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21
Membrane instantons from mirror symmetry
, 2007
"... We use mirror symmetry to determine and sum up a class of membrane instanton corrections to the hypermultiplet moduli space metric arising in Calabi–Yau threefold compactifications of type IIA strings. These corrections are mirror to the D1 and D(−1)–brane instantons on the IIB side and are given ex ..."
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Cited by 25 (7 self)
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We use mirror symmetry to determine and sum up a class of membrane instanton corrections to the hypermultiplet moduli space metric arising in Calabi–Yau threefold compactifications of type IIA strings. These corrections are mirror to the D1 and D(−1)–brane instantons on the IIB side and are given explicitly in terms of a single function in projective superspace. The corresponding four-dimensional effective action is completely fixed by the Euler number and the genus zero Gopakumar–Vafa invariants of the mirror Calabi–Yau.
Linear perturbations of quaternionic metrics
, 2008
"... We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifold M are in one-to-one correspondence with deformations ..."
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Cited by 20 (10 self)
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We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifold M are in one-to-one correspondence with deformations of its 4d + 4-dimensional hyperkähler cone S. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space ZS, with a suitable homogeneity condition that ensures that the hyperkähler cone property is preserved. Equivalently, we show that the deformations of M can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space ZM of M, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-Kähler metrics with d + 1 commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.
Conifold singularities, resumming instantons and nonperturbative mirror symmetry
, 2007
"... We determine the instanton corrected hypermultiplet moduli space in type IIB compactifications near a Calabi-Yau conifold point where the size of a two-cycle shrinks to zero. We show that D1-instantons resolve the conifold singularity caused by worldsheet instantons. Furthermore, by resumming the in ..."
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Cited by 17 (7 self)
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We determine the instanton corrected hypermultiplet moduli space in type IIB compactifications near a Calabi-Yau conifold point where the size of a two-cycle shrinks to zero. We show that D1-instantons resolve the conifold singularity caused by worldsheet instantons. Furthermore, by resumming the instanton series, we reproduce exactly the results obtained by Ooguri and Vafa on the type IIA side, where membrane instantons correct the hypermultiplet moduli space. Our calculations therefore establish that mirror symmetry holds non-perturbatively in the string coupling.
Linear perturbations of hyperkähler metrics
, 2009
"... We study general linear perturbations of a class of 4d real-dimensional hyperkähler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d + 1 variables, as opposed to the functions ..."
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Cited by 15 (6 self)
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We study general linear perturbations of a class of 4d real-dimensional hyperkähler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d + 1 variables, as opposed to the functions of d + 1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kähler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah-Hitchin manifold away from its negative mass Taub-NUT limit.
Quantum mirror symmetry and twistors
, 2009
"... Using the twistor approach to hypermultiplet moduli spaces, we derive the worldsheet, D(−1), and D1-instanton contributions to the generalized mirror map, relating Type IIA and Type IIB string theory compactified on generic mirror Calabi-Yau threefolds. For this purpose, we provide a novel descript ..."
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Cited by 13 (5 self)
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Using the twistor approach to hypermultiplet moduli spaces, we derive the worldsheet, D(−1), and D1-instanton contributions to the generalized mirror map, relating Type IIA and Type IIB string theory compactified on generic mirror Calabi-Yau threefolds. For this purpose, we provide a novel description of the twistor space underlying the Type IIB hypermultiplet moduli space where the SL(2, Z)-action is found to be free from quantum corrections. The extent to which instanton effects may resolve the perturbative singularities of the moduli space metric is discussed.
Recent results in four-dimensional non-perturbative string theory 1
, 710
"... Abstract. We review recent progress in understanding non-perturbative instanton corrections to the hypermultiplet moduli space in type II string compactifications on Calabi-Yau threefolds. 1. ..."
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Cited by 5 (2 self)
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Abstract. We review recent progress in understanding non-perturbative instanton corrections to the hypermultiplet moduli space in type II string compactifications on Calabi-Yau threefolds. 1.
Linear perturbations of quaternionic metrics I. The Hyperkähler case
, 2008
"... We study general linear perturbations of a class of 4d real-dimensional hyperkähler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d + 1 variables, as opposed to the functions ..."
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Cited by 5 (1 self)
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We study general linear perturbations of a class of 4d real-dimensional hyperkähler manifolds obtainable by the (generalized) Legendre transform method. Using twistor methods, we show that deformations can be encoded in a set of holomorphic functions of 2d + 1 variables, as opposed to the functions of d + 1 variables controlling the unperturbed metric. Such deformations generically break all tri-holomorphic isometries of the unperturbed metric. Geometrically, these functions generate the symplectomorphisms which relate local complex Darboux coordinate systems in different patches of the twistor space. The deformed Kähler potential follows from these data by a Penrose-type transform. As an illustration of our general framework, we determine the leading exponential deviation of the Atiyah-Hitchin manifold away from its negative mass Taub-NUT limit. In a companion paper, we extend these techniques to quaternionic-Kähler spaces with isometries.
Self-dual Einstein Spaces, Heavenly Metrics and
"... Abstract: Four-dimensional quaternion-Kähler metrics, or equivalently self-dual Ein-stein spaces M, are known to be encoded locally into one real function h subject to Przanowski’s Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion ..."
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Cited by 5 (2 self)
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Abstract: Four-dimensional quaternion-Kähler metrics, or equivalently self-dual Ein-stein spaces M, are known to be encoded locally into one real function h subject to Przanowski’s Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kähler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, as-sociated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kähler deformations ofM and, in the special case whereM has a Killing vector field, show that the corresponding variations of h are related to eigenmodes of the conformal Laplacian on M. We exemplify our findings on the four-sphere S4, the hyperbolic plane H4 and on the “universal hypermultiplet”, i.e. the hypermultiplet moduli space in type IIA string compactified on a rigid Calabi-Yau threefold. ar X iv
