Bosonic string theory
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.
Although bosonic string theory has many attractive features, it falls short as a viable physical model in two significant areas.
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:
|Bosonic string theory||Non-positive states|
|Open and closed, oriented||tachyon, graviton, dilaton, massless antisymmetric tensor|
|Open and closed, unoriented||tachyon, graviton, dilaton|
|Closed, oriented||tachyon, graviton, dilaton, antisymmetric tensor, U(1) vector boson|
|Closed, unoriented||tachyon, graviton, dilaton|
Note that all four theories have a negative energy tachyon () and a massless graviton.
The rest of this article applies to the closed, oriented theory, corresponding to borderless, orientable worldsheets.
Path integral perturbation theory
is the field on the worldsheet describing the embedding of the string in 25+1 spacetime; in the Polyakov formulation, is not to be understood as the induced metric from the embedding, but as an independent dynamical field. 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 . M is the worldsheet as a topological manifold parametrized by the coordinates. is the string tension and related to the Regge slope as .
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.
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 . A normalization factor is introduced to compensate overcounting from symmetries. While the computation of the partition function corresponds to the cosmological constant, the N-point function, including 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 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 .
h = 0
At tree-level, corresponding to genus 0, the cosmological constant vanishes: .
The four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude:
Where is the total momentum and , , are the Mandelstam variables.
h = 1
Genus 1 is the torus, and corresponds to the one-loop level. The partition function amounts to:
is a complex number with positive imaginary part ; , holomorphic to the moduli space of the torus, is any fundamental domain for the modular group acting on the upper half-plane, for example . is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure is simply the Poincaré metric which has PSL(2,R) as isometry group; the rest of the integrand is also invariant by virtue of and the fact that 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.
D'Hoker, Eric & Phong, D. H. (Oct 1988). "The geometry of string perturbation theory". Rev. Mod. Phys. American Physical Society. 60 (4): 917–1065. Bibcode:1988RvMP...60..917D. doi:10.1103/RevModPhys.60.917.
Belavin, A.A. & Knizhnik, V.G. (Feb 1986). "Complex geometry and the theory of quantum strings". ZhETF. 91 (2): 364–390. Bibcode:1986ZhETF..91..364B.