Mirror manifolds and topological field theory. (English) Zbl 0834.58013

Witten, Edward

Yau, Shing-Tung (ed.), Essays on mirror manifolds. Cambridge, MA: International Press. 120-159 (1992).

This is a detailed expository paper “devoted to sketching how some of the standard facts relevant to mirror symmetry and its applications can be naturally understood in the context of topological field theory”. On a Calabi-Yau manifold the author introduces two models of topological field theories called \(A\) model and \(B\) model. (Mirror symmetry relates the \(A\) model of one Calabi-Yau manifold to the \(B\) model of its mirror). The further contents of the paper may be seen from the following (partial) list of the section titles: Kähler manifolds and twisted manifolds; reduction to weak coupling; the ghost number anomaly; observables of the \(A\) model; evaluation of the path integrals; anomalies (\(B\)-model); observables (\(B\)-model); correlation functions; the fixed point theorem; relation to the “physical” model; closer look at the observables (both \(A\) and \(B\) models).
For the entire collection see [Zbl 0816.00010].

Reviewer: Yu.E.Gliklikh (Voronezh)

MSC:

58D99	Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
81T10	Model quantum field theories
81Q30	Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
32G13	Complex-analytic moduli problems
32J27	Compact Kähler manifolds: generalizations, classification

Keywords:

mirror symmetry; topological field theory