We show that the Ward identities corresponding to generalized BMS symmetries in scattering states built around degenerate vacua labeled by supertranslations and Diff(S^{2}) charges are equivalent to a a special class of double soft graviton theorems called consecutive double soft graviton theorems. We further argue that such double soft graviton theorems can be realized by considering nested Ward identities constructed out of two generalized BMS charges. In order to extend the analysis of the relationship of generalized BMS charge algebra with double soft theorems when the external states are massive, we consider the algebra of the generalized BMS vector fields at timelike infinity. We show that the vector fields at timelike infinity close under the modified Lie bracket proposed by Barnich et al. Google-meet link: https://meet.google.com/raa-wbwv-qcb

By a modular relation for a certain function F, we mean a relation governed by the map z → −1/z but not necessarily by z → z+1. Equivalently, the relation can be written in the form F(α) = F(β), where αβ = 1. There are many generalized modular relations in the literature such as the general theta transformation of the form F(w, α) = F(iw, β) or the Ramanujan-Guinand formula of the form F(z, α) = F(z, β) etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on SL2(Z), admits a beautiful generalization of the form F(z, w, α) = F(z, iw, β) obtained by Kesarwani, Moll and the speaker, that is, one can superimpose theta structure on it. In 2011, the speaker obtained a generalized modular relation involving infinite series of the Hurwitz zeta function ζ(z, a). It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on the generalized modular relation? While answering this question affirmatively, we were led to a surprising new generalization of ζ_(z, a). We show that this new zeta function, ζ_w(z, a), satisfies a beautiful theory. In particular, it is shown that ζ_w(z, a) can be analytically continued to Re(z) > −1, z ≠ 1. We also prove a two-variable generalization of Ramanujan’s formula which involves infinite series of ζ_w(z, a) and which is of the form F(z, w, α) = F(z, iw, β). This is joint work with Rahul Kumar. Google meet link: meet.google.com/ysw-raoa-bjr