Dual little strings and their partition functions

Abstract

We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds X N;M at a generic point in the Kähler moduli space. These manifolds engineer little string theories in five dimensions or lower and are dual to stacks of M5-branes probing a transverse orbifold singularity. Using the refined topological vertex formalism, we explicitly calculate a generic building block which allows us to compute the topological string partition function of X N;M as a series expansion in different Kähler parameters. Using this result, we give further explicit proof for a duality found previously in the literature, which relates X N;M ∼ X N 0 ;M 0 for NM ¼ N 0 M 0 and gcdðN; MÞ ¼ gcdðN 0 ; M 0 Þ.

From little string free energies towards modular graph functions

“…Rather than recounting all the properties of B (N,r) (s 1 ,s 2 ) found in [22], we shall instead give as an example the simplest case (namely N = 1) in the following, whose expansion will provide us with the general building blocks that are used in the remainder of this paper. 2 Due to the plethystic logarithm PLog, F N,1 only counts single particle BPS states. The former is defined as PLog Z N,1 ( a 1,...,N −1 , ρ, S, R; 1,2 ) = ∞ k=1 µ(k) k ln Z N,1 (k a 1,...,N , k S, k R; k 1,2 ), where µ is the Möbius function.…”

“…Furthermore, it was conjectured in [1] (and shown explicitly in a large number of examples) that the Calabi-Yau threefolds X N,M and X N ,M are dual 1 to one another if N M = N M and gcd(N, M ) = gcd(N , M ). It is therefore expected that the (non-perturbative) BPS-partition functions Z N,M (ω, 1,2 ) and Z N ,M (ω , 1,2 ) agree once a suitable duality map (conjectured in [1] for generic N and M ) for the Kähler parameters (mapping ω → ω ) is taken into account. 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.…”

“…It is therefore expected that the (non-perturbative) BPS-partition functions Z N,M (ω, 1,2 ) and Z N ,M (ω , 1,2 ) agree once a suitable duality map (conjectured in [1] for generic N and M ) for the Kähler parameters (mapping ω → ω ) is taken into account. 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 ).…”

“…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 ). This conjecture was proven at the level of the partition function for M = 1 in [12] and, by studying the Seiberg-Witten curve related to the Calabi-Yau geometry, in [15] for generic (M, N ) (however for vanishing parameters 1,2 ).…”