Fun with 2-Group Symmetry Po-Shen Hsin California Institute of Technology October 13, 2018 1803.09336 Benini, Córdova, PH
Global Symmetry in QFT • Global symmetry acts on operators and it leaves all correlation functions invariant. Global symmetry can have an ’t Hooft anomaly: in the presence of the background gauge field the partition function transforms by an overall phase. • Anomaly is invariant under the RG flow. Global symmetry and its anomaly provide non-perturbative tools to study the low energy quantum dynamics, which is often strongly coupled. • Applications to dualities and topological phases of matter. • Important to understand the complete global symmetry and its anomaly. Continuous and discrete symmetries, higher-form symmetries, two- group symmetry… 2
Outline • Review 0-form and 1-form symmetries in terms of symmetry defects. • 2-group symmetry. • Anomaly of 2-group symmetry. • Consistency condition on RG flow. 3
Ordinary 0-Form Global Symmetry [Gaiotto,Kapustin,Seiberg,Willett] • Generated by codimension-1 defects that obey group-law fusion 𝑉 𝐡 𝑉 𝐢𝐡 −𝟐 𝑉 𝐢 • Local operators are in representation of the symmetry group. 𝜚(𝑦) = (𝑆 𝐡 𝜚)(𝑦) 𝑉 𝐡 𝑉 𝐡 • The correlation functions of the symmetry defects are topological. • For continuous symmetry described by currents, the symmetry defect is 𝑉 𝐡 = exp 𝑗 ׯ⋆ 𝑘 . Topological property = current conservation 𝑒 ⋆ 𝑘 = 0 . • 1-form gauge field coupled to codimension-1 symmetry generator. 4
1-Form Global Symmetry [Kapustin,Seiberg], [Gaiotto,Kapustin,Seiberg,Willett] • Generated by codimension-2 defects that obey group-law fusion. Symmetry group must be Abelian. • Line operators transform by some charges under the symmetry group. • The correlation functions of the symmetry defects are topological. • 2-form gauge field coupled to codimension-2 symmetry generator. • Example: 4d Maxwell theory has 𝑉 1 × 𝑉(1) 1-form symmetry 𝐹 = 𝐺, 𝑁 =⋆ 𝐺, 𝐹 = 𝑒 ⋆ 𝑘 2 𝑁 = 0 . 𝑘 2 𝑘 2 𝑒 ⋆ 𝑘 2 • Example: 𝑇𝑉(𝑂) gauge theory. The Z 𝑂 center of gauge group assigns Z 𝑂 1-form charges to the Wilson lines. Gauging the Z 𝑂 1-form symmetry modifies the bundle to be the Z 𝑂 quotient 𝑇𝑉(𝑂)/Z 𝑂 . 5
2-Group Global Symmetry: Mixes 0-Form and 1-Form Symmetries [Baez,Lauda],[Baez,Schreiber],[Kapustin,Thorngren],[Sharpe],[Córdova, Dumitrescu,Intriligator],[Delcamp,Tiwari],[Benini,Córdova,PH ]… • 0-form symmetry 𝐻 . 1-form symmetry 𝐵 . • 0-form symmetry acts on 1-form symmetry 𝜍: 𝐻 → Aut 𝐵 . 3 𝐻, 𝐵 : 𝐻 × 𝐻 × 𝐻 → 𝐵 • Postnikov class 𝛾 ∈ 𝐼 𝜍 New 4-junction for symmetry defects. Non-associativity of 0-form symmetry defects. [Benini,Córdova , PH]… 6
2-Group Background Gauge Field [Baez,Lauda],[Baez,Schreiber],[Kapustin,Thorngren],[Sharpe],[Córdova, Dumitrescu,Intriligator],[Delcamp,Tiwari],[Benini,Córdova,PH ]… • Denote background 𝑌 for 0-form symmetry 𝐻 , and background 𝐶 2 for 1-form symmetry 𝐵 . 𝑌 is an 1-cocycle, 𝐶 2 is a 2-cochain that satisfies 𝜀 𝜍 𝐶 2 = 𝑌 ∗ 𝛾 . 3 (𝐶𝐻, 𝐵) is meaningful: 𝛾 → 𝛾 + 𝜀 𝜍 𝜇 2 , 𝐶 2 → 𝐶 2 + 𝑌 ∗ 𝜇 2 . Only 𝛾 ∈ 𝐼 𝜍 • Non-trivial 𝛾 : cannot gauge only the 0-form symmetry. • A 0-form gauge transform also produces a background for 1-form symmetry i.e. inserts a 1-form symmetry defect. • Can gauge only the 1-form symmetry, with 𝑌 = 0 . • If 𝛾 = 0 the 2-group symmetry factorizes into 0- and 1-form symmetries. 7
2-Group Background Gauge Field • If we turn off the background gauge field, then the 2-group symmetry means the correlation functions are invariant under 0-form and 1- form symmetry separately. • If we consider correlation functions with symmetry defects, then the 2-group symmetry implies a particular rule for fusing the 0-form symmetry defects. 8
2-Group Symmetry from Gauging a Subgroup in Mixed Anomaly (Green-Schwarz) [Tachikawa],[Córdova, Dumitrescu,Intriligator] • Two massless Dirac fermions in 4d, 𝑉 1 𝑌 × 𝑉 1 𝑍 0-form symmetry: 𝜔 1 𝜔 2 𝜔 3 𝜔 4 Weyl 5𝑒 𝑌 𝑒𝑌 𝑒𝑍 Mixed anomaly: 𝑙 2𝜌 , 𝑉 1 𝑌 1 -1 0 0 2𝜌 𝑉 1 𝑍 k k -k -k where 𝑙 is an integer. • Next we promote 𝑍 to be dynamical 𝑧 . Emergent 𝑉(1) 1-form symmetry generated by exp 𝑗ׯ 𝑒𝑧 . New background 𝐶 2 couples as 4𝑒 𝐶 2 𝑒𝑧/2𝜌 . Impose constraint on 𝐶 2 to maintain gauge invariance: 𝑌 𝑒𝑌 𝑒𝑧 2𝜌 = 0 ⇒ 𝑒𝐶 2 + 𝑙𝑌 𝑒𝑌 𝑒𝑧 𝑙 න 2𝜌 + න 𝑒𝐶 2 2𝜌 = 0 . 2𝜌 5𝑒 5𝑒 9
2-Group Symmetry from Gauging a Subgroup in Mixed Anomaly (Green-Schwarz) • Gauging 𝑉 1 𝑍 extends 𝑉 1 𝑌 by the emergent 1-form symmetry to become a 2-group symmetry: 𝐻 = 𝑉 1 𝑌 , 𝐵 = 𝑉(1) , 𝜍 = 1 , and the 𝑙 Postnikov class 𝛾 represented by − 2𝜌 𝑌𝑒𝑌 . • Analogous to Green-Schwarz mechanism. 𝑒𝑌 • The condition 𝑒𝐶 2 + 𝑙𝑌 2𝜌 = 0 modifies the gauge transformations 𝑌 → 𝑌 + 𝑒𝜇 0 𝑒𝑌 𝐶 2 → 𝐶 2 + 𝑒𝜇 1 − 𝑙𝜇 0 2𝜌 . Non-trivial background 𝑌 for 0-form symmetry also enforces a background 𝐶 2 for the 1-form symmetry. 10
2-Group Symmetry is Not An Anomaly for 0- Form Symmetry • Require the action 𝐶 2 ⋆ 𝑘 2 + 𝑌 ⋆ 𝑘 1 + ⋯ to be invariant under the 2- group gauge transformation implies the conservation of 0-form symmetry current 𝑘 1 is violated by a non-trivial operator 𝑘 2 , the 1- form symmetry current: 𝑙𝑒𝑌 𝑒 ⋆ 𝑘 1 = 𝑘 2 , 𝑒 ⋆ 𝑘 2 = 0. 2𝜌 • Partition function transforms under a 0-form gauge transformation by an operator insertion instead of a phase. • Not an ’t Hooft anomaly of the 0-form symmetry. 2-group symmetry cannot be ``canceled’’ by inflow. [Córdova,Dumitrescu,Intriligator],[Benini,Córdova, PH] 11
Example: QED3 with 2 Fermions of Charge 2 [Benini,Córdova,PH] • Wilson line of charge 1 is unbreakable and transforms under 𝐵 = Z 2 1-form symmetry corresponds to the Z 2 center in the gauge group. • Two free fermions have at least 𝑉 2 0-form symmetry, neglecting charge conjugation. After gauging 𝑉 1 , the basic monopole operator is dressed with 2 fermion zero modes, and thus the central Z 2 ⊂ 𝑉(2) symmetry that flips the sign of the two fermions does not act on any local operators. • Faithful 0-form symmetry 𝐻 = 𝑉(2)/Z 2 ≅ 𝑇𝑃 3 × 𝑉(1) . The 𝑉 1 is identified with the magnetic symmetry. 12
Example: QED3 with 2 Fermions of Charge 2 • Background 𝑌 for 𝐻 that is not a background for 𝑉(2) : non-trivial 𝑌 ∗ 𝑥 2 (𝐻) 𝑥 2 𝐻 = 𝑥 2 𝑇𝑃 3 + 𝑥 2 (𝑉(1)) is the Z 2 obstruction to lifting the bundle to a 𝑉(2) bundle. • The Z 2 : 𝜔 → −𝜔 in the quotient 𝐻 = 𝑉(2)/Z 2 can be identified with a Z 4 gauge rotation, since the fermions have charge 2. Backgrounds with non- trivial 𝑥 2 (𝐻) modifies the gauge bundle by a Z 4 quotient. [Benini,PH,Seiberg] • The Z 4 quotient requires background 𝐶 2 for Z 2 1-form symmetry 𝜀𝐶 2 = Bock 𝑌 ∗ 𝑥 2 𝐻 = 𝑌 ∗ Bock 𝑥 2 𝐻 . • 2-group symmetry with Postnikov class 𝛾 = Bock 𝑥 2 𝐻 = Bock 𝑥 2 𝑇𝑃 3 . 13
Enhanced 2-Group Symmetry at Low Energy • QED3 with 2 fermions of charge 2 can be obtained from the theory with charge 1 by gauging the Z 2 subgroup magnetic symmetry. • In the theory with charge 1, the 𝑉(1) magnetic symmetry is conjectured to enhance to 𝑇𝑉(2) at low energies, and the UV 0-form symmetry 𝑉(2) is conjectured to enhance to 𝑃(4) . [Xu,You], [PH,Seiberg],[Benini,PH,Seiberg],[Wang,Nahum,Metlitski,Xu,Senthil],[Córdova,PH,Seiberg] • In the theory with charge 2, the same conjecture implies there is an enhanced 2-group symmetry at low energies with 𝐻 IR = 𝑃 4 /Z 2 0- form symmetry, Z 2 1-form symmetry and the Postnikov class 𝛾 IR = Bock 𝑥 2 𝐻 IR = Bock 𝑥 2 𝑄𝑃 4 . 14
Anomaly for 2-Group Symmetry in the UV 4 • QED3 with two fermions of charge 2 has action σ 𝑘 𝑗 ത 𝜔 𝑘 𝛿𝐸 2𝑏 𝜔 𝑘 + 4𝜌 𝑏𝑒𝑏 , where we regularized the massless fermions. The theory is parity invariant. • For non-trivial 2-group background, the gauge bundle has a Z 4 quotient ׯ 𝑒𝑏 2𝜌 = 1 ෫ ∈ 𝑎 2 𝑁, Z 4 , 2 = 2෪ 𝑌 ∗ 𝑥 2 𝐻 4 ׯ 𝑍 2 mod Z , 𝑍 𝐶 2 − where tildes denote a lift to Z 4 cochains. The 2-group constraint implies 𝜀𝑍 2 = 0 and lift-independence. • The theory is not well-defined but has an anomaly for 2-group symmetry 4𝜌 𝑒𝑏𝑒𝑏 = 𝜌 4 න 4 න 𝑍 2 𝑍 2 . 4𝑒 4𝑒 15
Mass deformation • Give large positive masses to charge-2 fermions. The theory flows to 𝑉 1 4 . • The microscopic Z 2 1-form symmetry is enhanced to Z 4 . • The IR theory 𝑉 1 4 couples to the UV 2-group background using the background for the emergent Z 4 1-form symmetry in the IR: ෫ 2 = 2 ෪ 𝑌 ∗ 𝑥 2 𝐻 𝑍 𝐶 2 − . [Kapustin,Seiberg],[Gaiotto, • The Z 4 1- form symmetry has an ’t Hooft anomaly, Kapustin,Seiberg,Willett] 𝜌 𝜌 2 2 = න 4 𝑌 ∗ 𝑥 2 𝐻 2 − 𝜌𝐶 2 𝑌 ∗ 𝑥 2 𝐻 + 𝜌 𝐶 2 2 4 න 𝑍 , 4𝑒 4𝑒 where we omit tildes and use the continuous notation. Matches the anomaly in the UV. 16
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