**玻色弦理論**（英語：）是最早的弦論版本，約在1960年代晚期發展。其名稱由來是因為粒子譜中僅含有玻色子。

1980年代，在弦論的範疇下發現了超對稱；一個稱作超弦理論（超對稱弦理論）的新版本弦論成為了研究主題。儘管如此，玻色弦理論仍然是了解微擾弦理論的有用工具，並且超弦理論中的一些理論困難之處在玻色弦理論中已然現身。

疑難
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雖然玻色弦理論有許多吸引人的特質，其在成為物理模型理論有兩大缺陷：

1. 其只預測玻色子的存在，然而許多物理粒子為費米子。
2. 其預測了一種具有虛數質量的弦模式，暗示了此理論在迅子凝聚過程會有不穩定性。

類型
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有四種可能的玻色子弦理論，取決於是否允許開弦以及弦是否具有指定的可定向性。四種可能理論的光譜示意圖如下：

| 玻色弦理论 | 非正M^{2}狀態 |
| --- | --- |
| 可開弦定向 | 快子、引力子、脹子、無質量反對稱張量（massless antisymmetric tensor） |
| 可開弦無向 | 快子、引力子、脹子 |
| 閉弦定向 | 快子、引力子、脹子、反對稱張量（antisymmetric tensor）、U(1)、矢量玻色子 |
| 閉弦無向 | 快子、引力子、脹子 |

請注意，所有四種理論都有一個負能量快子 ({\displaystyle M^{2}=-{\frac {1}{\alpha '}}}) 和一個無質量引力子。

數學表示
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### 路徑積分表述

玻色子弦理論可以由路徑積分定義：

{\displaystyle 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)}

已隱藏部分未翻譯内容，歡迎參與翻譯。
{\displaystyle 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 {\displaystyle 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 {\displaystyle 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:

{\displaystyle 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:

{\displaystyle Z=\sum \_{h=0}^{\infty }\int {\frac {{\mathcal {D}}g\_{mn}{\mathcal {D}}X^{\mu }}{\mathcal {N}}}\exp(-I[g,X])}
{\displaystyle \left\langle V\_{i\_{1}}(k\_{1}^{\mu })\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\_{p}^{\mu })}
The perturbative series is expressed as a sum over topologies, indexed by the genus.
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

{\displaystyle 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 {\displaystyle h\geq 4}.

  

#### h = 0

At tree-level, corresponding to genus 0, the cosmological constant vanishes: {\displaystyle Z\_{0}=0}.

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

{\displaystyle 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

Fundamental domain for the modular group. 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:

{\displaystyle 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}; {\displaystyle {\mathcal {M}}\_{1}}, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group {\displaystyle PSL(2,\mathbb {Z} )} acting on the upper half-plane, for example {\displaystyle \left\{\tau \_{2}>0,|\tau |^{2}>1,-{\frac {1}{2}}<\tau \_{1}<{\frac {1}{2}}\right\}}. {\displaystyle \eta (\tau )} is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure {\displaystyle {\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 {\displaystyle \tau \_{2}\rightarrow |c\tau +d|^{2}\tau \_{2}} and the fact that {\displaystyle \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.

相關條目
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* 弦理論
* 快子

參考
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1. ↑  D'Hoker, Phong

### 參考文獻

D'Hoker, Eric & Phong, D. H.. Rev. Mod. Phys. (American Physical Society). Oct 1988, **60** (4): 917–1065. Bibcode:1988RvMP...60..917D. doi:10.1103/RevModPhys.60.917. 

Belavin, A.A. & Knizhnik, V.G.. ZhETF. Feb 1986, **91** (2): 364–390  [2022-06-18]. Bibcode:1986ZhETF..91..364B. （原始内容存档于2021-02-26）.

外部鏈接
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* How many string theories are there?
* PIRSA:C09001 - Introduction to the Bosonic String （页面存档备份，存于）

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* 论
* 编
弦理論 |
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| 基本对象 | 
* 弦
* 膜
* D膜
* S膜（英语：）
* NS5膜
* M2膜
* M5膜
* 黑膜（英语：）

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| 背景理論 | 
* 南部-后藤作用
* 泊里雅科夫作用量
* 陳-西蒙斯理論
* 共形场论
* 量子引力
* 卡鲁扎-克莱因理论
* 全像原理
* 万有理论
* 杨-米尔斯理论

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| 微扰弦理论 | 
* 玻色弦理論
* 超弦理論
* 第一型弦理論
* 第二型弦理論（第二A型 · 第二B型）
* 混合弦理論（SO(32)雜弦 · *E*8x*E*8雜弦）
* 弦拓扑
* 扭量弦理論（英语：）

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| 非微扰结果 | 
* 弦場論
* 弦論對偶性
* S對偶
* T對偶
* U對偶
* 镜像对称性
* M理论（入門）
* AdS/CFT对偶

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| 现象学 | 
* 弦唯象学
* 弦宇宙學
* 弦論地景
* 膜宇宙學

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| 数学方法 | 
* 陈–怕吞因素
* 摄动理论
* 巴塔林-維爾可維斯基代數
* 格爾斯滕哈伯代數
* 顶点算子代数
* 維拉宿代數
* 卡茨-穆迪代數
* 1/N展开

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| 几何 | 
* RNS体系（英语：）
* GS体系（英语：）
* GSO映射（英语：）
* 卡拉比—丘流形
* K3曲面
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* 紧化
* 軌形（英语：）
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* 錐形(弦论)（英语：）
* 塞伯格-維騰理論
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* 塞伯格-維騰映射
* 快子凝聚
* 微扰反常
* O(N)模型
* 非线性σ模型

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| 规范场论 | 
* 陈-西蒙斯形式
* 楊-米爾斯理論
* Wess-Zumino-Witten模型
* 纽结理论
* 瞬子
* BRST量子化
* BPS態
* 磁单极子

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| 超对称 | 
* 超引力
* 超空間
* 李超群（英语：）
* 超對稱楊-米爾斯理論

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| 理论家 | 
* 阿尔卡尼-哈米德
* 迈克尔·阿蒂亚
* 湯姆·班克斯
* 陈省身
* 戴克赫拉夫
* 朵夫（英语：）
* 費斯勒（英语：）
* 丹尼爾·弗里丹
* 蓋茲（英语：）
* Gliozzi（英语：）
* 迈克尔·格林 (物理学家)
* 布莱恩·葛林
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* 葛布澤（英语：）
* 哈維（英语：）
* 霍扎瓦（英语：）
* 加来道雄
* Klebanov（英语：）
* 南部阳一郎
* 孔采维奇
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* Martinec（英语：）
* 格雷·穆爾
* 莫特爾
* Nekrasov（英语：）
* Neveu（英语：）
* 奧利佛（英语：）
* 波尔钦斯基
* 泊里雅科夫
* 加來道雄
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* 皮埃尔·拉蒙
* Rohm（英语：）
* Scherk（英语：）
* 约翰·施瓦茨
* 内森·塞伯格
* Sen（英语：）
* Sơn（英语：）
* 安德鲁·施特罗明格
* 桑壯（英语：）
* 萨斯坎德
* 特·胡夫特
* Townsend（英语：）
* 瓦法
* 韦内齐亚诺
* 埃·弗尔林德
* 赫·弗尔林德（英语：）
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* Zaslow（英语：）
* 弗朗克·韦尔切克
* 文小刚
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* 历史
* 词汇

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