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.
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. [1] It may then have applications to the study of the QCD, the theory of strong interactions between quarks.[1] Another area of much research is two-dimensional string theory which provides simple toy models of string theory. There also exists a duality to the 3-dimensional Ising model.
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. 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.
See also
String theory, for general information about critical superstrings
Weyl anomaly
Central charge
Liouville gravity
References
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.
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String theory
Background
Strings History of string theory
First superstring revolution Second superstring revolution String theory landscape
Calabi-Yau-alternate
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 (physics) Twisted K-theory
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 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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