Branes and DAHA Representations

ISBN-13: 9783031281532 / Angielski

Du Pei; Ingmar Saberi; Peter Koroteev

Opis

In recent years, there has been an increased interest in exploring the connections between various disciplines of mathematics and theoretical physics such as representation theory, algebraic geometry, quantum field theory, and string theory. One of the challenges of modern mathematical physics is to understand rigorously the idea of quantization. The program of quantization by branes, which comes from string theory, is explored in the book.

This book provides a detailed description of the geometric approach to the representation theory of the double affine Hecke algebra (DAHA) of rank one. Spherical DAHA is known to arise from the deformation quantization of the moduli space of SL(2,C) flat connections on the punctured torus. The authors demonstrate the study of the topological A-model on this moduli space and establish a correspondence between Lagrangian branes of the A-model and DAHA modules.

The finite-dimensional DAHA representations are shown to be in one-to-one correspondence with the compact Lagrangian branes. Along the way, the authors discover new finite-dimensional indecomposable representations. They proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, modular tensor categories behind particular finite-dimensional representations with PSL(2,Z) action are identified. The relationship of Coulomb branch geometry and algebras of line operators in 4d N = 2* theories to the double affine Hecke algebra is studied further by using a further connection to the fivebrane system for the class S construction.

The book is targeted at experts in mathematical physics, representation theory, algebraic geometry, and string theory.

Spis treści

1 Introduction

      1.1 Background 

      1.2 Results 

      1.3 Structure

2 2d sigma-models and DAHA

  2.1  Higgs bundles and flat connections

  2.2  DAHA of rank one and its spherical algebra 

  2.3  Canonical coisotropic branes in A-models 

2.3.1 Spherical DAHA as the algebra of (Bcc , Bcc )-strings

  2.4  Lagrangian A-branes and modules of Oq(X) 

  2.5  (A, B, A)-branes for polynomial representations 22

  2.6  Branes with compact supports and object matching 26

  2.6.1  Generic fibers of the Hitchin fibration 27

  2.6.2  Irreducible components in singular fibers of type I2 28

  2.6.3  Moduli space of G-bundles 29

  2.6.4  Exceptional divisors 31

2.7 Bound

  2.7.1  At singular fiber of type I2 33

  2.7.2  At global nilpotent cone of type I0∗ 35

  3  3d theories and modularity 39

  3.1  DAHA and modularity 39

  3.1.1  SU(2): refined Chern-Simons and TQFT associated to Argyres-Douglas theory 43

  3.1.2  SU(N): higher rank generalization 45

  3.2  Relation to skein modules and MTC[M3] 47

  4  4d theories, fivebranes, and M-theory 49

  4.1  Coulomb branches of 4d N = 2∗ theories of rank one 50

  4.2  Algebra of line operators 53

  4.3  Including surface operator 57

  A  Glossary of symbols 62

  B  Basics of DAHA 63

  B.1  DAHA

  B.1.1  Double affine braid group and double affine Weyl group 64

  B.1.2  PBW theorem for DAHA 64

  B.1.3  Spherical subalgebra 65

  B.1.4  Braid group and SL(2, Z) action 65

  B.1.5  Polynomial representation of DAHA 66

  B.1.6  Symmetric bilinear form 67

  B.1.7  Degenerations 68

  B.2  DAHA of type A1 70

  B.2.1  Polynomial representation 70

  B.2.2  Functional representation 72

  B.2.3  Trigonometric Cherednik algebra of type A1 72

  B.2.4  Rational Cherednik algebra of type A1 73

states of branes and short exact sequences: morphism matching 33

–i–

63

C Quantum torus algebra 73

  C.1  Representations of quantum torus algebra 74

  C.1.1  Unitary representations 75

  C.1.2  Non-unitary representations 76

  C.1.3  Geometric viewpoint 77

  C.2  Branes for quantum torus algebra 78

  C.2.1  Cyclic representations 78

  C.2.2  Polynomial representations 79

  C.3  Symmetrized quantum torus 80

  C.3.1  Representation theory 81

  C.3.2  Corresponding branes 83

D 3d N = 4 theories and Cherednik algebras 84

  D.1  Coulomb branches of 3d N = 4 theories 84

  D.2  3d N = 4 Coulomb branches and Cherednik algebras 85

 

O autorze

Sergei Gukov is a professor of mathematics and theoretical physicist. Gukov graduated from Moscow Institute of Physics and Technology (MIPT) in Moscow, Russia before obtaining a doctorate in physics from Princeton University under the supervision of Edward Witten.[1]

He held a Long-term Prize fellowship of Clay Mathematics Institute at Harvard University (2001-2006) and during 2007-2008 was a member of the school of mathematics at the Institute for Advanced Study, Princeton. Since 2007, he has been professor of mathematics and theoretical physics at the California Institute of Technology (Caltech). Starting 2010, Gukov was elected as an external scientific member of the Max Planck Society at the MPIM, Bonn.

Peter Koroteev is a lecturer at the Department of Mathematics at Univeristy of California Berkeley. He got his PhD from University of Minnesota in 2012 under supervision of Prof. Arkady Vainshtein. Since then he worked as Postdoctoral Researcher at Perimeter Institute for Theoretical Physics 2012-2016 and as Visiting Assistant Professor at University of California, Davis, 2016-2019. Peter’s work is focused on the interplay between representation theory, algebraic geometry, and mathematical physics. He is also an instructor at Berkeley and Stanford Math Circles.

Satoshi Nawata currently serves as an associate professor of physics at Fudan University. He received a Bachelor of Science at Tokyo Institute of Technology. He went on to achieve a PhD from the University of Wisconsin-Milwaukee. Following this, Satoshi Nawata had the opportunity to hold a postdoctoral position at several notable institutions, including the Tata Institute of Fundamental Research, Perimeter Institute for Theoretical Physics, NIKHEF, University of Warsaw, and the California Institute of Technology among others. He also spent time as a visiting researcher at the Max Planck Institute for Mathematics and IHES. Since 2016, Satoshi Nawata has been imparting knowledge and driving research as an Associate Professor at Fudan University.