Bosonic string theory

Short description: 26-dimensional string theory

String theory

Fundamental objects

* String
* Brane
* D-brane

Perturbative theory

* Bosonic
* Superstring
* Type I
* Type II (IIA / IIB)
* Heterotic (SO(32) * E8×E8)

Non-perturbative results

* S-duality
* T-duality
* U-duality
* M-theory
* F-theory
* AdS/CFT correspondence

Phenomenology

* Phenomenology
* Cosmology
* Landscape

Mathematics

* Mirror symmetry
* Monstrous moonshine

Related concepts

* Theory of everything
* Conformal field theory
* Quantum gravity
* Supersymmetry
* Supergravity
* Twistor string theory
* N = 4 supersymmetric Yang–Mills theory
* Kaluza–Klein theory
* Multiverse
* Holographic principle

Theorists

* Aganagić
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* Kachru
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* Kallosh
* Kaluza
* Kapustin
* Klebanov
* Knizhnik
* Kontsevich
* Klein
* Linde
* Maldacena
* Mandelstam
* Marolf
* Martinec
* Minwalla
* Moore
* Motl
* Mukhi
* Myers
* Nanopoulos
* Naqvi
* Năstase
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* Ooguri
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* Polchinski
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* Rajaraman
* Ramond
* Randall
* Randjbar-Daemi
* Roček
* Rohm
* Scherk
* Schwarz
* Seiberg
* Sen
* Shenker
* Siegel
* Silverstein
* Sơn
* Staudacher
* Steinhardt
* Strominger
* Sundrum
* Susskind
* 't Hooft
* Townsend
* Trivedi
* Turok
* Vafa
* Veneziano
* Verlinde
* Verlinde
* Wess
* Witten
* Yau
* Yoneya
* Zamolodchikov
* Zamolodchikov
* Zaslow
* Zumino
* Zwiebach

* History
* Glossary

* v
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Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons 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.

## Contents

* 1 Problems
* 2 Types of bosonic strings
* 3 Mathematics
* 3.1 Path integral perturbation theory
* 3.1.1 h = 0
* 3.1.2 h = 1
* 4 See also
* 5 Notes
* 6 References
* 7 External links

## 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 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,[1] 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 open strings are allowed and whether strings have a specified orientation. Recall that a theory of open strings also must include closed strings; open strings can be thought as having their endpoints fixed on a D25-brane that fills all of spacetime. A specific orientation of the string means that only interaction corresponding to an 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:

Bosonic string theory | Non-positive M2 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 (M2=−1α′) 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[2] to be defined by the path integral quantization of the Polyakov action:

I0[g,X]=T8π∫Md2ξggmn∂mxμ∂nxνGμν(x)

xμ(ξ) 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 Gμν=δμν. M is the worldsheet as a topological manifold parametrized by the ξ coordinates. T is the string tension and related to the Regge slope as T=12πα′.

I0 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=I0+λχ(M)+μ02∫Md2ξg

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=∑h=0∞∫𝒟gmn𝒟Xμ𝒩exp⁡(−I[g,X])
⟨Vi1(k1μ)⋯Vip(kpμ)⟩=∑h=0∞∫𝒟gmn𝒟Xμ𝒩exp⁡(−I[g,X])Vi1(k1μ)⋯Vip(kpμ)
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 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′(ξ)=eσ(ξ)g(ξ)

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≥4.

#### h = 0

At tree-level, corresponding to genus 0, the cosmological constant vanishes: Z0=0.

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

A4∝(2π)26δ26(k)Γ(−1−s/2)Γ(−1−t/2)Γ(−1−u/2)Γ(2+s/2)Γ(2+t/2)Γ(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:

Z1=∫ℳ1d2τ8π2τ221(4π2τ2)12|η(τ)|−48

τ is a complex number with positive imaginary part τ2; ℳ1, holomorphic to the moduli space of the torus, is any fundamental domain for the modular group PSL(2,ℤ) acting on the upper half-plane, for example {τ2>0,|τ|2>1,−12<τ1<12}. η(τ) is the Dedekind eta function. The integrand is of course invariant under the modular group: the measure d2ττ22 is simply the Poincaré metric which has PSL(2,R) as isometry group; the rest of the integrand is also invariant by virtue of τ2->|cτ+d|2τ2 and the fact that η(τ) 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

1. ↑ Lovelace, Claud (1971), "Pomeron form factors and dual Regge cuts", Physics Letters B34 (6): 500–506, doi:10.1016/0370-2693(71)90665-4, Bibcode: 1971PhLB...34..500L .
2. ↑ D'Hoker, Phong

## References

"The geometry of string perturbation theory". Rev. Mod. Phys. (American Physical Society) 60 (4): 917–1065. Oct 1988. doi:10.1103/RevModPhys.60.917. Bibcode: 1988RvMP...60..917D.

Belavin, A.A.; Knizhnik, V.G. (Feb 1986). "Complex geometry and the theory of quantum strings". ZhETF 91 (2): 364–390. Bibcode: 1986ZhETF..91..364B. http://www.jetp.ac.ru/cgi-bin/index/e/64/2/p214?a=list. Retrieved 2015-04-24.

## External links

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

* v
* t
* e

String theory

Background|

* Strings
* History of string theory
* First superstring revolution
* Second superstring revolution
* String theory landscape

Theory|

* Nambu–Goto action
* Polyakov action
* Bosonic string theory
* Superstring theory
* Type I string
* Type II string
* Type IIA string
* Type IIB string
* Heterotic string
* N=2 superstring
* F-theory
* String field theory
* Matrix string theory
* Non-critical string theory
* Non-linear sigma model
* Tachyon condensation
* RNS formalism
* GS formalism

String duality|

* T-duality
* S-duality
* U-duality
* Montonen–Olive duality

Particles and fields|

* Graviton
* Dilaton
* Tachyon
* Ramond–Ramond field
* Kalb–Ramond field
* Magnetic monopole
* Dual graviton
* Dual photon

Branes|

* D-brane
* NS5-brane
* M2-brane
* M5-brane
* S-brane
* Black brane
* Black holes
* Black string
* Brane cosmology
* Quiver diagram
* Hanany–Witten transition

Conformal field theory|

* Virasoro algebra
* Mirror symmetry
* Conformal anomaly
* Conformal algebra
* Superconformal algebra
* Vertex operator algebra
* Loop algebra
* Kac–Moody algebra
* Wess–Zumino–Witten model

Gauge theory|

* Anomalies
* Instantons
* Chern–Simons form
* Bogomol'nyi–Prasad–Sommerfield bound
* Exceptional Lie groups (G2, F4, E6, E7, E8)
* ADE classification
* Dirac string
* p-form electrodynamics

Geometry|

* Kaluza–Klein theory
* Compactification
* Why 10 dimensions?
* Kähler manifold
* Ricci-flat manifold
* Calabi–Yau manifold
* Hyperkähler manifold
* K3 surface
* G2 manifold
* Spin(7)-manifold
* Generalized complex manifold
* Orbifold
* Conifold
* Orientifold
* Moduli space
* Hořava–Witten domain wall
* K-theory
* Twisted K-theory

Supersymmetry|

* Supergravity
* Superspace
* Lie superalgebra
* Lie supergroup

Holography|

* Holographic principle
* AdS/CFT correspondence

M-theory|

* Matrix theory
* Introduction to M-theory

String theorists|

* Aganagić
* Arkani-Hamed
* Atiyah
* Banks
* Berenstein
* Bousso
* Cleaver
* Curtright
* Dijkgraaf
* Distler
* Douglas
* Duff
* Ferrara
* Fischler
* Friedan
* Gates
* Gliozzi
* Gopakumar
* Green
* Greene
* Gross
* Gubser
* Gukov
* Guth
* Hanson
* Harvey
* Hořava
* Gibbons
* Kachru
* Kaku
* Kallosh
* Kaluza
* Kapustin
* Klebanov
* Knizhnik
* Kontsevich
* Klein
* Linde
* Maldacena
* Mandelstam
* Marolf
* Martinec
* Minwalla
* Moore
* Motl
* Mukhi
* Myers
* Nanopoulos
* Naqvi
* Năstase
* Nekrasov
* Neveu
* Nielsen
* van Nieuwenhuizen
* Novikov
* Olive
* Ooguri
* Ovrut
* Polchinski
* Polyakov
* Rajaraman
* Ramond
* Randall
* Randjbar-Daemi
* Roček
* Rohm
* Scherk
* Schwarz
* Seiberg
* Sen
* Shenker
* Siegel
* Silverstein
* Sơn
* Staudacher
* Steinhardt
* Strominger
* Sundrum
* Susskind
* 't Hooft
* Townsend
* Trivedi
* Turok
* Vafa
* Veneziano
* Verlinde
* Verlinde
* Wess
* Witten
* Yau
* Yoneya
* Zamolodchikov
* Zamolodchikov
* Zaslow
* Zumino
* Zwiebach

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