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Geometric transitions, Chern-Simons gauge theory and Veneziano type amplitudes

Tohru Eguchi, Hiroaki Kanno

TL;DR

The paper shows that all-genus topological string amplitudes for certain local Calabi-Yau manifolds arising from geometric transitions can be computed via a free-field operator framework in 2D CFT. By representing Schur and skew Schur functions with vertex operators and exploiting the boson-fermion correspondence, the authors recast Chern-Simons Hopf link invariants and topological vertices as vacuum expectations of products of vertex operators, yielding Veneziano-like amplitudes. A key result is the expression $K^{SU(N)}_{\{R_i\}} = \prod_{i=1}^N \dim_q R_i \cdot \langle 0 | \prod_{k=1}^N V_-^{[R_k^t]}(q) V_+^{[R_k]}(q) Q_k^{L_0} | 0 \rangle$, which reproduces Nekrasov's all-genus formula for $SU(N)$ gauge theories. The approach highlights a simple free-field structure underlying all-genus topological string amplitudes and suggests avenues for extending to compact Calabi-Yau and Fano geometries.

Abstract

We consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory. We introduce an operator technique of 2-dimensional CFT which greatly simplifies the computations. We in particular show that in the case of local Calabi-Yau manifolds described by toric geometry basic amplitudes are written as vacuum expectation values of a product vertex operators and thus appear quite similar to the Veneziano amplitudes of the old dual resonance models. Topological string amplitudes can be easily evaluated using vertex operator algebra.

Geometric transitions, Chern-Simons gauge theory and Veneziano type amplitudes

TL;DR

The paper shows that all-genus topological string amplitudes for certain local Calabi-Yau manifolds arising from geometric transitions can be computed via a free-field operator framework in 2D CFT. By representing Schur and skew Schur functions with vertex operators and exploiting the boson-fermion correspondence, the authors recast Chern-Simons Hopf link invariants and topological vertices as vacuum expectations of products of vertex operators, yielding Veneziano-like amplitudes. A key result is the expression , which reproduces Nekrasov's all-genus formula for gauge theories. The approach highlights a simple free-field structure underlying all-genus topological string amplitudes and suggests avenues for extending to compact Calabi-Yau and Fano geometries.

Abstract

We consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory. We introduce an operator technique of 2-dimensional CFT which greatly simplifies the computations. We in particular show that in the case of local Calabi-Yau manifolds described by toric geometry basic amplitudes are written as vacuum expectation values of a product vertex operators and thus appear quite similar to the Veneziano amplitudes of the old dual resonance models. Topological string amplitudes can be easily evaluated using vertex operator algebra.

Paper Structure

This paper contains 4 sections, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free fields and the vertex operator
  3. Sample calculation of Veneziano-like amplitudes
  4. Discussions

Figures (2)

  • Figure 1: Ladder diagram for $SU(N)$ gauge theory. Note that there are $N+1$ possible toric diagrams ($m= 0, \cdots N$). $m$ is related to the coefficients of the Chern-Simons coupling in five dimensions.
  • Figure 2: Veneziano-like diagram from cutting in half the $SU(N)$ ladder diagram. Note that we respect topology only and ignore the slopes of the edges.