# Solvable non-diagonal 2d CFTs

Some two-dimensional conformal field theories are exactly solvable, starting with the minimal models. Here we would like to explore a more recently discovered theory, which is supposed to exist for all values of the Virasoro algebra's central charge ${\displaystyle c}$ such that ${\displaystyle \Re c<13}$. (In contrast, minimal models exist for rational values which moreover obey ${\displaystyle c<1}$.)

## Motivations

• Getting a better idea of the space of consistent CFTs, by investigating one of the rare exactly solvable examples.
• Unifying known solvable CFTs such as Liouville theory and minimal models, by understanding the analytic properties of correlation functions as functions of the central charge and conformal dimensions.
• Potential applications to statistical physics.

## Type of project

Tools: A good understanding of the bootstrap approach to 2d CFT is required. Then the project involves analytic calculations and numerical checks of crossing symmetry, possibly based on existing Python code at GitLab.

Chances of success: The sought-after CFT is almost sure to exist, as some of its four-point functions can be shown to be crossing symmetric in some channels.[1]. However, completing the solution involves some guesswork, and is not a straightforward application of known methods.

Length and difficulty: Hard to tell, as this depends on some inspired guesswork. If the most obvious guesses worked, the main results could be obtained very quickly.

## Known results

• For ${\displaystyle c\leq 1}$ , the CFT is a limit of D-series minimal models.[1][2]
• The spectrum has a diagonal continuous sector, and a non-diagonal discrete (but non-rational) sector.[1][2]
• Degenerate fields exist, so that three-point structure constants can be computed analytically.[1][2]
• When the central charge tends to a rational value, some correlation functions reduce to correlation functions in D-series minimal models.[3]
• The theory differs from the critical 2d Potts model, although some of its four-point functions provide very good approximations of that model's connectivities.[4]. Unlike our theory, the Potts model is probably not analytically solvable with known methods.

## Solving the theory for complex central charges

The theory has been exactly solved for ${\displaystyle c\leq }$  by taking limits of D-series minimal models.[2] More general values of the central charge are however not accessible by this method: other techniques are needed.

The known analytic bootstrap techniques rely on assuming the existence of degenerate fields, and using crossing symmetry for determining how structure constants change under shifts of momentums by ${\displaystyle \beta }$  and ${\displaystyle \beta ^{-1}}$ , where ${\displaystyle \beta }$  is defined from the central charge by

${\displaystyle c=13-6\beta ^{2}-6\beta ^{-2}.}$

The resulting shift equations have unique smooth solutions for ${\displaystyle \beta ^{2}\in \mathbb {R} }$ , but we are now interested in the regime

${\displaystyle c\in \left\{\Re c<13\right\}-(-\infty ,1)\quad \iff \quad \beta ^{2}\in \left\{\Re \beta ^{2}>0\right\}-(0,\infty ).}$

In this regime, the shift equations have infinitely many solutions. Only one of these solutions is expected to provide the structure constants of the CFT. Maybe that solution is distinguished by its analytic properties. If we pick a solution, we can numerically test crossing symmetry of the corresponding four-point functions: only the correct solution is expected to pass this stringent test.

One should guess not only the structure constants, but also the spectrum in the diagonal channels. Some four-point functions coincide with four-point functions of Liouville theory, whose spectrum is known to be made of a continuum, plus a finite number of discrete terms. It is not clear how to determine the discrete terms in more general four-point functions. But any guess can be checked by numerically testing crossing symmetry.

## References

1. Migliaccio, Santiago; Ribault, Sylvain (2018). "The analytic bootstrap equations of non-diagonal two-dimensional CFT". Journal of High Energy Physics 2018 (5). doi:10.1007/jhep05(2018)169. ISSN 1029-8479.
2. Ribault, Sylvain (2019). "The non-rational limit of D-series minimal models". arXiv:1909.10784 [hep-th].
3. Ribault, Sylvain (2019-06-25). "On 2d CFTs that interpolate between minimal models". SciPost Physics (Stichting SciPost) 6 (6). doi:10.21468/scipostphys.6.6.075. ISSN 2542-4653.
4. Picco, Marco; Ribault, Sylvain; Santachiara, Raoul (2019-10-07). "On four-point connectivities in the critical 2d Potts model". SciPost Physics (Stichting SciPost) 7 (4). doi:10.21468/scipostphys.7.4.044. ISSN 2542-4653.