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

Sergei Alexandrov

Laboratoire Charles Coulomb Montpellier

Instantons, wall-crossing and quantum dilogarithm identities

S.A., B.Pioline arXiv:1511.02892

string compactifications

• n Calabi-Yau
• D-instantons

cluster varieties

Wall-crossing

N=2 SUSY gauge theories

string compactifications

• n Calabi-Yau
• D-instantons

cluster varieties

Wall-crossing

N=2 SUSY gauge theories

dilogarithm identities

Dilogarithm

Euler dilogarithm: Rogers dilogarithm: periodic sequence

Pentagon identity

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

Fourier transform

Pentagon identity in the integral form

string compactifications

• n Calabi-Yau
• D-instantons

cluster varieties

Wall-crossing

N=2 SUSY gauge theories

dilogarithm identities

string compactifications

• n Calabi-Yau
• D-instantons

cluster varieties

Wall-crossing

N=2 SUSY gauge theories

dilogarithm identities Quantum

quantization of in W background (deformation)

motivic

identities for

string compactifications

• n Calabi-Yau
• D-instantons

cluster varieties

Wall-crossing

N=2 SUSY gauge theories

dilogarithm identities Quantum

& theta-series

quantization of

• NS5-instantons
• n Taub-NUT

(geometric) identities for identities for

string compactifications

• n Calabi-Yau
• D-instantons

cluster varieties

Wall-crossing

N=2 SUSY gauge theories

dilogarithm identities Quantum

& theta-series

quantization of

• NS5-instantons
• n Taub-NUT

(geometric) identities for

Double quantization

complex Chern-Simons theory

identities for

N =2 SUSY, BPS spectrum and wall-crossing

BPS bound

moduli holomorphic prepotential

N =2 SUSY algebra:

central charge electromagnetic charge

N =2 SUSY, BPS spectrum and wall-crossing

BPS index

counts BPS states

• f charge

(with sign)

BPS bound

moduli holomorphic prepotential

N =2 SUSY algebra:

central charge electromagnetic charge

N =2 SUSY, BPS spectrum and wall-crossing

BPS index

counts BPS states

• f charge

(with sign)

BPS bound

moduli holomorphic prepotential

N =2 SUSY algebra:

central charge electromagnetic charge

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

BPS index

counts BPS states

• f charge

(with sign)

BPS bound

moduli holomorphic prepotential

N =2 SUSY algebra:

central charge electromagnetic charge bound state exists bound state does not exist position

• f walls

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

BPS index

counts BPS states

• f charge

(with sign)

Example: pure SU(2) N =2 SUYM

Seiberg,Witten ’94

wall BPS bound

moduli holomorphic prepotential

N =2 SUSY algebra:

central charge electromagnetic charge bound state exists bound state does not exist position

• f walls

is only piecewise constant because of walls of marginal stability in the moduli space across which it jumps discontinuously

Kontsevich-Soibelman solution

– algebra of symplectomorphisms of a torus

• – non-commutative KS operators

– BPS indices on the two sides of the wall

• The ordering is determined by the phase of the central charge

Kontsevich-Soibelman solution

Example: Pentagon relation

– algebra of symplectomorphisms of a torus

• – non-commutative KS operators

– BPS indices on the two sides of the wall

• The ordering is determined by the phase of the central charge

Calabi-Yau compactifications

Type II string theory compactification

• n a Calabi-Yau

N =2 supergravity in 4d coupled to matter moduli space The low energy effective action is completely determined by the geometry of

quaternion-Kähler special Kähler classically exact receives -corrections

Calabi-Yau compactifications

Type II string theory compactification

• n a Calabi-Yau

N =2 supergravity in 4d coupled to matter moduli space The low energy effective action is completely determined by the geometry of

quaternion-Kähler

Metric

• n

= classical + 1-loop + + D-brane instantons NS5-brane instantons

special Kähler DT invariants (BPS indices) D-branes wrapping cycles of CY NS5-branes wrapping CY classically exact receives -corrections

S.A.,Pioline,Saueressig, Vandoren ’08, S.A. ‘09

given by twistorial construction

Calabi-Yau compactifications

Type II string theory compactification

• n a Calabi-Yau

N =2 supergravity in 4d coupled to matter moduli space The low energy effective action is completely determined by the geometry of

quaternion-Kähler

Metric

• n

= classical + 1-loop + + D-brane instantons NS5-brane instantons

special Kähler DT invariants (BPS indices) D-branes wrapping cycles of CY NS5-branes wrapping CY classically exact receives -corrections

?

S.A.,Pioline,Saueressig, Vandoren ’08, S.A. ‘09

given by twistorial construction

Calabi-Yau compactifications

Type II string theory compactification

• n a Calabi-Yau

N =2 supergravity in 4d coupled to matter moduli space The low energy effective action is completely determined by the geometry of

quaternion-Kähler

Metric

• n

= classical + 1-loop + + D-brane instantons NS5-brane instantons

special Kähler DT invariants (BPS indices) D-branes wrapping cycles of CY NS5-branes wrapping CY classically exact receives -corrections

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

• D-instantons:

Instanton transition functions:

• NS5-instantons:

Twistorial construction

─ non-trivial bundle over

KS operator

Twistor space

holomorphic contact structure

holomorphic Darboux coordinates

The metric is determined by transition functions

.

generating contact transformations

• D-instantons:

Instanton transition functions:

• NS5-instantons:

Twistorial construction

─ non-trivial bundle over

• What is this function?
• Is it consistent with wall-crossing?

KS operator

Twistor space

holomorphic contact structure

holomorphic Darboux coordinates

The metric is determined by transition functions

.

generating contact transformations

• D-instantons:

Instanton transition functions:

• NS5-instantons:

Heisenberg symmetry & theta-series

is invariant under the action of the Heisenberg group

central element

classically

Lift to the twistor space

Heisenberg symmetry & theta-series

is invariant under the action of the Heisenberg group

central element

non-perturbatively

Lift to the twistor space

Heisenberg symmetry & theta-series

is invariant under the action of the Heisenberg group

central element

non-perturbatively

Lift to the twistor space

(D-instantons) (NS5-instantons)

General form:

─ abelian Fourier expansion ─ non-abelian Fourier expansion

Functions on invariant under the Heisenberg action (on twisted torus)

Heisenberg symmetry & theta-series

is invariant under the action of the Heisenberg group

central element

non-perturbatively

Lift to the twistor space

(D-instantons) (NS5-instantons)

General form:

─ abelian Fourier expansion ─ non-abelian Fourier expansion

Functions on invariant under the Heisenberg action (on twisted torus)

Theta series and wave functions

Theta series and wave functions

Theta series and wave functions

Fourier transform

Theta series and wave functions

wave function in -representation Fourier transform

Theta series and wave functions

wave function in -representation

topological string partition function in the real polarization

For NS5-instantons on

S.A.,Persson,Pioline ’10

S-duality of type IIB string theory Fourier transform

The problem

─ Rogers dilogarithm

where

• btained by requiring the

invariance of the contact 1-form

The action of KS operators: ─ induces a transformation of

The problem

─ Rogers dilogarithm

where

• btained by requiring the

invariance of the contact 1-form

The action of KS operators: ─ induces a transformation of The problem: What is this transformation? Applications: ● mutual consistency of D-brane and NS5-brane instantons

• geometric quantization of cluster varieties
• defines a line bundle over which describes N

N =2

gauge theory on Taub-NUT space

Dey,Neitzke ’14 Fock,Goncharov

• new quantum dilogarithm identities

The result

where Faddeev’s quantum dilogarithm

The result

where Faddeev’s quantum dilogarithm

Example: two simple cases pure electric charge pure magnetic charge

The result

where Faddeev’s quantum dilogarithm

Example: two simple cases pure electric charge pure magnetic charge Observation: The transformation is just a result of the change

• f representation for the wave function

The result

where Faddeev’s quantum dilogarithm

Example: two simple cases pure electric charge pure magnetic charge Observation: The transformation is just a result of the change

• f representation for the wave function

It can be realized by a unitary operator in the Hilbert space where lives but

New dilogarithm identities

Wall-crossing requires for a set of charges

New dilogarithm identities

Wall-crossing requires for a set of charges anti-homomorphism

New dilogarithm identities

Wall-crossing requires for a set of charges anti-homomorphism Example: new integral pentagon identity

New dilogarithm identities

Wall-crossing requires for a set of charges anti-homomorphism Example: new integral pentagon identity

New dilogarithm identities

Wall-crossing requires for a set of charges anti-homomorphism Example: new integral pentagon identity Theorem: In spanned by vector valued functions the following operator identity holds

Double quantization

• ld

new

Double quantization

• ld

new can one combine the two quantization parameters?

Double quantization

• ld

new Modular double: can one combine the two quantization parameters?

Double quantization

• ld

new Modular double: can one combine the two quantization parameters?

Double quantization

• ld

new Modular double: These operators satisfy can one combine the two quantization parameters?

Double quantization

• ld

new Modular double: Generate two sequences

• f operators of period 5

These operators satisfy can one combine the two quantization parameters?

Double quantization

• ld

new Modular double: Conjecture Generate two sequences

• f operators of period 5

These operators satisfy can one combine the two quantization parameters?

Conclusions

• We provided a quantization of KS symplectomorphisms realizing

them on generalized theta-functions.

• We derived a new integral pentagon identity with a discrete

quantization parameter k.

• We proposed a modular double construction which combines the

discrete and real quantization parameters. Do these constructions appear in integrable models or field theories?

This has applications in ● Calabi-Yau compactifications of string theory

• N =2 gauge theories on Taub-NUT
• quantization of cluster varieties

Andersen,Kashaev ’14

“Complex quantum Chern-Simons”