[Paper Review] On the partition sum of the NS five-brane

Robbert Dijkgraaf, Erik Verlinde+1 more|ArXiv.org|May 27, 2002

Black Holes and Theoretical Physics29 references61 citations

TL;DR

This paper computes the quantum-corrected Euclidean partition function of the Type IIA NS five-brane wrapped on a Calabi-Yau threefold in a double-scaled decoupling limit with a flat RR 3-form background. Using T-duality to map the five-brane to an ALE singularity in Type IIB, the classical flux sum is expressed as a theta-function, while quantum corrections arise from B-model topological string amplitudes, yielding a full partition function that satisfies holomorphic anomaly equations and links little string theory to topological string theory.

ABSTRACT

We study the Type IIA NS five-brane wrapped on a Calabi-Yau manifold X in a double-scaled decoupling limit. We calculate the euclidean partition function in the presence of a flat RR 3-form field. The classical contribution is given by a sum over fluxes of the self-dual tensor field which reduces to a theta-function. The quantum contributions are computed using a T-dual IIB background where the five-branes are replaced by an ALE singularity. Using the supergravity effective action we find that the loop corrections to the free energy are given by B-model topological string amplitudes. This seems to provide a direct link between the double-scaled little strings on the five-brane worldvolume and topological strings. Both the classical and quantum contributions to the partition function satisfy (conjugate) holomorphic anomaly equations, which explains an observation of Witten relating topological string theory to the quantization of three-form fields.

Motivation & Objective

• To compute the Euclidean partition function of the NS five-brane wrapped on a Calabi-Yau manifold in a double-scaled decoupling limit with a flat RR 3-form field.
• To understand the quantum corrections to the five-brane partition function, especially in the presence of a background RR field.
• To establish a direct link between the double-scaled little string theory on the five-brane and topological string theory via the partition function.
• To show that both classical and quantum contributions to the partition function satisfy conjugate holomorphic anomaly equations.
• To provide a framework for computing five-brane instanton corrections to hypermultiplet moduli spaces in type IIA Calabi-Yau compactifications.

Proposed method

• Use T-duality to map the IIA NS five-brane configuration on a Calabi-Yau to a Type IIB background with an ALE singularity of type $A_{k-1}$, eliminating five-branes.
• Compute the classical partition function as a sum over fluxes of the self-dual tensor field, expressed as a theta-function $Z_X^{cl} = \overline{\Theta_{\alpha,\beta}(x;z)}$.
• Identify quantum corrections via the supergravity effective action, where F-terms involving $\mathcal{F}_g(z,\bar{z})$ contribute to the free energy.
• Relate the quantum corrections to B-model topological string amplitudes $\mathcal{F}_g(z,\bar{z})$, which encode loop corrections in the effective string coupling $\lambda^{2g-2}$.
• Use the holomorphic anomaly equations of the B-model to show that both classical and quantum parts of the partition function satisfy conjugate holomorphic anomaly equations.
• Derive the BCOV equations in the new coordinates by transforming derivatives and expressing the wavefunction in terms of $x^i$, $\lambda$, and complex structure moduli $z^i$.

Experimental results

Research questions

• RQ1 How does the presence of a flat RR 3-form field affect the quantum corrections to the five-brane partition function in the double-scaled limit?
• RQ2 What is the precise form of the quantum-corrected partition function for the NS five-brane wrapped on a Calabi-Yau manifold?
• RQ3 How does T-duality relate the five-brane system in IIA to a topological string background in IIB, and what is the role of the ALE singularity?
• RQ4 Why do both classical and quantum contributions to the partition function satisfy holomorphic anomaly equations, and what is the physical significance of this?
• RQ5 Can the five-brane partition function be expressed as a product of a theta-function and a generating function of topological string amplitudes, and does this generalize the familiar $\theta/\eta$ form to Calabi-Yau manifolds?

Key findings

• The classical partition function is given by a theta-function $Z_X^{cl} = \overline{\Theta_{\alpha,\beta}(x;z)}$, which sums over fluxes of the RR 3-form field on the Calabi-Yau manifold.
• Quantum corrections are encoded in the generating function $Z_X^{qu} = \exp\left(\sum_{g>0} \mathcal{F}_g(z,\bar{z}) \lambda^{2g-2}\right)$, where $\mathcal{F}_g$ are B-model topological string amplitudes.
• The full partition function is the product $Z_X = Z_X^{cl} \cdot Z_X^{qu}$, combining classical flux sums and quantum loop corrections.
• In the case $X = K3 \times T^2$, the partition function reduces to $Z_{K3\times T^2} = \frac{\theta_{4,20}(\tau,\bar{\tau})}{\eta(\tau)^{24}}$, where the theta-function arises from fluxes and the eta-function from the $g=1$ loop amplitude.
• Both classical and quantum parts of the partition function satisfy conjugate holomorphic anomaly equations, explaining Witten's observation on the quantization of three-form fields.
• The result establishes a direct link between the double-scaled little string theory on the five-brane worldvolume and topological string theory via the partition function's structure.