The Mathematics of Fivebranes

TL;DR

The paper surveys how fivebranes illuminate connections between non-perturbative brane dynamics and deep mathematics, notably modular and automorphic forms, elliptic genera, and generalized Kac–Moody algebras, through the quantization of instanton moduli spaces and symmetric products. It develops generating functions for orbifold Euler numbers and elliptic genera, culminating in automorphic products $\Phi(\sigma,\tau,z)$ that count BPS states and serve as denominators in GMK algebras for groups like $SO(3,2;\mathbb{Z})\cong Sp(4;\mathbb{Z})$. The Calabi–Yau geometry controls the symmetry and growth of the partitions, with explicit forms for the K3 case and connections to the Narain lattice and T-/S-duality. Finally, the work presents a geometric picture of string interactions as deformations of $S^N X$ or instanton moduli, including a link to noncommutative instantons via the Nekrasov–Schwarz construction, highlighting a unifying framework for brane dynamics and number theory.

Abstract

Fivebranes are non-perturbative objects in string theory that generalize two-dimensional conformal field theory and relate such diverse subjects as moduli spaces of vector bundles on surfaces, automorphic forms, elliptic genera, the geometry of Calabi-Yau threefolds, and generalized Kac-Moody algebras.