1. Calabi-Yau compactifications: results, relations & problems
Sergei Alexandrov
Laboratoire Charles Coulomb
Montpellier
in collaboration with S.Banerjee, J.Manschot, D.Persson, B.Pioline, F.Saueressig, S.Vandoren

2. Calabi-Yau compactifications
N =2 gauge theories
wall-crossing
QK/HK
correspondence
Integrability
TBA
Toda hierarchy
and c=1 string
Quantum integrability
Calabi-Yau
compactifications
Fine mathematics
mock modularity
cluster algebras
Topological strings
NS5-instantons
top. wave functions =

3. Calabi-Yau compactifications
The goal: to find the complete non-perturbative effective action in 4d
for type II string theory compactified on arbitrary CY
Type II string theory
supergravity multiplet (metric)
compactification
on a Calabi-Yau
vector multiplets (gauge fields & scalars)
hypermultiplets (only scalars)
N =2 supergravity in 4d
coupled to matter
The low-energy effective action is determined by the metric on the moduli space parameterized by scalars of vector and hypermultiplets
Moduli space
– special Kähler (given by )
is classically exact
(no corrections in string coupling gs)
known
– quaternion-Kähler
receives all types of gs-corrections
unknown
The (concrete) goal: to find the non-perturbative geometry of

4. Quantum corrections + Tree level HM metric - corrections perturbative
includes -corrections
(perturbative + worldsheet instantons) +
In the image of the c-map
(captured by the prepotential )
- corrections
perturbative
corrections
non-perturbative
Instantons –
(Euclidean) world volumes
wrapping non-trivial cycles of CY
● D-brane instantons
● NS5-brane instantons
one-loop correction
controlled by
Antoniadis,Minasian,Theisen,Vanhove ’03,
Robles-Llana,Saueressig,Vandoren ’06,
S.A. ‘07
Axionic couplings break continuous isometries
At the end no continuous isometries remain

5. Instanton corrections
SL(2,Z)
mirror
e/m
duality
6 NS5
5 D4 ×
3 D2 ×
1 D0 ×
{D2, NS5}
{α’, D(-1), D1, , D5, NS5} D3
6 D5,NS5 ×
4 D3 ×
2 D1 ×
0 D(-1)
{A-D2, B-D2}
{α’, D(-1), D1, , D5} D3
Beyond this line the projective superspace approach is not applicable anymore
(not enough isometries)
{A-D2}
{α’, D(-1), D1}
Robles-Llana,Roček,
Saueressig,Theis,
Vandoren ’06
{α’, 1-loop }
using projective superspace approach
Twistor approach
S.A.,Pioline,Saueressig,Vandoren ’08

6. Twistor approach Holomorphicity
The problem: how to conveniently parametrize a QK manifold?
Quaternionic structure:
Twistor space
quaternion algebra of almost
complex structures t
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

7. Contact Hamiltonians and instanton corrections
gluing conditions
contact bracket
integral equations for “twistor lines”
metric on
Perturbative
metric
D-instanton
corrections
NS5-instanton
holomorphic
functions
It is sufficient to find
holomorphic functions corresponding to quantum corrections

8. ? Correspondence with integrable systems
Expansion in the patches around the north and south poles:
Dictionary
Darboux coordinates
fiber coordinate
spectral parameter
string equations
gluing 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
& hyperholomorphic
connection
QK/HK
correspondence
Haydys ’08
S.A.,Persson,Pioline ’11
Hitchin ‘12
Baker-Akhiezer
function
Quantum integrability?
Contact geometry of QK manifolds

9. sine-Liouville perturbation: Baker-Akhiezer function
Perturbative HM moduli space
Twistor description of at tree level
c-map
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 equation
sine-Liouville perturbation:
time-dependent background with a non-trivial tachyon condensate
NS5-brane
charge
quantization
parameter
Holom. function generating
NS5-brane instantons
One-fermion
wave function
Baker-Akhiezer function
of Toda hierarchy

10. D2-instantons in Type IIA
S.A.,Pioline,Saueressig,Vandoren ’08, S.A. ’09
D2-instantons in Type IIA
Holomorphic functions generating D2-instantons
― D-brane charge
generalized DT invariants
Equations for twistor lines coincide with equations of Thermodynamic Bethe Ansatz with an integrable S-matrix
Again integrability…
Analogous to the construction of the non-perturbative moduli space of 4d N =2 gauge theory compactified on S1
Gaiotto,Moore,Neitzke ’08
Example of
QK/HK correspondence

11. Instanton corrections in Type IIB
Type IIB formulation must be manifestly invariant under S-duality group
Quantum corrections in type IIB:
-corrections: w.s.inst.
SL(2,Z) duality
pert. gs-corrections:
instanton corrections: D(-1) D D D NS5
S.A.,Banerjee ‘14
using S-duality
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

12. S-duality in twistor space
S.A.,Banerjee ‘14
One must understand how S-duality is realized on twistor space and which constraints it imposes on
the twistorial construction
The idea: to add all images of the
D-instanton contributions under S-duality with a non-vanishing 5-brane charge
Non-linear holomorphic representation
of SL(2,Z) –duality group on
contact transformation
Condition for to carry
an isometric action of SL(2,Z)
Transformation property of
the contact bracket

13. Fivebrane instantons The input: Apply SL(2,Z) transformation
― D5-brane charge
Apply SL(2,Z) transformation
NS5-brane
charge
Compute fivebrane contact Hamiltonians
Encodes fivebrane instanton corrections to all orders of the instanton expansion
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

14. A-model topological wave function in
Open problems
● 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) =
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!