SCFTs associated to smooth K3 surfaces

The publication

Katrin Wendland
On Superconformal Field Theories Associated to Very Attractive Quartics
  • Frontiers in Number Theory, Physics and Geometry II, Pierre Cartier, Bernard Julia, Pierre Moussa, Piere Vanhoeve eds, Springer 2007, pp. 223-244
  • hep-th/0307066

explores explicit constructions of SCFTs associated to so-called very attractive quartic K3 hypersurfaces in CP3. In particular, I argue that orbifold techniques allow to construct such SCFTs which do not only carry the complex structure but also the normalized Kähler form induced by the embedding of the quartic in CP3.

I have generalized this result to yield a construction of SCFTs associated to a family of smooth quartic K3 surfaces in

Katrin Wendland
A Family of SCFTs hosting all very attractive relatives of the (2)4 Gepner model

The relevant family of quartics has four real parameters, and a three-parameter subfamily contains as a dense subset the very attractive ones which were considered in the previous work. This yields the first known family of SCFTs which simultaneously allows an explicit description in terms of smooth algebraic K3 surfaces, Landau-Ginzburg models, and orbifolds. It can be viewed as a pure complex structure deformation of the Gepner model (2)4. Hence this family of SCFTs is tailor made to apply and test modern techniques in SCFT, e.g. for constructing boundary states in SCFTs by means of matrix factorization. Indeed, as an application of my results, Brunner, Gaberdiel, and Keller have carried out first steps in this program, see hep-th/0603196 and also Schmidt-Colinet's hep-th/0701128. Possible further applications of my work on SCFTs associated to quartic K3 surfaces are concerned with the chiral de Rham complex.

At rational parameters, where the underlying K3 surfaces in our family of quartics in particular enjoy complex multiplication, the corresponding SCFTs are not necessarily rational. This contradicts one of the many beliefs on the relation between complex multiplication in geometry and rationality in CFT.

The second publication furthermore gives a relation between the Ricci-flat Kähler-Einstein metric on the Fermat quartic in CP3 on the one hand and an orbifold limit of such a metric on a closely related K3 surface on the other. This discovery will allow a numerical investigation of the Ricci-flat Kähler-Einstein metric on the Fermat quartic using the methods of hep-th/0506129. Moreover, one can hope that numerical investigations of SCFTs on smooth quartics could be carried out more generally, as an application of these methods.

One main mathematical ingredient to this work has been a construction by Hiroshi Inose, On certain Kummer surfaces, which can be realized as nonsingular quartic surfaces in P3, J. Fac. Sci. Uni. Tokyo IA 23 (1976), 545-560. Its generalization to other quartic K3 surfaces by Tetsuji Shioda is likely to imply generalizations of my results, which I plan to investigate.

Some of the relevant background from geometry is summarized in

Katrin Wendland
K3 en route From Geometry to Conformal Field Theory
  • lecture notes for the author's contribution to the 2013 Summer School “Geometric, Algebraic and Topological Methods for Quantum Field Theory” in Villa de Leyva, Colombia
  • arXiv:1503.08426 [math.DG]

Topics include the complex and Kähler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss toroidal superconformal field theories and their Z2-orbifolds, yielding a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is presented as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.