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19
Non-perturbative effects and the refined topological string
, 2013
"... The partition function of ABJM theory on the three-sphere has non-perturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi–Yau manifold known as lo ..."
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Cited by 23 (7 self)
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The partition function of ABJM theory on the three-sphere has non-perturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi–Yau manifold known as local P1 × P1, in the Nekrasov–Shatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi–Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi–Yau manifolds.
Wall-crossing of D4-branes using flow trees
, 2010
"... The moduli dependence of D4-branes on a Calabi-Yau manifold is studied using attractor flow trees, in the large volume limit of the Kähler cone. One of the moduli dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the fl ..."
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Cited by 11 (4 self)
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The moduli dependence of D4-branes on a Calabi-Yau manifold is studied using attractor flow trees, in the large volume limit of the Kähler cone. One of the moduli dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the flow parameters can be determined iteratively as function of the initial moduli, without explicit calculation of the flow of the moduli in the tree. Using this result, an indefinite quadratic form, which appears in the expression for the D4-D2-D0 BPS mass in the large volume limit, is proven to be positive definite for flow trees with 3 or less endpoints. The contribution of these flow trees to the BPS parti-tion function is therefore convergent. From non-primitive wall-crossing is deduced that the S-duality invariant partition function must be a generating function of the rational invariants Ω̄(Γ) = m|Γ Ω(Γ/m) m2 instead of the integer invariants Ω(Γ).
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
Instanton correction to the universal hypermultiplet and automorphic forms on SU(2
, 2009
"... Abstract: The hypermultiplet moduli space in Type IIA string theory compactified on a rigid Calabi-Yau threefold, corresponding to the “universal hypermultiplet”, is described at tree-level by the symmetric space SU(2,1)/(SU(2) × U(1)). To determine the quantum corrections to this metric, we posit ..."
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Cited by 4 (1 self)
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Abstract: The hypermultiplet moduli space in Type IIA string theory compactified on a rigid Calabi-Yau threefold, corresponding to the “universal hypermultiplet”, is described at tree-level by the symmetric space SU(2,1)/(SU(2) × U(1)). To determine the quantum corrections to this metric, we posit that a discrete subgroup of the continuous tree-level isometry group SU(2,1), namely the Picard modular group SU(2,1; Z[i]), must remain unbroken in the exact metric – including all perturbative and non-perturbative quantum corrections. Based on this assumption, we construct an SU(2,1; Z[i])-invariant, nonholomorphic Eisenstein series and analyze its non-Abelian Fourier expansion. We show that its Abelian and non-Abelian Fourier coefficients exhibit the expected form of instanton corrections due to Euclidean D2-branes wrapping special Lagrangian submanifolds, as well as Euclidean NS5-branes wrapping the entire Calabi-Yau threefold. Relying on the construction of quaternionic-Kähler manifolds M via their twistor space ZM, a CP 1 bundle over M, we conjecture that the exact contact potential on the twistor space of the
Arithmetic and Hyperbolic Structures in String Theory
, 2010
"... This monograph is an updated and extended version of the author’s PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in ..."
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Cited by 1 (1 self)
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This monograph is an updated and extended version of the author’s PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of a spacelike singularity (the “BKL-limit”). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of the theory. We develop in detail the relevant mathematics of Lorentzian Kac-Moody algebras and hyperbolic Coxeter groups, and explain with many examples how these structures are intimately connected with gravity. We also construct a geodesic sigma model invariant under the hyperbolic Kac-Moody group E10, and analyze to what extent its dynamics reproduces the dynamics of type II and eleven-dimensional supergravity.
J H E P12(2011)027
, 2011
"... Abstract: When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N = 2 string vacua and the Coulomb branch of rigid N = 2 gauge theories on R3 × S1 are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similar ..."
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Abstract: When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N = 2 string vacua and the Coulomb branch of rigid N = 2 gauge theories on R3 × S1 are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kähler manifolds with a quaternionic isometry and, on the other hand, hyperkähler manifolds with a rotational isometry, equipped with a canonical hyperholomorphic circle bundle and a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and cluster algebras.
