# Bosonic string theory

Bosonic string theory is the original version of string theory, developed in the late 1960s.

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 "toy model" to understand many general features of perturbative string theory, and string theory textbooks usually start with the bosonic string. The first volume of Polchinski's String Theory and Zwiebach's A First Course in String Theory are good examples.

## Problems

Although bosonic string theory has many attractive features, it falls short as a viable physical model in two significant areas and is forced to posit a 26 dimensional spacetime to remedy inconsistencies.

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 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.

## Mathematics

In bosonic string theory and in the Neveu–Schwarz sector of superstring theory, the action in a curved background (ignoring the Fradkin–Tseytlin term for dilaton coupling) can be constructed by covariantizing' the massless closed string vertex operator with respect to target-space reparameterization invariance. This procedure can also be used here after constructing the massless closed string vertex operator from the left-right' product of two massless open string vertex operators. The complete worldsheet action for the type-II superstring in a flat background in conformal gauge is a formula upon which Eric Sidewater (USA) made the first improvement upon which all subgroups (15) are factored in:

$A_N = \int D\mu \int D[X] \exp \left( -\frac{1}{2\tau\alpha} \int \partial_z X_\mu(z,\overline{z}) \partial_{\overline{z}} X^\mu(z,\overline{z}) \, dz^2 + i \sum_{i=1}^N k_{i \mu} X^\mu (z_i,\overline{z}_i) \right)$