Mock modular index of M2-M5 brane systems

We present Bogomolny-Prasad-Sommerfield (BPS) indices of the supergroup Wess-Zumino-Witten (WZW) models that live on intersecting M2-M5-brane systems. They can encode data of the stretched M2-branes between M5-branes and count the BPS states. They are generally expressed in terms of mock theta functions via the Kac-Wakimoto character formula of the affine Lie superalgebra. We give an explicit expression of the index for the $PSL(2|2{)}_{k=1}$ WZW model in terms of the second-order multivariable Appell-Lerch sum. It indicates that wall crossing occurs in the BPS state counting due to the $C$ field on the M5-branes.

Figure 1

A brane configuration with $N$ D3-branes and $M$ D3-branes terminating on a single NS5-brane at $x^6=0$ from left and right, respectively. The horizontal lines are the D3-branes extending in the $x^6$ direction. The sequences $\{x_i\}$ and $\{y_k\}$ of the heights of the D3-branes in the picture label the Ramond-Ramond (RR) charges of the D3-branes. In this example, $N=M=7$.

Figure 2

The type-IIB configuration of the ABJM model. Here, the $x^6$ direction is compact, and there are two stacks of D3-branes on the circle labeled by $\mathrm{D}{3}^{+}$ and $\mathrm{D}{3}^{-}$.

Figure 3

The type-IIB configuration of the ABJM model with two boundaries of M5- and $\mathrm{M}{5}^{\prime }$-branes which correspond to NS5- and $\mathrm{NS}{5}^{\prime }$-branes. The D3-branes are in finite boxes in the $(x^2,x^6)$ plane.

Figure 4

($N-d$) M2-branes attached to one of the M5-branes and $d$ M2-branes stretched between the two M5-branes. The vertical bold (respectively, dotted) line represents the M5- (respectively, $\mathrm{M}{5}^{\prime }$-)brane, and the horizontal lines denotes M2-branes in the $x^2$ direction. The M2-brane charges $\{x_i\}$ and $\{y_k\}$ are illustrated as the heights of the M2-branes. Here is the case with $N=7$ and $d=4$.

Figure 5

The operation of the right move and the left move on the brane configuration in Fig. 4. The right move $R_{2\to 3}$ lifts the second mode at 5 (shown in dotted red) to the third at 11 (shown in red), while the left move $L_{1\leftarrow 2}$ reduces the second mode of the location 5 (shown in dotted blue) to 0 (shown in blue).

Figure 6

The M2-M5 system for a typical module with $N=2$, $d=0$, for which no M2-brane is stretched between the M5-branes.

Figure 7

$N=d=2$ M2-branes stretched between the M5- and $\mathrm{M}{5}^{\prime }$-branes.

Figure 8

The Dynkin diagrams corresponding to (5.24). The white dot $○$ represents a simple even root, and the gray dot $\otimes$ represents a simple odd root of zero length.