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

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

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

known

• – quaternion-Kähler

• receives all types of gs-corrections

unknown

Calabi-Yau compactifications

Type II string theory

compactificationon 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

Quantum corrections

- corrections

Tree level HM metric

+ In the image of the c-map

(captured by the prepotential )

one-loop correctioncontrolled by

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

perturbativecorrections

non-perturbativecorrections

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

Instanton corrections

6 D5,NS5

5 ×

4 D3

3 ×

2 D1

1 ×

0 D(-1)

{α’, D(-1), D1}

{α’, 1-loop }

{A-D2}

{A-D2, B-D2} {α’, D(-1), D1, , D5}

{α’, D(-1), D1, , D5, NS5}{D2, NS5}

SL(2,)

mirror

e/m duality

mirror

mirror

SL(2,)

6 NS5

5 D4

4 ×

3 D2

2 ×

1 D0

0 × Robles-Llana,Roček,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

Twistor approach

Twistor space

t

The problem: how to conveniently parametrize a QK manifold?

• 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 Hamiltonians and instanton corrections

contact bracketgluing 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

Expansion in the patches around the north and south poles:

Correspondence with integrable systems

?

Dictionary

Darboux coordinates

fiber coordinate spectral parameter

string equationsgluing conditions

twistor fiber spectral curve

Lax & Orlov-Shulman operators

Generalizes the fact that hyperkähler spaces are complex integrable systems (Donagi,Witten ’95)

Manifestation of

with an isometry

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

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 equationsine-Liouville perturbation:

time-dependent background with a non-trivial tachyon condensate

NS5-brane charge

quantization parameter

Holom. function generatingNS5-brane instantons

One-fermion wave function

Baker-Akhiezer function

of Toda hierarchy

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 S1

Gaiotto,Moore,Neitzke ’08

Example ofQK/HK correspondence

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

Again integrability…

Instanton corrections in Type IIBType IIB formulation must be manifestly invariant under S-duality group

• pert. gs-corrections:

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

Quantum corrections in type IIB:

SL(2,) duality• -corrections: w.s.inst.

Robles-Llana,Roček,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

S-duality in twistor spaceS.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

Fivebrane instantons

NS5-brane chargeCompute 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

● 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!