Bosonic string theory
Bosonic string theory is the original version of String Theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory (supersymmetric string theory) became the real focus. Nevertheless, bosonic string theory remains a very useful model to understand many general features of perturbative string theory, and many theoretical difficulties of superstrings can actually already be found in the context of bosonic strings. Problems. Although bosonic string theory has  many attractive features, it falls short as a viable physical model in two significant areas. First, it predicts only the existence of bosons whereas many physical particles are fermions. Second, it predicts the existence of a mode of the string with imaginary mass, implying that the theory has an instability to a process known as "tachyon condensation". In addition, bosonic string theory in a general spacetime dimension displays inconsistencies due to the conformal anomaly. But, as was first noticed by Claud Lovelace, in a spacetime of 26 dimensions (25 dimensions of space and one of time), the critical dimension for  the theory, the anomaly cancels. This high dimensionality is not necessarily a problem for string theory, because it can be formulated in such a way that along the 22 excess dimensions spacetime is folded up to form a small torus or other compact manifold. This would leave only the familiar four dimensions of spacetime visible to low energy experiments. The existence of a critical dimension where the anomaly cancels is a general feature of all string theories. Types of bosonic strings. There are four possible bosonic string theories, depending on whether open strings are allowed and whether strings have a  specified orientation. Recall that a theory of open strings also must include closed strings; open strings can be thought as having their endpoints fixed on a D25-brane that fills all of spacetime. A specific orientation of the string means that only interaction corresponding to an orientable worldsheet are allowed (e.g., two strings can only merge with equal orientation). A sketch of the spectra of the four possible theories is as follows: Note that all four theories have a negative energy tachyon (formula_1) and a massless graviton. The rest of this article applies to the closed, oriented theory, corresponding to  borderless, orientable worldsheets. Mathematics. Path integral perturbation theory. Bosonic string theory can be said to be defined by the path integral quantization of the Polyakov action: formula_3 is the field on the worldsheet describing the embedding of the string in 25+1 spacetime; in the Polyakov formulation, formula_4 is not to be understood as the induced metric from the embedding, but as an independent dynamical field. formula_5 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 formula_6 . M is the worldsheet as a topological manifold parametrized by the formula_7 coordinates. formula_8 is the string tension and related to the Regge slope as formula_9. formula_10 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: 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: 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 formula_14. A normalization factor formula_15 is introduced to compensate overcounting from symmetries. While the computation of the partition function corresponds to the cosmological constant, the N-point function, including formula_16 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 formula_4 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 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 formula_19. h = 0. At tree-level, corresponding to genus 0, the cosmological constant vanishes: formula_20. The four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude: Where formula_22 is the total momentum and formula_23, formula_24, formula_25 are the Mandelstam variables. h = 1. Genus 1 is the torus, and corresponds to the one-loop level. The partition function amounts to: formula_27 is a complex number with positive imaginary part formula_28; formula_29, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group formula_30 acting on the  upper half-plane, for example formula_31. formula_32 is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure formula_33 is simply the Poincaré metric which has PSL(2,R) as isometry group; the rest of the integrand is also invariant by virtue of formula_34 and the fact that formula_32 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.