K-Decompositions and 3d Gauge Theories

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.Comment: 121 pages + 2 appendices, 80 figures; Version 2: reorganized mathematical perspective, swapped Sections 3 and

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)

モジュラー多様体の双有理幾何学とコンパクト化及びモジュラー形式の数論について

京都大学新制・課程博士博士(理学)甲第24385号理博第4884号新制||理||1699(附属図書館)京都大学大学院理学研究科数学・数理解析専攻(主査)准教授 伊藤 哲史, 教授 雪江 明彦, 教授 池田 保学位規則第4条第1項該当Doctor of ScienceKyoto UniversityDFA

Four-manifolds, geometries and knots

The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of such manifolds and knots. The first chapter is purely algebraic. The rest of the book may be divided into three parts: general results on homotopy and surgery (Chapters 2-6), geometries and geometric decompositions (Chapters 7-13), and 2-knots (Chapters 14-18). In many cases the Euler characteristic, fundamental group and Stiefel-Whitney classes together form a complete system of invariants for the homotopy type of such manifolds, and the possible values of the invariants can be described explicitly. The strongest results are characterizations of manifolds which fibre homotopically over S^1 or an aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to homeomorphism). As a consequence 2-knots whose groups are poly-Z are determined up to Gluck reconstruction and change of orientations by their groups alone. This book arose out of two earlier books "2-Knots and their Groups" and "The Algebraic Characterization of Geometric 4-Manifolds", published by Cambridge University Press for the Australian Mathematical Society and for the London Mathematical Society, respectively. About a quarter of the present text has been taken from these books, and I thank Cambridge University Press for their permission to use this material. The book has been revised in March 2007. For details see the end of the preface.Comment: This is the revised version published by Geometry & Topology Monographs in March 200

The geometry and physics of F-theory compactifications

In this PhD thesis we study the structure of gauge and gravitational anomalies in effective theories obtained by compactfication of F-theory on Calabi-Yau manifolds. In particular, we study the continuous local anomalies in 2D N = (0, 2) effective theories from elliptically fibered Calabi-Yau five-fold compactifications and discrete gauge anomalies in 6D N = (1, 0) theories from F-theory on genus-one fibrations of Calabi-Yau three-folds. Certain anomalies associated with these symmetries, induced at 1-loop in perturbative theories, can be cancelled by a corresponding generalized Green-Schwarz mechanism operating at the level of chiral fields in the effective theories. We derive closed expressions for types Green-Schwarz mechanisms in F-theory compactifications, as well as the gravitational and gauge anomalies. These expressions in both cases involve topological invariants of the underlying fibrations of Calabi-Yau manifolds. Cancellation of these anomalies in the effective theories predicts intricate topological identities which must hold on every corresponding Calabi-Yau manifold. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of local anomalies for continuous symmetry in 6D N = (1, 0) and 4D N = 1 theories obtained from F-theory

Ohio State University Bulletin

Classes available for students to enroll in during the 1968-1969 academic year for The Ohio State University