[Paper Review] Mirror Manifolds And Topological Field Theory

Edward Witten|ArXiv.org|Dec 19, 1991

Homotopy and Cohomology in Algebraic Topology14 references400 citations

TL;DR

This paper establishes a topological field theory framework for mirror symmetry in Calabi-Yau manifolds by introducing two twisted models—A and B—that compute correlation functions via counting holomorphic maps (A model) and periods of differential forms (B model). The key result is that Yukawa couplings in the physical sigma model coincide with observables in the twisted models, enabling exact computation of instanton-corrected amplitudes and providing a geometric foundation for the mirror map.

ABSTRACT

These notes are devoted to explaining aspects of the mirror manifold problem that can be naturally understood from the point of view of topological field theory. Basically this involves studying the topological field theories made by twisting $N=2$ sigma models. This is mainly a review of old results, except for the discussion in \S7 of certain facts that may be relevant to constructing the ``mirror map'' between mirror moduli spaces.

Motivation & Objective

• To clarify how mirror symmetry emerges from topological field theory in the context of Calabi-Yau manifolds.
• To demonstrate that correlation functions in the A and B models compute physical observables like Yukawa couplings via geometric invariants.
• To propose extended moduli spaces as a natural framework for understanding the mirror map between Calabi-Yau manifolds.
• To show that instanton corrections vanish for the metric in the A model through dimension counting and C*-action symmetry.
• To explore the role of the exponential map and linear structures on moduli spaces in both A and B models.

Proposed method

• Twisting the N=2 nonlinear sigma model on a Riemann surface Σ yields two topological field theories: the A model (twisted for holomorphic maps) and the B model (twisted for periods of differential forms).
• The A model correlation functions are computed by counting holomorphic maps Σ→X with constraints, reduced to virtual dimension zero via C*-action on genus zero surfaces with two marked points.
• The B model correlation functions are computed by evaluating periods of differential forms on X, corresponding to classical cohomological data.
• A fixed-point theorem in the Feynman path integral formalism explains the reduction of observables to classical geometry in both models.
• The physical Yukawa couplings are identified as matrix elements in the untwisted model that match observables in the A and B twisted models under genus zero conditions.
• Extended moduli spaces for A(X) and B(Y) are introduced, with the metric on B(Y) proposed as a key to understanding the mirror map.

Experimental results

Research questions

• RQ1 How can the A and B models of a Calabi-Yau manifold be derived from twisting the N=2 nonlinear sigma model?
• RQ2 Why do the A and B model correlation functions compute rational curve counts and periods, respectively?
• RQ3 In what sense do physical Yukawa couplings in the untwisted model coincide with observables in the twisted A and B models?
• RQ4 Why do instanton corrections vanish for the metric in the A model, despite the presence of nontrivial holomorphic maps?
• RQ5 How can the extended moduli spaces of the A and B models provide a deeper understanding of the mirror map?

Key findings

• The A model correlation functions are determined by counting holomorphic maps Σ→X with constraints, reduced to virtual dimension zero via C*-action on genus zero surfaces with two marked points.
• The B model correlation functions are computed by integrating differential forms over cycles in X, corresponding to periods of the holomorphic (3,0)-form in Calabi-Yau threefolds.
• Yukawa couplings in the physical sigma model coincide with observables in the A and B models, respectively, due to nonrenormalization theorems and genus zero conditions.
• Instanton corrections to the metric in the A model vanish because the virtual dimension of the moduli space of rational curves intersecting two cycles is negative (-2), implying generic emptiness after C*-quotient.
• The extended moduli space of the B model, though not yet fully understood, is conjectured to be the natural setting for the mirror map, with the metric on this space potentially determining the map up to isometry.
• The exponential map on the moduli space of the A model corresponds to a linear structure on ⊕ₙHⁿ(X,C), while its counterpart in the B model remains poorly understood and likely depends on the base point.