Editing Bosonic string theory

{{Short description|26-dimensional string theory}}
{{String theory|cTopic=Theory}}
'''Bosonic string theory''' is the original version of [[string theory]], developed in the late 1960s. It is so called because it contains only [[boson]]s 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.

== Problems ==

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 number|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]],<ref name="PR">{{citation|last=Lovelace|first=Claud|title=Pomeron form factors and dual Regge cuts|journal=Physics Letters|volume=B34|issue=6|year=1971|pages=500–506|bibcode=1971PhLB...34..500L|doi=10.1016/0370-2693(71)90665-4}}.</ref> 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.

== Types of bosonic strings ==

There are four possible bosonic string theories, depending on whether [[String (physics)#Closed and open strings|open strings]] are allowed and whether strings have a specified [[Orientability#Orientability of differentiable manifolds|orientation]]. A theory of open strings must also include closed strings, because open strings can be thought of as having their endpoints fixed on a [[D-brane|D25-brane]] that fills all of spacetime. A specific orientation of the string means that only interaction corresponding to an [[Orientability|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:

{| class="wikitable"
|-
! Bosonic string theory || Non-positive <math>M^2</math> states
|-
| Open and closed, oriented || tachyon, [[graviton]], [[dilaton]], massless antisymmetric tensor
|-
| Open and closed, unoriented || tachyon, graviton, dilaton
|-
| Closed, oriented || tachyon, graviton, dilaton, antisymmetric tensor, [[U(1)]] [[vector boson]]
|-
| Closed, unoriented || tachyon, graviton, dilaton
|}

Note that all four theories have a negative energy tachyon (<math>M^2 = - \frac{1}{\alpha'}</math>) and a massless graviton.

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

== Mathematics ==

=== Path integral perturbation theory ===

Bosonic string theory can be said<ref>D'Hoker, Phong</ref> to be defined by the [[Path integral formulation|path integral quantization]] of the [[Polyakov action]]:

: <math> 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) </math>

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

<math>I_0</math> has [[Diffeomorphism invariance|diffeomorphism]] and [[Weyl transformation|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]]:

: <math> I = I_0 + \lambda \chi(M) + \mu_0^2 \int_M d^2\xi \sqrt{g} </math>

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 (quantum field theory)|partition function]] and [[Correlation function (quantum field theory)|N-point function]]:

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

[[File:Sum over genera.png|thumb|right|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 manifold|Riemannian surfaces]] and are thus identified by a genus <math>h</math>. A normalization factor <math>\mathcal{N}</math> is introduced to compensate overcounting from symmetries. While the computation of the partition function corresponds to the [[cosmological constant]], the N-point function, including <math>p</math> 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 <math>g</math> path-integral in the partition function is ''a priori'' a sum over possible Riemannian structures; however, [[Quotient space (topology)|quotienting]] with respect to Weyl transformations allows us to only consider [[conformal structure]]s, that is, equivalence classes of metrics under the identifications of metrics related by

: <math> g'(\xi) = e^{\sigma(\xi)} g(\xi) </math>

Since the world-sheet is two dimensional, there is a 1-1 correspondence between conformal structures and [[complex manifold|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 <math>h \geq 4</math>.

<!-- The single most important quantity in first quantized bosonic string theory is the N-point scattering amplitude. This treats the incoming and outgoing strings as points, which in string theory are [[tachyon]]s, with momentum ''k''<sub>''i''</sub> which connect to a string world surface at the surface points ''z''<sub>''i''</sub>. It is given by the following [[functional integral]] over all possible embeddings of this 2D surface in 26 dimensions:<ref>Polchinski, Joseph. ''String Theory: Volume I''. Cambridge University Press, p. 173.</ref>

: <math> A_N = \int  D\mu \int  D[X] \exp \left( -\frac{1}{4\pi\alpha} \int \partial_z X_\mu(z,\overline{z}) \partial_{\overline{z}}  X^\mu(z,\overline{z}) \, dz^2 + i \sum_{i=1}^N  k_{i \mu} X^\mu (z_i,\overline{z}_i) \right) </math>

The functional integral can be done because it is a Gaussian to become:

: <math> A_N = \int  D\mu \prod_{0<i<j<N+1} |z_i-z_j|^{2\alpha k_i.k_j} </math>

This is integrated over the various points ''z''<sub>''i''</sub>. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. Also you need to make sure you are not integrating multiple times over different parameterizations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function):

: <math> A_4 = \frac{ \Gamma (-1+\frac12(k_1+k_2)^2) \Gamma (-1+\frac12(k_2+k_3)^2)  } { \Gamma (-2+\frac12((k_1+k_2)^2+(k_2+k_3)^2)) } </math>

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&nbsp;''&theta;''. 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: <math> Z_0 = 0 </math>.

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

: <math> 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)} </math>

Where <math>k</math> is the total momentum and <math>s</math>, <math>t</math>, <math>u</math> are the [[Mandelstam variables]].

==== h = 1 ====

[[File:ModularGroup-FundamentalDomain.svg|thumb|right|alt=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 Feynman diagram|one-loop level]]. The partition function amounts to:

: <math> 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} </math>

<math>\tau</math> is a [[complex number]] with positive imaginary part <math>\tau_2</math>; <math>\mathcal{M}_1</math>, holomorphic to the moduli space of the torus, is any [[fundamental domain]] for the [[modular group]] <math>PSL(2,\mathbb{Z})</math> acting on the [[upper half-plane]], for example <math> \left\{ \tau_2 > 0, |\tau|^2 > 1, -\frac{1}{2} < \tau_1 < \frac{1}{2}  \right\} </math>. <math>\eta(\tau)</math> is the [[Dedekind eta function]]. The integrand is of course invariant under the modular group: the measure <math> \frac{d^2 \tau}{\tau_2^2} </math> is simply the [[Poincaré metric]] which has [[SL2(R)|PSL(2,R)]] as isometry group; the rest of the integrand is also invariant by virtue of <math>\tau_2 \rightarrow |c \tau + d|^2 \tau_2 </math> and the fact that <math>\eta(\tau)</math> 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.

==See also==
*[[Nambu–Goto action]]
*[[Polyakov action]]

==Notes==
{{reflist}}

==References==
{{cite journal
|  title = The geometry of string perturbation theory
|author1=D'Hoker, Eric  |author2=Phong, D. H. |authorlink2=Duong Hong Phong
 |name-list-style=amp |  journal = Rev. Mod. Phys.
|  volume = 60
|  issue = 4
|  pages = 917–1065
|date=Oct 1988

|  publisher = American Physical Society
|bibcode = 1988RvMP...60..917D |doi = 10.1103/RevModPhys.60.917 }}

{{cite journal
 |title           = Complex geometry and the theory of quantum strings
 |author1         = Belavin, A.A.
 |author2         = Knizhnik, V.G.
 |name-list-style = amp
 |journal         = ZhETF
 |volume          = 91
 |issue           = 2
 |pages           = 364–390
 |date            = Feb 1986
 |url             = http://www.jetp.ac.ru/cgi-bin/index/e/64/2/p214?a=list
 |bibcode         = 1986ZhETF..91..364B
 |access-date     = 2015-04-24
 |archive-date    = 2021-02-26
 |archive-url     = https://web.archive.org/web/20210226154142/http://www.jetp.ac.ru/cgi-bin/index/e/64/2/p214?a=list
 |url-status      = dead
}}

==External links==
* [https://web.archive.org/web/20101008035958/http://superstringtheory.com/basics/basic5a.html How many string theories are there?]
* [http://pirsa.org/C09001 PIRSA:C09001 - Introduction to the Bosonic String]
{{String theory topics |state=collapsed}}

[[Category:String theory]]