Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

72 ’t hooft monopoles in lattice guts
Φ(x + Lĵ) = Ω † j (x)Φ∗ (x)Ω j (x),
U µ (x + Lĵ) = Ω † j (x)U∗ µ(x)Ω j (x + ˆµ), (4.44)
with SU(N) gauge transformations Ω j (x). We have started with the natural choice
of complex conjugation in each spatial direction. We will call these C-periodic
boundary conditions [89]. 2
As for ’t Hooft’s twisted boundary conditions, successive applications in different
directions should not depend on their order. This implies the equalities [89]
and
Ω † j (x + Lˆk)Ω T k (x)Φ(x)Ω∗ k (x)Ω j(x + Lˆk)
= Φ(x + Lĵ + Lˆk)
= Ω † k (x + Lĵ)ΩT j (x)Φ(x)Ω∗ j (x)Ω k(x + Lĵ),
Ω † j (x + Lˆk)Ω T k (x)U µ(x)Ω ∗ k (x + ˆµ)Ω j(x + Lˆk + ˆµ)
= U µ (x + Lĵ + Lˆk)
= Ω † k (x + Lĵ)ΩT j (x)U µ(x)Ω ∗ j (x + ˆµ)Ω k(x + Lĵ + ˆµ).
(4.45)
(4.46)
Since both fields are blind to center elements, Eq. (4.45) implies the cocycle condition
Ω ∗ i (x)Ω j(x + Lî) = z ij Ω ∗ j (x)Ω i(x + Lĵ),
z ij = e in ij
, (4.47)
where the Nth roots of unity z ij = z ∗ ji
formed by the antisymmetric ’twist tensor’
n ij = −n ji have the usual parametrization in terms of three Z N -valued numbers m i ,
n ij = 2π N ɛ ijkm k , m i ∈ Z N . (4.48)
Furthermore, Eq. (4.46) implies that the z ij have to be independent of position.
All choices of Ω i (x), Ω j (x) with the same twist z ij are gauge equivalent [47, 89],
and we therefore assume that we can choose the matrices Ω j to be independent of
position analogous to the standard ‘twist eaters’ in the case of ’t Hooft’s twisted
boundary conditions without charge conjugation [106].
Non-trivial twists are only possible for even N, which is revealed by applying
the cocycle condition (4.47) to the product Ω i Ω ∗ j Ω k [89]. On the one hand,
Ω i Ω ∗ j Ω k = z jk Ω i Ω ∗ k Ω j
but applying the condition in the opposite order yields
= z jk z ki Ω k Ωi ∗ Ω j
= z jk z ki z ij Ω k Ω ∗ j Ω i, (4.49)
Ω i Ω ∗ j Ω k = z ji Ω j Ω ∗ i Ω k
= z ji z ik Ω j Ω ∗ k Ω i
= z ji z ik z kj Ω k Ω ∗ j Ω i. (4.50)
2 In fact, in the terminology of Ref. [89], these correspond to C-periodic boundary conditions with
C = −1, and C = 1 would correspond to boundary conditions without complex conjugation.