M5/D4 brane partition function on a circle bundle

TL;DR

This paper computes and compares the partition functions of Abelian M5 and D4 branes on circle bundles, revealing their equivalence in zero modes and instanton contributions, and explores modular properties and singular fibrations.

Contribution

It demonstrates the equivalence of M5 and D4 brane partition functions including instanton effects, extending the understanding beyond BPS states and analyzing modular properties.

Findings

Zero mode partition functions of M5 and D4 branes agree.

Instanton partition functions match up to charge three.

KK modes are absent in the D4 brane, indicating differences in oscillator modes.

Abstract

We consider Abelian M5 brane on a six-manifold which we take as a circle bundle over a five-manifold $M$. We compute the zero mode part of the M5 brane partition function using Chern-Simons theory and Hamiltonian formulation respectively and find an agreement. We also show that the D4 brane on $M$ shares exactly the same zero mode partition function again using the Hamiltonian formulation. For the oscillator modes we find that KK modes associated with the circle compactification are missing from the D4 brane. By making an infinitesimal noncommutative deformation we have instanton threshold bound states. We explicitly compute the instanton partition function up to instanton charge three, and show a perfect match with a corresponding contribution inside the M5 brane partition function, thus providing a very strong supporting evidence that D4 brane is identical with M5 brane which extends beyond the BPS sector. We comment on the modular properties of the M5 brane partition function when compactified on $T^2$ times a four-manifold. We briefly discuss a case of a singular fibration.