Download Gauge theories in two dimensions and quantum integrable systems.

Two Dimensional Gauge Theories
and
Quantum Integrable Systems
Nikita Nekrasov
IHES
Imperial College
April 10, 2008
Based on
NN, S.Shatashvili, to appear
Prior work:
E.Witten, 1992;
A.Gorsky, NN; J.Minahan, A.Polychronakos;
M.Douglas; ~1993-1994; A.Gerasimov ~1993;
G.Moore, NN, S.Shatashvili ~1997-1998;
A.Losev, NN, S.Shatashvili ~1997-1998;
A.Gerasimov, S.Shatashvili ~ 2006-2007
We are going to relate
2,3, and 4 dimensional
susy gauge theories
with four supersymmetries
N=1 d=4
And quantum integrable systems
soluble by Bethe Ansatz techniques.
Mathematically speaking, the
cohomology, K-theory and elliptic
cohomology of various gauge theory
moduli spaces, like moduli of flat
connections and instantons
And quantum integrable systems
soluble by Bethe Ansatz techniques.
For example, we shall relate the
XXX Heisenberg magnet
and
2d N=2 SYM theory
with some matter
(pre-)History
In 1992 E.Witten studied two
dimensional Yang-Mills theory with
the goal to understand the relation
between the physical and
topological gravities in 2d.
(pre-)History
There are two interesting kinds
of
Two dimensional Yang-Mills
theories
Yang-Mills theories in 2d
(1)
Cohomological YM
= twisted N=2 super-Yang-Mills theory,
with gauge group G,
whose BPS (or TFT) sector is related to
the intersection theory on
the moduli space MG of
flat G-connections on
a Riemann surface
Yang-Mills theories in 2d
N=2 super-Yang-Mills theory
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Field content:
Yang-Mills theories in 2d
(2)
Physical YM =
N=0 Yang-Mills theory, with gauge group G;
The moduli space MG of flat G-connections
= minima of the action;
The theory is exactly soluble (A.Migdal) with the
help of the Polyakov lattice YM action
Yang-Mills theories in 2d
Physical YM
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Field content:
Yang-Mills theories in 2d
Witten found a way to map the BPS sector
of the N=2 theory to the N=0 theory.
The result is:
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills theories in 2d
Two dimensional Yang-Mills partition function is
given by the explicit sum
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills theories in 2d
In the limit
Quic kTime™ and a
TIFF (Unc ompres sed) decompress or
are needed to see this picture.
the partition function computes the
volume of MG
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills theories in 2d
Witten’s approach: add twisted
superpotential and its conjugate
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills theories in 2d
Take a limit
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
In the limit the fields
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
are infinitely massive and can be
integrated out:
one is left with the field content of
the physical YM theory
QuickTi me™ and a
TIFF ( Uncompressed) decompr essor
are needed to see thi s picture.
Yang-Mills theories in 2d
Both physical and cohomological Yang-Mills
theories define topological field theories (TFT)
Yang-Mills theories in 2d
Both physical and cohomological Yang-Mills
theories define topological field theories (TFT)
Vacuum states + deformations = quantum mechanics
YM in 2d and particles on a circle
Physical YM is explicitly equivalent to
a quantum mechanical model: free fermions on a
circle
Can be checked by a partition function on a two-torus
Gross
Douglas
YM in 2d and particles on a circle
Physical YM is explicitly equivalent to
a quantum mechanical model: free fermions on a
circle
States are labelled by the partitions, for G=U(N)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
YM in 2d and particles on a circle
For N=2 YM these free fermions on a circle
Label the vacua of the theory deformed by twisted
superpotential W
YM in 2d and particles on a circle
The fermions can be made interacting by adding a
localized matter: for example a time-like Wilson loop
in some representation V of the gauge group:
YM in 2d and particles on a circle
One gets Calogero-Sutherland (spin) particles on a circle
(1993-94) A.Gorsky,NN; J.Minahan,A.Polychronakos;
History
In 1997 G.Moore, NN and S.Shatashvili
studied integrals over
various hyperkahler quotients,
with the aim to understand
instanton integrals in
four dimensional gauge theories
History
In 1997 G.Moore, NN and S.Shatashvili studied
integrals over
Inspired by the work of H.Nakajima
various hyperkahler quotients,
with the aim to understand
instanton integrals in
four dimensional gauge theories
This eventually led to the derivation in 2002 of
the Seiberg-Witten solution of N=2 d=4 theory
Yang-Mills-Higgs theory
Among various examples, MNS
studied Hitchin’s moduli space MH
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills-Higgs theory
Unlike the case of two-dimensional
Yang-Mills theory where the moduli
space MG is compact,
Hitchin’s moduli space is noncompact
(it is roughly T*MG modulo subtleties)
and the volume is infinite.
Yang-Mills-Higgs theory
In order to cure this infnity in a reasonable
way MNS used the U(1) symmetry of MH
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
The volume becomes a DH-type expression:
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Where H is the Hamiltonian
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills-Higgs theory
Using the supersymmetry and localization
the regularized volume of MH
was computed with the result
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Yang-Mills-Higgs theory
Where the eigenvalues solve the equations:
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
YMH and NLS
The experts would immediately recognise the
Bethe ansatz (BA) equations for
the non-linear Schroedinger theory (NLS)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
NLS = large spin limit of the SU(2) XXX spin chain
YMH and NLS
Moreover the NLS Hamiltonians
are the 0-observables of the theory, like
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
The VEV of the observable
=
The eigenvalue of the Hamiltonian
YMH and NLS
Since 1997 nothing came out of
this result.
It could have been simply a
coincidence.
…….
History
In 2006
A.Gerasimov and S.Shatashvili
have revived the subject
YMH and interacting particles
GS noticed that YMH theory
viewed as TFT is equivalent to
the quantum Yang system:
N particles on a circle with
delta-interaction:
YMH and interacting particles
Thus: YM with the matter -fermions with pair-wise
interaction
History
More importantly,
GS suggested that TFT/QIS
equivalence is much more
universal
Today
We shall rederive the result of MNS from a
modern perspective
Generalize to cover virtually all BA soluble
systems both with finite and infinite spin
Suggest natural extensions of the BA equations
Hitchin equations
Solutions can be viewed as the susy field
configurations for
the N=2 gauged linear sigma model
For adjoint-valued linear fields
Hitchin equations
The moduli space MH of solutions is a
hyperkahler manifold
The integrals over MH are computed
by the correlation functions of
an N=2 d=2 susy gauge theory
Hitchin equations
The kahler form on MH comes from
twisted tree level superpotential
The epsilon-term
comes from
a twisted mass of the matter multiplet
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Generalization
Take an N=2 d=2 gauge theory with matter,
In some representation R
of the gauge group G
Generalization
Integrate out the matter fields,
compute the effective (twisted)
super-potential
on the Coulomb branch
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Mathematically speaking
Consider the moduli space MR of R-Higgs pairs
with gauge group G
Up to the action of the complexified gauge group GC
Mathematically speaking
Stability conditions:
Up to the action of the compact gauge group G
Mathematically speaking
Pushforward the unit class down to
the moduli space MG of GC-bundles
Equivariantly with respect to the action
of the global symmetry group K on MR
Mathematically speaking
The pushforward can be expressed in terms
of the Donaldson-like classes of
the moduli space MG
2-observables and 0-observables
Mathematically speaking
The pushforward can be expressed in terms
of the Donaldson-like classes of
the moduli space MG
2-observables and 0-observables
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Mathematically speaking
The masses are the equivariant parameters
For the global symmetry group K
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Vacua of the gauge theory
For G = U(N)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Due to quantization of the gauge flux
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Vacua of the gauge theory
For G = U(N)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Equations familiar from yesterday’s lecture
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
partitions
Vacua of the gauge theory
Familiar example: CPN model
Field content:
(N+1) chiral multiplet of charge
+1
Qi i=1, … , N+1
U(1) gauge group
Effective superpotential:
N+1
vacuum
Vacua of gauge theory
Another example:
Field
Gauge group:
content:
G=U(N)
Matter chiral multiplets:
1 adjoint,
mass
fundamentals,
mass
anti-fundamentals, mass
Vacua of gauge theory
Effective superpotential:
Vacua of gauge theory
Equations for vacua:
Vacua of gauge theory
Nonanomalo
us case:
Redefine:
Vacua of gauge theory
Vacua:
Gauge theory -- spin chain
Identical
to the
Bethe
ansatz
equation
s for
Gauge theory -- spin chain
Vacua =
eigensta
tes of
the
Hamilton
ian:
Table of dualities
XXX spin chain
SU(2)
L spins
N excitations
U(N) d=2 N=2
Chiral multiplets:
1 adjoint
L fundamentals
L anti-fund.
Special masses!
Table of dualities:
mathematically speaking
XXX spin chain
SU(2)
L spins
N excitations
(Equivariant)
Intersection theory
on MR for
Table of dualities
XXZ spin chain
SU(2)
L spins
N excitations
U(N) d=3 N=1
Compactified on a circle
Chiral multiplets:
1 adjoint
L fundamentals
L anti-fund.
Table of dualities:
mathematically speaking
XXZ spin chain
SU(2)
L spins
N excitations
Equivariant K-theory
of the moduli
space MR
Table of dualities
XYZ spin chain
SU(2), L = 2N spins
N excitations
U(N) d=4 N=1
Compactified on a 2-torus
= elliptic curve E
Chiral multiplets:
1 adjoint
L = 2N fundamentals
L = 2N anti-fund.
Masses = wilson loops
of the flavour group
= points on the Jacobian of E
Table of dualities:
mathematically speaking
XYZ spin chain
SU(2), L = 2N spins
N excitations
Elliptic genus of
the moduli
space MR
Masses = K bundle over E
= points on the BunK of E
Table of dualities
It is remarkable that the spin chain has
precisely those generalizations:
rational (XXX), trigonometric (XXZ) and elliptic (XYZ)
that can be matched to the 2, 3, and 4 dim cases.
Algebraic Bethe Ansatz
Faddeev et al.
The spin chain is solved algebraically
using certain operators,
Which obey exchange commutation
relations
Faddeev-Zamolodchikov algebra…
Algebraic Bethe Ansatz
The eigenvectors, Bethe vectors, are
obtained by applying these
operators to the « fake » vacuum.
ABA vs GAUGE THEORY
For the spin chain it is natural to fix L = total
number of spins
and consider various N = excitation levels
In the gauge theory context N is fixed.
ABA vs GAUGE THEORY
However, if the theory is embedded
into string theory via brane
realization
then changing N is easy:
bring in an extra brane.
Hanany-Hori’02
ABA vs GAUGE THEORY
Mathematically speaking
We claim that the Algebraic Bethe Ansatz is
most naturally related to the derived
category of the category of coherent
sheaves on some local CY
ABA vs STRING THEORY
THUS:
B
is for BRANE!
is for location!
More general spin chains
The SU(2) spin chain
has generalizations to
other groups and representations.
I quote the corresponding
Bethe ansatz equations
from N.Reshetikhin
General groups/reps
For simply-laced group H of rank r
General groups/reps
For simply-laced group H of rank r
Label representations of the Yangian of H
A.N.Kirillov-N.Reshetikhin modules
Cartan matrix of H
General groups/reps
from GAUGE THEORY
Take the Dynkin diagram corresponding to H
A simply-laced group of rank r
QUIVER GAUGE THEORY
Symmetries
QUIVER GAUGE THEORY
Symmetries
QUIVER GAUGE THEORY
Charged matter
Adjoint chiral multiplet
Fundamental chiral multiplet
Anti-fundamental chiral multiplet
Bi-fundamental chiral multiplet
QUIVER GAUGE THEORY
Matter fields: adjoints
QUIVER GAUGE THEORY
Matter fields:
fundamentals+anti-fundamentals
QUIVER GAUGE THEORY
Matter fields: bi-fundamentals
QUIVER GAUGE THEORY
Quiver gauge theory: full content
QUIVER GAUGE THEORY:
MASSES
Adjoints
i
QUIVER GAUGE THEORY:
MASSES
Fundamentals
Anti-fundamentals
i
a = 1, …. , Li
QUIVER GAUGE THEORY:
MASSES
Bi-fundamentals
i
j
QUIVER GAUGE THEORY
What is so special about these
masses?
QUIVER GAUGE THEORY
From the gauge theory point of view
nothing special…..
QUIVER GAUGE THEORY
The mass puzzle!
The mass puzzle
The Bethe ansatz -- like equations
Can be written for an arbitrary matrix
The mass puzzle
However the Yangian symmetry Y(H)
would get replaced by some ugly
infinite-dimensional « free » algreba
without nice representations
The mass puzzle
Therefore we conclude that our
choice of masses is dictated by
the hidden symmetry -- that of
the dual spin chain
The Standard Model has
many free parameters
Among them are the fermion masses
Is there a (hidden) symmetry principle
behind them?
The Standard Model has
many free parameters
In the supersymmetric models
we considered
the mass tuning
can be « explained »
using a duality to some
quantum integrable system
Further generalizations:
Superpotential
from prepotential
Tree level part
Induced
by twist
The N=2* theory on R2 X S2
Flux
superpotential
(Losev,NN, Shatashvili’97)
Superpotential
from prepotential
Magnetic flux
Electric flux
In the limit of vanishing S2 the magnetic flux should vanish
Instanton corrected BA
equations
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Effective S-matrix contains 2-body, 3-body, …
interactions
Instanton corrected BA
equations
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Instanton corrected BA
equations
The prepotential of the low-energy effective theory
Is governed by a classical (holomorphic) integrable system
Donagi-Witten’95
Liouville tori = Jacobians of Seiberg-Witten curves
Classical integrable system
vs
Quantum integrable system
That system is quantized when the gauge theory is subject to
the Omega-background
NN’02
NN,Okounkov’03
Braverman’03
Our quantum system is different!
Blowing up the two-sphere
Wall-crossing phenomena
(new states, new solutions)
Something for the future
Naturalness of our quivers
Somewhat unusual matter content
Branes at orbifolds typically lead to
smth like
Naturalness of our quivers
This picture would arise in the
sa(i)  0
BA for QCD
Faddeev-Korchemsky’94
limit
Naturalness of our quivers
Other quivers?
Naturalness of our quivers
Possibly with the help of K.Saito’s construction
CONCLUSIONS
1.
2.
We found the Bethe Ansatz equations
are the equations describing the
vacuum configurations of certain quiver
gauge theories in two dimensions
The duality to the spin chain requires
certain relations between the masses of
the matter fields to be obeyed. This
could have phenomenological
consequences.
CONCLUSIONS
3. The algebraic Bethe ansatz seems
to provide a realization of the brane
creation operators -- something of
major importance both for
topological and physical string
theories
4. Obviously this is a beginning of a
beautiful story….