玻色弦理论

玻色弦理论（Bosonic string theory）是最早的弦论版本，约在1960年代晚期发展。其名称由来是因为粒子谱中仅含有玻色子。
1980年代，在弦论的范畴下发现了超对称；一个称作超弦理论（超对称弦理论）的新版本弦论成为了研究主题。尽管如此，玻色弦理论仍然是了解微扰弦理论的有用工具，并且超弦理论中的一些理论困难之处在玻色弦理论中已然现身。
疑难.
虽然玻色弦理论有许多吸引人的特质，其在成为物理模型理论有两大缺陷：
类型.
有四种可能的玻色子弦理论，取决于是否允许开弦以及弦是否具有指定的可定向性。四种可能理论的光谱示意图如下：
请注意，所有四种理论都有一个负能量快子 (formula_1) 和一个无质量引力子。
数学表示.
路径积分表述.
玻色子弦理论可以由路径积分定义：
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.
"i" which connect to a string world surface at the surface points "z""i". It is given by the following functional integral over all possible embeddings of this 2D surface in 26 dimensions:
The functional integral can be done because it is a Gaussian to become:
Which is a beta function, known as Veneziano amplitude. It was this beta function which was apparently found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the Grassmann coordinates "θ". Since there are various ways this can be done this leads to different string theories.
When integrating over surfaces such as the torus, we end up with equations in terms of theta functions and elliptic functions such as the Dedekind eta function. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface. -->
h = 0.
At tree-level, corresponding to genus 0, the cosmological constant vanishes: formula_22.
The four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude:
Where formula_24 is the total momentum and formula_25, formula_26, formula_27 are the Mandelstam variables.
h = 1.
formula_29 is a complex number with positive imaginary part formula_30; formula_31, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group formula_32 acting on the upper half-plane, for example formula_33. formula_34 is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure formula_35 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_36 and the fact that formula_34 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.