Non-critical string theory: Difference between revisions

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{{Short description|Theory in physics}}

{{string theory}}

The '''non-critical string theory''' describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invariant background. However, there are recent developments which make possible

[[Non-critical string theory: Lorentz invariance|Lorentz invariant quantization]]

of string theory in 4-dimensional Minkowski space-time.{{Citation needed|date=August 2020}}

There are several applications of the non-critical string. Through the [[AdS/CFT correspondence]] it provides a holographic description of gauge theories which are asymptotically free.{{Citation needed|date=February 2007}}<ref name=":0">{{Cite journal|last=Kiritsis|first=Elias|s2cid=2236596|date=26 Jan 2009|title=Dissecting the string theory dual of QCD|journal=Fortschritte der Physik|volume=57|issue=5–7|pages=369–417|arxiv=0901.1772|doi=10.1002/prop.200900011|bibcode=2009ForPh..57..396K}}</ref> It may then have applications to the study of the [[Quantum chromodynamics|QCD]], the theory of strong interactions between [[quarks]].<ref name=":0" /> Another area of much research is two-dimensional string theory which provides simple [[toy model]]s of [[string theory]]. There also exists a [[string duality|duality]] to the 3-dimensional [[Ising model]].{{Citation needed|date=February 2007}}

== The critical dimension and central charge ==

In order for a [[string theory]] to be consistent, the [[worldsheet]] theory must be conformally invariant. The obstruction to [[conformal symmetry]] is known as the [[Weyl anomaly]] and is proportional to the [[central charge]] of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish. For the [[bosonic string]] this can be accomplished by a worldsheet theory consisting of 26 free [[Massless free scalar bosons in two dimensions|free bosons]]. Since each boson is interpreted as a flat spacetime dimension, the critical dimension of the bosonic string is 26. A similar logic for the [[superstring]] results in 10 free bosons (and 10 free [[fermions]] as required by worldsheet [[supersymmetry]]). The bosons are again interpreted as spacetime dimensions and so the critical dimension for the superstring is 10. A string theory which is formulated in the critical dimension is called a '''critical string'''.

The non-critical string is not formulated with the critical dimension, but nonetheless has vanishing Weyl anomaly. A worldsheet theory with the correct central charge can be constructed by introducing a non-trivial target space, commonly by giving an [[expectation value]] to the [[dilaton]] which varies linearly along some spacetime direction. (From the point of view of the worldsheet CFT, this corresponds to having a [[Massless free scalar bosons in two dimensions|background charge]].)
For this reason non-critical string theory is sometimes called the '''linear dilaton theory'''. Since the dilaton is related to the string [[coupling constant]], this theory contains a region where the coupling is weak (and so perturbation theory is valid) and another region where the theory is strongly coupled. For dilaton varying along a [[spacelike]] direction, the dimension of the theory is less than the critical dimension and so the theory is termed '''subcritical'''. For dilaton varying along a [[timelike]] direction, the dimension is greater than the critical dimension and the theory is termed '''supercritical'''. The dilaton can also vary along a [[lightlike]] direction, in which case the dimension is equal to the critical dimension and the theory is a critical string theory.

== Two-dimensional string theory ==

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** Vol. 1: An introduction to the bosonic string. {{ISBN|0-521-63303-6}}.

** Vol. 2: Superstring theory and beyond. {{ISBN|0-521-63304-4}}.

* {{cite journal | last=Polyakov | first=A.M. | title=Quantum geometry of bosonic strings | journal=Physics Letters B | publisher=Elsevier BV | volume=103 | issue=3 | year=1981 | issn=0370-2693 | doi=10.1016/0370-2693(81)90743-7 | pages=207–210| bibcode=1981PhLB..103..207P }}

* {{cite journal | last=Polyakov | first=A.M. | title=Quantum geometry of fermionic strings | journal=Physics Letters B | publisher=Elsevier BV | volume=103 | issue=3 | year=1981 | issn=0370-2693 | doi=10.1016/0370-2693(81)90744-9 | pages=211–213| bibcode=1981PhLB..103..211P }}

* {{cite journal | last1=Curtright | first1=Thomas L. | last2=Thorn | first2=Charles B. | title=Conformally Invariant Quantization of the Liouville Theory | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=48 | issue=19 | date=1982-05-10 | issn=0031-9007 | doi=10.1103/physrevlett.48.1309 | pages=1309–1313| bibcode=1982PhRvL..48.1309C }} [Erratum-ibid. 48 (1982) 1768].

* {{cite journal | last1=Gervais | first1=Jean-Loup | last2=Neveu | first2=André | title=Dual string spectrum in Polyakov's quantization (II). Mode separation | journal=Nuclear Physics B | publisher=Elsevier BV | volume=209 | issue=1 | year=1982 | issn=0550-3213 | doi=10.1016/0550-3213(82)90105-5 | pages=125–145| bibcode=1982NuPhB.209..125G }}

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