INSTANTON PARTITION FUNCTIONS Nikita Nekrasov IHES (Bures-sur-Yvette) & ITEP (Moscow)QUARKS-2008 May 25, 2008 Nikita Nekrasov IHES (Bures-sur-Yvette) &

INSTANTON PARTITION FUNCTIONS

INSTANTON PARTITION FUNCTIONS

Nikita Nekrasov

IHES (Bures-sur-Yvette) & ITEP (Moscow)

QUARKS-2008QUARKS-2008May 25, 2008May 25, 2008

Nikita Nekrasov

IHES (Bures-sur-Yvette) & ITEP (Moscow)

QUARKS-2008QUARKS-2008May 25, 2008May 25, 2008

Biased list of refsBiased list of refs

NN, NN, A.Aleksandrov~2008; NN, NN, A.Aleksandrov~2008; NN, A.Marshakov~2006;NN, A.Marshakov~2006;A.Iqbal, NN, A.Okounkov, C.Vafa~2004;A.Iqbal, NN, A.Okounkov, C.Vafa~2004;A.Braverman ~2004;A.Braverman ~2004;NN, A.Okounkov ~2003;NN, A.Okounkov ~2003;H.Nakajima, K.Yoshioka ~2003;H.Nakajima, K.Yoshioka ~2003;A.Losev, NN, A.Marshakov ~2002;A.Losev, NN, A.Marshakov ~2002;NN, 2002;NN, 2002;A.Schwarz, NN, 1998;A.Schwarz, NN, 1998;G.Moore, NN, S.Shatashvili ~1997-1998; G.Moore, NN, S.Shatashvili ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998;A.Losev, NN, S.Shatashvili ~1997-1998;A.Gerasimov, S.Shatashvili ~ 2006-2007A.Gerasimov, S.Shatashvili ~ 2006-2007

NN, NN, A.Aleksandrov~2008; NN, NN, A.Aleksandrov~2008; NN, A.Marshakov~2006;NN, A.Marshakov~2006;A.Iqbal, NN, A.Okounkov, C.Vafa~2004;A.Iqbal, NN, A.Okounkov, C.Vafa~2004;A.Braverman ~2004;A.Braverman ~2004;NN, A.Okounkov ~2003;NN, A.Okounkov ~2003;H.Nakajima, K.Yoshioka ~2003;H.Nakajima, K.Yoshioka ~2003;A.Losev, NN, A.Marshakov ~2002;A.Losev, NN, A.Marshakov ~2002;NN, 2002;NN, 2002;A.Schwarz, NN, 1998;A.Schwarz, NN, 1998;G.Moore, NN, S.Shatashvili ~1997-1998; G.Moore, NN, S.Shatashvili ~1997-1998; A.Losev, NN, S.Shatashvili ~1997-1998;A.Losev, NN, S.Shatashvili ~1997-1998;A.Gerasimov, S.Shatashvili ~ 2006-2007A.Gerasimov, S.Shatashvili ~ 2006-2007

Mathematical problem:counting


Integers: 1,2,3,….Integers: 1,2,3,….



Integers: 1,2,3,….Integers: 1,2,3,….



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

…


…




…


…

Mathematical problem:generating functions






Euler

Unexpected symmetryUnexpected symmetry

Dedekind eta

More structure:Arms, legs, and hooks

More structure:Arms, legs, and hooks

Growth processGrowth process

Plancherel measurePlancherel measure



Plane partitions of integers:((1));

((2)),((1,1)),((1),1);((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));….


((2)),((1,1)),((1),1);((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));….




((2)),((1,1)),((1),1);((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));

….


((2)),((1,1)),((1),1);((3)),((2,1)),((1,1,1)),((2),(1)),((1),(1),(1));

….



MacMahon

Mathematical problem:more structural countingMathematical problem:

more structural counting

Quantum gauge theory Quantum gauge theory Four dimensions

Quantum gauge theory Quantum gauge theory Four dimensions

Quantum sigma model Quantum sigma model Two dimensions

Quantum sigma model Quantum sigma model Two dimensions

InstantonsInstantons

Minimize Euclidean action in a given topology of the field configurations

Minimize Euclidean action in a given topology of the field configurations

Gauge instantons

(Almost) Kahler target sigma model

instantons

Counting InstantonsCounting Instantons

Approximation for ordinary theories. Sometimes exact results for

supersymmetric theories.

Counting InstantonsCounting Instantons

Approximation for ordinary theories. Sometimes exact results for

supersymmetric theories.

Instanton partition functions in four

dimensions


dimensionsSupersymmetric N=4 theory (Vafa-

Witten)Supersymmetric N=4 theory (Vafa-

Witten)


dimensions




Witten)

Transforms nicely under a (subgroup of) SL(2, Z)


dimensions




Witten)

Transforms nicely under a (subgroup of) SL(2, Z)

Hidden elliptic curve:


dimensions


dimensionsSupersymmetric N=2 theory (Donaldson-

Witten)

Supersymmetric N=2 theory (Donaldson-

Witten)

Intersection theory on the moduli space of gauge instantons


dimensions


dimensionsSupersymmetric N=2 theory (Donaldson-

Witten)

Supersymmetric N=2 theory (Donaldson-

Witten)Donaldson invariants of four-manifolds

Seiberg-Witten invariants of four-manifolds


dimensions


dimensionsSupersymmetric N=2 theory

On Euclidean space R4

Supersymmetric N=2 theory On Euclidean space R4


dimensions


dimensionsSupersymmetric N=2 theory

On Euclidean space R4

Boundary conditions at infinity SO(4) Equivariant theory

Supersymmetric N=2 theory On Euclidean space R4

Boundary conditions at infinity SO(4) Equivariant theory

Instanton partition function


Supersymmetric N=2 theory on Euclidean space R4



Supersymmetric pure N=2 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



Supersymmetric N=2 super YM theory with matter







Bundle of DiracZero modes

In the instantonbackground



Explicit evaluation using localization For pure super Yang-Mills theory:



Compactification of the instanton moduli space

to

Add point-like instantons + extra stuff





For G = U(N)



Perturbative part (contribution of a trivial connection)

For G = U(N)



Instanton part For G = U(N)

Sum over N-tuples of partitions



Generalized growth modelGeneralized growth model

















Instanton partition functionLimit shapeLimit shape

Emerging geometry



Emerging algebraic geometry



Emerging algebraic geometry NN+A.Okounkov



Seiberg-Witten geometry NN+A.Okounkov



Seiberg-Witten geometry

Integrability: Toda chain, Calogero-Moser particles, spin chains Hitchin system



The full instanton sum has ahidden

infinite dimensional symmetry algebra

The full instanton sum has ahidden

infinite dimensional symmetry algebra



Special rotation parametersSU(2) reduction

Special rotation parametersSU(2) reduction



Fourier transform (electric-magnetic duality)


QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.













Free fermion representation Free fermion representation



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



Free fermion representation Free fermion representation







Theory with matter in adjoint representaton

Theory with matter in adjoint representaton



That elliptic curve again



Abelian theory with matter in adjoint representaton:

back to hooks

Abelian theory with matter in adjoint representaton:

back to hooks



Amazingly this partition functionis also almost modular

Amazingly this partition functionis also almost modular



Full-fledged partition function:Generic rotations and fifth

dimensionK-theoretic version

Full-fledged partition function:Generic rotations and fifth

dimensionK-theoretic version



Free field representation:Infinite product formula

Free field representation:Infinite product formula



Free fields and modularity:Infinite product of theta functions

Free fields and modularity:Infinite product of theta functions



Free field representation:Second quantization representation






Bosons (+) and fermions (-)



Free fields? Where? What kind?Free fields? Where? What kind?

M-theory to the rescueM-theory to the rescue

The kind of instanton counting we encountered

occurs naturally in the theory ofD4 branes in IIA string theory

to which D0 branes (codimension 4 defects, just like

instantons)can bind

The kind of instanton counting we encountered

occurs naturally in the theory ofD4 branes in IIA string theory

to which D0 branes (codimension 4 defects, just like

instantons)can bind



D4 branes

D0’s

SU(4) rotation


D4 brane + D0’s become

Lift to M-theory

M5 brane wrapped on R4 X elliptic curve

Free fields = the tensor multiplet of (2,0) supersymmetry

The modularity of the partition function is theconsequence of the general covariance of the

six dimensional theory

NN+E.Witten

M-theory to the rescueM-theory to the rescueIn the limit

To visualize this boson deform R4 to Taub-Nut spaceThe tensor field gets a normalizable localized mode

The partition function becomes that of a free chiral bosonon elliptic curve

Higher dimensional perspective on the gauge

instanton counting


instanton counting

Complicated hook measure on

Partitions comes from simple

Uniform measure on plane (3d) partitions


instanton counting


instanton counting

Complicated hook measure on

Partitions comes from simple

Uniform measure on plane (3d) partitions

What is the physics of this relation?

Gauge theory = low energy limit of string theory compactification




X

Instanton partition function =

String instanton partition function







Instanton partition function for

gauge group G=

String instanton partition function for special X


gauge group G=


Local CY’sGeometric enigneeringKatz, Klemm, Vafa


gauge group G=



gauge group G=


Kontsevich’s moduli space of stable maps

String instanton partition function for CY X =

counting holomorphic curves on X





Gromov-Witten theory



Gromov-Witten theory

Counting holomorphic curves on X (GW theory)

= Counting equations

describing holomorphic curves (ideal sheaves)

Counting holomorphic curves on X (GW theory)

= Counting equations

describing holomorphic curves (ideal sheaves)

Counting equations describing holomorphic curves (ideal sheaves)

Donaldson-Thomas theory

Counting equations describing holomorphic curves (ideal sheaves)

Donaldson-Thomas theory

For special X, e.g. toric,Donaldson-Thomas theory

can be done using localization

=sum over fixed points

=toric ideal sheaves

For special X, e.g. toric,Donaldson-Thomas theory

can be done using localization

=sum over fixed points

=toric ideal sheaves

Simplest toric X = C3

toric ideal sheaves =

monomial ideals

Simplest toric X = C3

toric ideal sheaves =

monomial ideals

Monomial ideals =three dimensional

partitions


partitions


partitions


partitions

Topological vertexTopological vertex

Equivariant vertex(beyond CY)

Equivariant vertex(beyond CY)

K-theoreticEquivariant vertex

(beyond string theory & CY)

K-theoreticEquivariant vertex

(beyond string theory & CY)

The case of C3The case of C3

Contribution of a three dimensional partition





















The partition functionThe partition function



Counts bound states of D0’s and a D6 brane



The partition functionhas a free field realization

The partition functionhas a free field realization







The partition functionSpecial limits




If, in addition:



If, in addition:

Our good old MacMahon friend

The partition functionSecond quantization

The partition functionSecond quantization









Explanation via M-theoryExplanation via M-theoryType IIA realization

Explanation via M-theoryExplanation via M-theoryLift to M-theory

Explanation via M-theoryExplanation via M-theoryDeform TN to R4

R10 rotated over the circle: SU(5) rotation

Explanation via M-theoryExplanation via M-theoryFree fields =

linearized supergravity multiplet NN+E.Witten

Instanton partition functions

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

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

Documents

INSTANTON PARTITION FUNCTIONS Nikita Nekrasov IHES (Bures-sur-Yvette) & ITEP (Moscow)QUARKS-2008 May 25, 2008 Nikita Nekrasov IHES (Bures-sur-Yvette) &