# Phase Structure of (Quantum) Gauge Theory

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?