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Sergei Alexandrov. Twistor Approach to String Compactifications, NS5-branes & Integrability. Laboratoire Charles Coulomb Montpellier. reviews: S.A. arXiv:1111.2892 S.A., J.Manschot., D.Persson., B.Pioline arXiv:1304.0766.

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## Sergei Alexandrov

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**Sergei Alexandrov**Twistor Approach to String Compactifications,NS5-branes & Integrability Laboratoire Charles Coulomb Montpellier reviews: S.A. arXiv:1111.2892 S.A., J.Manschot., D.Persson., B.Pioline arXiv:1304.0766 S.A., S.Banerjee arXiv:1405.0291 arXiv:1403.1265**Plan of the talk**• Compactified string theories and their moduli spaces: HM moduli space, its symmetries and quantum corrections • Twistor description of Quaternion-Kähler spaces • Instanton corrections to the HM moduli space • NS5-branes & Integrability**The problem**N =2 supersymmetry Type II string theory compactification on a Calabi-Yau N =2 supergravity in 4d coupled to matter Superstring theory extended objects in 10-dimensional spacetime low energy limit & compactification Effective field theory in M(4) The goal:to find the complete non-perturbative effective action in 4d for type II string theory compactified on arbitrary CY**Motivation:**• Non-perturbative structure of string theory non-perturbative realization of S-duality and mirror symmetry • Preliminary step towards phenomenological models, applications to cosmology • BPS black holes • Relations to N =2 gauge theories, wall-crossing, topological strings… • Hidden integrability • Extremely rich mathematical structure**The low-energy action is determined by the geometry of the**moduli space parameterized by scalars of vector andhypermultiplets: • –special Kähler (given by ) • is classically exact • (no corrections in string couplinggs) • –quaternion-Kähler • receives all types of gs-corrections known unknown The goal: to find the non-perturbative geometry of N =2 supergravity vector multiplets (gauge fields & scalars) hypermultiplets (only scalars) supergravity multiplet (metric) given by holomorphic prepotential non-linear σ-model**complex structure moduli**periods of RR gauge potentials NS-axion (dual to the B-field) dilaton (string coupling ) Heisenberg symmetry (shifts RR-fields and NS-axion) Type IIA Type IIB Symplectic invariance (mixes the basis of 3-cycles) S-duality (inverts string coupling) form SL(2,R) group Mirror symmetry HM moduli space Coordinates on the moduli space ― fields in 4 uncompactified dimensions In Type IIA: Classical metricon ― image of the c-map (determined by ) Classical symmetries:**- corrections**perturbative + worldsheet instantons non-perturbative corrections perturbative corrections - corrections In the image of the c-map Captured by the prepotential Instantons – (Euclidean) world volumes wrapping non-trivial cycles of CY one-loop correction controlled by Antoniadis,Minasian,Theisen,Vanhove ’03 Robles-Llana,Saueressig,Vandoren ’06, S.A. ‘07 Quantum corrections Classical HM metric +**1 ×**3 D2 5 × + NS5-brane instantons S-duality Twistor approach Instanton corrections D-brane instantons Type IIA Type IIB 0 D(-1) 2 D1 4 D3 6 D5 S.A.,Pioline,Saueressig, Vandoren ’08 S.A. ‘09 S.A.,Banerjee ’14 Axionic couplings break continuous isometries At the end no continuous isometries remain**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 idea: one should work at the level of the twistor space Quaternionic structure: Twistor space quaternion algebra of almost complex structures z**transition functions**contact Hamiltonians gluing conditions gluing conditions contact bracket integral equations for “twistor lines” It is sufficient to find holomorphic functions corresponding to quantum corrections metric on Contact transformations and transition functions Two ways to parametrize contact transformations**Every D2 brane of charge**defines a ray on along ― central charge & The associated contact Hamiltonian generalized DT invariants Equations for twistor lines coincide with equations of Thermodynamic Bethe Ansatz with an integrable S-matrix Test: Penrose transform D-instantons in Type IIA**the twistor construction of**D-instanton corrected ― hyperkähler ― quaternion-Kähler D-instanton corrections to This is an example of a general construction: Haydys ’08,S.A.,Persson,Pioline ’11, Hitchin ‘12 with a Killing vector & hyperholomorphic connection QK/HK correspondence with a Killing vector Local c-map Rigid c-map Instanton corrections: BPS particles winding the circle NS5-brane instantons to break the isometry along NS axion and the correspondence with QK/HK correspondence the non-perturbative moduli space of 4d N=2 gauge theory compactified on S1 is similar to Gaiotto,Moore,Neitzke ’08**HM metric in Type IIB**SL(2,Z) duality • -corrections: w.s.inst. • pert. gs-corrections: • instanton corrections: D(-1) D1 D3 D5 NS5 Characteristic feature of type IIB formulation – SL(2,Z) symmetry • SL(2,R) symmetry • known relations between type IIA and type IIB fields • (mirror map) Classical theory: • break SL(2,R) to a discrete subgroup SL(2,Z) • generate corrections to the mirror map Quantum effects: One needs to understand how S-duality is realized on twistor space and which constraints it imposes on the twistorial construction**S-duality in twistor space**on fiber: Transformation property of the contact bracket Condition for to carry an isometric action of SL(2,Z) The problem: to find relevant for instanton corrections to the HM moduli space in Type IIB formulation S.A.,Banerjee ‘14 Non-linear holomorphic representation of S-duality on classical contact transformation**Instanton corrections in Type IIB**closely related to(0,4) elliptic genusandmock modular forms S.A.,Manschot,Pioline ‘12 Poisson resummation of Type IIA Darboux coordinates in the sector with Quantum corrections: • -corrections: w.s.inst. • pert. gs-corrections: • instanton corrections: D(-1) D1 D3 D5 NS5 brane charges:**The contact structure is invariant under full U-duality**group • One can evaluate the action on Darboux coordinates • and write down the integral equations on twistor lines • One can find explicitly the full quantum mirror map A-model topological wave function in the real polarization • Relation to the topological • string wave function Fivebrane instantons The idea: to add all images of the D-instanton contributions under S-dualitywith a non-vanishing 5-brane charge The input: S-duality acts linearly on contact Hamiltonians NS5-brane charge**NS-axion corresponds to the central element in the**Hiesenberg algebra – non-abelian Fourier expansion (“quantum torus”) • Universal hypermultiplet (S.A. ’12) quantization parameter NS5-brane charge • Wall-crossing: Conjecture In Type IIA formulation the NS5-brane instantons are encoded by there is a natural extension involving quantum dilog NS5-branes & Integrability ? = Inclusion of NS5-branes quantum deformation How NS5-brane instantons look like in type IIA? Holom. function generating NS5-brane instantons Baker-Akhiezer function of Toda hierarchy One-fermion wave function**Conclusions**Main results • Twistor description of instanton corrections to • the metric on the HM moduli space • Explicit realization of S-duality in the twistor space • Quantum mirror map • Relations to integrable structures Some open issues • 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? • (quantum dilog?) • Resolution of one-loop singularity • Resummation of divergent series over brane charges**D3-instantons I**modular form of weight (-1,0) ― indefinite theta series of weight weight mult. system modular form entering (0,4) elliptic genus formal Poisson resummation Maldacena,Strominger, Witten ‘97 ― S-duality transformation (holomorphic) In the sector with fixed consider the formal sum**D3-instantons II**shadow usually appears as a non-holomorphic completion of a holomorphic Mock modular form How to justify such a modification? The correction is given by holomorphic in twistor space! Type IIA construction is consistent with S-duality Effect of a holomorphic contact transformation The above calculation is very formal since the sum over charges is divergent! Darboux coordinates in the large volume limit: modular form of weight (-1,0) period integral

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