TY - BOOK
T1 - Branes and DAHA Representations
AU - Gukov, Sergei
AU - Koroteev, Peter
AU - Nawata, Satoshi
AU - Pei, Du
AU - Saberi, Ingmar
N1 - Funding Information:
Acknowledgements We would like to thank D. Ben-Zvi, M. Bullimore, I. Cherednik, M. Dedushenko, D. E. Diaconescu, T. Dimofte, P. Etingof, M. Fluder, D. Jordan, T. Kimura, A. Kir-illov Jr., M. Kontsevich, T. Q. T. Le, Y. Lekili, M. Mazzocco, V. Mikhaylov, A. Mellit, G. W. Moore, H. Nakajima, A. Neitzke, A. Oblomkov, T. Pantev, P. Putrov, M. Romo, P. Samuelson, A. Sanders, O. Schiffmann, Peng Shan, E. Sharpe, V. Shende, M. Sperling, Y. Soibelman, Y. Tachikawa, A. Tri-pathy, E. Vasserot, B. Webster, P. Wedrich, B. Williams, Wenbin Yan, Ke Ye, Y. Yoshida, R.D. Zhu for valuable discussion and correspondence. A portion of this work was performed • at the American Institute of Mathematics supported by a SQuaRE grant “New connections between link homologies and physics”, • at the Aspen Center for Physics supported by National Science Foundation grant PHY-1066293 during the program, “Boundaries and Defects in Quantum Field Theories” and PHY-1607611 during the workshop, “Quantum Knot Homology and Supersymmetric Gauge Theories”, • at the Kavli Institute for Theoretical Physics in Santa Barbara supported by the National Science Foundation under Grant No. NSF PHY-1748958 during the workshop “Quantum Knot Invariants and Supersymmetric Gauge Theories”, and • at the International Centre for Theoretical Sciences (ICTS) during the program, “Quantum Fields, Geometry and Representation Theory” (Code: ICTS/qftgrt2018/07). S.N. thanks the Center for Quantum Geometry of Moduli Spaces, Max Planck Institute for Mathematics Bonn, and IHES for hospitality. I.S. thanks Uppsala University, the Center for Quantum Geometry of Moduli Spaces, the Mathematisches Forschungsinstitut Oberwolfach, and Fudan University for hospitality. We are grateful to organizers of seminars, workshops and conferences which have provided us chances to present this work.
Funding Information:
The work of S.G., S.N., and D.P. was supported by the Walter Burke Institute for Theoretical Physics and the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632. The work of S.G. was also supported by the National Science Foundation under Grant No. NSF DMS 1664227. The work of S.N. was also supported by the center of excellence grant “Center for Quantum Geometry of Moduli Space” from the Danish National Research Foundation (DNRF95) and by NSFC Grant No.11850410428 and Fudan University Original Project (No. IDH1512092/002). The work of D.P. was partly supported by the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuantum) which received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 810573. DP’s work was also partly supported by research grant 42125 from VILLUM FONDEN. The work of I.S. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES Excellence Cluster), and by the Free State of Bavaria. P.K. is partially supported by the AMS Simons Travel Grant.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
U2 - 10.1007/978-3-031-28154-9
DO - 10.1007/978-3-031-28154-9
M3 - Monograph
AN - SCOPUS:85171308751
SN - 978-3-031-28153-2
T3 - SpringerBriefs in Mathematical Physics
BT - Branes and DAHA Representations
PB - Springer
ER -