sergei alexandrov
DESCRIPTION
Sergei Alexandrov. Calabi-Yau compactifications: results, relations & problems. Laboratoire Charles Coulomb Montpellier. in collaboration with S.Banerjee, J.Manschot, D.Persson, B.Pioline, F.Saueressig, S.Vandoren. N =2 gauge theories. Integrability. TBA Toda hierarchy and c=1 string - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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…
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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
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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
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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
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● 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!