Instantons, wall-crossing and quantum dilogarithm identities Sergei - PowerPoint PPT Presentation

Instantons, wall-crossing and quantum dilogarithm identities Sergei Alexandrov Laboratoire Charles Coulomb Montpellier S.A., B.Pioline arXiv:1511.02892

string compactifications on Calabi-Yau • D-instantons N=2 SUSY gauge cluster varieties theories Wall-crossing

string compactifications on Calabi-Yau • D-instantons N=2 SUSY gauge cluster varieties theories Wall-crossing dilogarithm identities

Dilogarithm Euler dilogarithm: Rogers dilogarithm: Pentagon identity periodic sequence Generic dilogarithm identity corresponding to a periodic sequence Applications: ● integrability (Y- systems, TBA, …) ● topological field theories (discretized path integrals) ● SUSY gauge theories (scattering amplitudes, BPS vacua,…)

Quantum dilogarithm Classical limit: Pentagon identity

Quantum dilogarithm Classical limit: Pentagon identity Pentagon identity in the integral form Fourier transform

string compactifications on Calabi-Yau • D-instantons N=2 SUSY gauge cluster varieties theories Wall-crossing dilogarithm identities

string compactifications on Calabi-Yau • D-instantons quantization of N=2 SUSY gauge cluster varieties theories in W background (deformation) Wall-crossing motivic Quantum dilogarithm identities identities for

string compactifications on Calabi-Yau • D-instantons • NS5-instantons quantization of N=2 SUSY gauge cluster varieties theories on Taub-NUT (geometric) Wall-crossing & theta-series Quantum dilogarithm identities identities for identities for

string compactifications on Calabi-Yau • D-instantons • NS5-instantons quantization of N=2 SUSY gauge cluster varieties theories on Taub-NUT (geometric) Wall-crossing & theta-series Quantum dilogarithm identities identities for identities for Double quantization complex Chern-Simons theory

N =2 SUSY, BPS spectrum and wall-crossing N =2 SUSY algebra: electromagnetic charge central charge holomorphic moduli prepotential BPS bound

N =2 SUSY, BPS spectrum and wall-crossing N =2 SUSY algebra: electromagnetic charge central charge holomorphic moduli prepotential BPS index counts BPS states BPS bound of charge (with sign)

N =2 SUSY, BPS spectrum and wall-crossing N =2 SUSY algebra: electromagnetic charge central charge holomorphic moduli prepotential BPS index counts BPS states BPS bound of charge (with sign) is only piecewise constant because of walls of marginal stability in the moduli space across which it jumps discontinuously

N =2 SUSY, BPS spectrum and wall-crossing N =2 SUSY algebra: electromagnetic charge central charge holomorphic moduli prepotential BPS index counts BPS states BPS bound of charge (with sign) is only piecewise constant because of walls of marginal stability in the moduli space across which it jumps discontinuously position of walls bound state bound state does not exist exists

N =2 SUSY, BPS spectrum and wall-crossing N =2 SUSY algebra: electromagnetic charge central charge holomorphic moduli prepotential BPS index counts BPS states BPS bound of charge (with sign) is only piecewise constant because of walls of marginal stability in the moduli space across which it jumps discontinuously Example: pure SU(2) N =2 SUYM position of walls wall bound state bound state does not exist exists Seiberg,Witten ’94

Kontsevich-Soibelman solution ● – non-commutative KS operators ● – algebra of symplectomorphisms of a torus ● – BPS indices on the two sides of the wall ● The ordering is determined by the phase of the central charge

Kontsevich-Soibelman solution ● – non-commutative KS operators ● – algebra of symplectomorphisms of a torus ● – BPS indices on the two sides of the wall ● The ordering is determined by the phase of the central charge Example: Pentagon relation

Calabi-Yau compactifications Type II string theory The low energy effective action is completely determined by the geometry of compactification moduli space on a Calabi-Yau N =2 supergravity in 4d coupled to matter special Kähler quaternion-Kähler receives -corrections classically exact

Calabi-Yau compactifications Type II string theory The low energy effective action is completely determined by the geometry of compactification moduli space on a Calabi-Yau N =2 supergravity in 4d coupled to matter special Kähler quaternion-Kähler receives -corrections classically exact Metric D-brane NS5-brane = classical + 1-loop + + on instantons instantons D-branes wrapping NS5-branes cycles of CY wrapping CY DT invariants (BPS indices)

Calabi-Yau compactifications Type II string theory The low energy effective action is completely determined by the geometry of compactification moduli space on a Calabi-Yau N =2 supergravity in 4d coupled to matter special Kähler quaternion-Kähler receives -corrections classically exact Metric D-brane NS5-brane = classical + 1-loop + + on instantons instantons given by twistorial D-branes wrapping NS5-branes construction cycles of CY wrapping CY S.A.,Pioline,Saueressig, Vandoren ’08, S.A. ‘09 DT invariants (BPS indices)

Calabi-Yau compactifications Type II string theory The low energy effective action is completely determined by the geometry of compactification moduli space on a Calabi-Yau N =2 supergravity in 4d coupled to matter special Kähler quaternion-Kähler receives -corrections classically exact Metric D-brane NS5-brane ? = classical + 1-loop + + on instantons instantons given by twistorial D-branes wrapping NS5-branes construction cycles of CY wrapping CY S.A.,Pioline,Saueressig, Vandoren ’08, S.A. ‘09 DT invariants (BPS indices)

Twistorial construction ─ non-trivial bundle over Twistor space holomorphic contact structure holomorphic Darboux coordinates

Twistorial construction ─ non-trivial bundle over Twistor space holomorphic contact structure holomorphic Darboux coordinates The metric is determined by transition functions generating contact transformations .

Twistorial construction ─ non-trivial bundle over Twistor space holomorphic contact structure holomorphic Darboux coordinates The metric is determined by transition functions generating contact transformations . Instanton transition functions: ● D-instantons: ● NS5-instantons:

Twistorial construction ─ non-trivial bundle over Twistor space holomorphic contact structure holomorphic Darboux coordinates The metric is determined by transition functions generating contact transformations . Instanton transition functions: ● D-instantons: KS operator ● NS5-instantons:

Twistorial construction ─ non-trivial bundle over Twistor space holomorphic contact structure holomorphic Darboux coordinates The metric is determined by transition functions generating contact transformations . Instanton transition functions: ● D-instantons: KS operator ● NS5-instantons: ● What is this function? ● Is it consistent with wall-crossing?

Heisenberg symmetry & theta-series is invariant under the action of the Heisenberg group central element Lift to the twistor space classically

Heisenberg symmetry & theta-series is invariant under the action of the Heisenberg group central element Lift to the twistor space non-perturbatively

Heisenberg symmetry & theta-series is invariant under the action of the Heisenberg group central element Lift to the twistor space non-perturbatively Functions on invariant under the Heisenberg action (on twisted torus) General form: ─ abelian Fourier expansion (D-instantons) ─ non -abelian Fourier expansion (NS5-instantons)

Heisenberg symmetry & theta-series is invariant under the action of the Heisenberg group central element Lift to the twistor space non-perturbatively Functions on invariant under the Heisenberg action (on twisted torus) General form: ─ abelian Fourier expansion (D-instantons) ─ non -abelian Fourier expansion (NS5-instantons)

Theta series and wave functions

Theta series and wave functions

Theta series and wave functions Fourier transform

Theta series and wave functions Fourier transform wave function in -representation

Theta series and wave functions Fourier transform wave function in -representation S.A.,Persson,Pioline ’10 For NS5-instantons on S-duality of type IIB string theory topological string partition function in the real polarization

The problem The action of KS operators: obtained by requiring the ─ invariance of the contact 1-form where ─ Rogers dilogarithm induces a transformation of

The problem The action of KS operators: obtained by requiring the ─ invariance of the contact 1-form where ─ Rogers dilogarithm induces a transformation of The problem: What is this transformation? Applications: ● mutual consistency of D-brane and NS5-brane instantons ● defines a line bundle over which describes N N =2 gauge theory on Taub-NUT space Dey,Neitzke ’14 ● geometric quantization of cluster varieties Fock,Goncharov ● new quantum dilogarithm identities

The result Faddeev’s quantum where dilogarithm