Chern-Simons gauge theory and the AdS(3) / CFT(2) correspondence

The bulk partition function of pure Chern-Simons theory on a three-manifold is a state in the space of conformal blocks of the dual boundary RCFT, and therefore transforms non-trivially under the boundary modular group. In contrast the bulk partition function of AdS_3 string theory is the modular-invariant partition function of the dual CFT on the boundary. This is a puzzle because AdS_3 string theory formally reduces to pure Chern-Simons theory at long distances. We study this puzzle in the context of massive Chern-Simons theory. We show that the puzzle is resolved in this context by the appearance of a chiral “spectator boson” in the boundary CFT which restores modular invariance. It couples to the conformal metric but not to the gauge field on the boundary. Consequently, we find a generalization of the standard Chern-Simons/RCFT correspondence involving “nonholomorphic conformal blocks” and nonrational boundary CFTs. These generalizations appear in the long-distance limit of AdS_3 string theory, where the role of the spectator boson is played by other degrees of freedom in the theory.

[1]  G. Moore,et al.  Search for a holographic dual to $\textit{AdS}_3\times S^3\times S^3\times S^1$ , 2004 .

[2]  G. Moore,et al.  The Search for a Holographic Dual to AdS(3)xS(3)xS(3)xS(1) , 2004, hep-th/0403090.

[3]  J. Maldacena,et al.  Wormholes in AdS , 2004, hep-th/0401024.

[4]  M. Freedman,et al.  A class of P,T-invariant topological phases of interacting electrons , 2003, cond-mat/0307511.

[5]  E. Sezgin,et al.  Yang-Mills-Chern-Simons Supergravity , 2003, hep-th/0305242.

[6]  D. Freed,et al.  Elliptic Cohomology: The M -theory 3-form and E 8 gauge theory , 2007 .

[7]  E. Witten.  SL(2;Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry , 2003, hep-th/0307041.

[8]  H. Samtleben,et al.  Kaluza-Klein supergravity on AdS3 x S3 , 2003, hep-th/0306202.

[9]  E. Sezgin,et al.  SU(2) reduction of six-dimensional (1,0) supergravity , 2002, hep-th/0212323.

[10]  J. Fuchs,et al.  TFT construction of RCFT correlators I: Partition functions , 2002, hep-th/0204148.

[11]  T. Gannon.  Boundary conformal field theory and fusion ring representations , 2001, hep-th/0106105.

[12]  J. Maldacena,et al.  D-brane charges in five-brane backgrounds , 2001, hep-th/0108152.

[13]  S. Mathur.  Gravity on $AdS_3$ and flat connections in the boundary CFT , 2001, hep-th/0101118.

[14]  S. Theisen,et al.  Cubic couplings in D=6 N=4b supergravity on AdS_3\times S^3 , 2000, hep-th/0007061.

[15]  J. Maldacena,et al.  Large N Field Theories, String Theory and Gravity , 1999, hep-th/9905111.

[16]  F. Larsen,et al.  U(1) Charges and Moduli in the D1-D5 System , 1999, hep-th/9905064.

[17]  D. Kutasov,et al.  More comments on string theory on AdS(3) , 1999, hep-th/9903219.

[18]  E. Witten.  AdS / CFT correspondence and topological field theory , 1998, hep-th/9812012.

[19]  X. Wen,et al.  Classification of disordered phases of quantum Hall edge states , 1997, cond-mat/9710208.

[20]  M. Manoliu.  Abelian Chern–Simons theory. I. A topological quantum field theory , 1996, dg-ga/9610001.

[21]  M. Manoliu.  Quantization of symplectic tori in a real polarization , 1996, dg-ga/9609012.

[22]  E. Thiran,et al.  Structuring the set of incompressible quantum Hall fluids , 1995, cond-mat/9505156.

[23]  J. Fröhlich,et al.  A classification of quantum Hall fluids , 1995, cond-mat/9503113.

[24]  A. Zee.  Quantum hall fluids , 1995, cond-mat/9501022.

[25]  C. Schweigert,et al.  Galois modular invariants of WZW models , 1994, hep-th/9410010.

[26]  F. Gabbiani,et al.  The Fractional Quantum Hall Effect, Chern-Simons Theory, and Integral Lattices , 1995 .

[27]  S. D. Pietra,et al.  Geometric quantization of Chern-Simons gauge theory , 1991 .

[28]  S. Carlip,et al.  THREE-DIMENSIONAL TOPOLOGICAL FIELD THEORIES AND STRINGS , 1991 .

[29]  Blok,et al.  Effective theories of the fractional quantum Hall effect: Hierarchy construction. , 1990, Physical review. B, Condensed matter.

[30]  Read.  Excitation structure of the hierarchy scheme in the fractional quantum Hall effect. , 1990, Physical review letters.

[31]  E. Witten,et al.  Topological gauge theories and group cohomology , 1990 .

[32]  Nathan Seiberg,et al.  LECTURES ON RCFT , 1989 .

[33]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[34]  G. Moore,et al.  Classical and quantum conformal field theory , 1989 .

[35]  V. Nair,et al.  U(1) Chern-Simons theory and c=1 conformal blocks , 1989 .

[36]  G. Moore,et al.  Taming the Conformal Zoo , 1989 .

[37]  G. Moore,et al.  Naturality in Conformal Field Theory , 1989 .

[38]  E. Verlinde,et al.  Modular Invariance and the Fusion Algebra , 1988 .

[39]  R. Jackiw,et al.  Topologically Massive Gauge Theories , 1982 .

[40]  Stanley Deser,et al.  Three-dimensional massive gauge theories , 1982 .