Circle compactification and ’t Hooft anomaly Tanizaki, Yuya; Misumi, Tatsuhiro; Sakai, Norisuke
The journal of high energy physics,
12/2017, Letnik:
2017, Številka:
12
Journal Article
Recenzirano
Odprti dostop
A
bstract
Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known ...exception is an ’t Hooft anomaly involving one-form symmetries as in pure SU(
N
) Yang-Mills theory at
θ
=
π
. Recent development about large-
N
volume independence, however, gives us a circumstantial evidence that ’t Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an ’t Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider
ℤ
N
-twisted
ℂ
P
N
−
1
sigma model and massless
ℤ
N
-QCD, and compute their anomalies explicitly.
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.
Lattice fermions as spectral graphs Yumoto, Jun; Misumi, Tatsuhiro
The journal of high energy physics,
02/2022, Letnik:
2022, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A
bstract
We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate ...the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized fourdimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.
A
bstract
We revisit 2D
= (2, 2) super Yang-Mills lattice formulation (Sugino model) to investigate its fermion action with two (Majorana) fermion flavors and exact chiral-U(1)
R
symmetry. We show ...that the reconcilement of chiral symmetry and absence of further species-doubling originates in the 4D clifford algebra structure of the action, where 2D two flavors are spuriously treated as a single 4D four-spinor with four 4D gamma matrices introduced into kinetic and Wilson terms. This fermion construction based on the higher-dimensional clifford algebra is extended to four dimensions in two manners: (1) pseudo-8D sixteen-spinor treatment of 4D four flavors with eight 8D gamma matrices, (2) pseudo-6D eight-spinor treatment of 4D two flavors with five out of six 6D gamma matrices. We obtain 4D four-species and two-species lattice fermions with unbroken subgroup of chiral symmetry and other essential properties. We discuss their relations to staggered and Wilson twisted-mass fermions. We also discuss their potential feedback to 4D super Yang-Mills lattice formulations.
Abstract We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S 1. We describe the Stokes graphs of a general potential problem as a network of ...Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.
A
bstract
We derive the semiclassical contributions from the real and complex bions in the two-dimensional ℂ
P
N
− 1
sigma model on ℝ×
S
1
with a twisted boundary condition. The bion configurations ...are saddle points of the complexified Euclidean action, which can be viewed as bound states of a pair of fractional instantons with opposite topological charges. We first derive the bion solutions by solving the equation of motion in the model with a potential which simulates an interaction induced by fermions in the ℂ
P
N
− 1
quantum mechanics. The bion solutions have quasi-moduli parameters corresponding to the relative distance and phase between the constituent fractional instantons. By summing over the Kaluza-Klein modes of the quantum fluctuations around the bion backgrounds, we find that the effective action for the quasi-moduli parameters is renormalized and becomes a function of the dynamical scale (or the renormalized coupling constant). Based on the renormalized effective action, we obtain the semiclassical bion contribution in a weak coupling limit by making use of the Lefschetz thimble method. We find in the supersymmetric case that the bion contribution vanishes as expected from supersymmetry. In non-supersymmetric cases, the non-perturbative contribution has an imaginary ambiguity which is consistent with the expected infrared renormalon ambiguity. Our results explicitly demonstrate that the complex bion can explain the infrared renormalon.
A
bstract
We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with
N
minima on
S
1
. We describe the Stokes graphs of a general potential problem as a network of ...Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of
N
-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even
N
and global inconsistency for odd
N
. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.
We propose a new class of tight-binding models where a flat band exists either gapped from or crossing right through a dispersive band on two-band (i.e., two sites/unit cell) tetragonal and honeycomb ...lattices. By imposing a condition on the hopping parameters for generic models with up to third-neighbor hoppings, we first obtain models having a rigorously flat band isolated from a dispersive band with a gap, which accommodate both rank reducing and non-rank reducing of the Hamiltonian. The models include Tasaki's flat-band models, but the present model generally has a nonzero flat-band energy whose gap from the dispersive band is controllable as well. We then modify the models by appropriately changing the second- or third-neighbor hoppings, leading to a new class of two-dimensional lattices where a (slightly warped) flat band pierces a dispersive one. As with the known flat-band models, the connectivity condition is satisfied in the present models, so that we have unusual, unorthogonalizable Wannier orbitals. We have also shown that the present flat-band model can be extended to three (or higher) dimensions. Implications on possible high-TC superconductivity are discussed when a repulsive electron-electron interaction is introduced, where the flat band is envisaged to be utilized as intermediate states in pair scattering processes.
A
bstract
The resurgence structure of the 2d
O
(
N
) sigma model at large
N
is studied with a focus on an IR momentum cutoff scale
a
that regularizes IR singularities in the semiclassical expansion. ...Transseries expressions for condensates and correlators are derived as series of the dynamical scale Λ (nonperturbative exponential) and coupling
λ
μ
renormalized at the momentum scale
μ
. While there is no ambiguity when
a >
Λ, we find for
a <
Λ that the nonperturbative sectors have new imaginary ambiguities besides the well-known renormalon ambiguity in the perturbative sector. These ambiguities arise as a result of an analytic continuation of transseries coefficients to small values of the IR cutoff
a
below the dynamical scale Λ. We find that the imaginary ambiguities are cancelled each other when we take all of them into account. By comparing the semiclassical expansion with the transseries for the exact large-
N
result, we find that some ambiguities vanish in the
a →
0 limit and hence the resurgence structure changes when going from the semiclassical expansion to the exact result with no IR cutoff. An application of our approach to the ℂ
P
N−
1
sigma model is also discussed. We find in the compactified model with the ℤ
N
twisted boundary condition that the resurgence structure changes discontinuously as the compactification radius is varied.
PDF
