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Sergei Alexandrov. Calabi-Yau compactifications: results, relations & problems. Laboratoire Charles Coulomb Montpellier. in collaboration with S.Banerjee, J.Manschot, D.Persson, B.Pioline, F.Saueressig, S.Vandoren. N =2 gauge theories. Integrability. TBA Toda hierarchy and c=1 string

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## Sergei Alexandrov

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**Sergei Alexandrov**Calabi-Yau compactifications:results, relations & problems Laboratoire Charles Coulomb Montpellier in collaboration with S.Banerjee, J.Manschot, D.Persson, B.Pioline, F.Saueressig, S.Vandoren**N =2 gauge theories**Integrability • TBA • Toda hierarchy • and c=1 string • Quantum integrability • wall-crossing • QK/HK • correspondence Fine mathematics Topological strings • mock modularity • cluster algebras NS5-instantons top. wave functions = Calabi-Yau compactifications**The low-energy effective action is determined by the metric**on the moduli spaceparameterized by scalars of vector and hypermultiplets Moduli space • –special Kähler (given by ) • is classically exact • (no corrections in string couplinggs) • –quaternion-Kähler • receives all types of gs-corrections known unknown The (concrete) goal: to find the non-perturbative geometry of 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 aCalabi-Yau vector multiplets(gauge fields & scalars) hypermultiplets(only scalars) N =2 supergravity in 4d coupled to matter**In the image of the c-map**(captured by the prepotential ) includes -corrections (perturbative + worldsheet instantons) Instantons – (Euclidean) world volumes wrapping non-trivial cycles of CY non-perturbative corrections perturbative corrections - corrections ● D-brane instantons one-loop correction controlled by ● NS5-brane instantons Antoniadis,Minasian,Theisen,Vanhove ’03, Robles-Llana,Saueressig,Vandoren ’06, S.A. ‘07 Quantum corrections Tree level HM metric + Axionic couplings break continuous isometries At the end no continuous isometries remain**mirror**SL(2,Z) mirror e/m duality mirror SL(2,Z) Beyond this line the projective superspace approach is not applicable anymore (not enough isometries) Twistor approach S.A.,Pioline,Saueressig,Vandoren ’08 Instanton corrections • 6 NS5 • 5 D4 • 4 × • 3 D2 • 2 × • 1 D0 • 0 × {D2, NS5} {α’, D(-1), D1, , D5, NS5} D3 • 6 D5,NS5 • 5 × • 4 D3 • 3 × • 2 D1 • 1 × • 0 D(-1) D3 {A-D2, B-D2} {α’, D(-1), D1, , D5} {A-D2} {α’, D(-1), D1} Robles-Llana,Roček, Saueressig,Theis, Vandoren ’06 {α’, 1-loop } using projective superspace approach**is aKähler manifold**• carriesholomorphic contact • structure • symmetries of can be lifted to • holomorphic symmetries of The geometry is determined by contact transformations between sets of Darboux coordinates Holomorphicity generated by holomorphic functions Twistor approach The problem: how to conveniently parametrize a QK manifold? Quaternionic structure: Twistor space quaternion algebra of almost complex structures t**integral equations for “twistor lines”**contact bracket Perturbative metric holomorphic functions It is sufficient to find holomorphic functions corresponding to quantum corrections D-instanton corrections holomorphic functions metric on NS5-instanton corrections holomorphic functions Contact Hamiltonians and instanton corrections gluing conditions**Dictionary**fiber coordinate spectral parameter twistor fiber spectral curve Darboux coordinates Lax & Orlov-Shulman operators with an isometry & hyperholomorphic connection gluing conditions string equations with an isometry ? Haydys ’08 S.A.,Persson,Pioline ’11 Hitchin ‘12 Manifestation of QK/HK correspondence Contact geometry of QK manifolds Quantum integrability? Correspondence with integrable systems Expansion in the patches around the north and south poles: Generalizes the fact that hyperkähler spaces are complex integrable systems (Donagi,Witten ’95) Baker-Akhiezer function**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) = sine-Liouville perturbation: time-dependent background with a non-trivial tachyon condensate Toda equation Holom. function generating NS5-brane instantons One-fermion wave function Baker-Akhiezer function of Toda hierarchy NS5-brane charge quantization parameter Perturbative HM moduli space c-map**Again integrability…**Analogous to the construction of the non-perturbative moduli space of 4d N=2 gauge theory compactified on S1 Equations for twistor lines coincide with equations of Thermodynamic Bethe Ansatz with an integrable S-matrix Gaiotto,Moore,Neitzke ’08 Example of QK/HK correspondence 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**Instanton corrections in Type IIB**SL(2,Z) duality using S-duality S.A.,Banerjee ‘14 Robles-Llana,Roček,Saueressig, Theis,Vandoren ’06 Twistorial description: from mirror symmetry In the one-instanton approximation the mirror of the type IIA construction is consistent with S-duality requires technique ofmock modular forms Gopakumar-Vafa invariants of genus zero manifestly S-duality formulation is unknown S.A.,Saueressig, ’09 S.A.,Manschot,Pioline ‘12 Type IIB formulation must be manifestly invariant under S-duality group Quantum corrections in type IIB: • -corrections:w.s.inst. • pert. gs-corrections: • instanton corrections:D(-1) D1 D3 D5 NS5**S-duality in twistor space**Transformation property of the contact bracket Non-linear holomorphic representation of SL(2,Z)–duality group on Condition for to carry an isometric action of SL(2,Z) contact transformation 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-dualitywith a non-vanishing 5-brane charge**Encodes fivebrane instanton corrections to all orders of the**instanton expansion Fivebrane instantons The input: ― D5-brane charge Apply SL(2,Z) transformation NS5-brane charge Compute fivebrane contact Hamiltonians • 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**?**= Inclusion of NS5-branes quantum deformation A-model topological wave function in the real polarization quantum dilog? Relation to the topological string wave function 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) THANK YOU!

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