String coupling ($g_s$)

The string coupling controls topology change and string splitting and joining. A constant background dilaton ${\Phi }_0$ sets $g_s=e^{{\Phi }_0}$.

For a connected oriented closed-string worldsheet of genus $g$ with $n$ external closed-string vertices, the perturbative amplitude carries $g_s^{2g-2+n}$. The exponent combines the dilaton factor $g_s^{-\chi }$ for Euler characteristic $\chi =2-2g$ with one $g_s$ from each normalized vertex.

The dilaton is a scalar closed-string background field coupled to the worldsheet Ricci scalar. Its constant mode determines the string coupling, while spacetime variation modifies the background-field equations.