[Paper Review] Comments on Worldsheet Description of the Omega Background

Yu Nakayama, Hirosi Ooguri|arXiv (Cornell University)|Jun 27, 2011

Black Holes and Theoretical Physics22 references4 citations

TL;DR

This paper proposes a worldsheet description of Nekrasov's partition function in the Omega background for non-self-dual fluxes, generalizing the known correspondence between Nekrasov partition functions and topological string amplitudes beyond the self-dual limit. Using heterotic string compactifications on $T^2 \times K3$, it identifies the role of FI terms and R-symmetry twists in generating the required worldsheet amplitudes, reproducing the correct $\epsilon_\pm$ dependence and confirming reflection symmetry in the partition function.

ABSTRACT

Nekrasov's partition function is defined on a flat bundle of R^4 over S^1 called the Omega background. When the fibration is self-dual, the partition function is known to be equal to the topological string partition function, which computes scattering amplitudes of self-dual gravitons and graviphotons in type II superstring compactified on a Calabi-Yau manifold. We propose a generalization of this correspondence when the fibration is not necessarily self-dual.

Motivation & Objective

• To generalize the correspondence between Nekrasov's partition function and topological string amplitudes beyond the self-dual Omega background.
• To identify the worldsheet realization of non-self-dual Omega backgrounds in heterotic string theory on $T^2 \times K3$.
• To clarify the role of FI terms and R-symmetry twists in generating the correct $\epsilon_+ \neq 0$ dependence in the partition function.
• To test the proposal via one-loop heterotic string amplitude computations and compare with Nekrasov's formula.
• To extend the geometric interpretation of the Omega background in type II string theory using vertex operator insertions.

Proposed method

• Construct a Melvin-type geometry with non-self-dual field strengths $\mathcal{F}$, parameterized by $\epsilon_+$ and $\epsilon_-$, to model the Omega background.
• Use heterotic string compactification on $T^2 \times K3$ to realize the Omega background, with FI terms in vector multiplets generating R-symmetry twists.
• Identify the worldsheet vertex operators for the FI terms and graviphotons, including $V_{D^A}$ and $\Psi_A$, to describe the topological twist.
• Compute the one-loop heterotic string amplitude with insertions of $(-1,-1)$ picture vertex operators and spectral flow operators $\rho(z)$, $\bar{\rho}(\bar{z})$.
• Derive the partition function via heat kernel methods and spectral functions, matching the form $\log Z = \int \frac{dt}{t} \frac{\mathrm{tr}(-1)^{2J_-^3 + 2J_+^3} e^{-4t\epsilon_- J_-^3 - 4t\epsilon_+ J_+^3}}{{\sinh}(\epsilon_- + \epsilon_+)t \cdot {\sinh}(\epsilon_- - \epsilon_+)t} e^{-\mu t}$.
• Use heterotic/type IIA duality to map the background to a type II superstring setup, where the Reeb vector geometrically realizes the $U(1)$ R symmetry.

Experimental results

Research questions

• RQ1 How can the Nekrasov partition function be generalized beyond the self-dual Omega background ($\epsilon_+ = 0$) in a worldsheet formulation?
• RQ2 What is the role of FI terms and R-symmetry twists in generating the $\epsilon_+ \neq 0$ dependence in the partition function?
• RQ3 How do vertex operator insertions in the heterotic string realize the topological twist required for the worldsheet description?
• RQ4 Can the one-loop heterotic string amplitude reproduce the full Nekrasov partition function for non-self-dual backgrounds?
• RQ5 How is the $\epsilon_+ \to -\epsilon_+$ reflection symmetry realized in the worldsheet theory?

Key findings

• The one-loop heterotic string amplitude with appropriate vertex operator insertions reproduces the correct $\epsilon_+$ and $\epsilon_-$ dependence of Nekrasov's partition function.
• The presence of the $-2\cosh(2\epsilon_+ t)$ factor in the amplitude is traced to the FI term, which induces the worldsheet R-symmetry twist.
• The worldsheet theory exhibits the expected reflection symmetry $\epsilon_\pm \to -\epsilon_\pm$, consistent with the enhanced $SU(2)$ R-symmetry in the field theory limit.
• The proposal successfully generalizes the correspondence between Nekrasov partition functions and topological string amplitudes to non-self-dual Omega backgrounds.
• The type II dual description realizes the $U(1)$ R symmetry geometrically via the Reeb vector, and the vertex operators for the graviphoton and FI terms match the required spectral flow and twist structures.
• The derived partition function matches the general formula $\log Z = \int \frac{dt}{t} \frac{\mathrm{tr}(-1)^{2J_-^3 + 2J_+^3} e^{-4t\epsilon_- J_-^3 - 4t\epsilon_+ J_+^3}}{{\sinh}(\epsilon_- + \epsilon_+)t \cdot {\sinh}(\epsilon_- - \epsilon_+)t} e^{-\mu t}$, confirming consistency with known results.