JHEP09(2018)076
We find these correspondences between high-T DW physics and low-T bulk and DW physics
quite fascinating. The matching of various anomalies and the rich DW physics uncovered
make these properties worth pointing out and pursuing further.
1
This paper is organized as follows. In section 2, we study the charge-q Schwinger model,
its discrete symmetries, its ’t Hooft anomalies, and the anomaly saturation. In section 3,
we review the DW solution in the high temperature SU(2) SYM theory and show that the
worldvolume of the DW is a charge-2 axial Schwinger model. We also discuss the anomaly
inflow and the manifestation of the anomaly on the DW. We conclude, in section 4, by
a discussion of the generalizations to QCD(adj) with a larger number of adjoint fermions
and a proposal to study the high-T domain walls on the lattice.
2 Discrete ’t Hooft anomalies in the charge-q Schwinger model
Consider the charge-q vector massless Schwinger model with Lagrangian
2
L = −
1
4e
2
f
kl
f
kl
+ i
¯
ψ
+
(∂
−
+ iqA
−
)ψ
+
+ i
¯
ψ
−
(∂
+
+ iqA
+
)ψ
−
, (2.1)
where k, l = 0, 1 are spacetime indices, ∂
±
≡ ∂
t
± ∂
x
, A
±
≡ A
t
± A
x
, t and x are the
two-dimensional Minkowski space coordinates, q ≥ 2 is an integer and e is the gauge
coupling. The spacetime metric is g
kl
= diag(+, −), and we further assume that space is
compactified on a circle of circumference L, with x ≡ x+L. The fields ψ
+
(ψ
−
) are the left
(right) moving components of the Dirac fermion and
¯
ψ
±
are the hermitean conjugate fields.
Our notation follows from that of [12] and, as in that reference, we impose antiperiodic
boundary conditions on ψ
±
around the spatial circle.
3
The major difference of our discussion from that in [11, 12]—where the model (2.1)
with q = 1 was solved exactly in Hamiltonian language for arbitrary values of L (see also
the textbook [16] which emphasizes the eL 1 limit)—is in the assumption that q > 1
and in the corresponding global issues and discrete anomalies that arise.
4
Understanding
the symmetry structure and anomalies of (2.1) is of interest from multiple points of view:
1. On its own, the charge-q vectorlike Schwinger model (2.1) is an interesting example
that provides an exactly solvable setting to study the manifestation of the recently
discovered mixed discrete 0-form/1-form ’t Hooft anomalies [4, 5].
2. Two-dimensional models closely related to (2.1) also appear within the framework
of four-dimensional gauge theories. We show in section 3 that the axial version of
1
The spirit of the correspondences outlined resembles those found in the high-T DWs of pure Yang-Mills
theory at θ = π [5] but the dynamics here appears richer.
2
The charge-q Schwinger model was also discussed in [6], but with no reference to anomalies.
3
We note that we could also follow [11] and take the fermions periodic, with no change in the results
regarding symmetry realizations and anomalies; also, the utility of Weyl fermion notation will become clear
further below.
4
We caution the reader against concluding that the value of q is irrelevant: we are considering a compact
U(1) theory with (light) dynamical charges with quantized charge q > 1. The theory can be probed with
nondynamical q = 1 charges. One can think of the latter as of very (infinitely) massive dynamical charges.
– 3 –
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