BPS Degeneracies and Superconformal Index in Diverse Dimensions

TL;DR

This work uncovers a unifying framework whereby BPS partition functions and superconformal indices are intrinsically linked across dimensions from 2 to 5. It shows that, for complex central charges in 2d/4d, a suitably specialized BPS partition function reproduces portions of the index, while for real central charges in 3d/5d the full index is captured by BPS data on the Coulomb branch, with refined topological strings providing a concrete computational tool. In 5d, the index can be computed as a squared topological-string amplitude, including sectors with 3d defects, and holds under flop equivalences; numerous toric CY examples illustrate precise matches to gauge-theory results and wall-crossing structures. The results suggest a deep OSV-like relation between BPS spectra and SCFT partition functions, offering a path toward reconstructing conformal data purely from BPS information across diverse dimensions. The work emphasizes refined topological-vertex methods, open/closed string amplitudes, and defect insertions as essential ingredients for computing indices in strongly coupled theories.

Abstract

We present a unifying theme relating BPS partition functions and superconformal indices. In the case with complex SUSY central charges (as in N=2 in d=4 and N=(2,2) in d=2) the known results can be reinterpreted as the statement that the BPS partition functions can be used to compute a specialization of the superconformal indices. We argue that in the case with real central charge in the supersymmetry algebra, as in N=1 in d=5 (or the N=2 in d=3), the BPS degeneracy captures the full superconformal index. Furthermore, we argue that refined topological strings, which captures 5d BPS degeneracies of M-theory on CY 3-folds, can be used to compute 5d supersymmetric index including in the sectors with 3d defects for a large class of 5d superconformal theories. Moreover, we provide evidence that distinct Calabi-Yau singularities which are expected to lead to the same SCFT yield the same index.

Figures (20)

• Figure 1: The geometry of Lagrangian brane on local $\mathbb{P}^{1}\times \mathbb{P}^{1}$. Here we have chosen the spectator brane to be $(p,q)$ with slope $p/q$. The CS level on the brane is at $k=q$. The Lagrangian brane is suspended from the spectator brane at either of the two points (denoted by black dots). The Coulomb branch parameter is labeled by $a$. Moreover the slope being $p/q$ affects how the effective FI terms $\xi_{\pm}=\xi_0+{a\over 2}\pm {ap\over 2q}$ change with $a$.
• Figure 2: A generic $(p,q)$ 5-brane web.
• Figure 3: The singular limit of the web gives a superconformal theory. In this case the theory has $SU(2)$ global symmetry at the superconformal point. In the M-theory compactification this corresponds to a 4-cycle ($\mathbb{P}^{1}\times \mathbb{P}^{1}$) shrinking to a point.
• Figure 4:
• Figure 5: The Newton polygon (a) and web diagram (b) of local $\mathbb{P}^{1}\times \mathbb{P}^{1}$. The Newton polygon has a unique triangulation therefore this geometry has only one phase.