Superalgebras and Superconformal Field Theory i n collaboration with Thomas Creutzig amp John Duncan Alberta Number Theory Days Banff 1719032017 Main Question Can a selfdual ID: 791032
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Slide1
Wolfgang Riedler
Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory
in collaboration with Thomas Creutzig & John Duncan
Alberta Number Theory Days
Banff, 17.-19.03.2017
Slide2Main Question
Can a self-dual vertex operator algebra (VOA) be identified with a bulk conformal field theory (CFT) in some sense?
Slide3Moonshine
Slide4Moonshine
Slide5Motivation
N=4 superconformal algebra with central charge 6 appears in all of these.
Slide6Vertex
Operator Super-Algebras
Slide7Vertex
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.
Slide8Representations:
Conformal Field Theory
Definition.
as given above is a potential bulk conformal field theory if is modular invariant.
Slide9Main 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.
Slide10…but we can do better.
Slide11…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.
Slide12Example: SCFT
of Type D
A connection between sigma models and Conway moonshine.
Slide13-
Fin -[EOT] – Eguchi,
Ooguri, Tachikawa. “Notes on the K3 Surface and the Mathieu Group M24”, Experiment. Math. Volume 20, Issue 1 (2011), 91-96.[MSV] – Malikov, Schechtman, Vaintrop. “Chiral de
Rham complex”, Comm. Math. Phys. 204 (1999), 439-473.[JD] – Duncan, Mack-Crane. “
Derived Equivalences of K3 Surfaces and Twined Elliptic Genera”
, Res. Math. Sci. (2016) 3:1.
Slide14A Classification Result
Theorem.
If is a self-dual -cofinite VOSA of CFT type with central charge c 12 then it is isomorphic to one of the following:
Slide15Example 1.5: Super CFT of Type D
Slide16Example 2: Super CFT of Type A
Slide17Example 3: Super CFT of
Gepner Type
Slide18Modularity
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).