A Conventions
377
A.5
List of Symbols
General:
• D: number of non-compact spacetime dimensions
• g: loop order for a scattering amplitude
• n: number of external closed string states
• x μ : spacetime non-compact coordinates
• σ a = (t, σ ): worldsheet coordinates
• g s : closed string coupling
• Z g = A g,0 : genus-g vacuum amplitude
• A g,n (k 1 , . . . , k n ) α 1 ,...,α n := A g,n ({k i }) {α i } : g-loop n-point scattering amplitude
for states with quantum numbers {k i , α i } (if connected, amputated Green functions for n ≥ 3)
• G g,n (k 1 , . . . , k n ) α 1 ,...,α n : g-loop n-point Green function for states with quantum
numbers {k i , α i }
• T ⊥
ab : traceless symmetric tensor or traceless component of the tensor T ab
• : generic (set of) matter field(s)
Hilbert spaces:
• H: generic Hilbert space (in general, Hilbert space of the matter plus ghost CFT)
• H ± = H ∩ ker b
±
0
• H 0 = H ∩ ker b
−
0 ∩ ker b
+
0
• H(Q B ): absolute cohomology of the operator Q B inside the space H
• H − (Q B ) = H(Q B ) ∩ H − : semi-relative cohomology of the operator Q B inside
the space H
• H 0 (Q B ) = H(Q B ) ∩ H 0 : relative cohomology of the operator Q B inside the
space H
• A: Grassmann parity of the operator or state A
Riemann surfaces:
• g: Riemann surface genus (number of holes/handles)
• n: number of bulk punctures/marked points
• g,n : genus-g Riemann surface with n punctures
• g = g,0 : genus-g Riemann surface
• M g,n : moduli space of genus-g Riemann surfaces with n punctures
• M g = M g,0 : moduli space of genus-g Riemann surfaces
• M g,n = dim M g,n
• M c
g,n = dim C M g,n
• M g = M g,0 = dim M g = dim ker P
†
1
• M c
g = M c
g,0 = dim C M g