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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 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.

Introduction.- 2d sigma-models and DAHA.- 3d theories and modularity.-4 d theories, fivebranes, and M-theory.- Appendix.

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.

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.

ISBN: **9783031281532**

Condizione: Nuovo

Collana: SpringerBriefs in Mathematical Physics

Dimensioni: 235 x 155 mm

Formato: Brossura

Illustration Notes:VI, 143 p. 42 illus., 25 illus. in color.

Pagine Arabe: 143

Pagine Romane: vi

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