The structure of 2D semi-simple field theories
type: Item
instance of: scholarly article
title: The structure of 2D semi-simple field theories
zbMATH Open document ID: 1248.53074
author: Constantin Teleman
published in: Inventiones Mathematicae
publication date: 2012-06-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/0712.0160
review text: Algebraic classifications of 2-dimensional family topological field theories (FTFT's) are presented. It leads to the reconstruction of Gromov-Witten invariants from the quantum cup-product at a single semisimple point and the first Chern class of the manifold. On a topological space \(X\), a FTFT is a symmetric monoidal transformation \(Z\) (i.e., multiplicative under disjoint unions) from \({\mathcal C}(X)\) to \({\mathcal F}(X)\), where objects of \({\mathcal C}\) and \({\mathcal F}\) are bundles of closed oriented 1-manifolds over \(X\) and flat complex vector bundles over \(X\), and morphisms are bundles of compact 2-bordisms, modulo boundary-fixing oriented homeomorphisms over \(X\), and the graded vector spaces \[ \text{Hom}_{{\mathcal F}(X)}(V, W)= H^*(X: \Hom_X(V, W)). \] In this paper, assuming \(A= Z(S^1)\) to be semisimple, classifications of the following four variants of FTFT's are given. 1. The simplest case, where the surfaces have parametrized boundaries. 2. The case boundaries allow free rotations. 3. The case surfaces are allowed to degenerate modally into the Lefschetz fibrations of algebraic geometry. 4. Deligne-Mumford theories (DMT's), which includes the Cohomological Field theories (CohFT's). \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)]. The transform from \({\mathcal C}(X)\) to \({\mathcal F}(X)\) of the types 1, 2 and 3, 4 are denoted by \(\widetilde Z\), \(Z\) and \(\overline Z\), respectively (\S1). Classifications of these theories are described in terms of the tautological classes on the moduli of surfaces. The key to the classifications are stability theorems of \textit{J. L. Harer} [Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)], cf. [\textit{N. V. Ivanov}, On stabilization of the homology of Teichmüller modular groups. Leningr. Math. J. 1, No. 3, 675--691 (1990); translation from Algebra Anal. 1, No. 3, 110--126 (1989; Zbl 0727.30036); Theorem 2.1] and Madsen and…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 6050512
zbMATH Keywords: 2D family topology field theory
zbMATH Keywords: Lefschetz fibration
zbMATH Keywords: Deligne-Mumford theory
zbMATH Keywords: cohomological field theory
zbMATH Keywords: Frobenius manifold
zbMATH Keywords: Gromov-Witten invariant
MaRDI profile type: Publication
OpenAlex ID: W2123872555
arXiv ID: 0712.0160
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DOI: 10.1007/S00222-011-0352-5
description: scientific article; zbMATH DE number 6050512