D3-instantons, mock theta series and twistors

Abstract

The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2, Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2, Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.

Hypermultiplet metric and D-instantons

“…The progress mentioned above was related with the developments of twistorial methods which provide an efficient parametrization of QK geometries [7][8][9]. Combining these methods with the symmetries expected to survive at quantum level, a large class of instanton corrections to the HM moduli space has been found [10][11][12][13][14][15][16][17] (see [18,19] for reviews). Although the description of few types of instantons remains still unknown, the complete non-perturbative picture for this class of compactifications seems to be already not far from our reach.…”

Indefinite theta functions and black hole partition functions

“…Indefinite theta functions ϑ(z; τ ) were defined and studied by Zwegers in [11], and are modified versions of the sums considered in [29]. They have found recent string theory applications in [23,25,[30][31][32]. Indefinite theta functions are based on quadratic forms Q : R r → R of signature (r − 1, 1), defined in terms of symmetric non-degenerate r × r matrices A with integer coefficients, Q(x) = 1 2 x T Ax.…”

Rigid limit for hypermultiplets and five-dimensional gauge theories

“…In other words, it is possible if the t A -dependence of the mirror map drops out when one restricts to worldsheet instantons with charges kÎ and D-instantons with charges from Γ rig . In fact, the quantum corrected mirror map is known only in the presence of worldsheet and D1-instantons [42,43] and D3-instantons in the JHEP01(2018)156 large volume limit [44,45] (i.e. when all Kähler moduli are taken to be large).…”

“…The intersections of Mori generators with the basis divisors are given by the matrix 44) so that the Kähler moduli are subject to the following constraints…”