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**Calabi-Yau compactifications: results, relations & problems**

Sergei Alexandrov Laboratoire Charles Coulomb Montpellier in collaboration with S.Banerjee, J.Manschot, D.Persson, B.Pioline, F.Saueressig, S.Vandoren

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**Calabi-Yau compactifications**

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

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**Calabi-Yau compactifications**

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

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**Quantum corrections + Tree level HM metric - corrections perturbative**

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

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**Instanton corrections**

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

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**Twistor approach Holomorphicity**

The problem: how to conveniently parametrize a QK manifold? Quaternionic structure: Twistor space quaternion algebra of almost complex structures t carries holomorphic contact structure symmetries of can be lifted to holomorphic symmetries of is a Kähler manifold Holomorphicity 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**

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

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**? Correspondence with integrable systems**

Expansion in the patches around the north and south poles: Dictionary Darboux coordinates fiber coordinate spectral parameter string equations gluing conditions twistor fiber spectral 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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**sine-Liouville perturbation: Baker-Akhiezer function**

Perturbative HM moduli space Twistor description of at tree level c-map 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**

S.A.,Pioline,Saueressig,Vandoren ’08, S.A. ’09 D2-instantons in Type IIA Holomorphic functions generating D2-instantons ― D-brane charge generalized DT invariants Equations for twistor lines coincide with equations of Thermodynamic Bethe Ansatz with an integrable S-matrix Again integrability… Analogous to the construction of the non-perturbative moduli space of 4d N =2 gauge theory compactified on S1 Gaiotto,Moore,Neitzke ’08 Example of QK/HK correspondence

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**Instanton corrections in Type IIB**

Type IIB formulation must be manifestly invariant under S-duality group Quantum corrections in type IIB: -corrections: w.s.inst. SL(2,Z) duality pert. gs-corrections: instanton corrections: D(-1) D D D NS5 S.A.,Banerjee ‘14 using S-duality 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

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**S-duality in twistor space**

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

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**Fivebrane instantons The input: Apply SL(2,Z) transformation**

― D5-brane charge Apply SL(2,Z) transformation NS5-brane charge Compute fivebrane contact Hamiltonians Encodes fivebrane instanton corrections to all orders of the instanton expansion 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

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**A-model topological wave function in**

Open problems ● 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) = 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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