Liouville field theory

In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional quantum field theory whose classical equation of motion resembles the Joseph Liouville's non-linear second order differential equation that appears in the classical geometrical problem of uniformizing Riemann surfaces.^[1]

The field theory is defined by the local action

$S = \frac{1}{4\pi } \int d^2x \sqrt{g} (g^{\mu \nu} \partial _\mu \phi \partial _{\nu} \phi + (b+b^{-1}) R \phi + 4\pi e^{2b\phi }),$

where $\partial _\mu = \partial /\partial x^\mu ,\ g_{\mu \nu}$ is the metric of the two-dimensional space on which the field theory is formulated, $R$ is the Ricci scalar of such space, and $b$ is a real coupling constant. The field $\phi$ is consequently dubbed the Liouville field.

The equation of motion associated to this action is

$\Delta \phi(x) = \frac {1}{2} (b+b^{-1}) R(x) + 4\pi b e^{2b\phi (x)}$

where $\Delta = g^{-1/2} \partial _{\mu} (g^{1/2} g^{\mu \nu} \partial_{\nu} )$ is the D'Alembert operator in such space (see also Laplace–Beltrami operator). In the case the metric of the space being the Euclidean metric, and using standard notation, it becomes the classical Liouville equation

$\left(\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} \right) \phi (x,y) = 4\pi b e^{2b \phi (x,y)}$

Liouville field theory is a conformal field theory that incarnates Weyl symmetry in a very special way.^[2] Its central charge $c$ is given in terms of the parameter appearing in the action through the expression $c=1+6(b+1/b)^{2}$. Liouville theory appears in the context of string theory when trying to formulate a non-critical version of the theory in the path integral formulation.^[3] Also in the string theory context, if coupled to a free bosonic field Liouville field theory can be thought of as the theory describing string excitations in a two-dimensional space(time).

Liouville field theory is one of the best understood examples of what is called a non-rational conformal field theory, for which some observables have been computed explicitly. Such is the case of two-point and three-point correlation functions of primary operators on the topology of the sphere.^[4][5] Explicit expressions for observables of the theory defined on other topologies, like the partition function on the torus and the one-point function on the disk, were also calculated in the recent years.

Liouville theory is also closely related to other problems in physics and mathematics, like two-dimensional quantum gravity, two-dimensional string theory, three-dimensional general relativity in negatively curved spaces, four-dimensional superconformal gauge theories, the uniformization problem of Riemann surfaces, and other problems in conformal mapping. It is also connected to other two-dimensional non-rational conformal field theories with affine symmetry, like the Wess–Zumino–Novikov–Witten theory for the group $SL(2,R)$, and, besides, it can be regarded as a special case (namely the case $N=1$) of the family of $A_{N}$ Toda field theories. Liouville theory also admits supersymmetric extension.^[6][7]

See also

• Liouville gravity
• Liouville's theorem
• Conformal field theory
• String theory

References

1. Lützen, J. Joseph Liouville, 1809–1882: Master of pure and applied mathematics, Springer 1990 ISBN 0387971807.
2. Jackiw, R. (2006). "Weyl symmetry and the Liouville theory". Theoretical and Mathematical Physics 148: 941. arXiv:hep-th/0511065. Bibcode:2006TMP...148..941J. doi:10.1007/s11232-006-0090-9.
3. Polyakov, A.M. (1981). "Quantum geometry of bosonic strings". Physics Letters B 103 (3): 207. Bibcode:1981PhLB..103..207P. doi:10.1016/0370-2693(81)90743-7.
4. Zamolodchikov, A.; Zamolodchikov, Al. (1996). "Conformal bootstrap in Liouville field theory". Nuclear Physics B 477 (2): 577. arXiv:hep-th/9506136. Bibcode:1996NuPhB.477..577Z. doi:10.1016/0550-3213(96)00351-3.
5. Dorn, H.; Otto, H.-J. (1992). "On correlation functions for non-critical strings with c⩽1 but d⩾1". Physics Letters B 291: 39. arXiv:hep-th/9206053. Bibcode:1992PhLB..291...39D. doi:10.1016/0370-2693(92)90116-L.
6. Teschner, J (2001). "Liouville theory revisited". Classical and Quantum Gravity 18 (23): R153. arXiv:hep-th/0104158. Bibcode:2001CQGra..18R.153T. doi:10.1088/0264-9381/18/23/201.
7. Nakayama, YU (2004). "Liouville Field Theory: A Decade After the Revolution". International Journal of Modern Physics A 19 (17n18): 2771. arXiv:hep-th/0402009. Bibcode:2004IJMPA..19.2771N. doi:10.1142/S0217751X04019500.