Beyond triality: dual quiver gauge theories and little string theories

Abstract

The web of dual gauge theories engineered from a class of toric Calabi-Yau threefolds is explored. In previous work, we have argued for a triality structure by compiling evidence for the fact that every such manifold X N,M (for given (N, M )) engineers three a priori different, weakly coupled quiver gauge theories in five dimensions. The strong coupling regime of the latter is in general described by Little String Theories. Furthermore, we also conjectured that the manifold X N,M is dual to X N ,M if N M = N M and gcd(N, M ) = gcd(N , M ). Combining this result with the triality structure, we currently argue for a large number of dual quiver gauge theories, whose instanton partition functions can be computed explicitly as specific expansions of the topological partition function Z N,M of X N,M . We illustrate this web of dual theories by studying explicit examples in detail. We also undertake first steps in further analysing the extended moduli space of X N,M with the goal of finding other dual gauge theories. the given conditions, the Kähler cones of the Calabi-Yau manifolds X N,M and X N ,M are part of a common extended moduli space. The relation (1.1) was explicitly shown for a large class of examples and passed highly non-trivial consistency checks for generic (N, M ). Moreover, it is expected that the partition functions Z N,M and Z N ,M agree upon taking into account the duality map implicit in (1.1). This fact was explicitly checked for gcd(N, M ) = 1 in [30].Secondly, in [32] the question was analysed what type of (low-energy) gauge theories can be engineered from a given Calabi-Yau manifold X N,M . By studying different series expansions of Z N,M (in terms of different sets of the Kähler parameters of X N,M ) it was found that the Kähler cone of X N,M contains three different regions that engineer five-dimensional quiver gauge theories with gauge groups( 1.2) respectively, where k = gcd(N, M ). It is expected that the UV completions of (at least some of) these gauge theories are LSTs. The relation among the three low-energy theories with gauge groups (1.2) was dubbed triality in [32], reflecting the fact that all three are engineered by the same Calabi-Yau threefold. It is important to realise that the mapping of the various gauge theory parameters is highly non-trivial among the members of the triality: indeed, the coupling constants, mass parameters and other Coulomb branch parameters are mixed in a highly non-trivial fashion when going from one theory to another.Combining the triality discussed in [32] with the observations of [31] that the Calabi-Yau manifold X N,M itself can be dualised to other manifolds in the sense of eq. (1.1) suggests an even more elaborate picture at the level of the low-energy theories: indeed it implies that a (circular) quiver gauge theory with N gauge nodes of type U (M ) (that is engineered by X N,M for arbitrary (N, M )) is dual to a whole web of other quiver gauge theories that have N gauge nodes of type U (M ) such that N M = N M and gcd(N, M ) = gcd(N , M ). In this p...

From little string free energies towards modular graph functions

“…In a series of works [1][2][3][4] it has been argued that a certain class of N = 1 supersymmetric little string theories (LSTs) [5][6][7][8][9][10][11][12][13][14][15] of A-type [16,17] exhibits an intricate web of dualities. These theories are engineered in M-theory by N parallel M5-branes spread out on a circle S 1 and probing a transverse Z M geometry.…”

“…This equality was shown explicitly in [2] for M = 1 and for general (N, M ) in [19] assuming a certain limit of two deformation parameters 1,2 that are required to render Z N,M (ω, 1,2 ) well defined. Combined with the triality of supersymmetric gauge theories argued in [3], this lead to the conjecture [4] of a vast web of dual gauge theories with gauge groups [U (N )] M with N M = N M and gcd(N, M ) = gcd(N , M ).…”

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

“…However, this description is in general not unique: fiber-base duality of X N,M (or general string S-duality) suggests that the theory [U (N )] M is dual to [U (M )] N , which implies the existence of an equivalent expansion of Z N,M in terms of a different subset of ω that matches the instanton series of the latter theory. Moreover, as was recently been pointed out in a series of works [12][13][14], there exist numerous other theories dual to [U (N )] M , each of which entailing a new (but equivalent) expansion of Z N,M . More precisely, based on geometric considerations related to the extended moduli space of X N,M it was conjectured [27] that the theory [U (N )] M is dual to [U (N )] M if N M = N M and gcd(N , M ) = gcd(N, M ).…”

Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums

“…This paper constitutes the second part in our study of symmetries in a class of supersymmetric quantum field theories, that are constructed from N parallel M5-branes spaced out on a circle and probing a transverse Z M orbifold background [2][3][4][5][6]. Continuing the work started in the companion paper [1] as well as [7], we focus on M = 1 and use recent insights into dualities among these configurations (which have been described in detail in [8][9][10][11]) to gain a better picture of the symmetries realised within a single such theory.…”

“…They thus correspond to a simplified version of string theory, in which notably gravity is decoupled. The study of such theories has attracted a lot of attention recently, with a focus on classifying them [14,15] and exploring their duality structures [9][10][11]. The study of the A-type LSTs has been particularly fruitful 1 due to the fact that if one considers them on R 4 × T 2 , the (refined) BPS partition function Z N,M (ω, 1,2 ) can be computed and analysed in a very explicit and direct fashion.…”