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Sergei Alexandrov Laboratoire Charles Coulomb Montpellier Calabi-Yau compactifications: results, relations & problems in collaboration with S.Banerjee, J.Manschot, D.Persson, B.Pioline, F.Saueressig, S.Vandoren

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Calabi-Yau compactifications Integrability TBA Toda hierarchy and c=1 string Quantum integrability N =2 gauge theories wall-crossing QK/HK correspondence Topological strings NS5-instantons top. wave functions = Fine mathematics mock modularity cluster algebras

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vector multiplets (gauge fields & scalars) supergravity multiplet (metric) hypermultiplets (only scalars) The low-energy effective action is determined by the metric on the moduli space parameterized by scalars of vector and hypermultiplets – special Kähler (given by ) is classically exact (no corrections in string coupling g s ) known – quaternion-Kähler receives all types of g s -corrections unknown Calabi-Yau compactifications Type II string theory compactification on a Calabi-Yau N =2 supergravity in 4d coupled to matter The (concrete) goal: to find the non-perturbative geometry of Moduli space The goal: to find the complete non-perturbative effective action in 4d for type II string theory compactified on arbitrary CY

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Quantum corrections - corrections Tree level HM metric + In the image of the c-map (captured by the prepotential ) one-loop correction controlled by Antoniadis,Minasian,Theisen,Vanhove ’03, Robles-Llana,Saueressig,Vandoren ’06, S.A. ‘07 perturbative corrections non-perturbative corrections includes -corrections (perturbative + worldsheet instantons) Instantons – (Euclidean) world volumes wrapping non-trivial cycles of CY ● D-brane instantons ● NS5-brane instantons Axionic couplings break continuous isometries At the end no continuous isometries remain

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Instanton corrections 6 D5,NS5 5 × 4 D3 3 × 2 D1 1 × 0 D(-1) {α’, D(-1), D1} {α’, 1-loop } {A-D2} {A-D2, B-D2}{α’, D(-1), D1,, D5} {α’, D(-1), D1,, D5, NS5}{D2, NS5} SL(2, ) mirror e/m duality mirror SL(2, ) 6 NS5 5 D4 4 × 3 D2 2 × 1 D0 0 × Robles-Llana,Roček, Saueressig,Theis, Vandoren ’06 Beyond this line the projective superspace approach is not applicable anymore (not enough isometries) using projective superspace approach Twistor approach S.A.,Pioline,Saueressig,Vandoren ’08 D3 D3

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Twistor approach Twistor space t The problem: how to conveniently parametrize a QK manifold? carries holomorphic contact structure symmetries of can be lifted to holomorphic symmetries of is a Kähler manifold Holomorphicity Quaternionic structure: quaternion algebra of almost complex structures The geometry is determined by contact transformations between sets of Darboux coordinates generated by holomorphic functions

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Contact Hamiltonians and instanton corrections contact bracket gluing conditions metric on It is sufficient to find holomorphic functions corresponding to quantum corrections integral equations for “twistor lines” Perturbative metric D-instanton corrections NS5-instanton corrections holomorphic functions holomorphic functions holomorphic functions

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Expansion in the patches around the north and south poles: Correspondence with integrable systems ? Dictionary Darboux coordinates fiber coordinatespectral parameter string equationsgluing conditions twistor fiberspectral curve Lax & Orlov-Shulman operators Generalizes the fact that hyperkähler spaces are complex integrable systems (Donagi,Witten ’95) Manifestation of with an isometry & hyperholomorphic connection QK/HK correspondence Haydys ’08 S.A.,Persson,Pioline ’11 Hitchin ‘12 Baker-Akhiezer function Quantum integrability? Contact geometry of QK manifolds

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Perturbative HM moduli space c-map Twistor description of at tree level 4d case: Universal hypermultiplet S.A. ‘12 A perturbed solution of Matrix Quantum Mechanics in the classical limit (c=1 string theory) QK geometry of UHM at tree level (local c-map in 4d) = Toda equation sine-Liouville perturbation: time-dependent background with a non-trivial tachyon condensate NS5-brane charge quantization parameter Holom. function generating NS5-brane instantons One-fermion wave function Baker-Akhiezer function of Toda hierarchy

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D2-instantons in Type IIA generalized DT invariants ― D-brane charge S.A.,Pioline,Saueressig, Vandoren ’08, S.A. ’09 Holomorphic functions generating D2-instantons Analogous to the construction of the non-perturbative moduli space of 4d N =2 gauge theory compactified on S 1 Gaiotto,Moore,Neitzke ’08 Example of QK/HK correspondence Equations for twistor lines coincide with equations of Thermodynamic Bethe Ansatz with an integrable S-matrix Again integrability…

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Instanton corrections in Type IIB Type IIB formulation must be manifestly invariant under S-duality group pert. g s -corrections: instanton corrections: D(-1) D1 D3 D5 NS5 Quantum corrections in type IIB: SL(2, ) duality -corrections: w.s.inst. Robles-Llana,Roček,Saueressig, Theis,Vandoren ’06 Gopakumar-Vafa invariants of genus zero Twistorial description: S.A.,Saueressig, ’09 requires technique of mock modular forms S.A.,Manschot,Pioline ‘12 from mirror symmetry In the one-instanton approximation the mirror of the type IIA construction is consistent with S-duality manifestly S-duality formulation is unknown S.A.,Banerjee ‘14 using S-duality

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S-duality in twistor space S.A.,Banerjee ‘14 Non-linear holomorphic representation of SL(2, ) –duality group on contact transformation Condition for to carry an isometric action of SL(2, ) Transformation property of the contact bracket The idea: to add all images of the D-instanton contributions under S-duality with a non-vanishing 5-brane charge One must understand how S-duality is realized on twistor space and which constraints it imposes on the twistorial construction

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Fivebrane instantons NS5-brane charge Compute fivebrane contact Hamiltonians The input: One can evaluate the action on Darboux coordinates and write down the integral equations on twistor lines The contact structure is invariant under full U-duality group ― D5-brane charge Apply SL(2, ) transformation Encodes fivebrane instanton corrections to all orders of the instanton expansion

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● Manifestly S-duality invariant description of D3-instantons ● NS5-brane instantons in the Type IIA picture Can the integrable structure of D-instantons in Type IIA be extended to include NS5-brane corrections? ● Resolution of one-loop singularity ● Resummation of divergent series over brane charges (expected to be regularized by NS5-branes) Open problems = Inclusion of NS5-branes quantum deformation ? quantum dilog ? A-model topological wave function in the real polarization Relation to the topological string wave function THANK YOU!

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