Path integral perturbation theory

Bosonic string theory can be said^[2] to be defined by the path integral quantization of the Polyakov action:

${\displaystyle I_{0}[g,X]={\frac {T}{8\pi }}\int _{M}d^{2}\xi {\sqrt {g}}g^{mn}\partial _{m}x^{\mu }\partial _{n}x^{\nu }G_{\mu \nu }(x)}$

${\displaystyle x^{\mu }(\xi )}$ is the field on the worldsheet describing the embedding of the string in 25+1 spacetime; in the Polyakov formulation, ${\displaystyle g}$ is not to be understood as the induced metric from the embedding, but as an independent dynamical field. ${\displaystyle G}$ is the metric on the target spacetime, which is usually taken to be the Minkowski metric in the perturbative theory. Under a Wick rotation, this is brought to a Euclidean metric ${\displaystyle G_{\mu \nu }=\delta _{\mu \nu }}$. M is the worldsheet as a topological manifold parametrized by the ${\displaystyle \xi }$ coordinates. ${\displaystyle T}$ is the string tension and related to the Regge slope as ${\displaystyle T={\frac {1}{2\pi \alpha '}}}$.

${\displaystyle I_{0}}$ has diffeomorphism and Weyl invariance. Weyl symmetry is broken upon quantization (Conformal anomaly) and therefore this action has to be supplemented with a counterterm, along with a hypothetical purely topological term, proportional to the Euler characteristic:

${\displaystyle I=I_{0}+\lambda \chi (M)+\mu _{0}^{2}\int _{M}d^{2}\xi {\sqrt {g}}}$

The explicit breaking of Weyl invariance by the counterterm can be cancelled away in the critical dimension 26.

Physical quantities are then constructed from the (Euclidean) partition function and N-point function:

${\displaystyle Z=\sum _{h=0}^{\infty }\int {\frac {{\mathcal {D}}g_{mn}{\mathcal {D}}X^{\mu }}{\mathcal {N}}}\exp(-I[g,X])}$

${\displaystyle \left\langle V_{i_{1}}(k_{1}^{\mu })\cdots V_{i_{p}}(k_{p}^{\mu })\right\rangle =\sum _{h=0}^{\infty }\int {\frac {{\mathcal {D}}g_{mn}{\mathcal {D}}X^{\mu }}{\mathcal {N}}}\exp(-I[g,X])V_{i_{1}}(k_{1}^{\mu })\cdots V_{i_{p}}(k_{p}^{\mu })}$

The perturbative series is expressed as a sum over topologies, indexed by the genus.

The discrete sum is a sum over possible topologies, which for euclidean bosonic orientable closed strings are compact orientable Riemannian surfaces and are thus identified by a genus ${\displaystyle h}$. A normalization factor ${\displaystyle {\mathcal {N}}}$ is introduced to compensate overcounting from symmetries. While the computation of the partition function corresponds to the cosmological constant, the N-point function, including ${\displaystyle p}$ vertex operators, describes the scattering amplitude of strings.

The symmetry group of the action actually reduces drastically the integration space to a finite dimensional manifold. The ${\displaystyle g}$ path-integral in the partition function is a priori a sum over possible Riemannian structures; however, quotienting with respect to Weyl transformations allows us to only consider conformal structures, that is, equivalence classes of metrics under the identifications of metrics related by

${\displaystyle g'(\xi )=e^{\sigma (\xi )}g(\xi )}$

Since the world-sheet is two dimensional, there is a 1-1 correspondence between conformal structures and complex structures. One still has to quotient away diffeomorphisms. This leaves us with an integration over the space of all possible complex structures modulo diffeomorphisms, which is simply the moduli space of the given topological surface, and is in fact a finite-dimensional complex manifold. The fundamental problem of perturbative bosonic strings therefore becomes the parametrization of Moduli space, which is non-trivial for genus ${\displaystyle h\geq 4}$.

h = 1

The shaded region is a possible fundamental domain for the modular group.

Genus 1 is the torus, and corresponds to the one-loop level. The partition function amounts to:

${\displaystyle Z_{1}=\int _{{\mathcal {M}}_{1}}{\frac {d^{2}\tau }{8\pi ^{2}\tau _{2}^{2}}}{\frac {1}{(4\pi ^{2}\tau _{2})^{12}}}\left|\eta (\tau )\right|^{-48}}$

${\displaystyle \tau }$ is a complex number with positive imaginary part ${\displaystyle \tau _{2}}$; ${\displaystyle {\mathcal {M}}_{1}}$, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group ${\displaystyle PSL(2,\mathbb {Z} )}$ acting on the upper half-plane, for example ${\displaystyle \left\{\tau _{2}>0,|\tau |^{2}>1,-{\frac {1}{2}}<\tau _{1}<{\frac {1}{2}}\right\}}$. ${\displaystyle \eta (\tau )}$ is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure ${\displaystyle {\frac {d^{2}\tau }{\tau _{2}^{2}}}}$ is simply the Poincaré metric which has PSL(2,R) as isometry group; the rest of the integrand is also invariant by virtue of ${\displaystyle \tau _{2}\rightarrow |c\tau +d|^{2}\tau _{2}}$ and the fact that ${\displaystyle \eta (\tau )}$ is a modular form of weight 1/2.

This integral diverges. This is due to the presence of the tachyon and is related to the instability of the perturbative vacuum.