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 $AdS_{3}$, and it is closely related to the Witten black hole SL(2)/U(1) and to the Bañados-Teitelboim-Zanelli black hole SL(2)/$\mathbb {Z}$. The model appears in recent works on the AdS/CFT correspondence.

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, including both analytic and numerical techniques. In addition, representation theory of the affine Lie algebra ${\widehat {\mathfrak {sl}}}_{2}$ 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.
• 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.
• The same authors also studied the correlations functions, and exactly determined some of them. However, the focus was on applications to the AdS/CFT correspondence, not on solving the model.
• The minisuperspace (large level) limit is solved. Some of the minisuperspace results are dictated by the ${\mathfrak {sl}}_{2}$ 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 $H_{3}^{+}$, and some results on the SL(2,R) WZW model can be obtained from the $H_{3}^{+}$ model by analytic continuation. However, this obscures the intrinsic algebraic features of the SL(2,R) WZW model. It would be better to study these algebraic features from first principles, in particular the fusion rules. The $H_{3}^{+}$ 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, 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 ${\widehat {\mathfrak {sl}}}_{2}/{\widehat {\mathfrak {u}}}_{1}$, which should help understand the boundary conditions in the SL(2)/U(1) model. (Known results on these boundary conditions are incomplete, in particular they do not account for the boundary condition that emerge from the lattice approach.)

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 $x$-basis makes four-point blocks depend on only one isospin variable (the cross-ratio), which should simplify their analytic and numerical computation.
• The $m$-basis makes the Wick rotation from $H_{3}^{+}$ transparent.
• The $\mu$-basis makes the $H_{3}^{+}$-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. This might be done either directly, using a version of the $H_{3}^{+}$-Liouville relation, , or indirectly, using Wick rotation from the $H_{3}^{+}$ 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 $m$-basis and in the $\mu$-basis.) As a warm-up, one may start with checking crossing symmetry in the $H_{3}^{+}$ model, which has not been done so far.

Some relevant literature

Articles on the AdS/CFT relation

The recent proof of the $AdS_{3}/CFT_{2}$ correspondence 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. (Section 3). This definition is consistent with the interpretation of $x$ as a spacetime coordinate for the boundary CFT. It is to be compared with earlier definitions.

Curiously, the $AdS_{3}/CFT_{2}$ correspondence relies on particular correlation functions that obey a quantization condition on the sum of the spins. (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 $H_{3}^{+}$-Liouville relation does not seem to directly match the $AdS_{3}/CFT_{2}$ with the Liouville condition.

The correspondence involves computing correlation functions that do not strictly belong to the WZW model, or at least that are not necessary for solving that model. In these correlation functions, spectral flow can be violated by more than one unit. It would be interesting to understand these correlation functions in terms of the WZW model. Computing the corresponding extended fusion rules, and relating them to the known WZW fusion rules, would also be interesting.