Heritage in transition

Abstract

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S-duality and refined BPS indices

“…Strong support for this conjecture comes from the local CY case O(K S ) → S, where generalized DT invariants supported on the base S are expected to be equal to VW invariants of S. In this case, the partition function (4.7) built out of the generating functions is expected to be modular, as a consequence of S-duality of N = 4 Yang-Mills theory. Unfortunately, it is not known at present how to derive the non-holomorphic completion from a gauge theory computation, 26 and the known completions for U(2) and U(3) VW partition functions on P 2 rely on expressing the holomorphic generating functions as Appell-Lerch sums and applying mathematical recipes to find their modular completions. Remarkably, we have shown that our natural construction reproduces exactly these modular completions (up to the sign flip discussed in footnote 24), without prior knowledge of the invariants.…”

“…In general [15], the difference h p,µ − h p,µ is a sum of products Ψ p 1 ,...pn,µ 1 ,...µn n i=1 h p i ,µ i with p = p 1 + • • • + p n , where Ψ p 1 ,...pn,µ 1 ,...µn is an indefinite theta series whose kernel can be written as an (n − 1)-times iterated Eichler integral of a Siegel-Narain theta series of signature (1, b 2 (Y) − 1) [18,24,25]. As a result, the holomorphic generating function h p,µ must transform non-homogeneously under SL(2, Z) like a vector-valued mock modular form of depth n − 1 [26]. Correspondingly, the modular completion h p,µ satisfies a holomorphic anomaly equation, sourced by a combination of the h p i ,µ i 's with i p a i = p a .…”