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：

- 其只预测玻色子的存在，然而许多物理粒子为费米子。
- 其预测了一种具有虚数质量的弦模式，暗示了此理论在迅子凝聚过程会有不稳定性。

## 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：

玻色弦理论｜非正$M^2$状态

可开弦定向｜快子、引力子、胀子、无质量反对称张量（massless antisymmetric tensor）

可开弦无向｜快子、引力子、胀子

闭弦定向｜快子、引力子、胀子、反对称张量（antisymmetric tensor）、U(1)、矢量玻色子

闭弦无向｜快子、引力子、胀子

请注意，所有四种理论都有一个负能量快子 ($M^2 = - \frac{1}{\alpha'}$) 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：

$$I_0[g,X] = \frac{T}{8\pi} \int_M d^2 \xi \sqrt{g} g^{mn} \partial_m x^\mu \partial_n x^\nu G_{\mu\nu}(x)$$

$x^\mu(\xi)$ is the field on the worldsheet describing the embedding of the string in 25+1 spacetime; in the Polyakov formulation, $g$ is not to be understood as the induced metric from the embedding, but as an independent dynamical field. $G$ 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 $G_{\mu\nu} = \delta_{\mu\nu}$. M is the worldsheet as a topological manifold parametrized by the $\xi$ coordinates. $T$ is the string tension and related to the Regge slope as $T = \frac{1}{2\pi\alpha'}$.

$I_0$ 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:

$$I = I_0 + \lambda \chi(M) + \mu_0^2 \int_M d^2\xi \sqrt{g}$$

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:

$$Z = \sum_{h=0}^\infty \int \frac{\mathcal{D}g_{mn} \mathcal{D}X^\mu}{\mathcal{N}} \exp ( - I[g,X] )$$

$$\left\langle V_{i_1} (k^\mu_1) \cdots V_{i_p}(k_p^\mu) \right\rangle = \sum_{h=0}^\infty \int \frac{\mathcal{D}g_{mn} \mathcal{D}X^\mu}{\mathcal{N}} \exp ( - I[g,X] ) V_{i_1} (k_1^\mu) \cdots V_{i_p} (k^\mu_p)$$

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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 $h$. A normalization factor $\mathcal{N}$ is introduced to compensate overcounting from symmetries. While the computation of the partition function corresponds to the cosmological constant, the N-point function, including $p$ 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 $g$ 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

$$g'(\xi) = e^{\sigma(\xi)} g(\xi)$$

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 $h \geq 4$.

#### h = 0

At tree-level, corresponding to genus 0, the cosmological constant vanishes: $Z_0 = 0$.

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

$$A_4 \propto (2\pi)^{26} \delta^{26}(k) \frac{\Gamma(-1-s/2) \Gamma(-1-t/2) \Gamma(-1-u/2)}{\Gamma(2+s/2) \Gamma(2+t/2) \Gamma(2+u/2)}$$

Where $k$ is the total momentum and $s$, $t$, $u$ are the Mandelstam variables.

#### h = 1

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

$$Z_1 = \int_{\mathcal{M}_1} \frac{d^2 \tau}{8\pi^2 \tau_2^2} \frac{1}{(4\pi^2 \tau_2)^{12}} \left| \eta(\tau) \right| ^{-48}$$

$\tau$ is a complex number with positive imaginary part $\tau_2$; $\mathcal{M}_1$, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group $PSL(2,\mathbb{Z})$ acting on the upper half-plane, for example $\left\{ \tau_2 > 0, |\tau|^2 > 1, -\frac{1}{2} < \tau_1 < \frac{1}{2} \right\}$. $\eta(\tau)$ is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure $\frac{d^2 \tau}{\tau_2^2}$ is simply the Poincaré metric which has PSL(2,R) as isometry group; the rest of the integrand is also invariant by virtue of $\tau_2 \rightarrow |c \tau + d|^2 \tau_2$ and the fact that $\eta(\tau)$ 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.

## 外部链接

- How many string theories are there?
- PIRSA:C09001 - Introduction to the Bosonic String