Circle compactification and ’t Hooft anomaly
HJE
Circle compacti cation and 't Hooft anomaly
Yuya Tanizaki 0 1 3 6 7
Tatsuhiro Misumi 0 1 2 3 4 5 7
Norisuke Sakai 0 1 2 3 5 7
0 Keio University
1 1-1 Tegata Gakuen-machi , Akita 010-8502 , Japan
2 iTHEMS , RIKEN
3 Upton , NY 11973 , U.S.A
4 Department of Mathematical Science, Akita University
5 Department of Physics, and Research and Education Center for Natural Sciences
6 RIKEN BNL Research Center, Brookhaven National Laboratory
7 2-1 Hirosawa , Wako, Saitama 351-0198 , Japan
Anomaly matching constrains low-energy physics of strongly-coupled eld theories, but it is not useful at nite temperature due to contamination from high-energy states. The known exception is an 't Hooft anomaly involving one-form symmetries as
Anomalies in Field and String Theories; Global Symmetries; Nonperturbative
-
in pure SU(N ) Yang-Mills theory at
=
. Recent development about large-N volume
independence, however, gives us a circumstantial evidence that 't Hooft anomalies can also
remain under circle compacti cations in some theories without one-form symmetries. We
develop a systematic procedure for deriving an 't Hooft anomaly of the circle-compacti ed
theory starting from the anomaly of the original uncompacti ed theory without one-form
symmetries, where the twisted boundary condition for the compacti ed direction plays a
pivotal role. As an application, we consider ZN -twisted CP N 1 sigma model and massless
ZN -QCD, and compute their anomalies explicitly.
Systematic procedure for anomaly with circle compacti cation
Comments on choice of the background holonomy
3.1 't Hooft anomaly and global inconsistency in two dimensions
ZN -twisted CP N 1 model and its anomaly
Comparison with previous studies and discussion
Four-dimensional 't Hooft anomaly of massless N - avor QCD
Massless ZN -QCD and its anomaly
Comparison with previous studies and discussion
1 Introduction Formalism 2 3
2.1
2.2
3.2
3.3
4.1
4.2
4.3
breaking classi es traditional phases of matter following Landau's characterization [1, 2].
In order to re ne the data of QFTs related to symmetry, one can try to promote global
symmetry to local gauge symmetry, but sometimes topology related to the symmetry gives
topological phases of matter pushes that notion further [6{10] and it is now applicable
also for systems with discrete symmetries, higher-form symmetries, and so forth, to derive
nontrivial consequences on vacuum structures [11{25].
Although anomaly matching is a powerful technique to study nonperturbative physics,
it cannot uncover details of dynamical aspects of QFTs and just provides us a consistency
condition. For example, SU(N ) Yang-Mills theory is believed to exhibit con nement in
four-dimensional spacetime, and an 't Hooft anomaly can tell us about some additional
information on vacuum assuming con nement, but it does not show how con nement can
happen. Analytic computation of con nement in four dimensions is currently impossible
because of its strong coupling nature. Still, it is found that the con nement of SU(N )
Yang-Mills theory is realizable on R
3
S1 by adding several massive adjoint fermions or
deformations of the action itself, and this con nement is calculable with reliable
semiclassical computations [26{31]. What is more interesting is that this semiclassical con nement is
argued to be adiabatically connected to the con nement in the strongly-coupled regime by
decompactifying the circle S1 especially in the large-N limit. This nding motivated many
studies of various asymptotically-free eld theories by compactifying one direction to a
circle with an appropriate boundary condition, and it is expected to map the strongly-coupled
dynamics into the semiclassical regime without losing its essential information [32{53].
In this paper, we would like to make a connection between these two recent
developments of nonperturbative QFTs; adiabatic circle compacti cation and 't Hooft anomaly
matching. If the vacuum structures of the original and circle-compacti ed theories are
really adiabatically connected, it is natural to think that both vacuum structures reproduce
the same 't Hooft anomaly matching condition. However, there is the following di culty
in this idea: anomaly is renormalization group invariant, and it is matched by the vacuum
or its low-energy excitations. Since other high-energy states do not produce the anomaly,
the e ect of anomaly would disappear once those high-energy states give dominant
contributions at
nite temperature. How can this observation be consistent with the story
about adiabatic continuity? In order to understand the situation better, we consider two
quick examples.
eld A, then
Let us consider a three-dimensional free Dirac fermion, which has a U(
1
) symmetry
and time-reversal symmetry T. Consider the partition function Z[A] under the U(
1
) gauge
Z[A] = jZ[A]j exp(i [A]=2):
Here, [A] is the eta invariant [11], which is roughly the U(
1
) level-1 Chern-Simons action
but is gauge-invariant modul (...truncated)