Sergei Alexandrov

Laboratoire Charles Coulomb Montpellier

reviews: S.A. arXiv:1111.2892S.A., J.Manschot., D.Persson.,B.Pioline arXiv:1304.0766

S.A., S.Banerjee arXiv:1405.0291 arXiv:1403.1265

Twistor Approach to String Compactifications,NS5-branes & Integrability

Plan of the talk

1. Compactified string theories and their moduli spaces: HM moduli space, its symmetries and quantum corrections

2. Twistor description of Quaternion-Kähler spaces

3. Instanton corrections to the HM moduli space

4. NS5-branes & Integrability

The problem

The goal: to find the complete non-perturbative effective action in 4d for type II string theory compactified on arbitrary CY

low energy limit

& compactification

Superstring theory

extended objects in 10-dimensional spacetime

Effective field theory in M(4)

Type II string theory

compactificationon a Calabi-Yau

N =2 supergravity in 4d coupled to matter

N =2 supersymmetry

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

vector multiplets(gauge fields & scalars)

N =2 supergravity

supergravity multiplet(metric)

hypermultiplets(only scalars)

non-linear σ-modelgiven by

holomorphic prepotential

The goal: to find the non-perturbative geometry of

The low-energy action is determined by the geometry of the moduli space parameterized by scalars of vector and hypermultiplets:

• – special Kähler (given by )• is classically exact(no corrections in string coupling gs)

known

• – quaternion-Kähler

• receives all types of gs-corrections

unknown

HM moduli spaceCoordinates on the moduli space ― fields in 4 uncompactified dimensions

complex structure moduli

periods of RR gauge potentials

NS-axion (dual to the B-field)

dilaton (string coupling )

In Type IIA:

Classical metric on ― image of the c-map (determined by )

Heisenberg symmetry(shifts RR-fields and NS-axion)

Symplectic invariance(mixes the basis of 3-cycles)

Type IIA

S-duality(inverts string coupling)

form SL(2,) group

Type IIB

Mirror symmetry

Classical symmetries:

Quantum corrections

- corrections - corrections

Classical HM metric

+

perturbative + worldsheet instantons

In the image of the c-map

Captured by the prepotential

one-loop correctioncontrolled by

Antoniadis,Minasian,Theisen,Vanhove ’03Robles-Llana,Saueressig,Vandoren ’06, S.A. ‘07

perturbativecorrections

non-perturbativecorrections

Instantons – (Euclidean) world volumes

wrapping non-trivial cycles of CY

Instanton corrections

D-brane instantonsType IIA Type IIB

NS5-brane instantons

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

1 ×

3 D2

5 ×

0 D(-1)

2 D1

4 D3

6 D5

Axionic couplings break continuous isometries

S-duality

At the end no continuous isometries remain

Twistor approach

S.A.,Banerjee ’14

Twistor approach

Twistor space

z

The idea: one should work at the level of the twistor space

• carries holomorphic contact structure

• symmetries of can be lifted toholomorphic symmetries of

• is a Kähler 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

Contact transformations and transition functionsTwo ways to parametrize contact transformations

transition functions

gluing conditions

contact Hamiltonians

contact bracket

gluing conditions

metric on

It is sufficient to find holomorphic functions corresponding

to quantum corrections

integral equations for “twistor lines”

D-instantons in Type IIA

Equations for twistor lines coincide with equations of Thermodynamic Bethe Ansatz with an integrable S-matrix

Test: Penrose transform

Every D2 brane of charge

defines a ray on along

― central charge

The associated contact Hamiltonian

&

generalized DT invariants

QK/HK correspondencethe non-perturbative moduli space of 4d N =2 gauge theory compactified on S1

Gaiotto,Moore,Neitzke ’08

― quaternion-Kähler ― hyperkähler

the twistor construction of D-instanton corrected

Instanton corrections: BPS particles winding the circle

D-instanton corrections to

This is an example of a general construction:

with a Killing vector

with a Killing vector & hyperholomorphic

connection

QK/HK correspondence

Haydys ’08, S.A.,Persson,Pioline ’11, Hitchin ‘12

Local c-map Rigid c-mapNS5-brane instantons to

break the isometry along NS axion and the correspondence with

is similar to

HM metric in Type IIB

• break SL(2,) to a discrete subgroup SL(2,)• generate corrections to the mirror mapQuantum effects:

Characteristic feature of type IIB formulation – SL(2,) symmetry

SL(2,) duality• pert. gs-corrections:

• instanton corrections: D(-1) D1 D3 D5 NS5

• -corrections: w.s.inst.

• SL(2,) symmetry• known relations between type IIA and type IIB fields (mirror map)

Classical theory:

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:

S.A.,Banerjee ‘14

Non-linear holomorphic representation of S-duality on classical

contact transformation

Condition for to carry an isometric action of SL(2,)

Transformation property of the contact bracket

The problem:

to find relevant for instanton corrections to the HM moduli space

in Type IIB formulation

Quantum corrections:

Instanton corrections in Type IIB

brane charges:

Poisson resummation of Type IIA Darboux coordinates in the sector with

• pert. gs-corrections:

• instanton corrections: D(-1) D1 D3 D5 NS5

• -corrections: w.s.inst.

closely related to (0,4) elliptic genus and mock modular forms

S.A.,Manschot,Pioline ‘12

Fivebrane instantonsThe idea: to add all images of the D-instanton contributions under S-duality with a non-vanishing 5-brane charge

NS5-brane charge

S-duality acts linearly on contact Hamiltonians

The input:

• The contact structure is invariant under full U-duality group• One can evaluate the action on Darboux coordinatesand 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

• NS-axion corresponds to the central element in the Hiesenberg algebra

– non-abelian Fourier expansion(“quantum torus”)

NS5-branes & Integrability

How NS5-brane instantons look like in type IIA?

=Inclusion of NS5-branes

quantum deformation

Conjecture In Type IIA formulation the NS5-brane

instantons are encoded by

Holom. function generatingNS5-brane instantons

Baker-Akhiezer functionof Toda hierarchy

• Universal hypermultiplet

NS5-brane charge

quantization parameter

(S.A. ’12)

One-fermion wave function

?

• Wall-crossing:

there is a natural extensioninvolving quantum dilog

Conclusions

• 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

• 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

Main results

Some open issues

D3-instantons IIn the sector with fixed consider the formal sum

modular formentering

(0,4) elliptic genus

weight

mult. system

Maldacena,Strominger,Witten ‘97

indefinite theta seriesof weight

― S-duality transformation (holomorphic)

formal Poisson resummation

modular form of weight (-1,0)

―

D3-instantons IIThe above calculation is very formal since the sum over charges is divergent!

Darboux coordinates in the large volume limit:

period integral

usually appears as a non-holomorphic completion of a holomorphic Mock modular form

shadow

modular form of weight (-1,0)

Type IIA construction is consistent with S-duality

How to justify such a modification?The correction is given by

holomorphic in twistor space!

Effect of a holomorphic contact transformation