Path integral perturbation theory Bosonic string theory can be said to be defined by the
path integral quantization of the
Polyakov action: : 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) x^\mu(\xi) is the field on the
worldsheet describing the most embedding of the string in 25 +1 spacetime; in the Polyakov formulation, g is not to be understood as the induced metric from the embedding, but as an independent dynamical field. 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 G_{\mu\nu} = \delta_{\mu\nu}. M is the worldsheet as a
topological manifold parametrized by the \xi coordinates. T is the string tension and related to the Regge slope as T = \frac{1}{2\pi\alpha'}. 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: : 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: : Z = \sum_{h=0}^\infty \int \frac{\mathcal{D}g_{mn} \mathcal{D}X^\mu}{\mathcal{N}} \exp ( - I[g,X] ) : \left\langle V_{i_1} (k^\mu_1) \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^\mu_p) 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 h. A normalization factor \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 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 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 : 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 h \geq 4.
i which connect to a string world surface at the surface points
zi. It is given by the following
functional integral over all possible embeddings of this 2D surface in 26 dimensions: : A_N = \int D\mu \int D[X] \exp \left( -\frac{1}{4\pi\alpha} \int \partial_z X_\mu(z,\overline{z}) \partial_{\overline{z}} X^\mu(z,\overline{z}) \, dz^2 + i \sum_{i=1}^N k_{i \mu} X^\mu (z_i,\overline{z}_i) \right) The functional integral can be done because it is a Gaussian to become: : A_N = \int D\mu \prod_{0 This is integrated over the various points
zi. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. Also you need to make sure you are not integrating multiple times over different parameterizations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function): : A_4 = \frac{ \Gamma (-1+\frac12(k_1+k_2)^2) \Gamma (-1+\frac12(k_2+k_3)^2) } { \Gamma (-2+\frac12((k_1+k_2)^2+(k_2+k_3)^2)) } Which is a
beta function, known as
Veneziano amplitude. It was this beta function which was apparently found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the Grassmann coordinates
θ. Since there are various ways this can be done this leads to different string theories. When integrating over surfaces such as the torus, we end up with equations in terms of
theta functions and elliptic functions such as the
Dedekind eta function. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface. --> ==== h = 0 ==== At tree-level, corresponding to genus 0, the cosmological constant vanishes: Z_0 = 0 . The four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude: : A_4 \propto (2\pi)^{26} \delta^{26}(k) \frac{\Gamma(-1-s/2) \Gamma(-1-t/2) \Gamma(-1-u/2)}{\Gamma(2+s/2) \Gamma(2+t/2) \Gamma(2+u/2)} Where k is the total momentum and s, t, u are the
Mandelstam variables. ==== h = 1 ==== Genus 1 is the torus, and corresponds to the
one-loop level. The partition function amounts to: : 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} \tau is a
complex number with positive imaginary part \tau_2; \mathcal{M}_1, holomorphic to the moduli space of the torus, is any
fundamental domain for the
modular group PSL(2,\mathbb{Z}) acting on the
upper half-plane, for example \left\{ \tau_2 > 0, |\tau|^2 > 1, -\frac{1}{2} . \eta(\tau) is the
Dedekind eta function. The integrand is of course invariant under the modular group: the measure \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 \tau_2 \rightarrow |c \tau + d|^2 \tau_2 and the fact that \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. ==See also==