Question: How to classify/characterize the phase structure of (quantum) gauge theory?
Gauge Theory (say with a gauge group $G_g$) is a powerful quantum field theoretic(QFT) tool to describe many-body quantum nature (because QFT naturally serves for understanding many-body quantum problem with (quasi-)particle creation/annihilation).
Classification of gauge theory shall be something profound, in a sense that gauge fields (p-form $A_\mu$, $B_{\mu\nu}$, or connections of $G_g$-bundle etc) are just mediators propagating the interactions between matter fields (fermion $\psi$, boson $\phi$). Thus, effectively, we may "integrate out" or "smooth over" the matter fields, to obtain an effective gauge theory described purely by gauge fields ($A_\mu$, $B_{\mu\nu}$, etc).
Characterization of gauge theory should NOT simply rely on its gauge group $G_g$, due to "Gauge symmetry is not a symmetry". We should not classify (its distinct or the same phases) or characterize (its properties) ONLY by the gauge group $G_g$. What I have been taught is that some familiar terms to describe the phase structure of (quantum) gauge theories, are:
(1) confined or deconfined
(2) gapped or gapless
(3) Higgs phase
(4) Coulomb phase
(5) topological or not.
(6) weakly-coupling or strongly-coupling
sub-Question A.:
Is this list above (1)-(6) somehow enough to address the phase structure of gauge theory? What are other important properties people look for to classify/characterize the phase structure of gauge theory? Like entanglement? How?
(for example, in 2+1D gapped deconfined weak-coupling topological gauge theory with finite ground state degeneracy on the $\mathbb{T}^2$ torus describes anyons can be classified/characterized by braiding statistics $S$ matrix (mutual statistics) and $T$ (topological spin) matrix.)
sub-Question B.:
Are these properties (1)-(6) somehow related instead of independent to each other?
It seems to me that confined of gauge fields implies that the matter fields are gapped? Such as 3+1D Non-Abelian Yang-Mills at IR low energy has confinement, then we have the Millennium prize's Yang–Mills(YM) existence and mass gap induced gapped mass $\Delta>0$ for the least massive particle, both(?) for the matter field or the gauge fields (glueball?). So confinement and gapped mass $\Delta>0$ are related for 3+1D YM theory. Intuitively, I thought confinement $\leftrightarrow$ gapped, deconfinement $\leftrightarrow$ gapless.
However, in 2+1D, condensed matter people study $Z_2$, U(1) spin-liquids, certain kind of 2+1D gauge theory, one may need to ask whether it is (1) confined or deconfined, (2) gapped or gapless, separate issues. So in 2+1D case, the deconfined can be gapped? the confined can be gapless? Why is that? Should one uses Renormalization group(RG) argument by Polyakov? how 2+1D/3+1D RG flow differently affect this (1) confined or deconfined, (2) gapped or gapless, separate issues?
sub-Question C.: are there known mathematical objects to classify gauge theory?
perhaps, say other than/beyond the recently-more-familiar group cohomology: either topological group cohomology $H^{d+1}(BG_g,U(1))$ using classifying space $BG_g$ of $G_g$, or Borel group cohomology $\mathcal{H}^{d+1}(G_g,U(1))$ recently studied in SPT and topological gauge theory and Dijkgraaf-Witten?
Best Answer
(1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is roughly the same as classifying phase structure of topologically ordered states. Some topologically ordered states are described by a group and can be related to a gauge theory. Some other topologically ordered states are not related to gauge theory.
(2) One way to classify "Phase Structure of (Quantum) Gauge Theory" is to classify topological terms in weak-coupling gauge theories. See
But the classification is not one-to-one: different topological terms and different gauge groups can correspond to the same gapped phase (with the same topological order).