[Paper Review] Topological Gravity as Large N Topological Gauge Theory

Rajesh Gopakumar, Cumrun Vafa|ArXiv.org|Feb 3, 1998

Black Holes and Theoretical Physics15 references8 citations

TL;DR

This paper proposes a duality between topological gravity on Calabi-Yau manifolds and large N topological Chern-Simons gauge theory, showing that the partition function of Kodaira-Spencer theory near singularities matches that of SU(∞) Chern-Simons theory on vanishing 3-cycles. The agreement is verified perturbatively for S³ and Lens spaces, with non-perturbative effects interpreted in the Type IIB picture, suggesting deeper equivalences in topological gravity-gauge systems.

ABSTRACT

We consider topological closed string theories on Calabi-Yau manifolds which compute superpotential terms in the corresponding compactified type II effective action. In particular, near certain singularities we compare the partition function of this topological theory (the Kodaira-Spencer theory) to $SU(\infty)$ Chern-Simons theory on the vanishing 3-cycle. We find agreement between these theories, which we check explicitly for the case of shrinking $S^3$ and Lens spaces, at the perturbative level. Moreover, the gauge theory has non-perturbative contributions which have a natural interpretation in the Type IIB picture. We provide a heuristic explanation for this agreement as well as suggest further equivalences in other topological gravity/gauge systems.

Motivation & Objective

• To establish a correspondence between topological gravity on Calabi-Yau manifolds and large N topological gauge theory.
• To compare the partition function of Kodaira-Spencer theory near singularities with that of SU(∞) Chern-Simons theory on vanishing 3-cycles.
• To verify the duality at the perturbative level for specific topologies like S³ and Lens spaces.
• To interpret non-perturbative contributions in the gauge theory within the Type IIB string theory framework.
• To suggest broader equivalences between topological gravity and gauge systems.

Proposed method

• The authors analyze topological closed string theory on Calabi-Yau manifolds to compute superpotential terms in the compactified type II effective action.
• They focus on the Kodaira-Spencer theory near singularities where the geometry features a vanishing 3-cycle.
• The partition function of this topological gravity theory is compared to that of SU(∞) Chern-Simons theory on the same 3-cycle.
• Perturbative agreement is explicitly checked for S³ and Lens space topologies using standard techniques in topological field theory.
• Non-perturbative contributions in the gauge theory are interpreted via dual Type IIB string theory constructions.
• A heuristic explanation for the duality is provided based on geometric transitions and large N limits.

Experimental results

Research questions

• RQ1 Does the partition function of topological gravity on Calabi-Yau manifolds near singularities match that of large N topological gauge theory?
• RQ2 Can the Kodaira-Spencer theory be described equivalently by SU(∞) Chern-Simons theory on a vanishing 3-cycle?
• RQ3 What is the nature of non-perturbative corrections in the gauge theory and how do they map to the IIB picture?
• RQ4 Are there broader dualities between topological gravity and topological gauge systems beyond the cases studied?
• RQ5 What is the geometric and physical mechanism underlying the proposed duality?

Key findings

• The partition function of Kodaira-Spencer theory on Calabi-Yau manifolds near singularities agrees with that of SU(∞) Chern-Simons theory on the vanishing 3-cycle at the perturbative level.
• Explicit agreement is confirmed for the case of a shrinking S³ and Lens spaces, supporting the duality in specific topological settings.
• Non-perturbative contributions in the gauge theory have a natural interpretation in terms of Type IIB string theory, suggesting a deeper physical origin.
• The duality is supported by a heuristic explanation based on geometric transitions and large N limits in topological field theory.
• The results suggest the existence of further equivalences between topological gravity and topological gauge systems in different backgrounds.
• The study provides a framework for understanding superpotential terms in compactified type II theories via large N gauge theory.