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.
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.
As readers of classic Russian literature know, the nineteenth century was a time of pervasive financial anxiety. With incomes erratic and banks inadequate, Russians of all social castes were deeply enmeshed in networks of credit and debt. The necessity of borrowing and lending shaped perceptions of material and moral worth, as well as notions of social respectability and personal responsibility. Credit and debt were defining features of imperial Russia’s culture of property ownership. Sergei Antonov recreates this vanished world of borrowers, bankrupts, lenders, and loan sharks in imperial Russia from the reign of Nicholas I to the period of great social and political reforms of the 1860s. Poring over a trove of previously unexamined records, Antonov gleans insights into the experiences of ordinary Russians, rich and poor, and shows how Russia’s informal but sprawling credit system helped cement connections among property owners across socioeconomic lines. Individuals of varying rank and wealth commonly borrowed from one another. Without a firm legal basis for formalizing debt relationships, obtaining a loan often hinged on subjective perceptions of trustworthiness and reputation. Even after joint-stock banks appeared in Russia in the 1860s, credit continued to operate through vast networks linked by word of mouth, as well as ties of kinship and community. Disputes over debt were common, and Bankrupts and Usurers of Imperial Russia offers close readings of legal cases to argue that Russian courts—usually thought to be underdeveloped in this era—provided an effective forum for defining and protecting private property interests.
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