Solving the SL(2,R) WZW model

The SL(2,R) WZW model may be the richest 2d CFT that can be solved analytically using known techniques of the analytic conformal bootstrap. Partial results suggest that we can compute the spectrum and three-point structure constants, but this has not been done completely. This project is to complete these computations (analytically), and to check crossing symmetry of four-point functions (numerically and/or analytically).

Motivations

The SL(2,R) WZW model has important applications, especially in string theory. It is the basis for the worldsheet approach to string theory in ${\displaystyle AdS_{3}}$, and it is closely related to the Witten black hole SL(2)/U(1) and to the Bañados-Teitelboim-Zanelli black hole SL(2)/${\displaystyle \mathbb {Z} }$. The model appears in recent works on the AdS/CFT correspondence.[1]

As a conformal field theory, the model has a spectrum with both a continuous sector and a discrete sector. This makes it richer that most CFTs that have been solved so far, in particular minimal models (discrete spectrums) and Liouville theory (continuous spectrum). The model comes with a continuous parameter called the level, and is expected to have interesting limits for special values of that parameter.

Type of project

Tools: Since we are focusing on the bootstrap approach, it is not necessary to understand the whole literature on the subject, which includes other techniques such as path integrals. On the other hand, a good understanding of the bootstrap approach to simpler models (such as Liouville theory) is needed,[2] including both analytic and numerical techniques. In addition, representation theory of the affine Lie algebra ${\displaystyle {\widehat {\mathfrak {sl}}}_{2}}$ will play an important role.

Chances of success: Very high in principle, since the main conceptual and technical difficulties have probably been solved already. However, the amount of work is large.

Length and difficulty: The project is a large undertaking, but it could be split in parts that could be of independent interest. The main difficulty is to master and synthetize a rather large amount of existing knowledge and techniques.

Known results

• There are two relevant review articles.[2][3]
• The spectrum of the model was determined by Maldacena and Ooguri, who in particular found the role of the spectral flow automorphism of the affine Lie algebra.[4]
• The same authors also studied the correlations functions, and exactly determined some of them.[5] However, the focus was on applications to the AdS/CFT correspondence, not on solving the model.
• The minisuperspace (large level) limit is solved.[6] Some of the minisuperspace results are dictated by the ${\displaystyle {\mathfrak {sl}}_{2}}$ symmetry and also hold for finite values of the level.

Work to be done

As a manifold, SL(2,R) is the Wick rotation of ${\displaystyle H_{3}^{+}}$, and some results on the SL(2,R) WZW model can be obtained from the ${\displaystyle H_{3}^{+}}$ model by analytic continuation. However, this obscures the intrinsic algebraic features of the SL(2,R) WZW model.[6] It would be better to study these algebraic features from first principles, in particular the fusion rules. The ${\displaystyle H_{3}^{+}}$ model may still be used for getting specific technical results, if need be.

Fusion rules

To compute fusion products of representations of the affine Lie algebra from first principles would be technically difficult, and is probably not necessary. Instead, it is probably possible to guess the fusion rules based on structural properties such as associativity, compatibility with spectral flow, and analyticity wrt the level. This has already been done to some extent,[2] and should be generalized to degenerate representations, which do not appear in the spectrum, but are used in the analytic bootstrap. Crossing symmetry of four-point functions will provide the ultimate proof of the validity of this approach.

As a bonus, knowing the fusion rules could help construct other theories with the same symmetry algebra, analogous to the various known solvable CFTs with Virasoro symmetry. And it is in principle straightforward to deduce fusion rules of the quotient ${\displaystyle {\widehat {\mathfrak {sl}}}_{2}/{\widehat {\mathfrak {u}}}_{1}}$, which should help understand the boundary conditions in the SL(2)/U(1) model. (Known results on these boundary conditions are incomplete,[7] in particular they do not account for the boundary condition that emerge from the lattice approach.[8])

Three-point structure constants

For completely solving the model, it is necessary to determine structure constants for all relevant types of representations: discrete, continuous, spectrally flowed, degenerate. And it would be interesting to do it in various relevant bases of fields, each basis having its own advantages:

• The ${\displaystyle x}$-basis makes four-point blocks depend on only one isospin variable (the cross-ratio), which should simplify their analytic and numerical computation.
• The ${\displaystyle m}$-basis makes the Wick rotation from ${\displaystyle H_{3}^{+}}$ transparent.
• The ${\displaystyle \mu }$-basis makes the ${\displaystyle H_{3}^{+}}$-Liouville relation simpler.

The clean way to compute structure constants in the analytic bootstrap approach involves computing four-point functions with degenerate fields. It might be possible to bypass that step and to use existing results, mostly obtained by Wick rotation, and to complete them using symmetry considerations and/or inspired guesses.

Analytic proof of crossing symmetry

The only realistic way to prove crossing symmetry is to build on crossing symmetry in Liouville theory.[9] This might be done either directly, using a version of the ${\displaystyle H_{3}^{+}}$-Liouville relation, [2], or indirectly, using Wick rotation from the ${\displaystyle H_{3}^{+}}$ model.

Numerical checks of crossing symmetry

Compared to numerical checks of crossing symmetry in Liouville theory and other CFTs with Virasoro symmetry only, the difficulty is that conformal blocks are now more complicated functions, depending on more variables. Moreover, the fields which propagate in a given channel should not be summed over their conformal dimensions only, but also over isospin variables. (Unless such variables are conserved, which is the case in the ${\displaystyle m}$-basis and in the ${\displaystyle \mu }$-basis.) As a warm-up, one may start with checking crossing symmetry in the ${\displaystyle H_{3}^{+}}$ model, which has not been done so far.

Some relevant literature

Articles on ${\displaystyle AdS_{3}/CFT_{2}}$

The recent proof of the ${\displaystyle AdS_{3}/CFT_{2}}$ correspondence[1][10] involves a study of the Ward identities for correlation functions of the WZW model. This starts with a definition of spectrally flowed fields by their OPEs with currents in ref.[1] (Section 3). This definition is consistent with the interpretation of ${\displaystyle x}$ as a spacetime coordinate for the boundary CFT. It is to be compared with earlier definitions.[11]

Curiously, the ${\displaystyle AdS_{3}/CFT_{2}}$ correspondence relies on particular correlation functions that obey a quantization condition on the sum of the spins.[1] (Eq. (1.2)). This condition is similar to the condition for correlation functions to have a Coulomb gas representation as finite-dimensional integrals in Liouville theory, although the ${\displaystyle H_{3}^{+}}$-Liouville relation does not seem to directly match the ${\displaystyle AdS_{3}/CFT_{2}}$ with the Liouville condition.[2]

References

1. Eberhardt, Lorenz; Gaberdiel, Matthias; Gopakumar, Rajesh (2019). "Deriving the AdS3/CFT2 Correspondence". arXiv:1911.00378 [hep-th].
2. Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th].
3. McElgin, Will (2015). "Notes on the SL(2,R) CFT". arXiv:1511.07256 [hep-th].
4. Maldacena, Juan; Ooguri, Hirosi (2001). "Strings in AdS3 and the SL(2,R) WZW model. I: The spectrum". Journal of Mathematical Physics (AIP Publishing) 42 (7): 2929–2960. doi:10.1063/1.1377273. ISSN 0022-2488.
5. Maldacena, Juan; Ooguri, Hirosi (2002-05-15). "Strings inAdS3and theSL(2,R)WZW model. III. Correlation functions". Physical Review D (American Physical Society (APS)) 65 (10). doi:10.1103/physrevd.65.106006. ISSN 0556-2821.
6. Ribault, Sylvain (2010). "Minisuperspace limit of the AdS 3 WZNW model". Journal of High Energy Physics (Springer Science and Business Media LLC) 2010 (4). doi:10.1007/jhep04(2010)096. ISSN 1029-8479.
7. Ribault, Sylvain; Schomerus, Volker (2004-02-06). "Branes in the 2D black hole". Journal of High Energy Physics (Springer Science and Business Media LLC) 2004 (02): 019–019. doi:10.1088/1126-6708/2004/02/019. ISSN 1029-8479.
8. Robertson, Niall F.; Jacobsen, Jesper Lykke; Saleur, Hubert (2019). "Conformally invariant boundary conditions in the antiferromagnetic Potts model and the SL(2, ℝ)/U(1) sigma model". Journal of High Energy Physics (Springer Science and Business Media LLC) 2019 (10). doi:10.1007/jhep10(2019)254. ISSN 1029-8479.
9. Teschner, J (2003). "A lecture on the Liouville vertex operators". International Journal of Modern Physics A 19 (2): 436–458. doi:10.1142/S0217751X04020567.
10. Eberhardt, Lorenz (2020-02-26). "AdS3/CFT2 at higher genus". arXiv.org. Retrieved 2020-05-12.
11. Ribault, Sylvain (2005-07-12). "Knizhnik-Zamolodchikov equations and spectral flow in AdS3 string theory". arXiv.org. doi:10.1088/1126-6708/2005/09/045. Retrieved 2020-05-12.