Abstract

We consider BPS-counting functions mathcal{Z} N,M of N parallel M5-branes probing a transverse ℤM orbifold geometry. These brane web configurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fibration over (affine) AN −1. We make this symmetry of mathcal{Z} N,M manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to infinitely many cancellations among states contributing to the partition function. The resulting (simplified) mathcal{Z} N,M can be written as a sum over weights forming a single irreducible representation of the Lie algebra mathfrak{a} N −1 (or its affine counterpart). We show this behaviour explicitly for an extensive list of examples for specific values of (N, M ) and generalise the arising pattern for generic configurations. Finally, for a particular compact M5-brane setup we use this form of the partition function to make the duality N ↔ M manifest.

Highlights

  • We consider BPS-counting functions ZN,M of N parallel M5-branes probing a transverse ZM orbifold geometry. These brane web configurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fibration over AN−1. We make this symmetry of ZN,M manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to infinitely many cancellations among states contributing to the partition function

  • Six dimensional superconformal field theories (SCFTs) along with their compactifications to lower dimensions have attracted a lot of attention in recent years: on the one hand, the dynamics of these theories display very rich structures which are interesting to explore in their own right

  • The partition function ZN,M (m = n, ) simplifies dramatically due to the fact that the corresponding supercharges obtain a non-trivial holonomy structure. This allows for infinitely many cancellations of different BPS-states contributing to the partition function, dramatically simplifying ZN,M (m = n, ): by studying a series of examples, we show that in the case of a non-compact brane configuration, the partition function becomes a polynomial of order M n2 in

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Summary

Introduction

Six dimensional superconformal field theories (SCFTs) along with their compactifications to lower dimensions have attracted a lot of attention in recent years: on the one hand, the dynamics of these theories display very rich structures which are interesting to explore in their own right. In this paper we generalise this observation to make the AN−1 (or affine AN−1) symmetry of the partition function ZN,M manifest and organise it according to irreducible (integrable) representations of the associated Lie algebra aN−1 (or affine aN−1) for certain choices of the deformation parameters: for simplicity, we consider the unrefined partition functions (i.e. we choose 1 = − 2 = ) and consider the case m = n with n ∈ N While the former enhances the supersymmetry to N = (4, 0), the latter choice does not change the superconformal algebra on the M-string world-sheet. Several supplementary computations as well as additional information on simple and affine Lie algebras and their representations are relegated to 5 appendices

Non-compact brane webs
Compact brane webs
Deformation parameters
Orbifold action and brane web parameters
Type II description
Toric Calabi-Yau manifolds
Supersymmetry
Compact and non-compact M-brane configurations
Particular values of the deformation parameters
Examples: non-compact brane configurations
Examples: compact brane configuration

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