Bosonic string theory

Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it only contains 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.

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,[1] 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.

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:

The rest of this article applies to the closed, oriented theory, corresponding to borderless, orientable worldsheets.

Bosonic string theory can be said[2] to be defined by the path integral quantization of the Polyakov action:

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 perturbative series is expressed as a sum over topologies, indexed by the genus.

The four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude:

The shaded region is a possible fundamental domain for the modular group.

Genus 1 is the torus, and corresponds to the one-loop level. The partition function amounts to:

This integral diverges. This is due to the presence of the tachyon and is related to the instability of the perturbative vacuum.

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