BPS degeneracies and superconformal index in diverse dimensions

We present a unifying theme relating BPS partition functions and superconformal indices. In the case with complex supersymmetry central charges (as in $\mathcal{N}=2$ in $d=4$ and $\mathcal{N}=(2,2)$ 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 $\mathcal{N}=1$ in $d=5$ (or the $\mathcal{N}=2$ in $d=3$), the BPS degeneracy captures the full superconformal index. Furthermore, we argue that refined topological strings, which capture five-dimensional (5d) BPS degeneracies of M theory on Calabi-Yau 3-folds, can be used to compute a 5d supersymmetric index including in the sectors with three-dimensional 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 Superconformal field theory yield the same index.

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 Chern-Simons 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 ${\xi }_{\pm }={\xi }_0+\frac{a}{2}\pm \frac{ap}{2q}$ 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

A local deformation of 5-brane web.

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.

Figure 6

The parameter $h$ is determined by the position of the external legs and is fixed. In the 5d gauge theory, $h$ is proportional to the inverse of the tree level gauge coupling and has dimensions of mass.

Figure 7

Toric and web diagram of local ${\mathbb{P}}^1\times {\mathbb{P}}^1$ blown up at one point.

Figure 8

The Newton polygon (a) and the web diagram (b) of local ${\mathbb{F}}_1$. In this case, there are two distinct triangulations of the Newton polygon corresponding to two different phases.

Figure 9

The web corresponding to local ${\mathbb{F}}_1$.

Figure 10

The two triangulations of toric diagram of local ${\mathbb{F}}_1$ and corresponding webs.

Figure 11

Local ${\mathbb{P}}^2$ blown up at one point.

Figure 12

The Newton polygon (a) and the web diagram (b) of local ${\mathbb{F}}_2$.

Figure 13

In this case, since there are two parallel legs, the area of the base curve does not change with the area of the fiber curve.

Figure 14

The web diagram of local ${\mathbb{P}}^2$.

Figure 15

Local Calabi-Yau 3-fold with two ${\mathbb{F}}_1$ divisors.

Figure 16

A Lagrangian brane on ${\mathbb{C}}^3$ with boundary condition given by .$\alpha$.

Figure 17

The geometry of the Lagrangian brane on local ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

Figure 18

The two choices for the Lagrangian brane on local ${\mathbb{P}}^1\times {\mathbb{P}}^1$.