Z2 Lattice Gerbe Theory
Des Johnston
CompPhys14 Leipzig, Nov 2014
Des Johnston
Lattice Gerbe Theory
1/15
Plan of talk
Gerbes?
Lattice Gauge, Gerbe Theory
Z2 Lattice Gerbe Theory
Des Johnston
Lattice Gerbe Theory
2/15
Gerbes?
(Abelian) Gauge theory, one forms:
S=
1
4g 2
Z
d4 xFµν F µν
Fµν = ∂[µ Aν]
(Abelian) Gerbe theory, two forms:
1
S= 2
4g
Z
dd x Hµνλ H µνλ
Hµνλ = ∂[µ Bνλ]
Des Johnston
Lattice Gerbe Theory
3/15
Lattice Gauge/Gerbe Theory
Gauge - Hamiltonian defined on plaquettes
Gerbe - Hamiltonian defined on cubets
→
Des Johnston
Lattice Gerbe Theory
4/15
Higher Abelian Gauge theory
General Framework

H=−

Y
X

Cn+1
U (Cn ) + c.c.
Cn ∈∂Cn+1
U (Cn ) = exp(iA(Cn )) live on the boundaries Cn of cells
Cn+1
Hamiltonian given by the sum of products of the U (Cn )
around the boundary of a Cn+1
Des Johnston
Lattice Gerbe Theory
5/15
Z2 Gauge theory
Many of the properties are visible already in simplest Z2
case
Symmetries, observables (loops)
*
Γ(L) =
+
Y
U (C1 )
C1 ∈L
If a confining transition exists
(
exp(−A(L)) β < βc
Γ(L) ∼
exp(−P (L)) β > βc
Des Johnston
Lattice Gerbe Theory
6/15
Z2 Gerbe theory
Play same game with Gerbe theory
Symmetries, observables (surfaces)
*
Γ(S) =
+
Y
U (C2 )
C2 ∈S
If a confining transition exists
(
exp(−V (S)) β < βc
Γ(S) ∼
exp(−A(S)) β > βc
Des Johnston
Lattice Gerbe Theory
7/15
So Far, so general
We can use Wegner’s results on duality for generalized
Ising models (1971) to say more
Lattice N d-dimensional hypercubes
d
Md,n model, Ns = n−1
N spins sited at the centres of the
(n − 1)-dimensional hypercubes
Hamiltonians Hdn , product of 2n spins on the
(n − 1)-dimensional faces of the Nb = nd N n-dimensional
hypercubes.
Md,3 are lattice Gerbe theories
Des Johnston
Lattice Gerbe Theory
8/15
Schematically
Gauge Md,2
H = −
X
H = −
X
U4
Gerbe Md,3
Des Johnston
U6
Lattice Gerbe Theory
9/15
(Generalized) Ising duality
∗
Md,n on d-dimensional hypercubic lattice and dual Md,d−n
on dual lattice
∗
Zd,n (β) ∼ Zd,d−n
(β ∗ )
tanh(β) = exp(−2β ∗ )
Duality also in field
∗
Zd,n (β, h) ∼ Zd,d−n+1
(β ∗ , h∗ )
tanh(β) = exp(−2h∗ )
tanh(h) = exp(−2β ∗ )
Des Johnston
Lattice Gerbe Theory
10/15
Consequences of (Generalized) Ising
duality d = 3
d = 3, use in field duality
∗
Z3,3 (β, 0) ∼ Z3,1
(∞, h∗ )
Dualize back
Z3,3 (β, 0) ∼ 2Ns cosh(β)Nb + sinh(β)Nb
Analytic, no transition
Des Johnston
Lattice Gerbe Theory
11/15
Consequences of (Generalized) Ising
duality d = 4, 6..
d = 4, Z2 Gerbe theory dual to standard Ising model
∗
Z4,3 (β) ∼ Z4,1
(β ∗ )
Continuous transition
d = 6, self-duality
∗
Z2n,n (β) ∼ Z2n,n
(β ∗ )
6d Z2 Gerbe theory
is self dual, familiar critical temperature
√
βc = 21 ln(1 + 2)
Des Johnston
Lattice Gerbe Theory
12/15
Making life more complicated
Gauge Higgs
H = −
X
H = −
X
U4 − λ
X
U6 − λ
X
U σ2
hiji
Gerbe Higgs
Des Johnston
U σ4
Lattice Gerbe Theory
13/15
Conclusions
Interest in formulating lattice Gerbe theories
Wegner has very thoughtfully calculated much of what is
needed to discuss Z2 lattice Gerbe theory
These shed some light on Abelian and non-Abelian Gerbe
theories
These models are curiously unsimulated.....
Des Johnston
Lattice Gerbe Theory
14/15
References
A. E. Lipstein and R. A. Reid-Edwards, Lattice Gerbe
Theory, JHEP 09 (2014) 034, [arxiv:1404.2634].
C. Omero, P. A. Marchetti and A. Maritan, J. Phys. A 16,
1465 (1983).
F. J. Wegner, J. Math. Phys. 12 2259 (1971).
F. J. Wegner, Duality in Generalized Ising Models,
[arXiv:1411.5815]
D. A. Johnston, Z2 lattice gerbe theory, Phys. Rev. D 90,
107701 (2014), [arXiv:1405.7890]
Des Johnston
Lattice Gerbe Theory
15/15