Bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose.
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.
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.
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.
Bosonic string theory can be said[2] to be defined by the path integral quantization of the Polyakov action:
is the field on the worldsheet describing the most 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.
Under a Wick rotation, this is brought to a Euclidean metric
is the string tension and related to the Regge slope as
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
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.
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.
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
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
Genus 1 is the torus, and corresponds to the one-loop level.
is a complex number with positive imaginary part
, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group
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
This is due to the presence of the tachyon and is related to the instability of the perturbative vacuum.
"Complex geometry and the theory of quantum strings".
