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(1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is about 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

• http://arxiv.org/abs/1212.1827 , Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases, Ling-Yan Hung, Xiao-Gang Wen
• http://arxiv.org/abs/1303.1803 Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders Xiao-Gang Wen; Phys. Rev. D 88, 045013 (2013)

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).

(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

• http://arxiv.org/abs/1212.1827 , Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases, Ling-Yan Hung, Xiao-Gang Wen
• http://arxiv.org/abs/1303.1803 Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders Xiao-Gang Wen; Phys. Rev. D 88, 045013 (2013)

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).

(1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is about 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 http://arxiv.org/abs/1212.1827 , Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases, Ling-Yan Hung, Xiao-Gang Wen ) 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).

(1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is about 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

• http://arxiv.org/abs/1212.1827 , Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases, Ling-Yan Hung, Xiao-Gang Wen
• http://arxiv.org/abs/1303.1803 Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders Xiao-Gang Wen; Phys. Rev. D 88, 045013 (2013)

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).

(1) To classify "Phase Structure of (Quantum) Gauge Theory" with a gap is about the same as classify 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 http://arxiv.org/abs/1212.1827 , Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases, Ling-Yan Hung, Xiao-Gang Wen ) 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).

(1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is about 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 http://arxiv.org/abs/1212.1827 , Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases, Ling-Yan Hung, Xiao-Gang Wen ) 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).