2D conformal field theories and holography. (English) Zbl 1071.81093
Summary: It is known that the chiral part of any 2D conformal field theory defines a 3D topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3D topological theory that arises is a certain ”square” of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3D gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev–Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev–Viro theory. We compute the components of these states in the basis in the Turaev–Viro Hilbert space given by colored 3-valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting ”holographic” perspective on conformal field theories in two dimensions.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 57R56 | Topological quantum field theories (aspects of differential topology) |
References:
| [1] | Frohlich J., Int. J. Mod. Phys. A 4 pp 5321– (1989) · Zbl 0708.57005 · doi:10.1142/S0217751X89002296 |
| [2] | Felder G., Phys. Rev. Lett. 84 pp 1659– (2000) · Zbl 1038.81520 · doi:10.1103/PhysRevLett.84.1659 |
| [3] | Felder G., Compos. Math. 131 pp 189– (2002) · Zbl 1002.81045 · doi:10.1023/A:1014903315415 |
| [4] | C. Schweigert, J. Fuchs, and J. Walcher, Conformal Field Theory, Boundary Conditions and Applications to String Theory, in ”Budapest 2000, Nonperturbative QFT methods and their applications,” 37–93, also available as hep-th/0011109. · Zbl 0995.81120 |
| [5] | DOI: 10.1016/0550-3213(90)90510-K · doi:10.1016/0550-3213(90)90510-K |
| [6] | Krasnov K., Class. Quantum Grav. 20 pp 4015– (2003) · Zbl 1048.83018 · doi:10.1088/0264-9381/20/18/311 |
| [7] | DOI: 10.1016/0040-9383(92)90015-A · Zbl 0779.57009 · doi:10.1016/0040-9383(92)90015-A |
| [8] | K. Gawedzki, I. Todorov, and P. Tran-Ngoc-Bich, ”Canonical quantization of the boundary Wess–Zumino–Witten model,” hep-th/0101170. |
| [9] | Fuchs J., Nucl. Phys. B 646 pp 353– (2002) · Zbl 0999.81079 · doi:10.1016/S0550-3213(02)00744-7 |
| [10] | Benedetti R., J. Knot Theory Ramif. 5 pp 427– (1996) · Zbl 0890.57029 · doi:10.1142/S0218216596000266 |
| [11] | DOI: 10.1007/BF01217730 · Zbl 0667.57005 · doi:10.1007/BF01217730 |
| [12] | DOI: 10.1016/0001-8708(84)90040-9 · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9 |
| [13] | Witten E., Commun. Math. Phys. 144 pp 189– (1992) · Zbl 0766.53068 · doi:10.1007/BF02099196 |
| [14] | DOI: 10.1016/0550-3213(88)90603-7 · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7 |
| [15] | DOI: 10.1007/BF01238857 · Zbl 0694.53074 · doi:10.1007/BF01238857 |
| [16] | Penner R. C., Commun. Math. Phys. 113 pp 299– (1987) · Zbl 0642.32012 · doi:10.1007/BF01223515 |
| [17] | DOI: 10.1016/0040-9383(94)00053-0 · Zbl 0866.57014 · doi:10.1016/0040-9383(94)00053-0 |
| [18] | DOI: 10.1007/BF01239527 · Zbl 0725.57007 · doi:10.1007/BF01239527 |
| [19] | Boulatov D. V., Mod. Phys. Lett. A 8 pp 3491– (1993) · Zbl 1021.81699 · doi:10.1142/S0217732393003913 |
| [20] | Fock V. V., Am. Math. Soc. Transl. 191 pp 67– (1999) · doi:10.1090/trans2/191/03 |
| [21] | Alekseev A. Yu., St. Petersbg. Math. J. 6 pp 241– (1995) |
| [22] | A. Yu. Alekseev, also available as hep-th/9311074. |
| [23] | Alekseev A. Yu., Commun. Math. Phys. 166 pp 99– (1995) · Zbl 0829.53028 · doi:10.1007/BF02101598 |
| [24] | A. Yu. Alekseev and V. Shomerus, ”Representation theory of Chern–Simons observables,” q-alg/9503016. |
| [25] | Alekseev A. Yu., Commun. Math. Phys. 172 pp 317– (1995) · Zbl 0842.58037 · doi:10.1007/BF02099431 |
| [26] | Alekseev A. Yu., Commun. Math. Phys. 174 pp 561– (1995) · Zbl 0842.58038 · doi:10.1007/BF02101528 |
| [27] | Buffenoir E., Class. Quantum Grav. 19 pp 4953– (2002) · Zbl 1021.83016 · doi:10.1088/0264-9381/19/19/313 |
| [28] | DOI: 10.1016/0550-3213(87)90418-4 · doi:10.1016/0550-3213(87)90418-4 |
| [29] | DOI: 10.1103/PhysRevLett.68.1795 · Zbl 0969.83502 · doi:10.1103/PhysRevLett.68.1795 |
| [30] | V. V. Fock, ”Dual Teichmuller spaces” dg-ga/9702018; · Zbl 0986.32007 · doi:10.1007/BF02557246 |
| [31] | Chekhov L., Theor. Math. Phys. 120 pp 1245– (1999) · Zbl 0986.32007 · doi:10.1007/BF02557246 |
| [32] | Kashaev R. M., Lett. Math. Phys. 43 pp 105– (1998) · Zbl 0897.57014 · doi:10.1023/A:1007460128279 |
| [33] | R. M. Kashaev, also available as q-alg/9705021. · Zbl 0897.57014 · doi:10.1023/A:1007460128279 |
| [34] | B. Ponsot and J. Teschner, ”Liouville bootstrap via harmonic analysis on a noncompact quantum group,” hep-th/9911110. |
| [35] | Guadagnini E., Nucl. Phys. B 433 pp 597– (1995) · Zbl 1020.81844 · doi:10.1016/0550-3213(94)00444-J |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
