String genus expansion

For a connected oriented closed-string worldsheet of genus

g

with

n

external closed-string vertices, the perturbative amplitude carries

g_{s}^{2 g - 2 + n}

. The exponent combines the dilaton factor

g_{s}^{- χ}

for Euler characteristic

χ = 2 - 2 g

with one

g_{s}

from each normalized vertex.

Table of contents
- String-loop effective action expansion String genus expansion
- Dilaton Euler-characteristic weighting String genus expansion

String-loop effective action expansion ( $S_{eff} = \sum_{g \geq 0} g_{s}^{2 g - 2} S_{g}$ )

Connected oriented closed string worldsheets of genus

g

give the spacetime effective-action term

g_{s}^{2 g - 2} S_{g}

. Each

S_{g}

has a separate low-energy derivative expansion in

α^{'}

. The string coupling orders quantum string loops, while

α^{'}

orders effects of finite string size.

Dilaton Euler-characteristic weighting ( $g_{s}^{- χ} = g_{s}^{2 g - 2}$ )

A constant dilaton contributes

Φ_{0} χ (Σ)

to the Euclidean worldsheet action by the Gauss-Bonnet theorem. With string coupling

g_{s} = e^{Φ_{0}}

and Euler characteristic

χ = 2 - 2 g

, the path-integral weight is

g_{s}^{2 g - 2}

. Canonically normalized external closed-string vertices add one power of

g_{s}

each.

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