Slide 1

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 Slide 2 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 Slide 3 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 Khler (given by ) is classically exact (no corrections in string coupling g s ) known quaternion-Khler 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 Slide 4 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 Slide 5 Instanton corrections 6 D5,NS5 5 4 D3 3 2 D1 1 0 D(-1) {, D(-1), D1} {, 1-loop } {A-D2} {A-D2, B-D2}{, D(-1), D1,, D5} {, 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,Roek, 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 Slide 6 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 Khler 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 Slide 7 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 Slide 8 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 hyperkhler 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 Slide 9 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 Slide 10 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 Slide 11 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,Roek,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 Slide 12 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 Slide 13 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 Slide 14 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!