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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 Lorentz invariant quantization of string theory in 4-dimensional Minkowski space-time.^{[citation needed]}

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]}^[1] It may then have applications to the study of the QCD, the theory of strong interactions between quarks.^[1]

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 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 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

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.

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 theories. These "matrix models" are understood as describing the dynamics of open strings lying on D-branes in these theories. Degrees of freedom associated with closed strings, and spacetime itself, appear as emergent phenomena, providing an important example of open string tachyon condensation in string theory.

References

^ ^a ^b Kiritsis, Elias (26 Jan 2009). "Dissecting the string theory dual of QCD". Fortschritte der Physik. 57 (5–7): 369–417. arXiv:0901.1772. Bibcode:2009ForPh..57..396K. doi:10.1002/prop.200900011. S2CID 2236596.

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.
Polyakov, A.M. (1981). "Quantum geometry of bosonic strings". Physics Letters B. 103 (3): 207–210. Bibcode:1981PhLB..103..207P. doi:10.1016/0370-2693(81)90743-7. ISSN 0370-2693.
Polyakov, A.M. (1981). "Quantum geometry of fermionic strings". Physics Letters B. 103 (3): 211–213. Bibcode:1981PhLB..103..211P. doi:10.1016/0370-2693(81)90744-9. ISSN 0370-2693.
Curtright, Thomas L.; Thorn, Charles B. (1982-05-10). "Conformally Invariant Quantization of the Liouville Theory". Physical Review Letters. 48 (19): 1309–1313. Bibcode:1982PhRvL..48.1309C. doi:10.1103/physrevlett.48.1309. ISSN 0031-9007. [Erratum-ibid. 48 (1982) 1768].
Gervais, Jean-Loup; Neveu, André (1982). "Dual string spectrum in Polyakov's quantization (II). Mode separation". Nuclear Physics B. 209 (1): 125–145. Bibcode:1982NuPhB.209..125G. doi:10.1016/0550-3213(82)90105-5. ISSN 0550-3213.

[:0-1] Kiritsis, Elias (26 Jan 2009). "Dissecting the string theory dual of QCD". Fortschritte der Physik. 57 (5–7): 369–417. arXiv:0901.1772. Bibcode:2009ForPh..57..396K. doi:10.1002/prop.200900011. S2CID 2236596.

[1]

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+{{string theory}}
-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]].
+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 [[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]].)
-== 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 string theory ==
-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.
+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.
+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]]

v t e String theory
Background	Strings Cosmic 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	Worldsheet 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 theory K-theory (physics) Twisted K-theory
Supersymmetry	Supergravity Eleven-dimensional supergravity Type I supergravity Type IIA supergravity Type IIB 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 Curtright Dijkgraaf Distler Douglas Duff Dvali Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey 't Hooft Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Sagnotti Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach