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Deformed Topological Partition Function and Nekrasov Backgrounds

I. Antoniadis, S. Hohenegger, K. S. Narain, T. R. Taylor

TL;DR

The paper studies deformations of the N=2 topological string via higher-dimensional F-terms $W^{2g}\Upsilon^n$, producing amplitudes that couple $(R^-)^2 (T^-)^{2g-2} (F^+)^{2n}$ with coefficients $F_{g,n}$ and proposing a refinement of the topological string aligned with Nekrasov's $\epsilon_{1,2}$-backgrounds. It computes these amplitudes in Type II as genus-$g$ topological correlators and shows independence from the Type II dilaton, implying exactness at genus $g$, while near conifold singularities the behavior is governed by a $c=1$ string with radius deformation. The heterotic dual at one loop reproduces the perturbative part of Nekrasov's partition function in the $\Omega$-background (up to a phase in $\epsilon_+$), and duality suggests that Type II results also capture non-perturbative heterotic contributions. Overall, the work connects refined topological strings to gauge-theory instanton physics, clarifying how $F_{g,n}$ encode Nekrasov-like deformations and how conifold and $c=1$ dynamics emerge in the dual string descriptions.

Abstract

A deformation of the N=2 topological string partition function is analyzed by considering higher dimensional F-terms of the type W^{2g}*Upsilon^n, where W is the chiral Weyl superfield and each Upsilon factor stands for the chiral projection of a real function of N=2 vector multiplets. These terms generate physical amplitudes involving two anti-self-dual Riemann tensors, 2g-2 anti-self-dual graviphoton field strengths and 2n self-dual field strengths from the matter vector multiplets. Their coefficients F_{g,n} generalizing the genus g partition function F_{g,0} of the topological twisted type II theory, can be used to define a generating functional by introducing deformation parameters besides the string coupling. Choosing all matter field strengths to be that of the dual heterotic dilaton supermultiplet, one obtains two parameters that we argue should correspond to the deformation parameters of the Nekrasov partition function in the field theory limit, around the conifold singularity. Its perturbative part can be obtained from the one loop analysis on the heterotic side. This has been computed in [1] and in the field theory limit shown to be given by the radius deformation of c=1 CFT coupled to two-dimensional gravity. Quite remarkably this result reproduces the gauge theory answer up to a phase difference that may be attributed to the regularization procedure. The type II results are expected to be exact and should also capture the part that is non-perturbative in heterotic dilaton.

Deformed Topological Partition Function and Nekrasov Backgrounds

TL;DR

The paper studies deformations of the N=2 topological string via higher-dimensional F-terms , producing amplitudes that couple with coefficients and proposing a refinement of the topological string aligned with Nekrasov's -backgrounds. It computes these amplitudes in Type II as genus- topological correlators and shows independence from the Type II dilaton, implying exactness at genus , while near conifold singularities the behavior is governed by a string with radius deformation. The heterotic dual at one loop reproduces the perturbative part of Nekrasov's partition function in the -background (up to a phase in ), and duality suggests that Type II results also capture non-perturbative heterotic contributions. Overall, the work connects refined topological strings to gauge-theory instanton physics, clarifying how encode Nekrasov-like deformations and how conifold and dynamics emerge in the dual string descriptions.

Abstract

A deformation of the N=2 topological string partition function is analyzed by considering higher dimensional F-terms of the type W^{2g}*Upsilon^n, where W is the chiral Weyl superfield and each Upsilon factor stands for the chiral projection of a real function of N=2 vector multiplets. These terms generate physical amplitudes involving two anti-self-dual Riemann tensors, 2g-2 anti-self-dual graviphoton field strengths and 2n self-dual field strengths from the matter vector multiplets. Their coefficients F_{g,n} generalizing the genus g partition function F_{g,0} of the topological twisted type II theory, can be used to define a generating functional by introducing deformation parameters besides the string coupling. Choosing all matter field strengths to be that of the dual heterotic dilaton supermultiplet, one obtains two parameters that we argue should correspond to the deformation parameters of the Nekrasov partition function in the field theory limit, around the conifold singularity. Its perturbative part can be obtained from the one loop analysis on the heterotic side. This has been computed in [1] and in the field theory limit shown to be given by the radius deformation of c=1 CFT coupled to two-dimensional gravity. Quite remarkably this result reproduces the gauge theory answer up to a phase difference that may be attributed to the regularization procedure. The type II results are expected to be exact and should also capture the part that is non-perturbative in heterotic dilaton.

Paper Structure

This paper contains 5 sections, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Higher Derivative Couplings in the String Effective Action
  3. The Amplitude in Type II String Theory
  4. The Amplitude in Heterotic String Theory
  5. Concluding remarks

Figures (1)

  • Figure 1: Schematic drawing of the reducible diagram: The gauge kinetic term $F_I.F_J$ is connected to $F_{g,n-1}$ via the exchange of the scalar $\phi$.