Enveloping Algebras and Geometric Representation Theory

Abstract. The study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras. The aim of the workshop was to bring together researchers from diverse but highly interrelated subjects to discuss new developments and bring forward the research in this whole area by fostering the scientific interaction

Mathematical Methods for 4d N=2 QFTs

In this work we study different aspects of 4d N = 2 superconformal field theories. Not only we accurately define what we mean by a 4d N = 2 superconformal field theory, but we also invent and apply new mathematical methods to classify these theories and to study their physical content. Therefore, although the origin of the subject is physical, our methods and approach are rigorous mathematical theorems: the physical picture is useful to guide the intuition, but the full mathematical rigor is needed to get deep and precise results. No familiarity with the physical concept of Supersymmetry (SUSY) is need to understand the content of this thesis: everything will be explained in due time. The reader shall keep in mind that the driving force of this whole work are the consequences of SUSY at a mathematical level. Indeed, as it will be detailed in part II, a mathematician can understand a 4d N = 2 superconformal field theory as a complexified algebraic integrable system. The geometric properties are very constrained: we deal with special K\ua8ahler geometries with a few other additional structures (see part II for details). Thanks to the rigidity of these structures, we can compute explicitly many interesing quantities: in the end, we are able to give a coarse classification of the space of "action" variables of the integrable system, as well as a fine classification -- only in the case of rank k = 1 -- of the spaces of "angle" variables. We were able to classify conical special K\ua8ahler geometries via a number of deep facts of algebraic number theory, diophantine geometry and class field theory: the perfect overlap between mathematical theorems and physical intuition was astonishing. And we believe we have only scratched the surface of a much deeper theory: we can probably hope to get much more information than what we already discovered; of course, a deeper study of the subject -- as well as its generalizations -- is required. A 4d N = 2 superconformal field theory can thus be defined by its geometric structure: its scaling dimensions, its singular fibers, the monodromy around them and so on. But giving a proper and detailed definition is only the beginning: one may be interested in exploring its physical content. In particular, we are interested in supersymmetric quantities such as BPS states, framed BPS states and UV line operators. These quantities, thanks to SUSY, can be computed independently of many parameters of the theory: this peculiarity makes it possible to use the language of category theory to analyze the aforementioned aspects. As it will be proven in part V, to each 4d N = 2 superconformal field theory we can associate a web of categories, all connected by functors, that describe the BPS states, the framed BPS states (IR) and the UV line operators. Hence, following the old ideas of \u2018t Hooft, it is possible to describe the phase space of gauge theories via categories, since the vacuum expectation values of such line operators are the order parameters of the confinement/deconfinement phase transitions. Mathematically, the (quantum) cluster algebra of Fomin and Zelevinski is the structure needed. Moreover, the analysis of BPS objects led us to a deep understanding of generalized S-dualities. Not only were we able to precisely define -- abstractly and generally -- what the S-duality group of a 4d N = 2 superconformal field theory should be, but we were also able to write a computer algorithm to obtain these groups in many examples (with very high accuracy)

Arithmetic and Hyperbolic Structures in String Theory

This monograph is an updated and extended version of the author's PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of a spacelike singularity (the "BKL-limit"). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be described in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of the theory. Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are described by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by automorphic forms on the double quotient G(Z)\G/K. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on non-holomorphic Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also show how these techniques can be applied to hypermultiplet moduli spaces in type II Calabi-Yau compactifications, and we provide a detailed analysis for the universal hypermultiplet.Comment: 346 pages, updated and extended version of the author's PhD thesi

Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with highlighted applications to string theory. v2: 375 pages. Substantially extended and small correction

Infinite Matrix Product States for long range SU(N) spin models

We construct 1D and 2D long-range SU(N) spin models as parent Hamiltonians associated with infinite matrix product states. The latter are constructed from correlators of primary fields in the SU(N) level 1 WZW model. Since the resulting groundstates are of Gutzwiller-Jastrow type, our models can be regarded as lattice discretizations of fractional quantum Hall systems. We then focus on two specific types of 1D spin chains with spins located on the unit circle, a uniform and an alternating arrangement. For an equidistant distribution of identical spins we establish an explicit connection to the SU(N) Haldane-Shastry model, thereby proving that the model is critical and described by a SU(N) level 1 WZW model. In contrast, while turning out to be critical as well, the alternating model can only be treated numerically. Our numerical results rely on a reformulation of the original problem in terms of loop models.Comment: 37 pages, 6 figure