Sergei Alexandrov
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Transcript of Sergei Alexandrov
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!