378
A Conventions
• K g,n : conformal Killing vector group of genus-g Riemann surfaces with n
punctures
• K g = K g,0 = ker P 1 : conformal Killing vector group of genus-g Riemann
surfaces
• K g,n = dim K g,n
• K c
g,n = dim C K g,n = dim C ker P 1
• K g = K g,0 = dim ker P 1
• K c
g K c
g,0 = dim C ker P 1
• ψ i : real basis of ker P 1 , CKV
• φ i : real basis of ker P
†
1 , real quadratic differentials
• (ψ K , ¯
ψ K ): complex basis of ker P 1 , (anti-)holomorphic CKV
• (φ I , ¯
φ I ): complex basis of ker P
†
1 , (anti-)holomorphic quadratic differentials
• t λ ∈ M g,n : real moduli of M g,n
• m ∈ M g,n : complex moduli of M g,n
• t i ∈ M g : real moduli of M g
• m I ∈ M g : complex moduli of M g
• z: coordinate on the Riemann surface
• w i : local coordinates around punctures
• z a : local coordinates away from punctures
• f i (w i ): transition functions from w i to z
• σ α : coordinate system on the left of the contour C α
• τ α : coordinate system on the right of the contour C α
CFT:
• V α (k; σ a ) := V k,α (σ a ): matter vertex operator with 7 momentum k and quantum
numbers α inserted at position σ a = (z, ¯
z)
• V α (k; σ a ): unintegrated vertex operator with momentum k and quantum numbers
α inserted at position σ a
• V α (k) =
d 2 σ
√ g V α (k; σ ): integrated vertex operator
• on-shell (closed bosonic string): V α (k; σ a ) = c ¯
cV α (k; σ a ) is a (0, 0)-primary,
with V α (k; σ a ) a (1, 1)-primary matter operator
•
O: operator O with zero-modes removed
• O † : Hermitian adjoint
• O ‡ : Euclidean adjoint
• O t : BPZ conjugation
• O 1 |O 2 : BPZ inner product
• O
‡
1 |O 2 : Hermitian inner product
• |0: SL(2, C) (conformal) vacuum
• |: energy vacuum (lowest energy state)
7 When the momentum and/or quantum numbers are not relevant, we remove them or simply index
the operators by a number.
A Conventions
• K g,n : conformal Killing vector group of genus-g Riemann surfaces with n
punctures
• K g = K g,0 = ker P 1 : conformal Killing vector group of genus-g Riemann
surfaces
• K g,n = dim K g,n
• K c
g,n = dim C K g,n = dim C ker P 1
• K g = K g,0 = dim ker P 1
• K c
g K c
g,0 = dim C ker P 1
• ψ i : real basis of ker P 1 , CKV
• φ i : real basis of ker P
†
1 , real quadratic differentials
• (ψ K , ¯
ψ K ): complex basis of ker P 1 , (anti-)holomorphic CKV
• (φ I , ¯
φ I ): complex basis of ker P
†
1 , (anti-)holomorphic quadratic differentials
• t λ ∈ M g,n : real moduli of M g,n
• m ∈ M g,n : complex moduli of M g,n
• t i ∈ M g : real moduli of M g
• m I ∈ M g : complex moduli of M g
• z: coordinate on the Riemann surface
• w i : local coordinates around punctures
• z a : local coordinates away from punctures
• f i (w i ): transition functions from w i to z
• σ α : coordinate system on the left of the contour C α
• τ α : coordinate system on the right of the contour C α
CFT:
• V α (k; σ a ) := V k,α (σ a ): matter vertex operator with 7 momentum k and quantum
numbers α inserted at position σ a = (z, ¯
z)
• V α (k; σ a ): unintegrated vertex operator with momentum k and quantum numbers
α inserted at position σ a
• V α (k) =
d 2 σ
√ g V α (k; σ ): integrated vertex operator
• on-shell (closed bosonic string): V α (k; σ a ) = c ¯
cV α (k; σ a ) is a (0, 0)-primary,
with V α (k; σ a ) a (1, 1)-primary matter operator
•
O: operator O with zero-modes removed
• O † : Hermitian adjoint
• O ‡ : Euclidean adjoint
• O t : BPZ conjugation
• O 1 |O 2 : BPZ inner product
• O
‡
1 |O 2 : Hermitian inner product
• |0: SL(2, C) (conformal) vacuum
• |: energy vacuum (lowest energy state)
7 When the momentum and/or quantum numbers are not relevant, we remove them or simply index
the operators by a number.