Multiple elliptic gamma functions associated to cones

Abstract

Abstract. We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of SLr(Z)-elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.

Twisting with a Flip (The Art of Pestunization)

“…The change of variables will make use of the projector P + ω defined in (23). Here we select the canonical choice for the function ω defined in (43) because it results in simpler expressions. We define the following cohomological fields:…”

“…We can describe the freedom parametrized by b µ via a two form U µν that satisfies P + ωcan U = U where the canonical choice for ω is as in(43). This is somewhat more general because a b µ that is singular at the fixed points can correspond to a smooth U , but it makes explicit expressions more complicated.…”

7D supersymmetric Yang-Mills on curved manifolds

“…In the above discussions we allowed for general Reebs R = (ω 1 , ω 2 , ω 3 , ω 4 ) within the dual moment map cone. However, the generalised quadruple sine functions are also defined for complex R as long as the real part can be rotated to lie in the dual cone [35]. Thinking of our quadruple sines as complex functions we can use factorisation results for these special functions and apply them to our perturbative partition function.…”

“…We will start by defining some additional notions. We refer the reader to [35] for a thorough treatment of generalised quadruple sine functions and their factorisations, some of which are also summarised in Appendix C.…”

7D supersymmetric Yang-Mills on a 3-Sasakian manifold

“…Similar arguments were used in [9] for 7D toric Sasaki-Einstein manifolds. These were based on analogous factorisation results for generalised quadruple sine functions [33] and could be interpreted in terms of toric data.…”