zbMATH — the first resource for mathematics

String theory on K3 surfaces. (English) Zbl 0931.14020
Greene, B. (ed.) et al., Mirror symmetry II. Cambridge, MA: International Press, AMS/IP Stud. Adv. Math. 1, 703-716 (1997).
The authors deduce a description of the moduli space \(\mathcal M\) of \(N=(4,4)\) string theories with a K3 surface as the target space in the form \(\Gamma\backslash G/H\) where \(G\) and \(H\) are Lie groups and \(\Gamma\) is a discrete group. In this paper the basic ideas are explained while the detailed mathematical exposition is presented in another paper. Two assumptions are made about the form of \(\mathcal M\): It is geodesically complete and a Hausdorff space. The starting point is the knowledge that the moduli space, \(\mathcal M\), of conformal field theories on a K3 surface is locally of the form \(\mathcal T^{4,20}=O(4,20)/(O(4)\times O(20))\) and so globally is expected to take the form \(\Gamma\backslash\mathcal T^{4,20}\). The group \(\Gamma\) is determined to be isomorphic to \(O(\Lambda^{4,20})\) where \(\Lambda^{4,20}\) denotes a lattice. The form of \(\Gamma\) follows from the analysis of the moduli space of complex structures, and it is further generated by integral shifts of \(B\)-fields on the target space, by complex conjugation, and also by the mirror map. Particular attention is paid to the closer analysis of the mirror symmetry including its geometrical interpretation using an \(S^2\times S^2\) bundle over \(\mathcal M\), \(\mathcal T_{q\bar{q}}^{4,20}=O^+(4,20)/(SO(2)\times SO(2)\times O(20))\).
For the entire collection see [Zbl 0905.00079].

14J28 \(K3\) surfaces and Enriques surfaces
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)