Polyakov Path Integral Prescription

For $D\ge 3$, let the target be $M={\mathbb{R}}^{1,D-1}$ with $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$. Fix a nonempty compact, connected, oriented smooth surface ${\mathrm{\Sigma }}_g$ without boundary. Give it a complex atlas, meaning a covering family of homeomorphisms from open subsets of ${\mathrm{\Sigma }}_g$ onto open subsets of $\mathbb{C}$ whose transition maps are holomorphic in both directions. Assume the Axiom of Choice for the genus convention: $g\ge 0$ is the unique handle number for which the underlying oriented surface is homeomorphic to the sphere with $g$ handles. The worldsheet fields are smooth maps $X\in C^{\mathrm{\infty }}({\mathrm{\Sigma }}_g,M)$ and smooth positive-definite Riemannian metrics $g_E$ on ${\mathrm{\Sigma }}_g$; the formal map domain includes maps that are not immersions. In units $\mathrm{\hslash }=c=1$, $[X^{\mu }]=L$, $[{\alpha }^{\mathrm{\prime }}]=L^2$, $T=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}$, and worldsheet coordinates are dimensionless.

The Euclidean action assigned to $(X,g_E)$ is

Here ${\mathrm{\partial }}_a$ is differentiation in a smooth local worldsheet coordinate chart. On cylinder coordinates the continuation is $\tau =-i{\tau }_E$; on general ${\mathrm{\Sigma }}_g$ the displayed expression is the formally continued local action, and no global time coordinate is assumed.

Let ${\mathrm{Met}}^{+}({\mathrm{\Sigma }}_g)$ be the set of smooth positive-definite metrics, ${\mathrm{Diff}}^{+}({\mathrm{\Sigma }}_g)$ the smooth orientation-preserving diffeomorphisms, and $\mathrm{Weyl}({\mathrm{\Sigma }}_g)=C^{\mathrm{\infty }}({\mathrm{\Sigma }}_g,\mathbb{R})$ the additive Weyl parameters. The action on fields is

Define $g_1\sim g_2$ when $g_2=e^{2\omega }{\varphi }^{\ast }{g}_1$ for such $\varphi ,\omega$, and define the formal conformal moduli set ${\mathcal{M}}_g={\mathrm{Met}}^{+}({\mathrm{\Sigma }}_g)\mathrm{/}\sim$. The prescription symbolically divides the integral over $(g_E,X)$ by the diffeomorphism and Weyl action, leaving the formal integration over ${\mathcal{M}}_g$.

For a fixed metric $g_E$, let $\mathrm{\nabla }$ be its Levi-Civita connection and define the linearized Faddeev--Popov operator on smooth vector fields by

where $v_b=g_{E,bc}{v}^c$; it is the trace-free part of the metric variation under the infinitesimal diffeomorphism $v$. Here $\mathrm{\Gamma }(S_0^2{T}^{\ast }{\mathrm{\Sigma }}_g)$ denotes the smooth symmetric covariant $2$-tensor fields $h$ with $g_E^{ab}{h}_{ab}=0$. Its kernel $\ker P_{g_E}$ is the conformal-Killing vector-field space, hence the $c$-ghost zero-mode space. Use the $L^2(g_E)$ pairings $⟨v,w⟩={\int }_{{\mathrm{\Sigma }}_g}{g}_{E,ab}{v}^a{w}^{b}d{\mathrm{vol}}_{g_E}$ and $⟨h,k⟩={\int }_{{\mathrm{\Sigma }}_g}{g}_E^{ac}{g}_E^{bd}{h}_{ab}{k}_{cd}d{\mathrm{vol}}_{g_E}$ on vector fields and trace-free symmetric tensors; the formal adjoint $P_{g_E}^{\ast }$ defines the $b$-ghost zero-mode space $\ker P_{g_E}^{\ast }$; the linearized moduli space is the quotient $\mathrm{coker}{P}_{g_E}=\mathrm{\Gamma }(S_0^2{T}^{\ast }{\mathrm{\Sigma }}_g)\mathrm{/}\mathrm{im}{P}_{g_E}\mathrm{.}$ The gauge-fixing determinant is represented formally by the Grassmann-odd symmetric-traceless tensor $b_{ab}$ and vector $c^a$ with action $S_{gh}=(2\pi {)}^{-1}{\int }_{{\mathrm{\Sigma }}_g}d{\mathrm{vol}}_{g_E}{b}_{ab}{\mathrm{\nabla }}^a{c}^b$, in local complex coordinates with chiral components $b_{zz},c^z$ and antiholomorphic counterparts. This specifies the zero-mode spaces but does not assert a global gauge slice or a measure. Since the target metric has a timelike coordinate, $S_E$ is not a positive quadratic form in $X$; no target-time integration contour is chosen.

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§5.1–5.4.3 and 6.1–6.2.3 (PDF file pp.115–145; printed pp.108–138); printed pp.108–138); printed pp.108–138); printed pp.108–138); printed pp.108–138); printed pp.108–138); printed pp.108–138); formal Faddeev–Popov prescription, ghost determinant, residual moduli, and path-integral limits.

Bosonic Polyakov Model is the adopted Lorentzian model whose formal Euclidean prescription is recorded here. Conformal Gauge and Residual Symmetry gives the local gauge convention, and Bosonic Bc Ghost System gives the corresponding gauge-fixed ghost sector. The Euclidean worldsheet is a compact Riemann surface under Riemann surfaces and holomorphic atlases. Its genus is taken from Genus and Euler characteristic of a compact Riemann surface, whose exact statement assumes The Axiom of Choice; AC enters through the compact-surface classification used to define genus. The imports are mathematical prerequisites only and do not construct a physical measure.

The quotient, its conformal moduli set, and the two linearized ghost zero-mode spaces are the exact formal objects used here; no finite-dimensional moduli-space structure or global gauge slice is asserted. The symbols $\mathcal{D}X$ and $\mathcal{D}g$ do not designate measures whose existence, positivity, or convergence has been proved. The target has one timelike coordinate, so worldsheet Euclidean continuation alone does not specify a convergent target-field Gaussian; a target-time contour is separate and unspecified. No sewing, amplitude, or all-genus measure theorem is asserted.