玻色弦つる理論りろん

玻色弦つる理論りろん（英語えいご：Bosonic string theory）是これ最早もはや的てき弦つる論ろん版本はんぽん，約やく在ざい1960年代ねんだい晚期ばんき發展はってん。其名稱めいしょう由來ゆらい是ぜ因いん為ため粒子りゅうし譜ふ中ちゅう僅含有がんゆう玻色子こ。

1980年代ねんだい，在ざい弦つる論ろん的てき範疇はんちゅう下か發現はつげん了りょう超ちょう對稱たいしょう；一いち個こ稱たたえ作づく超ちょう弦つる理論りろん（超ちょう對稱たいしょう弦つる理論りろん）的てき新しん版本はんぽん弦つる論ろん成なり為ため了りょう研究けんきゅう主題しゅだい。儘管如此，玻色弦つる理論りろん仍然是ぜ了解りょうかい微ほろ擾弦つる理論りろん的てき有用ゆうよう工具こうぐ，並なみ且超弦つる理論りろん中ちゅう的てき一些理論困難之處在玻色弦理論中已然現身。

疑うたぐ難がた

雖然玻色弦つる理論りろん有ゆう許多きょた吸引きゅういん人的じんてき特質とくしつ，其在成なり為ため物理ぶつり模型もけい理論りろん有ゆう兩りょう大だい缺陷けっかん：

1. 其只預あずか測はか玻色子こ的てき存在そんざい，然しか而許多た物理ぶつり粒子りゅうし為ため費ひ米子よなご。
2. 其預測はか了りょう一種いっしゅ具有ぐゆう虛數きょすう質量しつりょう的てき弦つる模も式しき，暗示あんじ了りょう此理論ろん在ざい迅子凝聚ぎょうしゅう過程かてい會かい有ゆう不穩ふおん定性ていせい。

類型るいけい

有ゆう四種可能的玻色子弦理論，取と決けつ於是否ひ允許いんきょ開ひらき弦つる以及弦つる是ぜ否ひ具有ぐゆう指定してい的てき可か定てい向性こうせい。四種可能理論的光譜示意圖如下：

玻色弦つる理り论	非ひ正せい$M^{2}$狀態じょうたい
可か開ひらき弦つる定てい向むかい	快かい子こ、引力いんりょく子こ、脹子、無む質量しつりょう反對稱はんたいしょう張はり量りょう（massless antisymmetric tensor）
可か開ひらき弦つる無む向むかい	快かい子こ、引力いんりょく子こ、脹子
閉弦定てい向むかい	快かい子こ、引力いんりょく子こ、脹子、反對稱はんたいしょう張はり量りょう（antisymmetric tensor）、U(1)、矢や量りょう玻色子こ
閉弦無む向むかい	快かい子こ、引力いんりょく子こ、脹子

請注意ちゅうい，所有しょゆう四種理論都有一個負能量快子 ($M^{2}=-{\frac {1}{\alpha '}}$) 和わ一いち個こ無む質量しつりょう引力いんりょく子こ。

數學すうがく表示ひょうじ

路みち徑みち積分せきぶん表ひょう述じゅつ

玻色子こ弦つる理論りろん可か以由ゆかり路みち徑みち積分せきぶん定義ていぎ：

$I_{0}[g,X]={\frac {T}{8\pi }}\int _{M}d^{2}\xi {\sqrt {g}}g^{mn}\partial _{m}x^{\mu }\partial _{n}x^{\nu }G_{\mu \nu }(x)$

已やめ隱かくれ藏ぞう部分ぶぶん未み翻譯ほんやく内容ないよう，歡迎かんげい參與さんよ翻譯ほんやく。

$x^{\mu }(\xi )$ 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:

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

$\left\langle V_{i_{1}}(k_{1}^{\mu })\cdots V_{i_{p}}(k_{p}^{\mu })\right\rangle =\sum _{h=0}^{\infty }\int {\frac {{\mathcal {D}}g_{mn}{\mathcal {D}}X^{\mu }}{\mathcal {N}}}\exp(-I[g,X])V_{i_{1}}(k_{1}^{\mu })\cdots V_{i_{p}}(k_{p}^{\mu })$

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

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

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

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:

$Z_{1}=\int _{{\mathcal {M}}_{1}}{\frac {d^{2}\tau }{8\pi ^{2}\tau _{2}^{2}}}{\frac {1}{(4\pi ^{2}\tau _{2})^{12}}}\left|\eta (\tau )\right|^{-48}$

$\tau$ 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.

相關そうかん條目じょうもく

• 弦つる理論りろん
• 快かい子こ

參考さんこう

參考さんこう文獻ぶんけん

D'Hoker, Eric & Phong, D. H. The geometry of string perturbation theory. 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. Complex geometry and the theory of quantum strings. ZhETF. Feb 1986, 91 (2): 364–390 [2022-06-18]. Bibcode:1986ZhETF..91..364B. （原始げんし内容ないよう存そん档于2021-02-26）. 

外部がいぶ鏈接

• How many string theories are there?
• PIRSA:C09001 - Introduction to the Bosonic String （页面存そん档备份，存そん于互联网档案あん馆）