1. Thomas Creutzig & John Duncan
Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory
Wolfgang Riedler
in collaboration with
Thomas Creutzig & John Duncan
Alberta Number Theory Days
Banff,

2. Main Question
Can a self-dual vertex operator algebra (VOA) be identified with a bulk conformal field theory (CFT) in some sense?

3. Moonshine

4. Moonshine

5. Motivation
N=4 superconformal algebra with central charge 6 appears in all of these.

6. Vertex Operator Super-Algebras

7. Vertex Operator Super-Algebras
Two remarks:
In what follows we only consider “nice” VOSAs.
Def. A VOSA is self-dual if it is rational and has a unique irreducible module.

8. Representations: Conformal Field Theory
Definition.
as given above is a potential bulk conformal field theory if is modular invariant.

9. Main Question
Can a self-dual vertex operator algebra (VOA) be identified with a bulk conformal field theory (CFT) in some sense?
Yes.
Proposition.
With W as above, if the S-matrix of is real and the eigenvalues of the action of
on W belong to then is modular invariant.

10. …but we can do better.

11. …but we can do better. Proposition.
With W as above, if the S-matrix of is real, the eigenvalues of on lie in and the eigenvalues of on lie in then the vector valued function is modular.

12. Example: SCFT of Type D
A connection between sigma models and Conway moonshine.

13. - Fin -
[EOT] – Eguchi, Ooguri, Tachikawa. “Notes on the K3 Surface and the Mathieu Group M24”, Experiment. Math. Volume 20, Issue 1 (2011),
[MSV] – Malikov, Schechtman, Vaintrop. “Chiral de Rham complex”, Comm. Math. Phys. 204 (1999),
[JD] – Duncan, Mack-Crane. “Derived Equivalences of K3 Surfaces and Twined Elliptic Genera”, Res. Math. Sci. (2016) 3:1.

14. A Classification Result
Theorem.
If is a self-dual cofinite VOSA of CFT type with central charge c then it is isomorphic to one of the following:

15. Example 1.5: Super CFT of Type D

16. Example 2: Super CFT of Type A

17. Example 3: Super CFT of Gepner Type

18. Modularity Theorem. [Zhu]
On the upper half plane the characters of a rational, C2-cofinite VOA converge to holomorphic functions. Moreover, the linear space spanned by the limits of characters is invariant under the action of SL2(Z).