Two-dimensional BF theory as a CFT Pavel Mnev University of Notre - PowerPoint PPT Presentation

Two-dimensional BF theory as a CFT Pavel Mnev University of Notre Dame, PDMI RAS The Art of Quantization (L.D. Faddeev 85 anniversary conference), PDMI RAS, May 27, 2019 Joint work with Andrey S. Losev and Donald R. Youmans, arXiv:1712.01186, 1902.02738

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary 2D BF theory - reminder Reminder. 2D BF theory: Fix G a group, Σ - a surface. � Σ � B, dA + 1 Action: S cl = 2 [ A, A ] � Fields: A ∈ Ω 1 (Σ , g ) , B ∈ Ω 0 (Σ , g ∗ )

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary 2D BF theory - reminder Reminder. 2D BF theory: Fix G a group, Σ - a surface. � Σ � B, dA + 1 � Action: S cl = 2[ A, A ] � �� � F A Fields: A ∈ Ω 1 (Σ , g ) , B ∈ Ω 0 (Σ , g ∗ ) Equations of motion: F A = 0 , d A B = 0

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary 2D BF theory - reminder Reminder. 2D BF theory: Fix G a group, Σ - a surface. � Σ � B, dA + 1 Action: S cl = 2 [ A, A ] � Fields: A ∈ Ω 1 (Σ , g ) , B ∈ Ω 0 (Σ , g ∗ ) Equations of motion: F A = 0 , d A B = 0 Gauge symmetry: A �→ g − 1 Ag + g − 1 dg , B �→ g − 1 Bg .

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Classical abelian theory Reminder. Abelian 2D BF theory: Fix Σ - a surface. � Action: S cl = Σ B dA Fields: A ∈ Ω 1 (Σ) , B ∈ Ω 0 (Σ) Equations of motion: dA = 0 , dB = 0 Gauge symmetry: A �→ A + dα , B �→ B

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Classical abelian theory Reminder. Abelian 2D BF theory: Fix Σ - a surface. � Action: S cl = Σ B dA Fields: A ∈ Ω 1 (Σ) , B ∈ Ω 0 (Σ) Equations of motion: dA = 0 , dB = 0 Gauge symmetry: A �→ A + dα , B �→ B Want to impose Lorenz gauge d ∗ A = 0

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Gauge-fixing Gauge-fixing. Faddeev-Popov (gauge-fixed) action � B dA + λ d ∗ A + b d ∗ dc S = λ – Lagrangle multiplier, b, c – ghosts

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Gauge-fixing Gauge-fixing. Faddeev-Popov (gauge-fixed) action � B dA + λ d ∗ A + b d ∗ dc S = λ – Lagrangle multiplier, b, c – ghosts A ⊕ R 2 F = Ω 1 ⊕ R [1] ⊕ R [ − 1] B,λ c b  ( A, B, λ, b, c ) – section of  � Σ

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Gauge-fixing Gauge-fixing. Faddeev-Popov (gauge-fixed) action � B dA + λ d ∗ A + b d ∗ dc S = λ – Lagrangle multiplier, b, c – ghosts A ⊕ R 2 F = Ω 1 ⊕ R [1] ⊕ R [ − 1] B,λ c b  ( A, B, λ, b, c ) – section of  � Σ �→ A dc �→ BRST operator Q : b λ �→ B, λ, c 0

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Gauge-fixing Gauge-fixing. Faddeev-Popov (gauge-fixed) action � � S = B dA + λ d ∗ A + b d ∗ dc = S cl + Q b d ∗ A � �� � Ψ − g . f . fermion λ – Lagrangle multiplier, b, c – ghosts A ⊕ R 2 F = Ω 1 ⊕ R [1] ⊕ R [ − 1] B,λ c b   ( A, B, λ, b, c ) – section of � Σ A �→ dc BRST operator Q : b �→ λ B, λ, c �→ 0

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields Complex fields: γ = λ + iB γ = λ − iB , ¯ 2 2

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields Complex fields: γ = λ + iB γ = λ − iB , ¯ ; also split A = a dz + ¯ a d ¯ z ���� ���� 2 2 a ¯ a

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields Complex fields: γ = λ + iB γ = λ − iB , ¯ ; also split A = a dz + ¯ a d ¯ z ���� ���� 2 2 a ¯ a Gauge-fixed action in terms of complex fields � − γ ¯ a + b ∂ ¯ S = 2 i ∂ a + ¯ γ ∂ ¯ ∂ c Σ

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields γ = λ − iB Complex fields: γ = λ + iB , ¯ ; also split A = a dz + ¯ a d ¯ z 2 2 Gauge-fixed action in terms of complex fields � � � γ ¯ a + b ∂ ¯ d 2 z S = 4 ∂a + ¯ γ∂ ¯ ∂c Σ

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields γ = λ − iB Complex fields: γ = λ + iB , ¯ ; also split A = a dz + ¯ a d ¯ z 2 2 Gauge-fixed action in terms of complex fields � � � γ ¯ a + b ∂ ¯ d 2 z S = 4 ∂a + ¯ γ∂ ¯ ∂c Σ ¯ ∂a = 0 = ∂ ¯ a ¯ Equations of motion: ∂γ = 0 = ∂ ¯ γ ∂ ¯ ∂b = 0 = ∂ ¯ ∂c

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields γ = λ − iB Complex fields: γ = λ + iB , ¯ ; also split A = a dz + ¯ a d ¯ z 2 2 Gauge-fixed action in terms of complex fields � � � γ ¯ a + b ∂ ¯ d 2 z S = 4 ∂a + ¯ γ∂ ¯ ∂c Σ ¯ ∂a = 0 = ∂ ¯ a ¯ Equations of motion: ∂γ = 0 = ∂ ¯ γ ∂ ¯ ∂b = 0 = ∂ ¯ ∂c a �→ − ∂c − ¯ ¯ a �→ ∂c BRST operator Q : b �→ γ + ¯ γ γ, ¯ γ, c �→ 0

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Complex fields Complex fields γ = λ − iB Complex fields: γ = λ + iB , ¯ ; also split A = a dz + ¯ a d ¯ z 2 2 Gauge-fixed action in terms of complex fields � � � γ ¯ a + b ∂ ¯ d 2 z S = 4 ∂a + ¯ γ∂ ¯ ∂c Σ ¯ ∂a = 0 = ∂ ¯ a ¯ Equations of motion: ∂γ = 0 = ∂ ¯ γ ∂ ¯ ∂b = 0 = ∂ ¯ ∂c a �→ − ∂c − ¯ ¯ a �→ ∂c BRST operator Q : b �→ γ + ¯ γ γ, ¯ γ, c �→ 0 Note: S is invariant under Weyl transformations of metric

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Stress-energy tensor Stress-energy tensor, BRST current � − 1 � Stress-energy tensor: T µν = − 1 δ δ √ g δg µν S = Q √ g δg µν Ψ � �� � G µν – Q - exact!

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Stress-energy tensor Stress-energy tensor, BRST current � − 1 � Stress-energy tensor: T µν = − 1 δ δ √ g δg µν S = Q √ g δg µν Ψ � �� � G µν – Q - exact! Explicitly: G tot = ( dz ) 2 a ∂b z ) 2 ¯ a ¯ +( d ¯ ∂b ���� � �� � ¯ G G

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Stress-energy tensor Stress-energy tensor, BRST current � − 1 � Stress-energy tensor: T µν = − 1 δ δ √ g δg µν S = Q √ g δg µν Ψ � �� � G µν – Q - exact! Explicitly: G tot = ( dz ) 2 a ∂b z ) 2 ¯ a ¯ +( d ¯ ∂b ���� � �� � ¯ G G T tot = QG tot z ) 2 (¯ ∂b ¯ a ¯ = ( dz ) 2 ( ∂b ∂c + a ∂γ ) + ( d ¯ ∂c + ¯ ∂ ¯ γ ) � �� � � �� � ¯ T T

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Stress-energy tensor Stress-energy tensor, BRST current � − 1 � Stress-energy tensor: T µν = − 1 δ δ √ g δg µν S = Q √ g δg µν Ψ � �� � G µν – Q - exact! Explicitly: G tot = ( dz ) 2 a ∂b z ) 2 ¯ a ¯ +( d ¯ ∂b ���� � �� � ¯ G G T tot = QG tot z ) 2 (¯ ∂b ¯ a ¯ = ( dz ) 2 ( ∂b ∂c + a ∂γ ) + ( d ¯ ∂c + ¯ ∂ ¯ γ ) � �� � � �� � ¯ T T Noether current for Q -symmetry: J tot = 2 i ( dz γ∂c γ ¯ − d ¯ z ¯ ∂c ) ���� � �� � ¯ J J

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Stress-energy tensor Stress-energy tensor, BRST current � − 1 � Stress-energy tensor: T µν = − 1 δ δ √ g δg µν S = Q √ g δg µν Ψ � �� � G µν – Q - exact! Explicitly: G tot = ( dz ) 2 a ∂b z ) 2 ¯ a ¯ +( d ¯ ∂b ���� � �� � ¯ G G T tot = QG tot z ) 2 (¯ ∂b ¯ a ¯ = ( dz ) 2 ( ∂b ∂c + a ∂γ ) + ( d ¯ ∂c + ¯ ∂ ¯ γ ) � �� � � �� � ¯ T T Noether current for Q -symmetry: J tot = 2 i ( dz γ∂c γ ¯ − d ¯ z ¯ ∂c ) ���� � �� � ¯ J J ∂G ∼ 0 ∼ ∂ ¯ ¯ G ∂T ∼ 0 ∼ ∂ ¯ ¯ T Conservation laws: ∂J ∼ 0 ∼ ∂ ¯ ¯ J

2D BF theory - reminder Abelian BF theory Non-abelian BF theory Summary Quantization Quantization (on C ) Fix Σ = C . Correlators � � Φ 1 ( z 1 ) · · · Φ n ( z n ) � = 1 e − 1 4 π S Φ 1 ( z 1 ) · · · Φ n ( z n ) Z fields – given by Wick’s lemma with propagators 1 1 � a ( z ) γ ( w ) � = z − w, � ¯ a ( z )¯ γ ( w ) � = w, z − ¯ ¯ � c ( z ) b ( w ) � = 2 log | z − w | + C Composite fields: Φ( z ) ∈ F z = { differential polynomials in fields a, γ, ¯ γ, b, c } / e . o . m . a, ¯ ( F z , · , Q ) – free cdga, with · the normally-ordered product. F z is freely generated by: ; b, c ; ¯ ∂ k b, ¯ ∂ k c, ¯ a, ¯ ∂ k b, ∂ k c, ∂ l a, ∂ l γ ∂ l ¯ ∂ l ¯ γ with k ≥ 1 , l ≥ 0 � �� � � �� � holom . sector antiholom . sector