Black hole Degeneracies from the effective action Guillaume Bossard - PowerPoint PPT Presentation

Black hole Degeneracies from the effective action Guillaume Bossard CPHT, Ecole Polytechnique Texas A & M, April 2018

Outline Heterotic CHL strings and U-duality Protected couplings beyond perturbation theory Decompactification limit and BPS black hole degeneracies [ G. Bossard, C. Cosnier-Horeau, B. Pioline, 1608.01660 ] [ G. Bossard, C. Cosnier-Horeau, B. Pioline, 1702.01926 ] [ G. Bossard, C. Cosnier-Horeau, B. Pioline, 1805.xxxxx ]

Motivations Exact black hole degeneracies from amplitude computations Pioline conjecture: 3D BPS protected amplitudes 4D BPS black holes Black hole degeneracies: Fourier coefficients of modular forms Provides G 3 ⊃ G 4 modular forms generating functions N = 8 supergravity: ∇ 6 R 4 couplings and 1 8 BPS black holes N = 6 supergravity: ∇ 2 R 4 couplings and 1 6 BPS black holes N = 4 supergravity: ∇ 2 F 4 couplings and 1 4 BPS black holes Automorphic representation of the coupling from supersymmetry G 3 nilpotent orbit associated to the black hole solution Explicit for N = 4 string vacua with a notion of S-duality Orbifolds with balanced frame shape: Fricke S-duality

Heterotic string theory on T 6 At a generic point in moduli space N = 4 supergravity with scalar manifold SL (2) / SO (2) × SO (6 , 22) / ( SO (6) × SO (22)) with electric and magnetic charges ( Q , P ) in Λ 6 , 22 ∼ I 6 , 6 ⊕ E 8 ⊕ E 8 = I Also realized as type II on K3 × T 2 Heterotic axion-dilaton S Complex structure U on T 2 in type IIB ahler structure T = B + iV on T 2 in type IIA K¨ T-duality O (6 , 22 , Z ) and S-duality SL (2 , Z ) (T-duality in type II)

CHL orbifolds Chauduri, Hockney, Lykken orbifolds: Z N automorphism of the lattice and shift on a circle. ex: Permutation of the two E 8 factors and Z 2 shift on the circle. 48 For N prime one gets SO (6 , N +1 − 2) with [ Persson, Volpato] Λ 6 , 22 = I I 6 , 6 ⊕ E 8 ⊕ E 8 Λ 6 , 14 = I 5 , 5 ⊕ I I 1 , 1 [ 2 ] ⊕ E 8 [ 2 ] I Λ 6 , 10 = I I 3 , 3 ⊕ I I 3 , 3 [ 3 ] ⊕ A 2 ⊕ A 2 I 3 , 3 ⊕ I Λ 6 , 6 = I I 3 , 3 [ 5 ] � − 4 � − 1 Λ 6 , 4 = I I 2 , 2 ⊕ I I 2 , 2 [ 7 ] ⊕ − 1 − 2   − 2 0 − 1 0 Λ 6 , 2 = I 1 , 1 ⊕ I I 1 , 1 [ 11 ] ⊕ I   0 − 2 0 − 1     − 1 0 − 6 0 − 1 − 6 0 0 → � N i =1 Λ so ( P , P ) → N ( P , P ). Λ[ N ] diagonal embedding Λ ֒ I I 1 , 1 [ N ] is the lattice of momentum m = 0[ N ] and n ∈ Z .

U-duality group At most: Γ ⊂ SL (2 , R ) × O (6 , p ) that preserves Λ ∗ 6 , p ⊕ Λ 6 , p At least: Heterotic T-duality O (6 , p , Z ) automorphism of Λ 6 , p that stabilizes Λ ∗ ˜ 6 , p / Λ 6 , p Type II T-duality I 1 , 1 [ N ] that stabilizes Λ ∗ / Λ Γ 1 ( N ) × Γ 1 ( N ) automorphism of Λ= I I 1 , 1 ⊕ I Γ 1 ( N ) × ˜ O (6 , p , Z ) ⊂ Γ

U-duality group At most: Γ ⊂ SL (2 , R ) × O (6 , p ) that preserves Λ ∗ 6 , p ⊕ Λ 6 , p At most: Heterotic T-duality O (6 , p , Z ) automorphism of Λ ∗ 6 , p Type II T-duality Γ 0 ( N ) × Γ 0 ( N ) ∪ F ⊗ F automorphism of Λ= I I 1 , 1 ⊕ I I 1 , 1 [ N ] Γ 1 ( N ) × ˜ O (6 , p , Z ) ⊂ Γ 0 ( N ) × O (6 , p , Z ) ⊂ Γ Fricke duality F ⊗ σ ∈ Γ � � � � � 0 � � 0 � − 1 − 1 − 1 0 0 √ √ − 1 √ √ N ⊗ N = ⊗ N N 0 1 0 0 0 N N

U-duality group N-modular lattice (1-modular is selfdual) Λ 6 , p There exists σ ∈ O (6 , p ) such that σ Λ ∗ √ 6 , p = Λ 6 , p N then Fricke S-duality √ � � − σ S → − 1 N σ − 1 Q √ ( Q , P ) → P , , NS N

1/2 BPS couplings For SL (2) × SO (6 , p ) with k = p +2 24 = 2 N +1 � � 1 � d 4 x √− g log( S k − 1 2 | ∆ k ( S ) | 2 ) 8 R µνσρ R µνσρ +3 t 8 F a F b F a F b 8 π 2 � d 4 x √− gF abcd ( φ ) t 8 F a F b F c F d + with the weight k Γ 0 ( N ) cusp form ∆ k ( τ ) = η k ( τ ) η k ( N τ ) and � d 2 τ 1 F abcd ( φ ) = ∆ k ( τ )Γ Λ 6 , p [ P abcd ] τ 2 Γ 0 ( N ) \H + 2 with (function of V ( φ ) ∈ SO (6 , p ) / ( SO (6) × SO ( p )) through the left and right projections, with Q 2 L − Q 2 R = ( Q , Q ) and Q 2 L + Q 2 R = | V ( φ ) Q | 2 ) � � � 3 3 e i πτ Q 2 τ Q 2 Γ Λ6 , p [ P abcd ] = τ 3 L − i π ¯ Q La Q Lb Q Lc Q Ld − δ ( ab Q Lc Q Ld ) + (4 πτ 2 ) 2 δ ( ab δ cd ) R 2 2 πτ 2 Q ∈ Λ6 , p

1/2 BPS couplings ∆ k ( τ ) = η k ( τ ) η k ( N τ ) is Fricke invariant √ ∆ k ( − 1 N τ ) k ∆ k ( τ ) N τ ) = ( − i and so the effective action is invariant under S → − 1 NS and V ( φ ) → V ( φ ) σ , and � d 2 τ 1 F abcd ( φ σ ) = ∆ k ( τ )Γ σ Λ 6 , p [ P abcd ] τ 2 Γ 0 ( N ) \H + 2 � d 2 τ 1 ∆ k ( τ )Γ √ = 6 , p [ P abcd ] N Λ ∗ τ 2 Γ 0 ( N ) \H + 2 = F abcd ( φ )

Heterotic string theory on T 7 At a generic point in moduli space N = 8 supergravity with scalar manifold SO (8 , 24) / ( SO (8) × SO (24)) with U-duality group, the automorphism group O (8 , 24 , Z ) of I 8 , 8 ⊕ E 8 ⊕ E 8 I [ Sen] Not the solitons, monodromy of the scalars modulo O (8 , 24 , Z ) In general Γ 4 × O (7 , 1 + p , Z ) ֒ → O (8 , 2 + p , Z ) the automorphism group of the N-modular non-perturbative lattice Λ 8 , p +2 = I I 1 , 1 ⊕ I I 1 , 1 [ N ] ⊕ Λ 6 , p

Non-pertubative couplings Supersymmetry implies differential equations O (8 , 2 + p , Z ) invariance Boundary conditions: cusps (perturbative g s → 0 and decompactification R → ∞ ) singularities (surfaces of enhanced gauge symmetry) Determines the non-perturbative BPS couplings 1/2 BPS ( ∇ φ ) 4 : 1-loop � d 2 τ 1 F αβγδ ( φ ) = ∆ k ( τ )Γ Λ 7 , p +1 [ P αβγδ ] τ 2 Γ 0 ( N ) \H + 2 1/4 BPS ( ∇ 2 φ )( ∇ φ ) 2 : 2-loop � d 6 Ω 1 G αβ,γδ ( φ ) = Φ k − 2 (Ω)Γ Λ 7 , p +1 [ P αβ,γδ ] ( det Ω 2 ) 3 Γ 0 ( N ) \H +

Non-pertubative couplings Supersymmetry implies differential equations O (8 , 2 + p , Z ) invariance Boundary conditions: cusps (perturbative g s → 0 and decompactification R → ∞ ) singularities (surfaces of enhanced gauge symmetry) Determines the non-perturbative BPS couplings [ Obers Pioline] 1/2 BPS ( ∇ φ ) 4 : � d 2 τ 1 F abcd ( φ ) = ∆ k ( τ )Γ Λ 8 , p +2 [ P abcd ] τ 2 Γ 0 ( N ) \H + 2 1/4 BPS ( ∇ 2 φ )( ∇ φ ) 2 : � d 6 Ω 1 G ab , cd ( φ ) = Φ k − 2 (Ω)Γ Λ 8 , p +2 [ P ab , cd ] ( det Ω 2 ) 3 Γ 0 ( N ) \H +

At large radius � � � � 3 F (8 , p +2) R 2 log( S k 2 | ∆ k ( S ) | 4 ) − 2 k log R δ ( αβ δ γδ ) + ˆ F (6 , p ) = − αβγδ ( φ ) αβγδ 2(2 π ) 2 P ( ℓ ) 2 � � αβδγ ( Q L , P L ) K 2 − ℓ (2 π R M ( Q R , P R )) R 4 ¯ e 2 π i ( Q · a 1+ P · a 2) +4 c ( Q , P ) R 2 ℓ M ( Q R , P R ) 2 − ℓ ( Q , P ) ∈ Λ ∗ ℓ =0 6 , p ⊕ Λ6 , p Q ∧ P =0 � � � e 2 π i ( P · a 2+ M 1( ψ − 1 2 a 1 · a 2)+( M 2 − a 1 · P + 1 F M 1 2 ( a 1 · a 1) M 1) S 1) P − M 1 a 1 , M 2 − a 1 · P + 1 + 2 ( a 1 · a 1) M 1 αβγδ M 1 � =0 , M 2 P ∈ Λ6 , p with � 2 π � 3 P ( ℓ ) ˜ 2 � αβγδ ( P L ) 2 − ℓ αβγδ ( P , M 2 ) = 4 ( R 2 S 2 ) 3 ¯ F M 1 c ( M 1 , M 2 , P ) 2 − ℓ ( S cl ) K 3 ( R 2 S 2 ) ℓ S cl ℓ =0 ∆ k ( τ ) = � 1 m ≥− 1 c k ( m ) e 2 π im τ ) and the instanton measure ( � � � � � � − gcd( NQ 2 , P 2 , Q · P ) − gcd( NQ 2 , P 2 , Q · P ) c k ( Q , P ) = ¯ c k + c k d 2 Nd 2 d ≥ 1 d ≥ 1 ( Q , P ) / d ∈ Λ ∗ ( Q , P ) / d ∈ Λ 6 , p ⊕ N Λ ∗ 6 , p ⊕ Λ 6 , p 6 , p

At large radius � � � � 3 F (8 , p +2) R 2 log( S k 2 | ∆ k ( S ) | 4 ) − 2 k log R F (6 , p ) δ ( αβ δ γδ ) + ˆ = − αβγδ ( φ ) αβγδ 2(2 π ) 2 P ( ℓ ) 2 � � αβδγ ( Q L , P L ) K 2 − ℓ (2 π R M ( Q R , P R )) R 4 ¯ e 2 π i ( Q · a 1+ P · a 2) +4 c ( Q , P ) R 2 ℓ M ( Q R , P R ) 2 − ℓ ( Q , P ) ∈ Λ ∗ ℓ =0 6 , p ⊕ Λ6 , p Q ∧ P =0 � � � e 2 π i ( P · a 2+ M 1( ψ − 1 2 a 1 · a 2)+( M 2 − a 1 · P + 1 F M 1 2 ( a 1 · a 1) M 1) S 1) P − M 1 a 1 , M 2 − a 1 · P + 1 + 2 ( a 1 · a 1) M 1 αβγδ M 1 � =0 , M 2 P ∈ Λ6 , p with � 2 π � 3 P ( ℓ ) 2 ˜ � 2 − ℓ αβγδ ( P L ) αβγδ ( P , M 2 ) = 4 ( R 2 S 2 ) 3 ¯ F M 1 c ( M 1 , M 2 , P ) K 3 2 − ℓ ( S cl ) ( R 2 S 2 ) ℓ S cl ℓ =0 ∆ k ( τ ) = � 1 m ≥− 1 c k ( m ) e 2 π im τ ) and the instanton measure ( − gcd( NQ 2 , P 2 , Q · P ) � � � � c k ( Q , P ) = ¯ c k nd 2 d ≥ 1 n | N ( Q , P ) / d ∈ Λ ∗ 6 , p [ n ] ⊕ Λ 6 , p [ n ]

At large radius � 3 � � ˆ E 1 ( NS ) + ˆ � E 1 ( S ) + k F (8 , p +2) R 2 F (6 , p ) δ ( αβ δ γδ ) + ˆ = 2 π log R αβγδ ( φ ) αβγδ 2 π N + 1 P ( ℓ ) 2 � � αβδγ ( Q L , P L ) K 2 − ℓ (2 π R M ( Q R , P R )) R 4 ¯ e 2 π i ( Q · a 1+ P · a 2) +4 c ( Q , P ) R 2 ℓ M ( Q R , P R ) 2 − ℓ ( Q , P ) ∈ Λ ∗ ℓ =0 6 , p ⊕ Λ6 , p Q ∧ P =0 � � � e 2 π i ( P · a 2+ M 1( ψ − 1 2 a 1 · a 2)+( M 2 − a 1 · P + 1 F M 1 2 ( a 1 · a 1) M 1) S 1) P − M 1 a 1 , M 2 − a 1 · P + 1 + 2 ( a 1 · a 1) M 1 αβγδ M 1 � =0 , M 2 P ∈ Λ6 , p with � 2 π � 3 P ( ℓ ) ˜ 2 � αβγδ ( P L ) 2 − ℓ αβγδ ( P , M 2 ) = 4 ( R 2 S 2 ) 3 ¯ F M 1 c ( M 1 , M 2 , P ) K 3 2 − ℓ ( S cl ) ( R 2 S 2 ) ℓ S cl ℓ =0 ∆ k ( τ ) = � 1 m ≥− 1 c k ( m ) e 2 π im τ ) and the instanton measure ( � − gcd( NQ 2 , P 2 , Q · P ) � � � c k ( Q , P ) = ¯ c k nd 2 d ≥ 1 n | N ( Q , P ) / d ∈ Λ ∗ 6 , p [ n ] ⊕ Λ 6 , p [ n ]