## Overview

- Bridges the gap between geometry and representation theory
- Gives a description of new indecomposable finite-dimensional representations of DAHA
- Presents tensor categories for line operators
- This book is open access, which means that you have free and unlimited access

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 48)

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## About this book

This open access 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.

This is an open access book.

## Keywords

## Table of contents (4 chapters)

## Authors and Affiliations

## About the authors

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.

Du Pei is currently an associate professor of quantum mathematics at the University of Southern Denmark. He obtained his Ph.D. in theoretical physics in 2016 from Caltech, and has conducted research in theoretical physics and mathematical physics at Harvard University, MSRI, Caltech and Aarhus University. His main interests concern the geometric, algebraic and categorical properties of quantum field theory.

Ingmar Saberi is a Senior Researcher at Ludwig-Maximilians-Universität München, Germany at Ilka Brunner's research group. Previously he was a postdoc at the University of Heidelberg, Germany in Johannes Walcher's group at the Mathematisches Institut. Ingmar got his PhD at California Institute of Technology under supervision of Prof. Sergei Gukov.

## Bibliographic Information

Book Title: Branes and DAHA Representations

Authors: Sergei Gukov, Peter Koroteev, Satoshi Nawata, Du Pei, Ingmar Saberi

Series Title: SpringerBriefs in Mathematical Physics

DOI: https://doi.org/10.1007/978-3-031-28154-9

Publisher: Springer Cham

eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

Copyright Information: The Author(s) 2023

Softcover ISBN: 978-3-031-28153-2Published: 29 August 2023

eBook ISBN: 978-3-031-28154-9Published: 28 August 2023

Series ISSN: 2197-1757

Series E-ISSN: 2197-1765

Edition Number: 1

Number of Pages: VI, 143

Number of Illustrations: 17 b/w illustrations, 25 illustrations in colour

Topics: Mathematical Physics, Algebraic Geometry, Quantum Physics