Exact Results
inSupersymmetric Lattice Gauge Theories
Kazutoshi Ohta (Meiji Gakuin University)

Work in progress with S. Matsuura and T. Misumi (Keio Univ.)

JSPS/RFBR workshop, 3/6/2014

Introduction
The localization reduces the path integral to finite dimensional multiple
integrals or sums, and solves exactly (non-perturbatively) someproblems in field theories:

Z2dYM ==Z
Z

D

DAD e

Tr

R

DAD D · · · e
R

122

12i F

TrER2i F

122+ g12 SSYM++= eTr i F
N =(2,2)SYM
RX=(1-loop dets) eTr i F

122+

fixed points

In this sense, 2d (SUSY-)YM theory (and also 3d (SUSY-)CS theory) isintegrable (exactly solvable).

Introduction
Question:!
Does this integrable structure still hold in lattice gauge theory (ondiscretized space-time)?
Answer: YES!!
We can construct exactly solvable 2d gauge theories on the lattice.

On S2, S3, ...

On simplicial
complexToday, I explain how to apply localization method to the lattice gauge
theories and give some exact results.

Harish-Chandra Itzykson-Zuber Integral
Let us first consider the so-called Harich-Chandra Itzykson-Zuber(HCIZ) integral for a lesson:

ZHCIZ =whereZ

DU eHHCIZ=✓2⇡◆N (N1)/2deti,j e ai bj(a) (b)HHCIZ = TrAU BU †A, B : (constant) Hermite matrices
U : N ⇥ N unitary matrix(a),(b) : Vandermonde determinants of eigenvalues
Y(a) =(ai aj )i<jProof by localization
We can perform HCIZ integral exactly.!
Why? ⇒ This is because the localization works
Phase space (coadjoint action orbit) is isomorphic to a coset!
M=U(N)/U(1)N, which possesses a symplectic 2-form ω (KirillovKostant-Souriau 2-form). The Hamiltonian HHCIZ generates a vector field
V with ω.

dHHCIZ◆V ! = 0

We rewrite (identifying the right-invariant one form with a fermion)

ZHCIZ =1
where ! = Tr
2

†
[UBU,
R

1(b)
R]

Z

DU D R e

HHCIZ +!

Proof by localization
The exponent is invariant (Q(βHHCIZ-ω)=0) under the following“supersymmetry” (BRST symmetry)

QU = i R U,

Q R =i A+i R

RWe can also introduce Q-exact “action”:† 2

Tr[A, U BU ] + · · ·Q⌅ =Using these, if we deform the integral by Q-exact action

ZHCIZ =1(b)Z

DU D R e

HHCIZ +!

1Q⌅
g2

we can show that the integral is independent of g (⇒ WKB(1-loop) exact)

Proof by localization
The saddle points (fixed points) are given by the equation[A, U BU † ] = 0 and
⇒ U =

R =0

(permutation (Weyl group))

Evaluating the integral around the saddle points, we finally obtain

ZHCIZ ==✓✓

2⇡
2⇡

◆N (N◆N (N1)/21)/2| |X
1( 1)e(b)(a)1-loop det

HHCIZ +!

X
1| |( 1) e(a) (b)fixed points
Pi ai b (i)Migdal-Kazakov model
Let us next extend the HCIZ integral to the multi-matrix model on thelattice (induced QCD), that is;

ZMK =whereSMK =Z Y
Xhxyi

DUxy D

Trx UxytSMKex

Px TrV (x)†y UxyThis model is known as the Migdal-Kazokov model (1992).!
(The relation to 2d YM is also discussed in [Caselle-D’Adda-LorenzoMagnea-Panzeri],[Kharchev-Marshakov-Mironov-Morozov].)SUSY on lattice
The “action” (+symplectic 2-form) of MK model is invariant under thefollowing supersymmetry on the 2d lattice (generalization of the Suginomodel)

QUxy = xy ,Q x=0
Q ¯ x = ⌘x ,QYx = i[ x , x ],Q

xy = UxyQ⌘x = i[ ¯ x ,Q x = Yx

yx Uxyx]

where we have defined
We also findnilpotent on gauge invariant operators!
(⇒ equivariant cohomology)

SUSY action
The action of 2d N=(2,2) SUSY YM on the lattice is written in Q-exactform by:where #x ~ W(U)-W(U)† ~ F#ν are moment maps (superpotentialconstraints) associated with each loops (faces).

Localization in SUSY lattice
The partition function of the supersymmetric MK model deformed bythe Sugino action

ZsMK ==Z YZ Y

DUxy D
DUxy DxDxDt(SMK !)
exy
xy · · · ePt(SMK !)TrV ( )P

TrV ( )

is independent of the coupling g, since the action is Q-exact andQ(SMKQTrV (!) = 0x) = 0

whereSo the integral becomes WKB (1-loop) exact wrt the SUSY action

1Sg 2 Sugino

1-loop determinant
To evaluate the 1-loop determinant, we fix the gauge by:( U(N) → U(1)N )Then, we obtain the 1-loop determinant of the Sugino model:

where the subscripts mean that the determinants come from eachvariables, and S, L and F stand for sets of sites (vertices), links (edges)
and loops (faces), respectively.

Exact result at fixed points
The fixed points are classified again by the permutations (Weyl group):and(⇒ Dµ (x) = 0 )Then we get at the fixed points

ZsMK =X0Y{ xy } hxyi

( 1)| xy |Z Yid iYi<j( it`)ejPi2isPi V ( i)where χ=s-l+f, and s, l and f are respectively the number of vertices(sites), edges (links) and faces, which come from the number of eachvariables (matrices).

ExamplesOn one plaquette;] of ¯ and c, c̄ = 4] of Uxy = 4] of (or µ) = 1Y( ij)

1~ diski<jOn a tetrahedron;] of ¯ and c, c̄ = 4] of Uxy = 6] of (or µ) = 4Yi<j( i2)j~ sphereComparison with 2d YM
2d YM partition function on a general Riemann surface (Migdal)

XZ2d YM =Y{ni }2ZN i<j(ninj ) e

g2 A P2ni i
2m 2
One branch (trivial permutation) of sMK model (with V = 2 x )Z YYP 2m(t`+ 2 s) i iZsMK =d i ( i)ej

Remarks:ii<j✤Multiple integrals still remain because of flat directions of SUSY theory(not fixed points but fixed lines)!

✤The partition function is independent of the simplicial decomposition but
depends only on the topology (and area) (⇒ 2d YM is almost topological)

Conclusion and Discussion
Results:✤ We exactly evaluated the partition function of the MK model under therestricted symmetry (constraints)!
✤

Reversing the logic, we exactly calculated a vev of physical observable in
2d SUSY YM theory on the lattice!

✤We also found other observables and useful Ward-Takahashi identities
Problems:✤Application to other integrable systems (spin chain, etc.)!

✤Relation to (or realization in) string/M theory or gravity (topologicalinvariants, etc. in mathematics)

Our model is closely related to quiver gauge theory, deconstruction, etc...