Abstract
Abstract In type IIB string compactifications on a Calabi-Yau threefold, the hypermultiplet moduli space $ {\mathcal{M}_H} $ must carry an isometric action of the modular group SL(2, $ \mathbb{Z} $ ), inherited from the S-duality symmetry of type IIB string theory in ten dimensions. We investigate how this modular symmetry is realized at the level of the twistor space of $ {\mathcal{M}_H} $ , and construct a general class of SL(2, $ \mathbb{Z} $ )-invariant quaternion-Kähler metrics with two commuting isometries, parametrized by a suitably covariant family of holomorphic transition functions. This family should include $ {\mathcal{M}_H} $ corrected by D3-D1-D(-1)-instantons (with five-brane corrections ignored) and, after taking a suitable rigid limit, the Coulomb branch of five-dimensional $ \mathcal{N} = {2} $ gauge theories compactified on a torus, including monopole string instantons. These results allow us to considerably simplify the derivation of the mirror map between type IIA and IIB fields in the sector where only D1-D(-1)-instantons are retained.
Highlights
Methods in the physics literature [7,8,9,10], and further tailored for string theory applications in [11, 12]
We investigate how this modular symmetry is realized at the level of the twistor space of MH, and construct a general class of SL(2, Z)-invariant quaternion-Kahler metrics with two commuting isometries, parametrized by a suitably covariant family of holomorphic transition functions
By lifting the QK space to its twistor space, a CP 1 bundle endowed with canonical complex and contact structures, this method provides an efficient parametrization of the quaternion-Kahler metric in terms of a set of holomorphic transition functions between local Darboux coordinate systems, which play a role similar to the holomorphic prepotential of special Kahler geometry
Summary
We briefly recall some basic facts about the hypermultiplet moduli space MH in type II string theory compactified on a CY threefold. Identifies type IIA compactified on a Calabi-Yau threefold Y and type IIB compactified on the mirror threefold Y , the moduli space MH has two equivalent, dual descriptions. On either side, it governs the dynamics of. The complex moduli za = ba + ita parametrizing the moduli space KC(Y) of complex structures on Y or the complexified Kahler structure of KK(Y ) — these two spaces describe the VM moduli space of the dual type II theory compactified on the same CY threefold, and are identified by classical mirror symmetry; 3. On the type IIB side, the ten-dimensional string coupling τ2 ≡ 1/gs and the RR axion τ1 ≡ c0 combine into the ten-dimensional axio-dilaton field τ = τ1 + iτ. We refer to the latter field basis as the type IIA coordinates, and to the former as the type IIB coordinates
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