A meromorphic extension of the 3D index

Abstract

Using the locally compact abelian group T × Z, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2-3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a strict angle structure, our meromorphic function can be expanded into a Laurent power series whose coefficients are formal power series in q with integer coefficients that coincide with the 3D index of (Dimofte et al. in Adv Theor Math Phys 17(5):975-1076, 2013). Our meromorphic function can be computed explicitly from the matrix of the gluing equations of a triangulation, and we illustrate this with several examples.

New Pentagon Identities Revisited

“…We obtain a new solution to the pentagon identity in terms of gamma function (15) by considering q → 1 limit of the index. This solution is different from the one in literature (17). Pentagon identity (13) is equivalent to the "strongly coupled" regime of Faddeev-Volkov model.…”

“…Here we define pentagon identity in the integral form 2. For the case of ideal triangulation, see[7,17] …”

On the asymptotics of the meromorphic 3D-index

“…This paper concerns a certain extension of the 3D-index which was introduced by Garoufalidis and Kashaev [GK19] and which we will call the meromorphic 3D-index. The extension associates to every ideal triangulation of an oriented 3-manifold with toroidal boundary a meromorphic function which has 2 variables for every boundary component.…”

Twisted Neumann--Zagier matrices

“…One of the main discoveries of Neumann-Zagier is that the matrix (A|B) becomes, after some minor modifications, the upper part of a symplectic matrix with integer entries [NZ85]. The symplectic property of the NZ matrices and of the corresponding gluing equations define a linear symplectic structure on a vector space whose quantization leads to a plethora of quantum invariants that include the loop invariants of Dimofte and the first author [DG13,DG18], the 3D-index in both the original formulation of Dimofte-Gaiotto-Gukov [DGG14,DGG13] as well as the state-integral formulation of Kashaev and the first author [GK19] and Kashaev-Luo-Vartanov state-integral [KLV16,AGK]. All of those invariants are defined using the NZ matrices of a suitable ideal triangulation, and their topological invariance follows by proving that they are unchanged under Pachner 2-3 moves.…”

“…It turns out that the twisted NZ matrices have twisted symplectic properties which come from topology and using them one can give twisted versions of the above mentioned invariants, i.e., of the loop invariants [DG13,DG18], the 3D-index in [DGG14,DGG13] and [GK19] and KLV state-integral [KLV16,AGK]. A key property of such a twisted invariant is that it determines the corresponding (untwisted) invariant of all cyclic covers.…”