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

2. 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

3. 10-dimensional spacetime
The problem
Superstring theory
extended objects in
10-dimensional spacetime
low energy limit
& compactification
Effective field
theory in M(4)
Type II string theory
compactification
on a Calabi-Yau
N =2 supergravity in 4d
coupled to matter
N =2 supersymmetry
The goal: to find the complete non-perturbative effective action in 4d
for type II string theory compactified on arbitrary CY

4. 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

5. N =2 supergravity vector multiplets (gauge fields & scalars)
hypermultiplets
(only scalars)
supergravity multiplet
(metric)
given by
holomorphic prepotential
non-linear σ-model
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
The goal: to find the non-perturbative geometry of

6. HM moduli space
Coordinates 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 )
Classical
symmetries:
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,R) group
Type IIB
Mirror symmetry

7. Quantum corrections + Classical HM metric - corrections - corrections
perturbative
corrections
non-perturbative
perturbative + worldsheet instantons
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

8. Instanton corrections
D-brane instantons
Type IIA
Type IIB ×
3 D2 ×
0 D(-1)
2 D1
4 D3
6 D5
NS5-brane instantons +
S-duality
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
Twistor approach

9. Twistor approach Holomorphicity
The idea: one should work at the level of the twistor space
Quaternionic structure:
Twistor space
quaternion algebra of almost
complex structures z
carries holomorphic contact
structure
symmetries of can be lifted to
holomorphic symmetries of
is a Kähler manifold
Holomorphicity
The geometry is determined by contact transformations
between sets of Darboux coordinates
generated by holomorphic functions

10. Contact transformations and transition functions
Two ways to parametrize contact transformations
transition functions
gluing conditions
contact Hamiltonians
contact bracket
gluing conditions
integral equations for “twistor lines”
It is sufficient to find
holomorphic functions corresponding to quantum corrections
metric on

11. D-instantons in Type IIA
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

12. QK/HK correspondence
the non-perturbative moduli space of 4d N =2 gauge theory compactified on S1
the twistor construction of
D-instanton corrected
is similar to
― quaternion-Kähler
― hyperkähler
Gaiotto,Moore,Neitzke ’08
This is an example of a general construction:
with a Killing vector
& hyperholomorphic
connection
QK/HK correspondence
Haydys ’08, S.A.,Persson,Pioline ’11, Hitchin ‘12
NS5-brane instantons to
break the isometry along NS axion and the correspondence with
Local c-map
Rigid c-map
Instanton corrections:
BPS particles winding the circle
D-instanton corrections to

13. Characteristic feature of type IIB formulation – SL(2,Z) symmetry
HM metric in Type IIB
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:
pert. gs-corrections:
instanton corrections: D(-1) D D D NS5
-corrections: w.s.inst.
SL(2,Z) duality
One needs to understand how S-duality is realized on twistor space and which constraints it imposes on the twistorial construction

14. S-duality in twistor space
S.A.,Banerjee ‘14
on fiber:
Condition for to carry
an isometric action of SL(2,Z)
Transformation property of
the contact bracket
Non-linear holomorphic representation
of S-duality on classical
The problem:
to find relevant for instanton corrections to the HM moduli space in Type IIB formulation
contact transformation

15. Instanton corrections in Type IIB
Quantum corrections:
closely related to (0,4) elliptic genus and mock modular forms
S.A.,Manschot,Pioline ‘12
-corrections: w.s.inst.
pert. gs-corrections:
instanton corrections: D(-1) D D D NS5
brane charges:
Poisson resummation of
Type IIA Darboux
coordinates in the
sector with

16. A-model topological wave function in the real polarization
Fivebrane instantons
The idea: to add all images of the D-instanton contributions under
S-duality with a non-vanishing 5-brane charge
The input:
S-duality acts linearly on contact Hamiltonians
NS5-brane
charge
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

17. NS5-branes & Integrability= ?
How NS5-brane instantons
look like in type IIA?
Inclusion of
NS5-branes
quantum
deformation
NS-axion corresponds to the central element in the Hiesenberg algebra
– non-abelian Fourier expansion
(“quantum torus”)
Universal hypermultiplet
NS5-brane
charge
quantization
parameter
(S.A. ’12)
Holom. function generating
NS5-brane instantons
Baker-Akhiezer function
of Toda hierarchy
One-fermion
wave function
Wall-crossing:
there is a natural extension
involving quantum dilog
Conjecture
In Type IIA formulation the NS5-brane instantons are encoded by

18. 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

19. D3-instantons I ― In the sector with fixed consider the formal sum
modular form
of weight (-1,0) ―
modular form
entering
(0,4) elliptic genus
weight
mult. system
Maldacena,Strominger,
Witten ‘97
indefinite theta series
of weight
― S-duality transformation
(holomorphic)
formal Poisson resummation

20. Type IIA construction is
D3-instantons II
The above calculation is very formal since the sum over charges is divergent!
Darboux coordinates in the large volume limit:
shadow
usually appears as a non-holomorphic
completion of a holomorphic Mock modular form
modular form
of weight (-1,0)
period integral
How to justify such a modification?
The correction is given by
holomorphic in twistor space!
Effect of a holomorphic
contact transformation
Type IIA construction is
consistent with S-duality