Symmetric orbifold theories from little string residues

We study a class of little string theories (LSTs) of A type, described by $N$ parallel M5-branes spread out on a circle and which in the low energy regime engineer supersymmetric gauge theories with $U(N)$ gauge group. The Bogomol’nyi-Prasad-Sommerfield (BPS) states in this setting correspond to M2-branes stretched between the M5-branes. Generalizing an observation made by Ahmed et al. [Bound states of little strings and symmetric orbifold conformal field theories, Phys. Rev. D 96, 081901 (2017).], we provide evidence that the BPS counting functions of special subsectors of the latter exhibit a Hecke structure in the Nekrasov-Shatashvili (NS) limit; i.e., the different orders in an instanton expansion of the supersymmetric gauge theory are related through the action of Hecke operators. We extract $N$ distinct such reduced BPS counting functions from the full free energy of the LST with the help of contour integrals with respect to the gauge parameters of the $U(N)$ gauge group. Physically, the states captured by these functions correspond to configurations where the same number of M2-branes is stretched between some of these neighboring M5-branes, while the remaining M5-branes are collapsed on top of each other and a particular singular contribution is extracted. The Hecke structures suggest that these BPS states form the spectra of symmetric orbifold conformal field theories. We show, furthermore, that to leading instanton order (in the NS limit) the reduced BPS counting functions factorize into simpler building blocks. These building blocks are the expansion coefficients of the free energy for $N=1$ and the expansion of a particular function, which governs the counting of BPS states of a single M5-brane with single M2-branes ending on it on either side. To higher orders in the instanton expansion, we observe new elements appearing in this decomposition whose coefficients are related through a holomorphic anomaly equation.

Figure 1

$N$ parallel M5-branes (orange) with $(n_1,\dots ,n_N)$ M2-branes (blue) stretched between them. The distances between the M5-branes are $t_{1,\dots ,N}$.

Figure 2

Web diagram of $X_{N,1}$.

Figure 3

Web diagram of $X_{N,1}$ labeled by the parameters $({\widehat{a}}_1,\dots ,{\widehat{a}}_N,S,R)$.

Figure 4

Brane configuration for extracting ${\mathcal{C}}_i^{N,(r)}$.

Figure 5

Web diagram of $X_{2,1}$.

Figure 6

Brane web configurations made up of $N=2$ M5-branes (drawn in orange) spaced out on a circle, with various M2-branes (drawn in red and blue) stretched between them. (a) An equal number $\ell$ of M2-branes is stretched between the M5-branes on either side of the circle. Configurations of this type are relevant for the computation of ${\mathcal{C}}_1^{N=2,(r)}$. (b) $n_1$ M2-branes are stretched on one side of the circle and $n_2$ ($\ne n_1$) on the other side of the circle. Configurations of this type are relevant for the contributions ${\mathcal{C}}_2^{N=2,(r)}$.

Figure 7

Web diagram of $X_{3,1}$.

Figure 8

Brane-web configurations made up of $N=3$ M5-branes (drawn in orange) spaced out on a circle, with various M2-branes (drawn in red, blue, and green) stretched between them. (a) An equal number $\ell$ of M2-branes is stretched between any two neighboring M5-branes. Configurations of this type are relevant for ${\mathcal{C}}_1^{N=3,(r)}$. (b) $\ell$ M2-branes are stretched between M5-branes 2 and 3 as well as 3 and 1, while $n\ne \ell$ M2-branes between M5-branes 1 and 2. Configurations of this type are relevant for ${\mathcal{C}}_2^{N=3,(r)}$. (c) Different numbers $n_{1,2,3}$ (with $n_i\ne n_j$ for $i\ne j$) of M2-branes are stretched between any of the neighboring M5-branes. Configurations of this type are relevant for ${\mathcal{C}}_3^{N=3,(r)}$.

Figure 9

Web diagram of $X_{4,1}$.

Figure 10

Brane-web configurations made up of $N=4$ M5-branes (shown in orange) spaced out on a circle, with various M2-branes (shown in red and blue) stretched between them. (a) An equal number $\ell$ of M2-branes is stretched between any two neighboring M5-branes. Configurations of this type are relevant for the computation of ${\mathcal{C}}_1^{N=4,(r)}$. (b) $\ell$ M2-branes are stretched between M5-branes 2 and 3, 3 and 4, and 4 and 1, while $n\ne \ell$ M2-branes are stretched between M5-branes 1 and 2. Configurations of this type are relevant for the computation of ${\mathcal{C}}_2^{N=4,(r)}$.

Figure 11

Brane-web configurations made up of $N=4$ M5-branes (shown in orange) spaced out on a circle, with various M2-branes (shown in red and blue) stretched between them. (a) An equal number $\ell$ of M2-branes is stretched between the neighboring M5-branes 3 and 4 as well as 4 and 1, while different numbers of M2-branes $n_1\ne n_2\ne \ell$ are stretched between the M5-branes 1 and 2 as well as 2 and 3. Configurations of this type are relevant for the computation of ${\mathcal{C}}_3^{N=4,(r)}$. (b) An equal number $\ell$ of M2-branes is stretched between the neighboring M5-branes 2 and 3 as well as 4 and 1, while different numbers of M2-branes $n_1\ne n_3\ne \ell$ are stretched between the M5-branes 1 and 2 as well as 3 and 4. Configurations of this type are relevant for the computation of ${\tilde{\mathcal{C}}}_{3,(2s,0)}^{N=4,(r)}$.