physics knowledge and ways of thinking, fosters my independent study, as well as tolerates my capriciousness and digressions on research. Besides conduct of research, I also got greatly enjoyed and influenced by Arkady's optimistic and humorous personality during countless discussions and debates with him. I also want to express my special appreciation and thanks to Misha Shifman, who co- advised me and provided many insightful ideas on research and precious suggestions on my physics career during these years. Misha introduced me the topic on heterotic sigma model years ago and led me to this long but fascinating journey, while at that time I was not able to appreciate the beauty of the topic yet. I would like to thank Sasha Voronov. Arkady introduced me to Sasha. I learned from him how to calculate characteristic classes of the Grassmannian manifolds which was a key step in my research. I thank Sasha, Ron Poling and Misha Voloshin for serving on my thesis committee as well. And I have also greatly benefited from interactions with faculty, postdocs and students at department of physics and mathematics. Finally, words cannot express my gratitude to my family|my grandma, my father and especially my mom during these years. It is because of your endless supports, considerations and forgiveness that I can devote myself to my beloved physics and mathematics. i Dedication To those who held me up over the years ii Abstract Two-dimensional = (0; 1) and (0; 2) supersymmetric sigma models can be mainly N obtained in two ways: non-minimal heterotic deformation of = (1; 1) and (2; 2) sigma N models, and minimal construction which contains only (0; 1) or (0; 2) supermultiplets. The former deformed models with = (0; 2) supersymmetries emerge as low-energy world N sheet theories on non-Abelian strings supported in some = 1 four-dimensional Yang- N Mills theories. The latter, on the other hand, can be regarded as the elementary building blocks to construct generic = (0; 1) or (0; 2) chiral models. N In the thesis, we will study both types of sigma models. We start with the deformed heterotic sigma models with = (0; 2) supersymmetries. Our investigation is around the N calculation of NSVZ exact β-function of the heterotic models through instanton technique, and also verifies it by straightforward two-loop calculation and the \Konishi anomaly" of the hypercurrent. Finally, we also consider isometries on their target spaces, and show that the heterotic deformation is free of isometry and holonomy anomalies. Then we turn to analysis of a more fundamental minimal construction of chiral sigma models with = (0; 1) and (0; 2) supersymmetries. These minimal models with only N (left) chiral fermions may intrinsically suffer from chiral anomalies that will render the theories mathematically inconsistent. We focus on two important examples, the minimal O(N) and CP(N 1) models, and calculate their isometry anomalies. We show that the − CP(N 1) models with N > 2 has non-removable chiral anomalies, while the O(N) models − are anomaly free and thus exist quantum mechanically. We also disclose a relation between isometry anomalies in these non-linear sigma models (NLσM) and gauge anomalies in gauged linear sigma models (GLσM). Finally, we reveal a relation on anomaly correspondence between NLσM and GLσM to minimal models on homogeneous spaces. We interpret these anomalies more from geomet- ric perspectives and relate them to the characteristic classes of the target spaces. Through explicit calculation of anomalous fermionic effective action, we show how to add a series of local counterterms to remove the anomalies. We eventually reach a result that the remedy iii procedure is equivalent to require the target spaces of theories with trivial first Pontrya- gin class, and thus demonstrate Moore and Nelson's consistency condition in the case of homogeneous spaces. More importantly, we find that local counterterms further constrain \curable" models and make some of them flow to non-trivial infrared superconformal fixed point. We also discuss a interesting relation between = (0; 1) and (0; 2) supersymmetric N sigma model and gauge theories in the spirit of 't Hooft anomaly matching condition. iv Contents Acknowledgementsi Dedication ii Abstract iii List of Tables viii List of Figures ix 1 Introduction1 1.1 Background....................................1 1.2 Supersymmetry and 2d=4d correspondence..................2 1.3 Perspective from mathematical physics.....................4 1.4 Overview and Summary.............................5 2 N =(0; 2) Deformation of (2; 2) Sigma Models: Geometric Structure, Holomorphic Anomaly and Exact β Functions7 2.1 Introduction....................................7 2.2 Heterotic N = (0; 2) models in two-dimensions................ 11 2.2.1 Formulation of the model........................ 11 2.2.2 Geometry of heterotic deformation................... 14 2.2.3 Holomorphy and its breaking...................... 16 2.3 Beta functions.................................. 19 2.3.1 Generalities................................ 19 v 2.3.2 Beta functions at one loop........................ 20 2.4 Beyond one loop from instanton calculus.................... 22 2.5 Explicit two-loop calculations.......................... 25 2.5.1 Two-loop β function for g2 ....................... 25 2.5.2 Anomalous dimensions at two loops.................. 26 2.5.3 Beta functions and fixed point in ρ ................... 29 2.6 Isometries of the model............................. 29 2.7 Supercurrent multiplet.............................. 33 2.7.1 Hypercurrent in the undeformed N = (2; 2) theory......... 33 2.7.2 Hypercurrent in the heterotic N = (0; 2) theory........... 34 2.7.3 Analog of the Konishi anomaly and beta function........... 36 2.8 Conclusion.................................... 38 3 Isometry Anomalies in Minimal N = (0; 1) and N = (0; 2) Sigma Models 40 3.1 Introduction.................................... 40 3.2 O(N) Sigma Model............................... 44 3.3 CP(N −1) Sigma Model............................ 49 3.3.1 CP(1) is a special case......................... 53 3.3.2 A closer look at the correspondence between isometry and gauge anomalies................................. 54 3.4 Dual formalism for the O(N) Model...................... 60 3.5 Conclusions.................................... 62 4 Anomalies in Minimal N = (0; 1) and N = (0; 2) Sigma Models on Homogeneous Spaces 65 4.1 Introduction and summary........................... 65 4.2 Isometry anomalies................................ 68 4.2.1 Sigma models on M through gauge formulation........... 69 4.2.2 N = (0; 1) supersymmetric sigma model on M ........... 74 4.2.3 Isometry anomalies of sigma model on M ............... 79 4.3 Holonomy anomalies............................... 85 4.3.1 Anomaly matching condition...................... 85 vi 4.3.2 Comments on counterterms....................... 91 4.3.3 Examples................................. 96 4.3.4 Topological origin of anomaly cancellation............... 109 4.4 The determinant line bundle of homogeneous space sigma models...... 112 4.4.1 A digression on K¨ahlersigma model anomaly in Fujikawa's method........................... 113 4.4.2 Global vs local anomalies from geometric point of view........ 117 4.4.3 The determinant line bundle analysis.................. 119 4.5 Conclusion and outlooks............................. 124 References 127 Appendix A. Notation 135 Appendix B. Calculation of ∆κL 137 Appendix C. Vielbeins and Anomalies in CP(N −1) 140 vii List of Tables 2.1 Two-loop calculation for the g2 correction. The labeling of the diagrams follows that in Fig. 2.2. ................................... 27 4.1 The metric used in fermion path integral measure in curved indices. ........ 115 viii List of Figures 2.1 One-loop finite correction to the canonical coupling g. The wave line denotes the background field A. The solid line denotes the propagator of B, while the solid line with a wavy line superposed denotes that of . We shall follow the same notation B throughout this chapter. .............................. 17 2.2 Two-loop correction to the canonical coupling g. The dashed line denotes the quantum propagator of A. ............................. 26 2.3 One-loop correction to wave function renormalization of R and ζR. ........ 26 2.4 Two-loop corrections in the wavefunction renormalization of ζR . ......... 27 2.5 Two-loop corrections to the wavefunction renormalization of R . ......... 28 2.6 One-loop diagrams for vL current. The dots denote the vL currents, dashed lines are quantum A fields while wavy lines refer to the background A. ......... 36 3.1 The wavy lines denote external spin-connection fields !R, and solid lines denote chiral fermion L. ................................. 47 3.2 The sphere denotes S2N−1, while the solid line below is for CP(N 1). The vertical − circles are U(1)-fibers, each of which
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