Nonperturbative Corrections to 4D String Theory Effective Actions fromSL(2,Z)Duality and Supersymmetry

Abstract

We find the D(−1) and D1-brane instanton contributions to the hypermultiplet moduli space of type IIB string compactifications on Calabi-Yau threefolds. These combine with known perturbative and worldsheet instanton corrections into a single modular invariant function that determines the hypermultiplet low-energy effective action. 04.65.+e The absence of a complete nonperturbative formulation of string theory is its main shortcoming as a fullfledged quantum theory unifying all known fundamental interactions. Empirically, perturbative weakly coupled string theory does not give a detailed description of our universe; consequently, the understanding of nonperturbative phenomena is essential to making possible a detailed confrontation of string theory and experiment. Generally, it is difficult to obtain exact information about nonperturbative structures. However, in special cases such as those presented here, the symmetries and dualities of the theory are powerful enough to fix the exact couplings in the low-energy effective action.The examples we consider are provided by type II string compactifications on Calabi-Yau threefolds (CY), where the four-dimensional effective actions are constrained by N = 2 supersymmetry. The massless fields are components of a supergravity multiplet, vector multiplets, or hypermultiplets. Their scalar fields parametrize moduli spaces M VM and M HM , respectively, which locally form a direct product [1]. The special geometry of M VM is determined by a holomorphic function F (X) [2]. The exact expression for this function includes perturbative and worldsheet instanton corrections in the inverse string tension α ′ , which can in principle be computed by mirror symmetry, see e.g. [3]. (See Fig. 1 for details.) On the other hand, the string coupling constant g s is set by the vacuum expectation value of the dilaton, whose four-dimensional reduction belongs to a hypermultiplet. Thus M HM receives both perturbative and nonperturbative g s corrections. Building on earlier work [4,5], the perturbative corrections have recently been understood in [6]. The nonperturbative corrections arise in the IIA case from Euclidean D2 or NS5-branes wrapping around supersymmetric three-cycles or the entire CY, respectively, and in the IIB case from D(−1)-instantons as well as D1, D3, D5, and NS5-branes wrapping holomorphic cycles in the CY [7]. Little is known about summing up such corrections -see however [8,9] for some results in the limit where gravity decouples.In this letter, we use the constraints from supersymmetry and the SL(2, Z) duality symmetry of IIB string theory to determine the full D(−1) and D1-brane instanton corrected low-energy effective action for hypermultiplets in type IIB compactifications on CY. This provides a large class of four-dimensional N = 2 supergravity theories where exact results are obtained to all orders in both α ′ and g s ; such results were not available in four dimensions previously.Similar ideas were applied in [10,11,12,13] to obtain instanton corrections to highe...

$$\mathbb{C}^2/\mathbb{Z}_{n}$$ Fractional Branes and Monodromy

“…Assuming that monodromy around L 3 is trivial, we have a perfect agreement with (40). This is also consistent with the general statement that monodromy at an orbifold point has to be associated with B-field components other than the blow-up mode(s) of the orbifold, in this case D 2 .…”

D3-instantons, mock theta series and twistors

“…To this end, note that the derivative ∂ξ [0] a G m,n (t ′ ) transforms, like the transition function (B.8), with an overall factor of (cξ 0 (t ′ ) + d) −1 . Using the property 1 − z 2 → cξ 0 + d cτ + d (1 − z 2 ), (B.10) 22 We refrain from writing the full non-linear transformation of G ± m,n , since we are interested only in the linear approximation in this paper. The transformation property required for modular invariance at the non-linear level can be found in [36].…”

“…Using these techniques, combining earlier results on the twistorial description of the perturbative metric and of D1-D(-1)-instantons [22][23][24][25] and taking inspiration from a similar construction in the context of gauge theories [26], a general construction of the Dinstanton corrected HM moduli space was laid out in [27,28], in terms of the generalized Donaldson-Thomas (DT) invariants Ω(γ; z) which count D-instantons. For reasons that will become clear below, we refer to the construction of [27,28] as the 'type IIA' construction.…”

Linear Perturbations of Hyperkähler Metrics

“…As the Kähler classes of X (in type IIA, the complex structure) are taken to infinity, these instanton effects are much smaller than the ones already taken into account. Thus, it is natural to treat them as linear perturbations away from the metric found in [37]. This was carried out for the "universal hypermultiplet" in [1] using techniques germane to four-dimensional QK manifolds, and our aim here and in [3] is to develop similar techniques valid in any dimension.…”

“…More recently, the S-duality of type IIB string theory was used to derive instanton corrections to the hypermultiplet moduli space in general type II compactifications on a Calabi-Yau X , for a subset of instanton configurations which preserve the toric isometries [37,38]: namely D(−1), F1 and D1 instantons in type IIB, or D2-branes in an appropriate Lagrangian sublattice in H 3 (X, Z) (i.e. "wrapping A-cycles only", for a suitable choice of symplectic basis of A and B-cycles) in type IIA.…”