Title: Two%20Dimensional%20Gauge%20Theories%20and%20Quantum%20Integrable%20Systems
1Two Dimensional Gauge Theoriesand Quantum
Integrable Systems
- Nikita Nekrasov
- IHES
- Imperial College
- April 10, 2008
2Based 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
3We are going to relate 2,3, and 4 dimensional
susy gauge theorieswith four supersymmetries
N1 d4
- And quantum integrable systems
- soluble by Bethe Ansatz techniques.
4Mathematically 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.
5- For example, we shall relate the
- XXX Heisenberg magnet
- and
- 2d N2 SYM theory
- with some matter
6(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.
7(pre-)History
- There are two interesting kinds of
- Two dimensional Yang-Mills theories
8Yang-Mills theories in 2d
- (1)
-
- Cohomological YM
- twisted N2 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
9Yang-Mills theories in 2d
- N2 super-Yang-Mills theory
Field content
10Yang-Mills theories in 2d
- (2)
- Physical YM
- N0 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
11Yang-Mills theories in 2d
Field content
12Yang-Mills theories in 2d
- Witten found a way to map the BPS sector of the
N2 theory to the N0 theory. - The result is
13Yang-Mills theories in 2d
- Two dimensional Yang-Mills partition function is
given by the explicit sum
14Yang-Mills theories in 2d
- In the limit
- the partition function computes the volume of MG
15Yang-Mills theories in 2d
- Wittens approach add twisted superpotential and
its conjugate
16Yang-Mills theories in 2d
In the limit the fields are infinitely massive
and can be integrated out one is left with the
field content of the physical YM theory
17Yang-Mills theories in 2d
- Both physical and cohomological Yang-Mills
- theories define topological field theories (TFT)
18Yang-Mills theories in 2d
- Both physical and cohomological Yang-Mills
- theories define topological field theories (TFT)
Vacuum states deformations quantum mechanics
19YM 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
20YM 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 GU(N)
21YM in 2d and particles on a circle
For N2 YM these free fermions on a circle
Label the vacua of the theory deformed by twisted
superpotential W
22YM 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
23YM 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
24History
- 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
25History
- 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
- This eventually led to the derivation in 2002 of
- the Seiberg-Witten solution of N2 d4 theory
Inspired by the work of H.Nakajima
26Yang-Mills-Higgs theory
- Among various examples, MNS studied Hitchins
moduli space MH
27Yang-Mills-Higgs theory
- Unlike the case of two-dimensional
- Yang-Mills theory where the moduli space MG is
compact, - Hitchins moduli space is non-compact
- (it is roughly TMG modulo subtleties) and the
volume is infinite.
28Yang-Mills-Higgs theory
- In order to cure this infnity in a reasonable way
MNS used the U(1) symmetry of MH
The volume becomes a DH-type expression
Where H is the Hamiltonian
29Yang-Mills-Higgs theory
- Using the supersymmetry and localization
- the regularized volume of MH
- was computed with the result
30Yang-Mills-Higgs theory
- Where the eigenvalues solve the equations
31YMH and NLS
- The experts would immediately recognise the
- Bethe ansatz (BA) equations for
- the non-linear Schroedinger theory (NLS)
NLS large spin limit of the SU(2) XXX spin chain
32YMH and NLS
- Moreover the NLS Hamiltonians
- are the 0-observables of the theory, like
The VEV of the observable The eigenvalue of
the Hamiltonian
33YMH and NLS
- Since 1997 nothing came out of
- this result.
- It could have been simply a coincidence.
- .
34In 2006 A.Gerasimov and S.Shatashvili have
revived the subject
35YMH 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
36YMH and interacting particles
- Thus YM with the matter -- fermions with
pair-wise interaction
37History
- More importantly,
- GS suggested that TFT/QIS equivalence is much
more universal
38Today
- 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
39Hitchin equations
- Solutions can be viewed as the susy field
configurations for - the N2 gauged linear sigma model
For adjoint-valued linear fields
40Hitchin equations
- The moduli space MH of solutions is a hyperkahler
manifold - The integrals over MH are computed by the
correlation functions of - an N2 d2 susy gauge theory
41Hitchin equations
- The kahler form on MH comes from
- twisted tree level superpotential
- The epsilon-term comes
from - a twisted mass of the matter multiplet
42Generalization
- Take an N2 d2 gauge theory with matter,
- In some representation R
- of the gauge group G
43Generalization
- Integrate out the matter fields,
- compute the effective (twisted)
- super-potential
- on the Coulomb branch
44Mathematically speaking
- Consider the moduli space MR of R-Higgs pairs
- with gauge group G
Up to the action of the complexified gauge group
GC
45Mathematically speaking
Up to the action of the compact gauge group G
46Mathematically 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
47Mathematically speaking
- The pushforward can be expressed in terms of the
Donaldson-like classes of - the moduli space MG
- 2-observables and 0-observables
48Mathematically speaking
- The pushforward can be expressed in terms of the
Donaldson-like classes of - the moduli space MG
- 2-observables and 0-observables
49Mathematically speaking
- The masses are the equivariant parameters
- For the global symmetry group K
50Vacua of the gauge theory
For G U(N)
- Due to quantization of the gauge flux
51Vacua of the gauge theory
For G U(N)
- Equations familiar from yesterdays lecture
partitions
52Vacua of the gauge theory
- Familiar example CPN model
(N1) chiral multiplet of charge 1 Qi i1, ,
N1 U(1) gauge group
Field content
Effective superpotential
N1 vacuum
53Vacua of gauge theory
Another example
- Gauge group
- GU(N)
- Matter chiral multiplets
- 1 adjoint, mass
- fundamentals, mass
- anti-fundamentals, mass
Field content
54Vacua of gauge theory
Effective superpotential
55Vacua of gauge theory
Equations for vacua
56Vacua of gauge theory
Non-anomalous case
Redefine
57Vacua of gauge theory
Vacua
58Gauge theory -- spin chain
Identical to the Bethe ansatz equations for spin
XXX magnet
59Gauge theory -- spin chain
Vacua eigenstates of the Hamiltonian
60Table of dualities
- XXX spin chain
- SU(2)
- L spins
- N excitations
U(N) d2 N2 Chiral multiplets 1 adjoint L
fundamentals L anti-fund.
Special masses!
61Table of dualities mathematically speaking
- XXX spin chain
- SU(2)
- L spins
- N excitations
(Equivariant) Intersection theory on MR for
62Table of dualities
- XXZ spin chain
- SU(2)
- L spins
- N excitations
U(N) d3 N1 Compactified on a circle Chiral
multiplets 1 adjoint L fundamentals L anti-fund.
63Table of dualities mathematically speaking
- XXZ spin chain
- SU(2)
- L spins
- N excitations
Equivariant K-theory of the moduli space MR
64Table of dualities
- XYZ spin chain
- SU(2), L 2N spins
- N excitations
U(N) d4 N1 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
65Table 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
66Table 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.
67Algebraic Bethe Ansatz
Faddeev et al.
- The spin chain is solved algebraically using
certain operators, - Which obey exchange commutation relations
Faddeev-Zamolodchikov algebra
68Algebraic Bethe Ansatz
- The eigenvectors, Bethe vectors, are obtained by
applying these operators to the fake vacuum.
69ABA 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.
70ABA 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-Hori02
71ABA 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
72ABA vs STRING THEORY
is for location!
73More 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
74General groups/reps
- For simply-laced group H of rank r
75General 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
76General groups/repsfrom GAUGE THEORY
- Take the Dynkin diagram corresponding to H
- A simply-laced group of rank r
77QUIVER GAUGE THEORY
78QUIVER GAUGE THEORY
79QUIVER GAUGE THEORYCharged matter
Adjoint chiral multiplet
Fundamental chiral multiplet
Anti-fundamental chiral multiplet
Bi-fundamental chiral multiplet
80QUIVER GAUGE THEORY
81QUIVER GAUGE THEORY
- Matter fields
- fundamentalsanti-fundamentals
82QUIVER GAUGE THEORY
- Matter fields bi-fundamentals
83QUIVER GAUGE THEORY
- Quiver gauge theory full content
84QUIVER GAUGE THEORY MASSES
i
85QUIVER GAUGE THEORY MASSES
- Fundamentals
- Anti-fundamentals
i
a 1, . , Li
86QUIVER GAUGE THEORY MASSES
j
i
87QUIVER GAUGE THEORY
- What is so special about these masses?
88QUIVER GAUGE THEORY
- From the gauge theory point of view nothing
special..
89QUIVER GAUGE THEORY
90The mass puzzle
- The Bethe ansatz -- like equations
Can be written for an arbitrary matrix
91The mass puzzle
- However the Yangian symmetry Y(H) would get
replaced by some ugly infinite-dimensional
free algreba without nice representations
92The mass puzzle
- Therefore we conclude that our choice of masses
is dictated by the hidden symmetry -- that of the
dual spin chain
93The Standard Model has many free parameters
- Among them are the fermion masses
- Is there a (hidden) symmetry principle behind
them?
94The 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
95Further generalizationsSuperpotential from
prepotential
Tree level part
Flux superpotential (Losev,NN, Shatashvili97)
Induced by twist
The N2 theory on R2 X S2
96Superpotential from prepotential
Magnetic flux
Electric flux
In the limit of vanishing S2 the magnetic flux
should vanish
97Instanton corrected BA equations
Effective S-matrix contains 2-body, 3-body,
interactions
98Instanton corrected BA equations
99Instanton corrected BA equations
The prepotential of the low-energy effective
theory Is governed by a classical (holomorphic)
integrable system
Donagi-Witten95
Liouville tori Jacobians of Seiberg-Witten
curves
100Classical integrable systemvsQuantum integrable
system
That system is quantized when the gauge theory is
subject to the Omega-background
NN02 NN,Okounkov03 Braverman03
Our quantum system is different!
101Blowing up the two-sphere
- Wall-crossing phenomena
- (new states, new solutions)
Something for the future
102Naturalness of our quivers
- Somewhat unusual matter content
- Branes at orbifolds typically lead to smth like
103Naturalness of our quivers
- This picture would arise in the
- sa(i) ? 0
- limit
BA for QCD Faddeev-Korchemsky94
104Naturalness of our quivers
105Naturalness of our quivers
- Possibly with the help of K.Saitos construction
106CONCLUSIONS
- 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.
107CONCLUSIONS
- 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.