Simon Catterall- Exact lattice supersymmetry
Transcript of Simon Catterall- Exact lattice supersymmetry
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Exact lattice supersymmetry
Simon Catterall
Syracuse University
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Lattice SUSY
Old problem.
Difficult. SUSY extends Poincar broken bydiscretization.
Folklore: Impossible to put SUSY on lattice exactly.
Leads to (very) difficult fine tuning lots of relevant
SUSY breaking counterterms...
Way out!
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Motivations ?
Rigorous definition of SUSY QFT - like lattice QCD.
Dynamical SUSY breaking. Predicting soft terms inMSSM ...
Gauge-gravity duality ? Eg. large N strongly coupledN= 4 SYM and type II string theory in 5d AdS.
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New ideas
Topological twisting
Orbifolding/deconstruction (D. B. Kaplan, M. Unsl, A.Cohen, ...)
Focus on former. Emphasizes geometry. Continuumlimit clear.
Warning: Tricks work only for no. SUSYs Q multiple 2D
... In D = 4 unique theory: N= 4 SYM
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Example: Twisting in 2D
Simplest theory contains 2 fermions iGlobal symmetry: SOLorenz(2) SOR(2)Twist: decompose under diagonal subgroup
Consider fermions as matrix
i
Natural to expand:
=
2I+ + 1212
scalar, vector and tensor (twisted) components!
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Twisted supersymmetry
Twisted theory has scalar SUSY Q.
{Q, Q} = p implies:
Q2
= 0{Q,Q} = p
Plausible: S = Q(, )
Basic idea of lattice theory: discretize twisted formulation,
exact (scalar) SUSY only requires Q2 = 0
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Example: Q = 4 SYM in 2D
In twisted form (adjoint fields AH generators)
S =1
g2QTr F + [D,D] 1
2d
Q A =
Q = 0
Q A = 0
Q = FQ = d
Q d = 0
Note: complexified gauge field A = A + iB
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Action
Q-variation, integrate d:
S =1
g2Tr FF + 1
2[D,D]
2 D[] DRewrite as
S = 1g2
TrF2 + 2BDDB [B, B]2 + LF
where
LF =
122
D2 iB2 D1 + iB1D1 iB1 D2 iB2
1
2
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Moral
Twisting changes spins fields:
Scalars become vectors. Naturally embedded incomplexified connection
Fermions integer spins. Form components ofKhler-Dirac field.
Twisted entire Lorentz symmetry with R-symmetry
maximal twist. Necessary for lattice.
Flat space - twisting just change of variables.
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Lattice ?
A(x) U(n). Complexified Wilson links.
Natural fermion assignment on sites, links, 12diagonal links of cubic lattice.
Fields pick up non-standard U(N) gaugetransformations:
(x) G(x)(x)G(x)
(x) G(x)(x)G(x + )
(x) G(x + + )(x)G(x)
U(x) G(x)U(x)G(x + )
U(x) G(x + )U(x)G(x)
Choice of orientations ensure G.IExact lattice supersymmetry p.
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Lattice supersymmetry
As in continuum:
Q U =
Q = 0Q U = 0
Q = FL
Q = d
Q d = 0
Note: Q2 = 0 still.
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Derivatives
D(+) f(x) = U(x)f(x + ) f(x)U(x + )
D()
f(x) = f(x)U(x) U(x )f(x )
For U(x) = 1 + A(x) + . . . reduce to adjoint covariantderivatives
F = D(+) U(x) = U(x)U(x + ) U(x)U(x + )
Remarkably satisfy exact Bianchi identity:
D(
+)F = 0
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Recap
Discretize twisted version of continuum SYM
Need subgroup of R-symmetry to match SO(D).
Ensures all fermions represented by integer spin forms.Natural map to lattice.
In flat space: twisted formulation completely equivalent
to usual theoryAbsence of fermion doubling twisted fermions fill outKhler-Dirac field (like staggered quarks)
Lattice theory G.I, possesses exact Q and a point groupsymmetry which is subgroup of twisted rotationalsymmetry.
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Bonuses
Topological subsector:
< O(x1) . . . O(xN) > independent of coupling g2, and
points x1 . . . xN if QO = 0. Eg
< O >
g2=< Q(O) >= 0
Novel gauge invariance properties of lattice theorystrongly constrains possible counter terms reducessubstantially fine tuning needed to get full SUSY in
continuum limit.
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Q = 16 SYM in 4D
Twist: diagonal subgroup of SOLorenz(4) SOR(4)
Again after twisting regard fermions as 4 4 matrix.
To represent 10 bosons of N= 4 theory with complexconnections is most natural in five dimensions.
Fermion counting requires multiplet (, a, ab) where
a, b = 1 . . . 5Action contains same Q-exact term as for Q = 4 plusnew Q-closed piece.
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Details
Dimensional reduction to 4D A5 plus imag parts ofA, = 1 . . . 4 yield 6 scalars of N= 4
Fermions: ab
,
a
S = Q 18
abcdedeDcab
Twisted action reduces to Marcus topological twist of
N= 4 (GL-twist). Equivalent to usual theory in flatspace.
Identical to Q = 16 orbifold action (Kaplan, Unsl)
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Transition to lattice
Introduce cubic lattice with unit vectorsia =
ia, a = 1 . . . 4. Additional vector
5 = (1,1,1,1).
Notice:
a a = 0. Needed for G.I.
Assign fields to links in cubic lattice (plus diagonals). Egab(x) lives on link from (x + a + b) x.
Derivatives similar to Q = 4. eg
D(+)a f(x) = Ua(x)f(x + a) f(x)Ua(x)
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Si l i
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Simulations
Integrate out fermions. Resulting Pf [MF(A)] simulatedusing RHMC alg. (lattice QCD)
Use pbc SUSY exact. Z = W Witten index -
Q-invariance exhibits topological invariance W.
Preliminary results from single core code. Parallel codenow finished..
Test SUSY, I.R divergences, check sign problems.D = 2 with Q = 4 and D = 4 with Q = 16.
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S t i W d id tit
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Supersymmetric Ward identity
Q-exactness ensures that lnZpbc = 0
where = 1g2LD
V4D
Ensures: < SB >=12 V(N
2
1)(nbosons 1)Example: D = 0 SU(2)
SB exact
1.0 4.40(2) 4.5
10.0 4.47(2) 4.5
100.0 4.49(1) 4.5
SB exact
1.0 13.67(4) 13.5
10.0 13.52(2) 13.5
100.0 13.48(2) 13.5
Q = 4 Q = 16
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V t bilit fl t di ti
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Vacuum stability - flat directions
Is integration over moduli space [B, B] = 0 divergent ?
0 5 10 15 20 25 30lambda
0
0.5
1
1.5
2
2.5
3
P(lambda)
Q=4 D=0 SU(2)
-2 -1 0 1 2 3 4 5lambda
0
2
4
6
8
10
P(lambda)
Q=16 D=0 SU(2)
Q = 4 Q = 16
D = 0. SU(2). Periodic bcs. Eigenvalues of UU 1Scalars localized close to origin. Power law tails.p(Q = 4) 3, p(Q = 16) 15 (Staudacher et al.)
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Pfaffian phase
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Pfaffian phase
Simulation uses |P f(U)|. Measure phase (U).
< O >=< Oe >phase quenched
< e
>phase quenched
SU(2) D = 2: 42.
Q SqB SB SeB cos 4 70.61(4) 65(5) 72.0 -0.016(6)
16 214.7(4) 214.6(3) 216.0 0.999994(3)
< ei(U) >phase quenched pbc= W = 0 for Q = 4 ?
SUSY breaking (Tong et. al) ?
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Fermion eigenvalue distribution
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Fermion eigenvalue distribution
SU(2) D = 2: 22
Q = 4 Q = 16
Non-zero density for Q = 4 close to origin linked to logdivergence of < 2 > ?Potential Goldstino ?
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N 4 SYM i f di i
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N = 4 SYM in four dimensions
Initial results encouraging: 6000 trajs on SU(2) 24 lattice(1000 hrs)SB/S
exactB = 0.98 < cos() >= 0.98(1)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
lambda
0
2
4
6
8
10
P(lambda)
Q=16 D=4 SU(2)
Larger lattices currently under study using parallel code.
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Applications: holography
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Applications: holography
0 2 4 6 8 10 12
T
0
5
10
15
20
(1
/N2
)E/T
SU(5) quenched
SU(3)
SU(5)
SU(8)
black hole
Example of gauge-gravity duality: Thermodynamics of
N, T 0 AdS5 black hole reproduced by N= 4 SYMtheory reduced to D = 1
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Renormalization
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Renormalization
Lattice symmetries:
Gauge invariance
Q-symmetry.Point group symmetry - eg. natural lattice for N= 4 isA4.
Exact fermionic shift symmetry.Conclusion: Renormalized action contains same operatorsas bare theory except for SUSY mass term.
Examine flows at 1-loop - in progress (with J. Giedt)
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Future
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Future
Nonperturbative exploration N= 4 YM. Tests ofAdSCFT. Supersymmetric Wilson loops.
But what residual fine tuning needed to get full SUSY
as a 0 ?
Dimensional reductions duality between strings withDp-branes and (p + 1)-SYM ?
Add fermions in fundamental .. (Matsuura, Sugino inD = 2 recently).
Break N= 4 to N= 1 a la Strassler ..
Exact lattice supersymmetry p.
Marcus twist
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Marcus twist
Reduces to:
S = Tr FF +1
2D,D
2+
1
2,
2+ (D)
(D)
D[] D [, ]
D , D ,
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Vacuum stability - trace mode
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Vacuum stability - trace mode
Correspondance to continuum requiresU = 1 + aA + O(a2).
For U(N) this is not true < 1NTr U(x)U(x) > 0.5
det(U(x)U(x)) 0!
Vacuum instability det(UU) eB0 implies B0
0 50 100 150 200 250 300MC time
-1
0
1
2
3
4
5
6
7
8
9
10
scalareigenvalue
lambda1
lambda2
Q=4 D=0 U(2) m=0.1
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Truncation
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Truncation
Cannot cure with mass m2
Tr (UU I)2
m
0.01 0.45(2)0.1 0.57(6)
0.5 0.38(2)
SB(eB0U) e4B
0S(U) any {U}
Exponential effective potential for B0.
Fix ? - truncate to SU(N) S 1N2 O(a)Also removes exact 0 mode in fermion op.
Exact lattice supersymmetry p.