Supersymmetry for brane diagrams and bow varieties

TL;DR

This work characterizes when affine type A bow diagrams yield non-empty bow varieties by translating the problem into supersymmetric type IIB brane configurations through Hanany--Witten transitions. It provides a finite-step algorithm built from an infinite family of supersymmetry inequalities and a stratum-condition reformulation in terms of affine Lie algebra weights, thereby linking bow variety non-emptiness to dominant weights and level considerations. The results include a constructive procedure to realize supersymmetric brane diagrams and explicit points in $\,\mathcal{M}_{0,0}$, with a discussion of S-duality and extensions to types B, C, and D. The approach integrates combinatorial bow data, geometric invariant theory, and representation-theoretic weights (via generalized Young diagrams and level-rank duality) to yield a comprehensive framework for testing supersymmetry and non-emptiness, and offers practical algorithms and examples for implementation.

Abstract

We provide combinatorial and numerical criteria to characterize affine type A bow diagrams giving rise to a non-empty bow variety. The key idea is to prove that such diagrams correspond to supersymmetric brane systems in type IIB string theory, allowing us to reformulate the problem in purely combinatorial terms. To achieve this, we characterize supersymmetry for affine type A brane systems (and, by extension, for types B, C, and D) using Hanany--Witten transitions. This leads to a finite-step algorithm that decides whether a given affine type A bow or brane diagram is supersymmetric, which consists in checking a finite set of inequalities, so providing a numerical criterion for non-emptiness. Finally, we provide a different perspective by introducing a further criterion in terms of weights of affine Lie algebras. Along the way, we also prove that increasing dimension vectors between two consecutive x-points or arrows in a bow diagram (not necessarily of type A) preserves the properties of generating non-empty bow varieties.

Figures (7)

• Figure 1: Vertical segments are NS5-branes, diagonal segments are D5-branes and horizontal segments are D3-branes. Black dots denote extreme points of D3-branes.
• Figure 2: A fine type A brane diagram on the left and an affine type A brane diagram on the right.
• Figure 3: Bow diagram associated with a brane diagram.
• Figure 4: Bow diagram of affine type A on the left-hand side and its simplified version on the right-hand side.
• Figure 5: Equivalence between affine and finite type A bow diagrams.

Theorems & Definitions (83)

• Theorem : See Theorem \ref{['maintheorem']}
• Theorem : see Theorem \ref{['supersymmetricincrementsforbows']}, Corollary \ref{['corollarysupersymmetricincrementsforbow']}
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8