In this work with Dr. Madeeha Khalid we study explicit constructions of superconformal field theories associated to Borcea-Voisin (BV) 3-folds. The latter are examples of Calabi-Yau threefolds which are obtained by an orbifold procedure from the product of a K3 surface with an elliptic curve. A K3 surface can enter the Borcea-Voisin construction if and only if it admits an anti-symplectic involution. Such K3 surfaces have been classified by Nikulin in 1979, when he showed that they form 75 distinct connected families.
We have proved that precisely one of the 75 families in Nikulin's list contains Kummer surfaces generically. Moreover, we can show that for each K3 surface in this family, the geometric interpretation as a Kummer surface is compatible with the orbifold action by the anti-symplectic automorphism of the BV construction. Hence for these K3 surfaces, the BV construction can be lifted to the conformal field theory level by the methods already developed in my previous publications.
We can moreover extend this result to other orbifold limits of K3 as well as the smooth quartics investigated in my work on SCFTs associated to smooth K3 surfaces.
Altogether this means that we are able to calculate certain families of SCFTs on Calabi-Yau threefolds explicitly, along with their algebro-geometric interpretations. We are currently investigating the implications of our constructions. For example, we are interested in questions concerning mirror symmetry, quantum corrections and string-string dualities, which we expect to be able to address in much more concrete terms than has been possible before, given the classes of examples that we can now generate.