Frobenius Algebra Structures in Topological Quantum Field Theory
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Information Geometry and Frobenius Algebra
Information geometry and Frobenius algebra Ruichao Jiang, Javad Tavakoli, Yiqiang Zhao Oct 2020 Abstract We show that a Frobenius sturcture is equivalent to a dually flat sturcture in information geometry. We define a multiplication structure on the tangent spaces of statistical manifolds, which we call the statisti- cal product. We also define a scalar quantity, which we call the Yukawa term. By showing two examples from statistical mechanics, first the clas- sical ideal gas, second the quantum bosonic ideal gas, we argue that the Yukawa term quantifies information generation, which resembles how mass is generated via the 3-points interaction of two fermions and a Higgs boson (Higgs mechanism). In the classical case, The Yukawa term is identically zero, whereas in the quantum case, the Yukawa term diverges as the fu- gacity goes to zero, which indicates the Bose-Einstein condensation. 1 Introduction Amari in his own monograph [1] writes that major journals rejected his idea of dual affine connections on the grounds of their nonexistence in the mainstream Riemannian geometry. Later on, this dualistic structure turned out to be useful in information science, e.g. EM algorithms. The aim of this paper is to show the Frobenius algebra structure in information geometry and thus give a physical meaning for the Amari-Chentsov tensor. In information geometry, instead of the usual Levi-Civita connection, a non- metrical connections and its dual ∗ are used. Associated with these two connections is a mysterious∇ tensor, called∇ the Amari-Chentsov tensor (AC ten- arXiv:2010.05110v1 [math.DG] 10 Oct 2020 sor) and a one-parameter family of flat connections, called the α-connections. -
Quantum Cohomology of Lagrangian and Orthogonal Grassmannians
QUANTUM COHOMOLOGY OF LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS HARRY TAMVAKIS Abstract. This is the written version of my talk at the Mathematische Ar- beitstagung 2001, which reported on joint work with Andrew Kresch (papers [KT2] and [KT3]). The Classical Theory For the purposes of this talk we will work over the complex numbers, and in the holomorphic (or algebraic) category. There are three kinds of Grassmannians that we will consider, depending on their Lie type: • In type A,letG(k; N)=SLN =Pk be the Grassmannian which parametrizes N k-dimensional linear subspaces of C . • In type C,letLG = LG(n; 2n)=Sp2n=Pn be the Lagrangian Grassmannian. • In types B and D,letOG = OG(n; 2n +1)=SO2n+1=Pn denote the odd or- thogonal Grassmannian, which is isomorphic to the even orthogonal Grassmannian OG(n +1; 2n +2)=SO2n+2=Pn+1. We begin by defining the Lagrangian Grassmannian LG(n; 2n), and will discuss the Grassmannians for the other types later in the talk. Let V be the vector 2n V space C , equipped with a symplectic form. A subspace Σ of is isotropic if the restriction of the form to Σ vanishes. The maximal possible dimension of an isotropic subspace is n, and in this case Σ is called a Lagrangian subspace. The Lagrangian Grassmannian LG = LG(n; 2n) is the projective complex manifold which parametrizes Lagrangian subspaces Σ in V . There is a stratification of LG by Schubert varieties Xλ, one for each strict n ` ` λ partition λ =(λ1 >λ2 > ··· >λ` > 0) with λ1 6 . The number = ( )isthe n length of λ, and we let Dn denote the set of strict partitions λ with λ1 6 .To describe Xλ more precisely, fix an isotropic flag of subspaces Fi of V : 0 ⊂ F1 ⊂ F2 ⊂···⊂Fn ⊂ V with dim(Fi)=i for each i,andFn Lagrangian. -
Frobenius Algebras and Planar Open String Topological Field Theories
Frobenius algebras and planar open string topological field theories Aaron D. Lauda Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB UK email: [email protected] October 29, 2018 Abstract Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the category of 2-dimensional cobordisms can be described as the free symmetric monoidal category on a commutative Frobenius algebra, 2Thick is shown to be the free monoidal category on a noncommutative Frobenius algebra, hence justifying this choice of data in the Moore-Segal axioms. Our formalism is inherently categorical allowing us to generalize this result. As a stepping stone towards topological mem- brane theory we define a 2-category of open strings, planar open string worldsheets, and isotopy classes of 3-dimensional membranes defined by diffeomorphisms of the open string worldsheets. This 2-category is shown to be the free (weak monoidal) 2-category on a ‘categorified Frobenius al- gebra’, meaning that categorified Frobenius algebras determine invariants of these 3-dimensional membranes. arXiv:math/0508349v1 [math.QA] 18 Aug 2005 1 Introduction It is well known that 2-dimensional topological quantum field theories are equiv- alent to commutative Frobenius algebras [1, 16, 29]. This simple algebraic characterization arises directly from the simplicity of 2Cob, the 2-dimensional cobordism category. In fact, 2Cob admits a completely algebraic description as the free symmetric monoidal category on a commutative Frobenius algebra. -
Gromov--Witten Invariants and Quantum Cohomology
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 116, No. 4, November 2003, pp. 459–476. Printed in India Gromov–Witten invariants and quantum cohomology AMIYA MUKHERJEE Stat-Math Division, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India E-mail: [email protected] Dedicated to Professor K B Sinha on the occasion of his 60th birthday Abstract. This article is an elaboration of a talk given at an international confer- ence on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December 20–23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov–Witten invariants. Of course there are many important aspects that are not discussed here. Keywords. J-holomorphic curves; moduli spaces; Gromov–Witten classes; quantum cohomology. 1. Introduction Quantum cohomology is a new mathematical discipline influenced by the string theory as a joint venture of physicists and mathematicians. The notion was first proposed by Vafa [V], and developed by Witten [W] and others [B]. The theory consists of some new approaches to the problem of constructing invariants of compact symplectic manifolds and algebraic varieties. The approaches are related to the ideas of a (1 + 1)-dimensional topological quantum field theory, which indicate that the general principle of constructing invariants should be as follows: The invariants of a manifold M should be obtained by integrating cohomology classes over certain moduli space M associated to M. In our case the manifold is a symplectic manifold (M,ω), and the moduli space M is the space of 1 certain J-holomorphic spheres σ: CP M in a given homology class A H2(M;Z). -
Duality and Integrability in Topological String Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by SISSA Digital Library International School of Advanced Studies Mathematical Physics Sector Academic Year 2008–09 PhD Thesis in Mathematical Physics Duality and integrability in topological string theory Candidate Andrea Brini Supervisors External referees Prof. Boris Dubrovin Prof. Marcos Marino˜ Dr. Alessandro Tanzini Dr. Tom Coates 2 Acknowledgements Over the past few years I have benefited from the interaction with many people. I would like to thank first of all my supervisors: Alessandro Tanzini, for introducing me to the subject of this thesis, for his support, and for the fun we had during countless discussions, and Boris Dubrovin, from whose insightful explanations I have learned a lot. It is moreover a privilege to have Marcos Mari˜no and Tom Coates as referees of this thesis, and I would like to thank them for their words and ac- tive support in various occasions. Special thanks are also due to Sara Pasquetti, for her help, suggestions and inexhaustible stress-energy transfer, to Giulio Bonelli for many stimulating and inspiring discussions, and to my collaborators Renzo Cava- lieri, Luca Griguolo and Domenico Seminara. I would finally like to acknowledge the many people I had discussions/correspondence with: Murad Alim, Francesco Benini, Marco Bertola, Vincent Bouchard ;-), Ugo Bruzzo, Mattia Cafasso, Hua-Lian Chang, Michele Cirafici, Tom Claeys, Andrea Collinucci, Emanuel Diaconescu, Bertrand Ey- nard, Jarah Evslin, Fabio Ferrari Ruffino, Barbara -
Relative Floer and Quantum Cohomology and the Symplectic Topology of Lagrangian Submanifolds
Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds Yong-Geun Oh Department of Mathematics University of Wisconsin Madison, WI 53706 Dedicated to the memory of Andreas Floer Abstract: Floer (co)homology of the symplectic manifold which was originally introduced by Floer in relation to the Arnol’d conjecture has recently attracted much attention from both mathematicians and mathematical physicists either from the algebraic point of view or in relation to the quantum cohomology and the mirror symmetry. Very recent progress in its relative version, the Floer (co)homology of Lagrangian submanifolds, has revealed quite different mathematical perspective: The Floer (co)homology theory of Lagrangian submanifolds is a powerful tool in the study of symplectic topology of Lagrangian submanifolds, just as the classical (co)homology theory in topology has been so in the study of differential topology of submanifolds on differentiable manifolds. In this survey, we will review the Floer theory of Lagrangian submanifolds and explain the recent progress made by Chekanov and by the present author in this relative Floer theory, which have found several applications of the Floer theory to the symplectic topology of Lagrangian submanifolds which demonstrates the above perspective: They include a Floer theoretic proof of Gromov’s non-exactness theorem, an optimal lower bound for the symplectic disjunction energy, construction of new symplectic invariants, proofs of the non-degeneracy of Hofer’s distance on the space of Lagrangian submanifolds in the cotangent bundles and new results on the Maslov class obstruction to the Lagrangian embedding in Cn. Each of these applications accompanies new development in the Floer theory itself: localizations, semi-infinite cycles and minimax theory in the Floer theory and a spectral sequence as quantum cor- rections and others. -
1. Eigenvalues of Hermitian Matrices and Schubert Calculus
AROUND THE HORN CONJECTURE L MANIVEL Abstract We discuss the problem of determining the p ossible sp ectra of a sum of Hermitian matrices each with known sp ectrum We explain how the Horn conjecture which gives a complete answer to this question is related with algebraic geometry symplectic geometry and representation theory The rst lecture is an introduction to Schubert calculus from which one direction of Horns conjecture can b e deduced The reverse direction follows from an application of geometric invariant theory this is treated in the second lecture Finally we explain in the third lecture how a version of Horns problem for sp ecial unitary matrices is related to the quantum cohomology of Grassmannians Notes for Gael VI I I CIRM March Eigenvalues of hermitian matrices and Schubert calculus The problem Let A B C b e complex Hermitian n by n matrices Denote the set of eigenvalues or sp ectrum of A by A A A and similarly by B and C the sp ectra n of B and C The main theme of these notes is the following question Suppose that A B C What can be their spectra A B C There are obvious relations like trace C trace A trace B or C A B But a complete answer to this longstanding question was given only recently and combines works and ideas from representation theory symplectic and algebraic geometry Weyls inequalities There are various characterizations of the eigenvalues of Hermitian ma trices many of which are variants of the minimax principle Let j b e the standard Hermitian n pro duct on C If s n r -
Arxiv:Math/9805094V2
FROBENIUS MANIFOLD STRUCTURE ON DOLBEAULT COHOMOLOGY AND MIRROR SYMMETRY HUAI-DONG CAO & JIAN ZHOU Abstract. We construct a differential Gerstenhaber-Batalin-Vilkovisky alge- bra from Dolbeault complex of any closed K¨ahler manifold, and a Frobenius manifold structure on Dolbeault cohomology. String theory leads to the mysterious Mirror Conjecture, see Yau [29] for the history. One of the mathematical predictions made by physicists based on this con- jecture is the formula due to Candelas-de la Ossa-Green-Parkes [4] on the number of rational curves of any degree on a quintic in CP4. Recently, it has been proved by Lian-Liu-Yau [15]. The theory of quantum cohomology, also suggested by physicists, has lead to a better mathematical formulation of the Mirror Conjecture. As explained in Witten [28], there are two topological conformal field theories on a Calabi-Yau manifold X: A theory is independent of the complex structure of X, but depends on the K¨aher form on X, while B theory is independent of the K¨ahler form of X, but depends on the complex structure of X. Vafa [25] explained how two quantum cohomology rings and ′ arise from these theories, and the notion of mirror symmetry could be translatedR R into the equivalence of A theory on a Calabi-Yau manifold X with B theory on another Calabi-Yau manifold Xˆ, called the mirror of X, such that the quantum ring can be identified with ′. For a mathematicalR exposition in termsR of variation of Hodge structures, see e.g. Morrison [19] or Bertin-Peters [2]. There are two natural Frobenius algebras on any Calabi-Yau n-fold, A(X)= n Hk(X, Ωk), B(X)= n Hk(X, Ω−k), ⊕k=0 ⊕k=0 where Ω−k is the sheaf of holomorphic sections to ΛkTX. -
Frobenius Algebras and Their Quivers
Can. J. Math., Vol. XXX, No. 5, 1978, pp. 1029-1044 FROBENIUS ALGEBRAS AND THEIR QUIVERS EDWARD L. GREEN This paper studies the construction of Frobenius algebras. We begin with a description of when a graded ^-algebra has a Frobenius algebra as a homo- morphic image. We then turn to the question of actual constructions of Frobenius algebras. We give a, method for constructing Frobenius algebras as factor rings of special tensor algebras. Since the representation theory of special tensor algebras has been studied intensively ([6], see also [2; 3; 4]), our results permit the construction of Frobenius algebras which have representa tions with prescribed properties. Such constructions were successfully used in [9]. A special tensor algebra has an associated quiver (see § 3 and 4 for defini tions) which determines the tensor algebra. The basic result relating an associated quiver and the given tensor algebra is that the category of represen tations of the quiver is isomorphic to the category of modules over the tensor algebra [3; 4; 6]. One of the questions which is answered in this paper is: given a quiver J2, is there a Frobenius algebra T such that i2 is the quiver of T; that is, is there a special tensor algebra T = K + M + ®« M + . with associated quiver j2 such that there is a ring surjection \p : T —> T with ker \p Ç (®RM + ®%M + . .) and Y being a Frobenius algebra? This question was completely answered under the additional constraint that the radical of V cubed is zero [8, Theorem 2.1]. The study of radical cubed zero self-injective algebras has been used in classifying radical square zero Artin algebras of finite representation type [11]. -
Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary
Applications and Combinatorics in Algebraic Geometry Frank Sottile Summary Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its appli- cations. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its application-friendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators. The intellectual merits of this project include the development of applications of al- gebraic geometry and of combinatorial and combinatorial aspects of algebraic geometry. Specifically, Sottile will work to develop the theory and properties of orbitopes from the perspective of convex algebraic geometry, continue to investigate linear precision in geo- metric modeling, and apply the quantum Schubert calculus to linear systems theory. -
On the Frobenius Manifold
Frobenius manifolds and Frobenius algebra-valued integrable systems Dafeng Zuo (USTC) Russian-Chinese Conference on Integrable Systems and Geometry 19{25 August 2018 The Euler International Mathematical Institute 1 Outline: Part A. Motivations Part B. Some Known Facts about FM Part C. The Frobenius algebra-valued KP hierarchy Part D. Frobenius manifolds and Frobenius algebra-valued In- tegrable systems 2 Part A. Motivations 3 KdV equation : 4ut − 12uux − uxxx = 0 3 2 2 Lax pair: Lt = [L+;L];L = @ + 2 u tau function: u = (log τ)xx Bihamiltonian structure(BH): ({·; ·}2 =) Virasoro algebra) n o n o ( ) R R 3 f;~ ~g = 2 δf @ δg dx; f;~ ~g = −1 δf @ + 2u @ + 2 @ u δg dx: 1 δu @x δu 2 2 δu @x3 @x @x δu Dispersionless =) Hydrodynamics-type BH =) Frobenius manifold ······ 4 Natural generalization Lax pair/tau function/BH: KdV =) GDn / DS =) KP (other types) (N) {·; ·}2: Virasoro =) Wn algbra =) W1 algebra Lax pair/tau function/BH: dKdV =) dGDn =) dKP (N) {·; ·}2: Witt algebra =) wn algbra =) w1 algebra FM: A1 =) An−1 =) Infinite-dimensional analogue (?) 5 Other generalizations: the coupled KdV 4vt − 12vvx − vxxx = 0; 4wt − 12(vw)x − wxxx = 0: τ1 tau function: v = (log τ0)xx; w = ( )xx: τ0 R. Hirota, X.B.Hu and X.Y.Tang, J.Math.Anal.Appl.288(2003)326 BH: A.P.Fordy, A.G.Reyman and M.A.Semenov-Tian-Shansky,Classical r-matrices and compatible Poisson brackets for coupled KdV systems, Lett. Math. Phys. 17 (1989) 25{29. W.X.Ma and B.Fuchssteiner, The bihamiltonian structure of the perturbation equations of KdV Hierarchy. -
On the De Rham Cohomology of Algebraic Varieties
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. ROBIN HARTSHORNE On the de Rham cohomology of algebraic varieties Publications mathématiques de l’I.H.É.S., tome 45 (1975), p. 5-99 <http://www.numdam.org/item?id=PMIHES_1975__45__5_0> © Publications mathématiques de l’I.H.É.S., 1975, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES by ROBIN HARTSHORNE CONTENTS INTRODUCTION .............................................................................. 6 CHAPTER I. — Preliminaries .............................................................. n § i. Cohomology theories ............................................................... 9 § 2. Direct and inverse images .......................................................... 10 § 3. Hypercohomology ................................................................. 11 § 4. Inverse limits...................................................................... 12 § 5. Completions ...................................................................... 18 §