Non-critical string theory: Difference between revisions

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{{Short description|Theory in physics}}

The '''non-critical string theory''' describes the relativistic string without enforcing the critical dimension. Because in these cases, the non-critical string theory is not suitable to describe the fundamental nature of interactions, as it does not have the property of consistently quantized Lorenz symmetry and massless excitations. However, there are several applications for such a theory. The most prominent application is the phenomological description of the confinement of strong interaction by [[QCD string]]s. There also exists a [[duality]] to the 3-dimensional [[Ising model]].

{{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

== The Critical Dimension and Central Charge ==

[[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" />

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 [[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 intepreted as spacetime dimensions and so the critical dimension for the superstring is 10.

== The critical dimension and central charge ==

The non-critical string is not formulated with the critical dimension, but nonetheless has vanishing Weyl anomaly. A worldsheet theory with the correct central cahrge can be constructed by introducing a non-trivial target space, commonly by giving a spatially varying [[expectation value]] to the [[dilaton]]. Since the dilaton is related to the string [[coupling constant]], this leads to a region where the coupling is weak(and so perturbation theory is valid) and another region where the theory is strongly coupled.

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 intepretedinterpreted 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 cahrgecharge can be constructed by introducing a non-trivial target space, commonly by giving a spatially varyingan [[expectation value]] to the [[dilaton]]. Sincewhich thevaries dilatonlinearly isalong relatedsome tospacetime thedirection. string(From [[couplingthe constant]],point thisof leadsview to a region whereof the couplingworldsheet isCFT, weak(andthis socorresponds perturbationto theoryhaving isa valid)[[Massless andfree anotherscalar regionbosons wherein thetwo theorydimensions|background is strongly coupledcharge]].)

== Two-dimensional String Theory ==

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 Stringstring Theorytheory ==

Perhaps the most studied example of non-critical string theory is that with two-dimensional target space. While clearly not of phenomenological interest, string theories in two- dimensions serve as important toy models. They allow one to probe interesting concepts which would be computationally intractable in a more realistic scenario. Additionally, these models often have fully non-perturbative descriptions in the form of the quantum mechanics of large matrices.

These models often have fully non-perturbative descriptions in the form of the quantum mechanics of large matrices. Such a description known as the c=1 matrix model captures the dynamics of [[bosonic string theory]] in two dimensions. Of much recent interest are matrix models of the two-dimensional [[Type 0 string theory|Type 0 string theories]]. These "matrix models" are understood as describing the dynamics of [[open string (physics)|open string]]s lying on [[D-branes]] in these theories. Degrees of freedom associated with [[closed string]]s, and [[spacetime]] itself, appear as emergent phenomena, providing an important example of open string [[tachyon condensation]] in string theory.

== See also ==

{{physics-stub}}

* [[String theory]], for general information about critical superstrings

* [[Weyl anomaly]]

* [[Central charge]]

* [[Liouville gravity]]

== References ==

{{Reflist}}

{{Refbegin}}

* [[Joseph Polchinski|Polchinski, Joseph]] (1998). ''String Theory'', Cambridge University Press. A modern textbook.

** 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| 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| 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| 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| volume=209 | issue=1 | year=1982 | issn=0550-3213 | doi=10.1016/0550-3213(82)90105-5 | pages=125–145| bibcode=1982NuPhB.209..125G }}

{{Refend}}

{{String theory topics |state=collapsed}}

[[Category:String theory]]