Symmetries in A-type little string theories. Part I. Reduced free energy and paramodular groups

Abstract

We analyse the symmetries of a class of A-type little string theories that are engineered by N parallel M5-branes with M2-branes stretched between them. This paper deals with the so-called reduced free energy, which only receives contributions from the subset of the BPS states that carry the same charges under all the Cartan generators of the underlying gauge algebra. We argue (and check explicitly in a number of examples) that the former is invariant under the paramodular group Σ N ⊂ Sp(4, Q), which gets extended to a subgroup of Sp(4, R) in the Nekrasov-Shatashvili-limit. This extension agrees with the observation made in [18] that these BPS states form a symmetric orbifold CFT. Furthermore, we argue that Σ N (along with other symmetries) places strong constraints on the BPS counting function that governs the intersection between the M5-and M2-branes. * b.bastian@uu.nl † s.hohenegger@ipnl.in2p3.fr

Diagrammatic expansion of non-perturbative little string free energies

“…With the help of the toric diagram of X N,1 (which can directly be inferred from the M-brane web), the latter can be computed in an algorithmic fashion using the (refined) topological vertex [33,34]. This intrinsically geometric description, together with the very explicit form in which Z N,1 (or the related free energy F N,1 ) can be presented, have proven very fruitful and have aided in unravelling numerous interesting and surprising structures and symmetries [24][25][26][35][36][37][38][39][40][41][42]. Here we shall not present all the observations made in recent studies (see [42] for a more complete review), but only recount those that are relevant for the current work: the Calabi-Yau manifolds X N,1 are only a subset of a two-parameter class of manifolds, labelled X N,M , which can be used to describe orbifolds of the M5-brane configurations mentioned above [24].…”

“…It was furthermore argued in [39] that this web of dual gauge theories in fact also implies highly non-trivial (and intrinsically non-perturbative) symmetries for individual theories: focusing on the case of the U(N ) gauge theories engineered by X N,1 , invariance of the free energy F N,1 was shown under a particular dihedral symmetry group. 3 The implications of this symmetry were further explored in [40,41]: by studying series expansions of the single-particle 4 free energy F plet N,1 for N = 2, 3 and partially 4, characteristic patterns were observed, which together with the above mentioned symmetries allowed to conjecture a resummation of the former in an intriguing fashion: although being based on a limited (and a priori formal) series expansion, this allowed to write different contributions of F plet N,1 in terms of generating functions of multiple divisor sums introduced in [44] (see appendix A.4 for their definitions). In the case of N = 2 this was further shown to be equivalent to using JHEP04(2021)275 generalised Eisenstein series as described in [45].…”

“…Analysing the form of F plet N,1 proposed in [40,41] in the so-called unrefined limit (i.e. the limit in which Z N,1 captures the unrefined topological string partition function of X N,1 ) it was further observed in [1] that it can be re-written in a fashion that strongly resembles a Feynman diagrammatic decomposition.…”

“…, tN , m, τ ), which is closer to a description of the system in terms of M5-and M2-branes, as explained in the Introduction: t1,...,N denote the distances between the M5-branes that are spread out on a circle of radius ρ = N i=1 ti and compactified on a circle of radius τ , while m corresponds to a mass deformation from the perspective of a certain low energy gauge theory description. As explained in [39,40] (see also [42]) the two bases are related through a linear transformation (with the identification string. 7 From the perspective of the LST, they can be thought of as regularisation parameters that are necessary to render the non-perturbative partition function well defined.…”

Symmetric Orbifold Theories from Little String Residues

“…The current paper is a continuation of the analysis of the symmetries and structures discovered in [42,57,46,45]: focusing on the NS-limit of the free energy, we analyse the form of the free energy that was found in [42,57,46] and which is implied by the property (i) above. We observe new subsectors of the BPS-states that show a similar Hecke structure as discussed under (ii) above.…”