INSTANTON%20PARTITION%20FUNCTIONS

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INSTANTON PARTITION FUNCTIONS

• Nikita Nekrasov
• IHES (Bures-sur-Yvette) ITEP (Moscow)
• QUARKS-2008
• May 25, 2008

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Biased list of refs

• NN, NN, A.Aleksandrov2008
• NN, A.Marshakov2006
• A.Iqbal, NN, A.Okounkov, C.Vafa2004
• A.Braverman 2004
• NN, A.Okounkov 2003
• H.Nakajima, K.Yoshioka 2003
• A.Losev, NN, A.Marshakov 2002
• NN, 2002
• A.Schwarz, NN, 1998
• G.Moore, NN, S.Shatashvili 1997-1998
• A.Losev, NN, S.Shatashvili 1997-1998
• A.Gerasimov, S.Shatashvili 2006-2007

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Mathematical problemcounting

• Integers 1,2,3,.

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Mathematical problemcounting

• Integers 1,2,3,.

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Mathematical problemcounting

• Partitions of integers
• (1) (2) (1,1) (3) (2,1)
  (1,1,1)

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Mathematical problemcounting

• Partitions of integers
• (1) (2) (1,1) (3) (2,1)
  (1,1,1)

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Mathematical problemgenerating functions
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Mathematical problemgenerating functions
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Mathematical problemgenerating functions
Euler
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Unexpected symmetry
Dedekind eta
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More structureArms, legs, and hooks
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Growth process
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Plancherel measure
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Mathematical problemcounting

• Plane partitions of integers
• ((1))
• ((2)),((1,1)),((1),1)
• ((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1)).

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Mathematical problemcounting

• Plane partitions of integers
• ((1))
• ((2)),((1,1)),((1),1)
• ((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1)).

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Mathematical problemgenerating functions
MacMahon
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Mathematical problemmore structural counting
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Quantum gauge theory
Four dimensions
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Quantum gauge theory
Four dimensions
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Quantum sigma model
Two dimensions
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Quantum sigma model
Two dimensions
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Instantons

• Minimize Euclidean action in a given topology of
  the field configurations

Gauge instantons
(Almost) Kahler target sigma model instantons
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Counting Instantons
Approximation for ordinary theories. Sometimes
exact results for supersymmetric theories.
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Counting Instantons
Approximation for ordinary theories. Sometimes
exact results for supersymmetric theories.
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Instanton partition functions in four dimensions

• Supersymmetric N4 theory (Vafa-Witten)

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Instanton partition functions in four dimensions

• Supersymmetric N4 theory (Vafa-Witten)

Transforms nicely under a (subgroup of) SL(2, Z)
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Instanton partition functions in four dimensions

• Supersymmetric N4 theory (Vafa-Witten)

Transforms nicely under a (subgroup of) SL(2, Z)
Hidden elliptic curve
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Instanton partition functions in four dimensions

• Supersymmetric N2 theory
• 
  (Donaldson-Witten)

Intersection theory on the moduli space of gauge
instantons
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Instanton partition functions in four dimensions

• Supersymmetric N2 theory
• 
  (Donaldson-Witten)

Donaldson invariants of four-manifolds
Seiberg-Witten invariants of four-manifolds
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Instanton partition functions in four dimensions

• Supersymmetric N2 theory
• On Euclidean space R4

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Instanton partition functions in four dimensions

• Supersymmetric N2 theory
• On Euclidean space R4
• Boundary conditions at infinity
• SO(4) Equivariant theory

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Instanton partition function
Supersymmetric N2 theory on Euclidean space R4
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Instanton partition function
Supersymmetric pure N2 super YM theory on
Euclidean space R4
Degree Element of the ring of fractions of
H(BH) H G X SO(4), G - the gauge group
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Instanton partition function
Supersymmetric N2 super YM theory with matter
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Instanton partition function
Supersymmetric N2 super YM theory with matter
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Instanton partition function
Supersymmetric N2 super YM theory with matter
Bundle of Dirac Zero modes In the
instanton background
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Instanton partition function
Explicit evaluation using localization
For pure super Yang-Mills theory
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Instanton partition function
Compactification of the instanton moduli space
to
Add point-like instantons extra stuff
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Instanton partition function
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Instanton partition function
For G U(N)
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Instanton partition function
Perturbative part (contribution of a trivial
connection) For G U(N)
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Instanton partition function
Instanton part For G U(N)
Sum over N-tuples of partitions
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Instanton partition function

• Generalized growth model

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Instanton partition function

• Generalized growth model

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Instanton partition function

• Generalized growth model

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Instanton partition function

• Generalized growth model

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Instanton partition function

• Generalized growth model

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Instanton partition function

• Generalized growth model

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Instanton partition function

• Limit shape

Emerging geometry
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Instanton partition function

• Limit shape

Emerging algebraic geometry
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Instanton partition function

• Limit shape

NNA.Okounkov
Emerging algebraic geometry
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Instanton partition function

• Limit shape

NNA.Okounkov
Seiberg-Witten geometry
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Instanton partition function

• Limit shape

Seiberg-Witten geometry
Integrability Toda chain, Calogero-Moser
particles, spin chains
Hitchin system
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Instanton partition function

• The full instanton sum has a
• hidden
• infinite dimensional symmetry algebra

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Instanton partition function

• Special rotation parameters
• SU(2) reduction

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Instanton partition function

• Fourier transform
• (electric-magnetic duality)

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Instanton partition function

• Fourier transform
• (electric-magnetic duality)

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Instanton partition function

• Free fermion representation

J(z) form level 1 affine su(N) current algebra
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Instanton partition function

• Free fermion representation

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Instanton partition function

• Theory with matter in
• adjoint representaton

That elliptic curve again
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Instanton partition function

• Abelian theory with matter in
• adjoint representaton
• back to hooks

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Instanton partition function

• Amazingly this partition function
• is also almost modular

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Instanton partition function

• Full-fledged partition function
• Generic rotations and fifth dimension
• K-theoretic version

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Instanton partition function

• Free field representation
• Infinite product formula

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Instanton partition function

• Free fields and modularity
• Infinite product of theta functions

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Instanton partition function

• Free field representation
• Second quantization representation

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Instanton partition function

• Free field representation
• Second quantization representation

Bosons () and fermions (-)
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Instanton partition function

• Free fields? Where? What kind?

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M-theory to the rescue

• The kind of instanton counting
• we encountered
• occurs naturally in the theory of
• D4 branes in IIA string theory
• to which D0 branes
• (codimension 4 defects, just like instantons)
• can bind

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M-theory to the rescue
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M-theory to the rescue
SU(4) rotation
D4 branes
D0s
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M-theory to the rescue
Lift to M-theory
M5 brane wrapped on R4 X elliptic curve
D4 brane
D0s become
NNE.Witten
Free fields the tensor multiplet of (2,0)
supersymmetry
The modularity of the partition function is
the consequence of the general covariance of
the six dimensional theory
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M-theory to the rescue
In the limit
The partition function becomes that of a free
chiral boson on elliptic curve
To visualize this boson deform R4 to Taub-Nut
space The tensor field gets a normalizable
localized mode
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Higher dimensional perspective on the gauge
instanton counting
Complicated hook measure on Partitions comes from
simple Uniform measure on plane (3d) partitions
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Higher dimensional perspective on the gauge
instanton counting
Complicated hook measure on Partitions comes from
simple Uniform measure on plane (3d) partitions
What is the physics of this relation?
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Gauge theory low energy limit of string theory
compactification
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Gauge theory low energy limit of string theory
compactification
X
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Instanton partition function String instanton
partition function
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Instanton partition function String instanton
partition function
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Instanton partition function forgauge group G
String instanton partition function for special X
Local CYs Geometric enigneering Katz, Klemm, Vafa
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Instanton partition function forgauge group G
String instanton partition function for special X
Kontsevichs moduli space of stable maps
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String instanton partition function for CY X
counting holomorphic curves on X
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String instanton partition function for CY X
counting holomorphic curves on X Gromov-Witten
theory
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Counting holomorphic curves on X (GW theory)
Counting equations describing holomorphic curves
(ideal sheaves)
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Counting equations describing holomorphic curves
(ideal sheaves)Donaldson-Thomas theory
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For special X, e.g. toric,Donaldson-Thomas
theorycan be done using localizationsum over
fixed points toric ideal sheaves
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Simplest toric X C3toric ideal sheaves
monomial ideals
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Monomial ideals three dimensional partitions
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Monomial ideals three dimensional partitions
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Topological vertex
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Equivariant vertex(beyond CY)
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K-theoreticEquivariant vertex(beyond string
theory CY)
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The case of C3

• Contribution of a
• three dimensional partition

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The case of C3

• Contribution of a
• three dimensional partition

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The case of C3

• Contribution of a
• three dimensional partition

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The case of C3

• The partition function

Counts bound states of D0s and a D6 brane
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The partition functionhas a free field
realization
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The partition functionSpecial limits
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The partition functionSpecial limits
If, in addition
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The partition functionSpecial limits
If, in addition
Our good old MacMahon friend
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The partition functionSecond quantization
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Explanation via M-theory
Type IIA realization
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Explanation via M-theory
Lift to M-theory
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Explanation via M-theory
Deform TN to R4
R10 rotated over the circle SU(5) rotation
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Explanation via M-theory
Free fields linearized supergravity multiplet
NNE.Witten
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Instanton partition functions

• Generalize most known special functions
  (automorphic forms)
• Obey interesting differential and difference
  equations
• Relate combinatorics, algebra, representation
  theory and geometry string theory and gauge
  theory
• Might teach us about the nature
• 
  of M-theory