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Register a new userElliptic hypergeometric integrals and t Hooft anomaly matching conditions
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Elliptic hypergeometric integrals describe superconformal indices of 4d supersymmetric field theories. We show that all t Hooft anomaly matching conditions for Seiberg dual theories can be derived from $SL(3,mathbb{Z})$-modular transformation properties of the kernels of dual indices.
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We study the phase diagram of two-flavor massless two-color QCD (QC$_2$D) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin Roberge--Weiss (RW) point, two-flavor QC$_2$D enjoys the $mathbb{Z}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a $mathbb{Z}_2$ t Hooft anomaly of this system, which involves the $mathbb{Z}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrains the possible phase diagram at any temperatures and quark chemical potentials at the isospin RW point, and we compare it with previous results obtained by chiral effective field theory and lattice simulations. We also point out an interesting similarity of two-flavor massless QC$_2$D with $(2+1)$d quantum anti-ferromagnetic systems.
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Eulers and Selbergs beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.
Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Millers transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions. The series solutions constructed for these hypergeometric systems recover the well known results in literature.
We study the discrete chiral- and center-symmetry t Hooft anomaly matching in the charge-$q$ two-dimensional Schwinger model. We show that the algebra of the discrete symmetry operators involves a central extension, implying the existence of $q$ vacua, and that the chiral and center symmetries are spontaneously broken. We then argue that an axial version of the $q$$=$$2$ model appears in the worldvolume theory on domain walls between center-symmetry breaking vacua in the high-temperature $SU(2)$ ${cal N}$$=$$1$ super-Yang-Mills theory and that it inherits the discrete t Hooft anomalies of the four-dimensional bulk. The Schwinger model results suggest that the high-temperature domain wall exhibits a surprisingly rich structure: it supports a non-vanishing fermion condensate and perimeter law for spacelike Wilson loops, thus mirroring many properties of the strongly coupled four-dimensional low-temperature theory. We also discuss generalizations to theories with multiple adjoint fermions and possible lattice tests.
The dependence of the energies of axially symmetric monopoles of magnetic charges 2 and 3, on the Higgs self-interaction coupling constant, is studied numerically. Comparing the energy per unit topological charge of the charge-2 monopole with the energy of the spherically symmetric charge-1 monopole, we confirm that there is only a repulsive phase in the interaction energy between like monopoles
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