SLIDE 1
Black hole Degeneracies from the effective action Guillaume Bossard - - PowerPoint PPT Presentation
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
SLIDE 2
SLIDE 3
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 G3 ⊃ G4 modular forms generating functions N = 8 supergravity: ∇6R4 couplings and 1
8 BPS black holes
N = 6 supergravity: ∇2R4 couplings and 1
6 BPS black holes
N = 4 supergravity: ∇2F 4 couplings and 1
4 BPS black holes
Automorphic representation of the coupling from supersymmetry G3 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
SLIDE 4
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 Λ6,22 ∼ = I I6,6 ⊕E8 ⊕E8 Also realized as type II on K3×T 2 Heterotic axion-dilaton S Complex structure U on T 2 in type IIB K¨ ahler structure T = B + iV onT 2 in type IIA T-duality O(6, 22, Z) and S-duality SL(2, Z) (T-duality in type II)
SLIDE 5
CHL orbifolds
Chauduri, Hockney, Lykken orbifolds: ZN automorphism of the lattice and shift on a circle. ex: Permutation of the two E8 factors and Z2 shift on the circle. For N prime one gets SO(6,
48 N+1 − 2) with
[ Persson, Volpato]
Λ6,22 = I I6,6 ⊕ E8 ⊕ E8 Λ6,14 = I I5,5 ⊕ I I1,1[2] ⊕ E8[2] Λ6,10 = I I3,3 ⊕ I I3,3[3] ⊕ A2 ⊕ A2 Λ6,6 = I I3,3 ⊕ I I3,3[5] Λ6,4 = I I2,2 ⊕ I I2,2[7] ⊕
−4 −1 −1 −2
- Λ6,2
= I I1,1 ⊕ I I1,1[11] ⊕
−2 −1 −2 −1 −1 −6 −1 −6
Λ[N] diagonal embedding Λ ֒ → N
i=1 Λ so (P, P) → N(P, P).
I I1,1[N] is the lattice of momentum m = 0[N] and n ∈ Z.
SLIDE 6
U-duality group
At most: Γ ⊂ SL(2, R) × O(6, p) that preserves Λ∗
6,p ⊕ Λ6,p
At least: Heterotic T-duality ˜ O(6, p, Z) automorphism of Λ6,p that stabilizes Λ∗
6,p/Λ6,p
Type II T-duality
Γ1(N)×Γ1(N) automorphism of Λ=I I1,1⊕I I1,1[N] that stabilizes Λ∗/Λ
Γ1(N) × ˜ O(6, p, Z) ⊂ Γ
SLIDE 7
U-duality group
At most: Γ ⊂ SL(2, R) × O(6, p) that preserves Λ∗
6,p ⊕ Λ6,p
At most: Heterotic T-duality O(6, p, Z) automorphism of Λ∗
6,p
Type II T-duality
Γ0(N)×Γ0(N)∪F⊗F automorphism of Λ=I I1,1⊕I I1,1[N]
Γ1(N) × ˜ O(6, p, Z) ⊂ Γ0(N) × O(6, p, Z) ⊂ Γ Fricke duality F ⊗ σ ∈ Γ
- −1
√ N
√ N
- ⊗
- −1
√ N
√ N
- =
−1 N
- ⊗
−1 N
1
SLIDE 8
U-duality group
N-modular lattice (1-modular is selfdual) Λ6,p There exists σ ∈ O(6, p) such that Λ∗
6,p =
σ √ N Λ6,p then Fricke S-duality (Q, P) →
- − σ
√ N P, √ Nσ−1Q
- ,
S → − 1 NS
SLIDE 9
1/2 BPS couplings
For SL(2) × SO(6, p) with k = p+2
2
=
24 N+1
− 1 8π2
- d4x√−g log(Sk
2 |∆k(S)|2)
1 8RµνσρRµνσρ+3t8F aF bFaFb
- +
- d4x√−gFabcd(φ)t8F aF bF cF d
with the weight k Γ0(N) cusp form ∆k(τ) = ηk(τ)ηk(Nτ) and Fabcd(φ) =
- Γ0(N)\H+
d2τ τ 2
2
1 ∆k(τ)ΓΛ6,p[Pabcd] with (function of V (φ) ∈ SO(6, p)/(SO(6) × SO(p)) through the left and right projections, with
Q2
L − Q2 R = (Q, Q) and Q2 L + Q2 R = |V (φ)Q|2)
ΓΛ6,p [Pabcd ] = τ3
2
- Q∈Λ6,p
- QLaQLbQLc QLd −
3 2πτ2 δ(abQLc QLd) + 3 (4πτ2)2 δ(abδcd)
- eiπτQ2
L−iπ ¯ τQ2 R
SLIDE 10
1/2 BPS couplings
∆k(τ) = ηk(τ)ηk(Nτ) is Fricke invariant ∆k(− 1 Nτ ) = (−i √ Nτ)k∆k(τ) and so the effective action is invariant under S → − 1
NS and
V (φ) → V (φ)σ, and Fabcd(φσ) =
- Γ0(N)\H+
d2τ τ 2
2
1 ∆k(τ)ΓσΛ6,p[Pabcd] =
- Γ0(N)\H+
d2τ τ 2
2
1 ∆k(τ)Γ√
NΛ∗
6,p[Pabcd]
= Fabcd(φ)
SLIDE 11
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 I8,8 ⊕ E8 ⊕ E8
[ Sen]
Not the solitons, monodromy of the scalars modulo O(8, 24, Z) In general Γ4 × O(7, 1 + p, Z) ֒ → O(8, 2 + p, Z) the automorphism group of the N-modular non-perturbative lattice Λ8,p+2 = I I1,1 ⊕ I I1,1[N] ⊕ Λ6,p
SLIDE 12
Non-pertubative couplings
Supersymmetry implies differential equations O(8, 2 + p, Z) invariance Boundary conditions: cusps (perturbative gs → 0 and decompactification R → ∞) singularities (surfaces of enhanced gauge symmetry) Determines the non-perturbative BPS couplings 1/2 BPS (∇φ)4: 1-loop Fαβγδ(φ) =
- Γ0(N)\H+
d2τ τ 2
2
1 ∆k(τ)ΓΛ7,p+1[Pαβγδ] 1/4 BPS (∇2φ)(∇φ)2: 2-loop Gαβ,γδ(φ) =
- Γ0(N)\H+
d6Ω (detΩ2)3 1 Φk−2(Ω)ΓΛ7,p+1[Pαβ,γδ]
SLIDE 13
Non-pertubative couplings
Supersymmetry implies differential equations O(8, 2 + p, Z) invariance Boundary conditions: cusps (perturbative gs → 0 and decompactification R → ∞) singularities (surfaces of enhanced gauge symmetry) Determines the non-perturbative BPS couplings
[ Obers Pioline]
1/2 BPS (∇φ)4: Fabcd(φ) =
- Γ0(N)\H+
d2τ τ 2
2
1 ∆k(τ)ΓΛ8,p+2[Pabcd] 1/4 BPS (∇2φ)(∇φ)2: Gab,cd(φ) =
- Γ0(N)\H+
d6Ω (detΩ2)3 1 Φk−2(Ω)ΓΛ8,p+2[Pab,cd]
SLIDE 14
At large radius
F (8,p+2)
αβγδ
= R2
- −
3 2(2π)2
- log(S k
2 |∆k(S)|4) − 2k log R
- δ(αβδγδ) + ˆ
F (6,p)
αβγδ(φ)
- +4
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0
R4 ¯ c(Q, P)
2
- ℓ=0
P(ℓ)
αβδγ(QL, PL)
R2ℓ K2−ℓ (2πRM(QR, PR)) M(QR, PR)2−ℓ e2πi(Q·a1+P·a2) +
- M1=0,M2
P∈Λ6,p
F M1
αβγδ
- P−M1a1,M2−a1·P+ 1
2 (a1·a1)M1
- e2πi(P·a2+M1(ψ− 1
2 a1·a2)+(M2−a1·P+ 1 2 (a1·a1)M1)S1)
with
F M1
αβγδ(P, M2) = 4 (R2S2)3 ¯
c(M1, M2, P)
2
- ℓ=0
˜ P(ℓ)
αβγδ(PL)
(R2S2)ℓ 2π Scl 3
2 −ℓ
K 3
2 −ℓ(Scl)
and the instanton measure (
1 ∆k(τ) = m≥−1 ck(m)e2πimτ)
¯ ck(Q, P) =
- d≥1
(Q,P)/d ∈Λ∗
6,p⊕Λ6,p
ck
- − gcd(NQ2,P2,Q·P)
d2
- +
- d≥1
(Q,P)/d ∈Λ6,p⊕NΛ∗
6,p
ck
- − gcd(NQ2,P2,Q·P)
Nd2
SLIDE 15
At large radius
F (8,p+2)
αβγδ
= R2
- −
3 2(2π)2
- log(S k
2 |∆k(S)|4) − 2k log R
- δ(αβδγδ) + ˆ
F (6,p)
αβγδ(φ)
- +4
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0
R4 ¯ c(Q, P)
2
- ℓ=0
P(ℓ)
αβδγ(QL, PL)
R2ℓ K2−ℓ (2πRM(QR, PR)) M(QR, PR)2−ℓ e2πi(Q·a1+P·a2) +
- M1=0,M2
P∈Λ6,p
F M1
αβγδ
- P−M1a1,M2−a1·P+ 1
2 (a1·a1)M1
- e2πi(P·a2+M1(ψ− 1
2 a1·a2)+(M2−a1·P+ 1 2 (a1·a1)M1)S1)
with
F M1
αβγδ(P, M2) = 4 (R2S2)3 ¯
c(M1, M2, P)
2
- ℓ=0
˜ P(ℓ)
αβγδ(PL)
(R2S2)ℓ 2π Scl 3
2 −ℓ
K 3
2 −ℓ(Scl)
and the instanton measure (
1 ∆k(τ) = m≥−1 ck(m)e2πimτ) ¯ ck(Q, P) =
- n|N
- d≥1
(Q,P)/d ∈Λ∗
6,p[n]⊕Λ6,p[n]
ck
- − gcd(NQ2,P2,Q·P)
nd2
SLIDE 16
At large radius
F (8,p+2)
αβγδ
= R2 3 2π ˆ E1(NS) + ˆ E1(S) N + 1 + k 2π log R
- δ(αβδγδ) + ˆ
F (6,p)
αβγδ(φ)
- +4
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0
R4 ¯ c(Q, P)
2
- ℓ=0
P(ℓ)
αβδγ(QL, PL)
R2ℓ K2−ℓ (2πRM(QR, PR)) M(QR, PR)2−ℓ e2πi(Q·a1+P·a2) +
- M1=0,M2
P∈Λ6,p
F M1
αβγδ
- P−M1a1,M2−a1·P+ 1
2 (a1·a1)M1
- e2πi(P·a2+M1(ψ− 1
2 a1·a2)+(M2−a1·P+ 1 2 (a1·a1)M1)S1)
with
F M1
αβγδ(P, M2) = 4 (R2S2)3 ¯
c(M1, M2, P)
2
- ℓ=0
˜ P(ℓ)
αβγδ(PL)
(R2S2)ℓ 2π Scl 3
2 −ℓ
K 3
2 −ℓ(Scl)
and the instanton measure (
1 ∆k(τ) = m≥−1 ck(m)e2πimτ) ¯ ck(Q, P) =
- n|N
- d≥1
(Q,P)/d ∈Λ∗
6,p[n]⊕Λ6,p[n]
ck
- − gcd(NQ2,P2,Q·P)
nd2
SLIDE 17
1/2 BPS dyons
The 1/2 BPS dyons (Q, P) with Q ∧ P = 0 have degeneracy ¯ ck(Q, P) for primitive charges in either Λ∗
6,p ⊕ Λ6,p (n = 1) or
Λ6,p ⊕ NΛ∗
6,p (n = N).
M(QR, PR) is the 1/2 BPS mass of such dyons M(QR, PR) =
- 2(QR + SPR)(QR + ¯
SPR) S2 and R = e2U|∞ with ds2 = −e2Udt2 + e−2Udxidxi RM(QR, PR) = Scl(Q, P, φ|∞) Instantons are complex saddle points in the path integral
SO(8, p + 2)/(SO(6, 2) × SO(2, p)) ⊂ SO(8, p + 2, C)/(SO(8, C) × SO(p + 2, C))
SLIDE 18
Genus 2
The moduli space of genus 2 Riemann surfaces M2,0 ∼ = Sp(4, Z)\Sp(4, R)/U(2) M1,0 ×Z2 M1,0 The Siegel upper half plan minus the separating degeneration locus. Ω = Ω1 + iΩ2, with Ω2 a 2 by 2 positive matrix, with separating degeneration for Ω =
τ1 0 0 τ2
- .
Genus 2 Narain partition function with symmetrisation
ΓΛq,p [Pab,cd] = det(Ω2)
q 2
- Qi ∈Λq,p
- εijεklQL(aiQLb)kQL(cjQLd)l −
3 2π Ω−1ij 2
δab,QLciQLdj +
3 (4π)2detΩ2 δab,δcd
- eiπΩij QLi QLj −iπ ¯
Ωij QRi QRj
SLIDE 19
Siegel modular forms
Φk−2|γ = det(CΩ + D)2−kΦk−2
- (AΩ + B)(CΩ + D)−1
= Φk−2(Ω) with γ an element A B C D
- satisfying to
ABT = BAT , DC T = CDT , ADT − BC T = 1 Sp(4, Z) for A, B, C, D are 2 by 2 matrices over Z. Γ0(N) ⊂ Sp(4, Z) for C = 0[N]. Γ1(N) ⊂ Γ0(N) for A = 1[N] and D = 1[N]. Γ(N) ⊂ Γ1(N) for B = 0[N].
SLIDE 20
Combining twisted sectors
The observation is that the genus 2 amplitude is gouverned by a single function invariant under Γ0(N) for the lattice Λ7,p+1 = I I1,1[N] ⊕ ˜ Λ6,p = I I1,1 ⊕ Λ6,p , we have
[ D’Hoker and Phong]
Z 00
00
- =
- γ∈Sp(4)/Γ0(N)
- 1
Φk−2 ΓI
I1,1⊕˜ Λ6,p[Pab,cd]
- γ
Z 00
01
- =
- γ∈Γe1/Γ0,e1(N)
- 1
Φk−2 ΓI
I1,1⊕˜ Λ6,p[e
2πi N n2Pab,cd]
- γ
and
1 N2
- hi ,gi ∈Z/(NZ)
Z h1h2
g1g2
- =
- γ∈Sp(4)/Γ0(N)
- 1
Φk−2 1 N2 ΓI
I1,1 ⊕ ˜ Λ6,p[Pab,cd] +
- γ′ ∈ Γ0(N)/Γ0,e1(N)
ΓI
I1,1 ⊕ ˜ Λ6,p[e
2πi N n2Pab,cd]
- γ′
- γ
SLIDE 21
Combining twisted sectors
The observation is that the genus 2 amplitude is gouverned by a single function invariant under Γ0(N) for the lattice Λ7,p+1 = I I1,1[N] ⊕ ˜ Λ6,p , we have
1 N2
- hi ,gi ∈Z/(NZ)
Z h1h2
g1g2
- =
- γ∈Sp(4)/Γ0(N)
- 1
Φk−2 1 N2 ΓI
I1,1 ⊕ ˜ Λ6,p[Pab,cd] +
- γ′ ∈ Γ0(N)/Γ0,e1(N)
ΓI
I1,1 ⊕ ˜ Λ6,p[e
2πi N n2Pab,cd]
- γ′
- γ
=
- γ∈Sp(4)/Γ0(N)
- 1
Φk−2 ΓI
I1,1[N] ⊕ ˜ Λ6,p[Pab,cd]
- γ
and so
Gab,cd =
- Sp(4)\H+
d6Ω (detΩ2)3 1 N2
- hi ,gi ∈Z/(NZ)
Z h1h2
g1g2
- =
- Γ0(N)\H+
d6Ω (detΩ2)3 1 Φk−2 ΓΛ7,p+1 [Pab,cd ]
SLIDE 22
Combining twisted sectors
Φ10 from
[ D’Hoker and Phong] .
Computes Φ6 explicitly in Z2 model. Consistancy requires Γ0(N) invariance the splitting at the seperating degeneration Φk−2 τ1 ǫ
ǫ τ2
- = −(2πǫ)2∆k(τ1)∆k(τ2) + O(ǫ3)
The pole at the maximal degeneration at large Ω2
1 Φk−2(Ω) ∼ e−πitr Ω
- 2 −1
−1 2
- + 2e−πitr Ω
2 0 0 0
- + 2e−πitr Ω
0 0 0 2
- + 2e−πitr Ω
- 2 −2
−2 2
- determines Φk−2 as the known function associated
to the twisted orbifold K3 elliptic genus
SLIDE 23
At large radius
G (8,p+2)
αβ,γδ
= R4
- − 1
2π (δαβδγδ − δα(γδδ)β) ˆ E1(NS) + ˆ E1(S) N + 1 2 − 1 4 δαβ, N ˆ E1(NS) − ˆ E1(S) N2 − 1 G (6,p)
γδ (φ) + N ˆ
E1(S) − ˆ E1(NS) N2 − 1 G (6,p)
γδ (φσ)
- +
G (6,p)
αβ,γδ(φ)
- −6
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0
R5 ¯ Gαβ,(Q, P, φ)
1
- ℓ=0
P(ℓ)
δγ(QL, PL)
Rℓ K1−ℓ (2πRM(QR, PR)) M(QR, PR)2−ℓ e2πi(Q·a1+P·a2) +2
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0
R7 ¯ c(Q, P)
2
- ℓ=0
P(ℓ)
αβ,δγ(QL, PL)
Rℓ B 3
2 −[ ℓ 2 ]
- 2R2
S2
- 1
S1 S1 |S|2 Q2
R
QRPR QRPR P2
R
- |QR ∧ PR|
3 2 −[ ℓ 2 ]
e2πi(Q·a1+P·a2) +
- M1=0,M2
P∈Λ6,p
G M1
αβ,γδ
- P−M1a1,M2−a1·P+ 1
2 (a1·a1)M1
- e2πi(P·a2+M1(ψ− 1
2 a1·a2)+(M2−a1·P+ 1 2 (a1·a1)M1)S1)
SLIDE 24
At large radius
Matches the heteroric perturbative result
− 1 2π (δαβδγδ − δα(γδδ)β) ˆ E1(NS) + ˆ E1(S) N + 1 2 − 1 4 δαβ, N ˆ E1(NS) − ˆ E1(S) N2 − 1 G (6,p)
γδ (φ) + N ˆ
E1(S) − ˆ E1(NS) N2 − 1 G (6,p)
γδ (φσ)
- +
G (6,p)
αβ,γδ(φ)
∼ − 1 2πg 4
s
(δαβδγδ − δα(γδδ)β) − 1 4g 2
s
δαβ,G (6,p)
γδ (φ) +
G (6,p)
αβ,γδ(φ)
It is consistant with T-duality in type II perturbation theory
− 1 2π (δαβδγδ − δα(γδδ)β) ˆ E1(NU) + ˆ E1(U) N + 1 2 − 1 4 δαβ, N ˆ E1(NU) − ˆ E1(U) N2 − 1 G (6,p)
γδ (φ) + N ˆ
E1(U) − ˆ E1(NU) N2 − 1 G (6,p)
γδ (φσ)
- +
G (6,p)
αβ,γδ(φ)
∼ 1 g 4
s
G (4,p−2)
αβ,γδ (φ) + 2N
3g 2
s
(δαβδγδ − δα(γδδ)β) (ˆ
E1(NU)−ˆ E1(U))(ˆ E1(NT)−ˆ E1(T)) N2−1
− 1 4g 2
s
δαβ,
ˆ E1(NU)+ˆ E1(U)+ˆ E1(NT)+ˆ E1(T) N+1
- G (4,p−2)
γδ
(φ) − 1 2π (δαβδγδ − δα(γδδ)β) ˆ
E1(NU)+ˆ E1(U) N+1
2 + 2
ˆ E1(NU)ˆ E1(NT)+ˆ E1(U)ˆ E1(T) N+1
+ ˆ
E1(NT)+ˆ E1(T) N+1
2
SLIDE 25
At large radius
Matches the heteroric perturbative result on T 6
− 1 2π (δαβδγδ − δα(γδδ)β)ˆ E1(S) 2 − 1 4 δαβ, ˆ E1(S)G (6,22)
γδ (φ) +
G (6,22)
αβ,γδ(φ)
∼ − 1 2πg 4
s
(δαβδγδ − δα(γδδ)β) − 1 4g 2
s
δαβ,G (6,22)
γδ (φ) +
G (6,22)
αβ,γδ(φ)
and type II perturbation theory on T 2 × K3
− 1 2π (δαβδγδ − δα(γδδ)β)ˆ E1(U) 2 − 1 4 δαβ, ˆ E1(U)G (6,22)
γδ (φ) +
G (6,22)
αβ,γδ(φ)
∼ 1 g 4
s
G (4,20)
αβ,γδ(φ) −
1 4g 2
s
δαβ, ˆ E1(U) + ˆ E1(T) G (4,20)
γδ (φ)
− 1 2π (δαβδγδ − δα(γδδ)β)ˆ E1(U) + ˆ E1(T)2
SLIDE 26
Fourier coefficients
For detΩ2 large enough
Ck−2 Q2
QP QP P2 , Ω2
- =
- (R/Z)3+iΩ2
d3Ω eiπ(Q, P)ΩQ
P
- Φk−2(Ω)
= C F
k−2
Q2
QP QP P2
- +
- γ∈GL(2,Z)
ck(− (dQ−bP)2
2
)ck(− (aP−cQ)2
2
) ×
- −
δ(tr ( 0 1/2 1/2
- γ⊺Ω2γ))
2π
+ (dQ−bP)·(aP−cQ)
2
(sign(tr ( 0
1/2 1/2
- γ⊺Ω2γ)) − sign((dQ − dP) · (aP − cQ)))
- and
Φk−2 = (i √ N)kΦk−2
-
−1 1 1 1
twisted elliptic genus of K3 orbifold
- Ck−2
Q2
QP QP P2 , Ω2
- = 1
N
- (R/Z)2×R/NZ+iΩ2
d3Ω eiπ(Q, P)ΩQ
P
- Φk−2(Ω)
=
- C F
k−2
Q2
QP QP P2
- +
- γ∈Z2⋉Γ0(N)
ck(− N(dQ−bP)2
2
)ck(− (aP−cQ)2
2
) ×
- −
δ(tr ( 0 1/2 1/2
- γ⊺Ω2γ))
2π
+ (dQ−bP)·(aP−cQ)
2
(sign(tr ( 0
1/2 1/2
- γ⊺Ω2γ)) − sign((dQ − dP) · (aP − cQ)))
SLIDE 27
Fourier coefficients
The integral
- P2
d3Ω2 (detΩ2)
3 2 −s e
−πtr
- Ω−1
2
R2AT
1 S2
- 1
S1 S1 |S|2
- A+2Ω2A−1
- Q2
R
QRPR QRPR P2
R
- A−T
- =
2
- RdetA
2|QR ∧ PR| s Bs
- 2R2
S2
- 1
S1 S1 |S|2
- Q2
R
QRPR QRPR P2
R
- ,
that behaves as e−2πRM(Q,P) in the limit R → ∞, where
M(Q, P) =
- 2 |QR +SPR |2
S2
+ 4|QR ∧ PR|
The saddle point is at (for A and B symmetric matrices, detBdetA B−1 = Tr(AB)A − ABA)
Ω2 = R
M(Q, P) A⊺ 1 S2
- 1
S1 S1 |S|2
- +
1 |PR ∧QR |
- |PR|2
−PR · QR −PR · QR |QR|2
A .
SLIDE 28
Fourier coefficients
The integral
- P2
d3Ω2 (detΩ2)
3 2 −s Ck−2
- Q2
QP QP P2 , Ω2
- e
−πtr
- Ω−1
2 R2AT 1 S2
- 1
S1 S1 |S|2
- A+2Ω2A−1
- Q2
R
QRPR QRPR P2
R
- A−T
- = 2Ck−2
- Q2
QP QP P2 , Ω⋆
2
- RdetA
2|QR ∧ PR| s Bs
- 2R2
S2
- 1
S1 S1 |S|2
- Q2
R
QRPR QRPR P2
R
- + O(e−2πR(M(Q1,P1)+M(Q2,P2)))
that behaves as e−2πRM(Q,P) in the limit R → ∞, where away from the walls of marginal stability
M(Q, P) < M(Q1, P1) + M(Q2, P2) .
Contour prescription at A = 1
[ Cheng Verlinde]
Ω⋆
2 =
R
M(Q, P)
- 1
S2
- 1
S1 S1 |S|2
- +
1 |PR ∧QR |
- |PR|2
−PR · QR −PR · QR |QR|2
.
SLIDE 29
Fourier coefficients
¯ c(Q, P) =
- A∈M2(Z)/GL(2,Z)
A−1Q P
- ∈Λ6,p⊕Λ6,p
detA Ck−2
- A−1Q2 QP
QP P2
- A−⊺, A⊺Ω⋆
2A
- +
- A∈M2,0(N)/[Z2⋉Γ0(N)]
A−1Q P
- ∈Λ∗
6,p⊕Λ6,p
detA Ck−2
- A−1Q2 QP
QP P2
- A−⊺, A⊺Ω⋆
2A
- +
- A∈M2(Z)/GL(2,Z)
A−1 Q P/N
- ∈Λ∗
6,p⊕Λ∗ 6,p
detA Ck−2
- A−1NQ2
QP QP P2/N
- A−⊺, A⊺Ω⋆
2A
SLIDE 30
Exact degeneracy
For Heterotic string theory on T 6 ¯ c(Q, P) =
- A∈M2(Z)/GL(2,Z)
A−1Q P
- ∈Λ6,22⊕Λ6,22
detA C10
- A−1Q2 QP
QP P2
- A−⊺, A⊺Ω⋆
2A
- For a primitive electromagnetic charge (Q, P),
SL(2, Z) × O(6, 22, Z) permits to chose a representative Q = qe1 , P = pe1 + e2 , Q ∧ P = qe1 ∧ e2 for primitive vectors e1 and e2 in Λ6,22 and q, p integers. ¯ c(Q, P) =
- d|q
d C10 Q2/d2 QP/d
QP/d P2
- ,
d 0
0 1
- Ω⋆
2
d 0
0 1
- reproduces the
[ Dabholkar Gomes Murthy]
generalisation of
[ Dijkgraaf Verlinde Verlinde]
(mod contour prescription).
SLIDE 31
Exact degeneracy
For twisted charges with Q⊂Λ∗
6,p and P⊂Λ6,p
¯ c(Q, P) =
- A∈M2,0(N)/[Z2⋉Γ0(N)]
A−1Q P
- ∈Λ∗
6,p⊕Λ6,p
detA Ck−2
- A−1Q2 QP
QP P2
- A−⊺, A⊺Ω⋆
2A
- For both Q and P primitive and gcd(Q ∧ P) = 1,
Q = e1 + re2 , P = e2 , Q ∧ P = e1 ∧ e2 for primitive vectors e1 ∈ Λ∗
6,p and e2 ∈ Λ6,p and
¯ c(Q, P) = Ck−2 Q2 QP
QP P2
- , Ω⋆
2
- reproduces
[ Jatkar Sen] .
SLIDE 32
Supersymmetry Ward identities
1/2 BPS coupling
- d8θf (W a+) gives the constraints
(Wavefrontset: nilpotent orbit Λ1 ∈ so8,p+2, Υ1 ∈ so8)
D[e
[ˆ eDf ] ˆ f ]Fabcd
= 0 , D[e
ˆ aFa]bcd = 0 ,
D(e
ˆ gDf )ˆ g Fabcd
= − 3
2 δef Fabcd − 4 δe)(a Fbcd)(f + 3 δ(ab Fcd)ef .
1/4 BPS coupling d12θf (W a+1, W a+2) gives the constraints (Wavefrontset: nilpotent orbit 2Λ2 ∈ so8,p+2, 2Υ2 ∈ so8)
D[a1
ˆ aGa2|b|,a3]c = 0 ,
D[a1
[ˆ a1Da2 ˆ a2]Ga3]b,cd = 0 ,
D[a1
[ˆ a1Da2 ˆ a2Da3] ˆ a3]Gcd,ef = 0 ,
and the inhomogeneous equation
D(e
ˆ aDf )ˆ aGab,cd = − 5 2 δef Gab,cd − δe)(aGb)(f ,cd − δe)(cGd)(f ,ab + 3 2 δab,Gcd,ef
− π Fab(e
gFf )cdg − Fe)a(c gFd)bg(f
- .
SLIDE 33
Supersymmetry Ward identities
D(e
ˆ aDf )ˆ aGab,cd + 5 2 δef Gab,cd + δe)(aGb)(f ,cd + δe)(cGd)(f ,ab + 3 2 δab,Gcd,ef
= −π Fab(e
gFf )cdg − Fe)a(c gFd)bg(f
- Consistant with
G (8,p+2)
αβ,γδ
∼ R4
- − 1
2π (δαβδγδ − δα(γδδ)β) ˆ E1(NS) + ˆ E1(S) N + 1 2 − 1 4 δαβ, N ˆ E1(NS) − ˆ E1(S) N2 − 1 G (6,p)
γδ (φ) + N ˆ
E1(S) − ˆ E1(NS) N2 − 1 G (6,p)
γδ (φσ)
- +
G (6,p)
αβ,γδ(φ)
- and
F (8,p+2)
αβγδ
∼ R2 3 2π ˆ E1(NS) + ˆ E1(S) N + 1 δ(αβδγδ) + ˆ F (6,p)
αβγδ(φ)
SLIDE 34
Supersymmetry Ward identities
D(e
ˆ aDf )ˆ aGab,cd + 5 2 δef Gab,cd + δe)(aGb)(f ,cd + δe)(cGd)(f ,ab + 3 2 δab,Gcd,ef
= −π Fab(e
gFf )cdg − Fe)a(c gFd)bg(f
- Consistant with
G(8,p+2) αβ,γδ ∼ 2
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0 R7 ¯ c(Q, P) 2
- ℓ=0
P(ℓ) αβ,δγ (QL, PL) Rℓ B 3 2 −[ ℓ 2 ]
- 2R2
S2
- 1
S1 S1 |S|2 Q2 R QR PR QR PR P2 R
- |QR ∧ PR |
3 2 −[ ℓ 2 ] e2πi(Q·a1+P·a2)
and Q
P
- = A
1 0
0 0
- A−1Q
P
- + A
0 0
0 1
- A−1Q
P
- F (8,p+2)
αβγδ
∼ 4
- (Q,P)∈Λ∗
6,p⊕Λ6,p Q∧P=0
R4 ¯ c(Q, P)
2
- ℓ=0
P(ℓ)
αβδγ(QL, PL)
R2ℓ K2−ℓ (2πRM(QR, PR)) M(QR, PR)2−ℓ e2πi(Q·a1+P·a2)
SLIDE 35