A
bstract
We consider the SU(
N
) Yang-Mills theory, whose topological sectors are restricted to the instanton number with integer multiples of
p
. We can formulate such a quantum field theory ...maintaining locality and unitarity, and the model contains both 2
π
-periodic scalar and 3-form gauge fields. This can be interpreted as coupling a topological theory to Yang-Mills theory, so the local dynamics becomes identical with that of pure Yang-Mills theory. The theory has not only ℤ
N
1-form symmetry but also ℤ
p
3-form symmetry, and we study the global nature of this theory from the recent ’t Hooft anomaly matching. The computation of ’t Hooft anomaly incorporates an intriguing higher-group structure. We also carefully examine that how such kinematical constraint is realized in the dynamics by using the large-
N
and also the reliable semiclassics on ℝ
3
×
S
1
, and we find that the topological susceptibility plays a role of the order parameter for the ℤ
p
3-form symmetry. Introducing a fermion in the fundamental or adjoint representation, we find that the chiral symmetry becomes larger than the usual case by ℤ
p
, and it leads to the extra
p
vacua by discrete chiral symmetry breaking. No dynamical domain wall can interpolate those extra vacua since such objects must be charged under the 3-form symmetry in order to match the ’t Hooft anomaly.
A
bstract
We consider a class of quantum field theories and quantum mechanics, which we couple to ℤ
N
topological QFTs, in order to classify non-perturbative effects in the original theory. The ℤ
N
...TQFT structure arises naturally from turning on a classical background field for a ℤ
N
0- or 1-form global symmetry. In SU(
N
) Yang-Mills theory coupled to ℤ
N
TQFT, the non-perturbative expansion parameter is exp
−S
I
/N
= exp
−
8
π
2
/g
2
N
both in the semi-classical weak coupling domain and strong coupling domain, corresponding to a fractional topological charge configurations. To classify the non-perturbative effects in original SU(
N
) theory, we must use PSU(
N
) bundle and lift configurations (critical points at infinity) for which there is no obstruction back to SU(
N
). These provide a refinement of instanton sums: integer topological charge, but crucially fractional action configurations contribute, providing a TQFT protected generalization of resurgent semi-classical expansion to strong coupling. Monopole-instantons (or fractional instantons) on
T
3
×
S
L
1
can be interpreted as tunneling events in the ’t Hooft flux background in the PSU(
N
) bundle. The construction provides a new perspective to the strong coupling regime of QFTs and resolves a number of old standing issues, especially, fixes the conflicts between the large-
N
and instanton analysis. We derive the mass gap at
θ
= 0 and gaplessness at
θ
=
π
in
CP
1
model, and mass gap for arbitrary
θ
in
CP
N−
1
, N
≥ 3 on ℝ
2
.
A
bstract
We study a 3d lattice gauge theory with gauge group U(1)
N−
1
⋊ S
N
, which is obtained by gauging the S
N
global symmetry of a pure U(1)
N−
1
gauge theory, and we call it the semi-Abelian ...gauge theory. We compute mass gaps and string tensions for both theories using the monopole-gas description. We find that the effective potential receives equal contributions at leading order from monopoles associated with the entire SU(
N
) root system. Even though the center symmetry of the semi-Abelian gauge theory is given by ℤ
N
, we observe that the string tensions do not obey the
N
-ality rule and carry more detailed information on the representations of the gauge group. We find that this refinement is due to the presence of non-invertible topological lines as a remnant of U(1)
N−
1
one-form symmetry in the original Abelian lattice theory. Upon adding charged particles corresponding to
W
-bosons, such non-invertible symmetries are explicitly broken so that the
N
-ality rule should emerge in the deep infrared regime.
A
bstract
The dimension of the Hilbert space of QFT scales exponentially with the volume of the space in which the theory lives, yet in supersymmetric theories, one can define a graded dimension ...(such as the supersymmetric index) that counts just the number of bosonic minus fermionic ground states. Can we make this observation useful in non-supersymmetric QFTs in four dimensions? In this work, we construct
symmetry graded state sums
for a variety of non-supersymmetric theories. Among the theories we consider is one that is remarkably close to QCD: Yang-Mills theory with
N
f
=
N
c
fundamental Dirac fermions and one adjoint Weyl fermion, QCD(F/adj). This theory can be obtained from SQCD by decoupling scalars and carry exactly the same anomalies. Despite the existence of fundamental fermions, the theory possess an exact 0-form color-flavor center (CFC) symmetry for a particular grading/twist under which Polyakov loop is a genuine order parameters. By a two-loop analysis, we prove that CFC-symmetry remains unbroken at small
β
due to grading. Chiral symmetry is spontaneously broken within the domain of validity of semi-classics on ℝ
3
×
S
1
in a pattern identical to
N
f
=
N
c
SQCD on ℝ
4
and the two regimes are adiabatically connected. The vacuum structures of the theory on ℝ
4
and ℝ
3
×
S
1
are controlled by the same mixed ’t Hooft anomaly condition, implying a remarkable persistent order.
A
bstract
This work examines non-perturbative dynamics of a 2-dimensional QFT by using discrete ’t Hooft anomaly, semi-classics with circle compactification and bosonization. We focus on charge-
q N
...-flavor Schwinger model, and also Wess-Zumino-Witten model. We first apply the recent developments of discrete ’t Hooft anomaly matching to theories on ℝ
2
and its compactification to ℝ×
S
L
1
. We then compare the ’t Hooft anomaly with dynamics of the models by explicitly constructing eigenstates and calculating physical quantities on the cylinder spacetime with periodic and flavor-twisted boundary conditions. We find different boundary conditions realize different anomalies. Especially under the twisted boundary conditions, there are
Nq
vacua associated with discrete chiral symmetry breaking. Chiral condensates for this case have fractional
θ
dependence e
i
θ
/
Nq
, which provides the
Nq
-branch structure with soft fermion mass. We show that these behaviors at a small circumference cannot be explained by usual instantons but should be understood by “quantum” instantons, which saturate the BPS bound between classical action and quantum-induced effective potential. The effects of the quantum-instantons match the exact results obtained via bosonization within the region of applicability of semi-classics. We also argue that large-
N
limit of the Schwinger model with twisted boundary conditions satisfy volume independence.
A
bstract
There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral ...formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.
A
bstract
We study low-energy dynamics of SU(
N
)
K
chiral quiver gauge theories in connection with
N
= 1 super Yang-Mills (SYM) theory, and quantum chromodynamics with bi-fundamental fermions ...(QCD(BF)). These theories can be obtained by ℤ
K
orbifold projections of
N
= 1 SU(
N K
) SYM theory, but the perturbative planar equivalence does not extend nonperturbatively for
K ≥
3. In order to study low-energy behaviors, we analyze these systems using ’t Hooft anomaly matching and reliable semiclassics on ℝ
3
× S
1
. Thanks to ’t Hooft anomaly that involves 1-form center symmetry and discrete chiral symmetry, we predict that chiral symmetry must be spontaneously broken in the confinement phase, and there exist
N
vacua. Theories with even
K
possess a physical
θ
angle despite the presence of massless fermions, and we further predict the
N
-branch structure associated with it; the number of vacua is enhanced to 2
N
at
θ
=
π
due to spontaneous
C P
breaking. Both of these predictions are explicitly confirmed by reliable semiclassics on R
3
× S
1
with the double-trace deformation. Symmetry and anomaly of odd-
K
theories are the same as those of the
N
= 1 SYM, and the ones of even-
K
theories are same as those of QCD(BF). We unveil why there exists universality between vector-like and chiral quiver theories, and conjecture that their ground states can be continuously deformed without quantum phase transitions. We briefly discuss anomaly inflow on the domain walls connecting the vacua of the theory and possible anomaly matching scenarios.
A
bstract
We study quantum chromodynamics including the two-index symmetric or anti-symmetric quark (QCD(Sym/ASym)) on small ℝ
2
× T
2
with a suitable magnetic flux. We first discuss the ’t Hooft ...anomaly of these theories and claim that discrete chiral symmetry should be spontaneously broken completely to satisfy the anomaly matching condition. The
T
2
compactification with the magnetic flux preserves the ’t Hooft anomaly, and the 2d effective theory is constrained by the same anomaly of 4d QCD(Sym/ASym). We demonstrate the spontaneous breakdown of chiral symmetry using the dilute gas of center vortices, which confirms the prediction of the ’t Hooft anomaly. We also find that each vacuum maintains the charge conjugation symmetry, and this gives affirmative support for the nonperturbative large-
N
orientifold equivalence between QCD(Sym/ASym) and
N
= 1 supersymmetric SU(
N
) Yang-Mills theory.
PDF
