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The Weight of $Spin(32)/\mathbb{Z}_2$ Little Strings: T-duality and Hasse Diagrams

Hamza Ahmed, Florent Baume, Paul-Konstantin Oehlmann

Abstract

We study the worldvolume theories of stacks of $Spin(32)/\mathbb{Z}_2$ heterotic NS5-branes probing a transverse singularity $\mathfrak{g}$. We revisit and extend the original classification by Blum and Intriligator, and show that the resulting 6d Little String Theories (LSTs) are naturally labeled by affine dominant coweights of the singularity $\mathfrak{g}$. This in turn enables us to efficiently arrange these theories into groupings satisfying all known necessary conditions to be T-dual. Using this formulation, we then study the partial order of those coweights, and extend a recently proposed slice-subtraction algorithm to construct Hasse diagrams for LSTs directly from their six-dimensional generalized quivers, allowing us to probe certain properties of their Higgs branch. Along the way, we exploit these techniques to show that the number of duality orbits at maximal flavor rank is determined by the center of the transverse singularity $\mathfrak{g}$, and provide a simplified proof of a monotonicity theorem for this class of theories. Finally, we show how some of our techniques can be extended to other classes of 6d theories, such as Type-II LSTs and SCFTs.

The Weight of $Spin(32)/\mathbb{Z}_2$ Little Strings: T-duality and Hasse Diagrams

Abstract

We study the worldvolume theories of stacks of heterotic NS5-branes probing a transverse singularity . We revisit and extend the original classification by Blum and Intriligator, and show that the resulting 6d Little String Theories (LSTs) are naturally labeled by affine dominant coweights of the singularity . This in turn enables us to efficiently arrange these theories into groupings satisfying all known necessary conditions to be T-dual. Using this formulation, we then study the partial order of those coweights, and extend a recently proposed slice-subtraction algorithm to construct Hasse diagrams for LSTs directly from their six-dimensional generalized quivers, allowing us to probe certain properties of their Higgs branch. Along the way, we exploit these techniques to show that the number of duality orbits at maximal flavor rank is determined by the center of the transverse singularity , and provide a simplified proof of a monotonicity theorem for this class of theories. Finally, we show how some of our techniques can be extended to other classes of 6d theories, such as Type-II LSTs and SCFTs.

Paper Structure

This paper contains 48 sections, 120 equations, 1 figure, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Spin(32)/Z2 Heterotic Little String Theories
  3. Notation and conventions:
  4. Pictorial representation of LSTs:
  5. Anomalies of Spin(32)/Z2 LSTs
  6. "One-loop" contribution:
  7. Cancellation of quartic gauge anomalies:
  8. Green--Schwarz term and two-group invariants:
  9. Higgs-branch dimension:
  10. Extra abelian flavor symmetries:
  11. Classification of Spin(32)/Z2 LSTs and Coweights
  12. The McKay correspondence:
  13. The Frobenius--Schur indicator:
  14. Embeddings of the finite group:
  15. Recovering the quiver:
  16. ...and 33 more sections

Figures (1)

  • Figure 4.1: Part of the Hasse diagram of $Spin(32)/\mathbb{Z}_2$ LSTs with $\mathfrak{g}=\mathfrak{so}_{10}$ at fixed value of $Q$ following the partial order of finite coweights. Theories in a blue box indicate are the representative of a duality orbit. Each arrow indicates the subtracted quiver corresponding to the slice $\mathcal{S}$. The integers associated with each LST indicate the total flavor rank or the change from the theory with trivial coweight, e.g. $\Delta_0\kappa_R(\mu^\vee) = \kappa_R(\mu^\vee) - \kappa_R(\varnothing)$.

Theorems & Definitions (1)

  • Definition 1: T-duality of LSTs