Higher anomalies, higher symmetries, and cobordisms III: QCD matter phases anew

Abstract

We discuss the topological terms, the global symmetries and their 't Hooft anomalies of pure gauge theories in various dimensions, with dynamical gauge group G, the Lorentz symmetry group G Lorentz , and the internal global symmetry G e,[1] × G m, [d−3] which consists of 1-form electric center symmetry G e,[1] and (d − 3) form magnetic symmetry G m, [d−3] . The topological terms are determined by the cobordism invariants (Ω d ) G ′ where G ′ is the group extension of G Lorentz by G, which also characterize the invertible TQFTs or SPTs with global symmetry G ′ . The 't Hooft anomalies are determined by the cobordism invariants (Ω d+1 ) G ′′ where G ′′ is the symmetry extension of G Lorentz by the higher form symmetry G e,[1] × G m, [d−3] . Different symmetry extensions correspond to different fractionalizations of G Lorentz quantum numbers on the symmetry defects of G e,[1] × G m, [d−3] . We compute the cobordism groups/invariants described above for G = U(1), SU(2) and SO(3) in d ≤ 5, thus systematically classifies all the topological terms and the 't Hooft anomalies of d dimensional quantum gauge theories with the above gauge groups. December 2019W −

Generalized ’t Hooft anomalies on non-spin manifolds

“…(3.20) 17 Derivative and field strength insertions may be required in the precise definition of O,Õ. These, however, do not affect the U (1)B and Z8 quantum numbers of relevance here.…”

“…These, however, do not affect the U (1)B and Z8 quantum numbers of relevance here. 18 The IR composite only couples to the gravitational and baryon number backgrounds (2.7), hence Q c = Q f = 0. In addition, since the baryon charge of O is 3, the formula for the index (2.11) has to be modified by multiplying QB by 3 2 and taking dim R f dimRc = 1.…”

“…The study of these generalized anomalies is a currently active area of research with contributions coming from the high-energy, condensed matter, and mathematical communities. We do not claim to be in command of all points of view and only give a list of references written from a (largely) high-energy physics perspective and pertaining to theories somewhat similar to the ones discussed in this paper [7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Deconfinement on axion domain walls

“…Here, L is the S 1 circumference and g is the SU(N c ) gauge coupling, frozen at a scale of order Further, we have dimensionally reduced the four dimensional 2-form gauge field B c(2),4d = A c(1) ∧ dx 3 L + B c (2) into a 3d 1-form A c(1) and a 3d 2-form B c(2) gauge fields. 16 13 Generically, they will lead to an exponentially small shift of the holonomy φ away from the Z…”

“…We refer to these constraints as "generalized 't Hooft anomalies." Their study is evolving too rapidly to allow us to do justice to all interesting aspects currently investigated; we only note that complementary aspects of theories closely related to the ones we consider here are the subject of [6][7][8][9][10][11][12][13][14][15][16][17][18].…”