A Hiker's Guide to K3.¶Aspects of N= (4,4) Superconformal Field Theory¶with Central Charge c= 6

Abstract

We study the moduli space M of N = (4, 4) superconformal field theories with central charge c = 6. After a slight emendation of its global description we find the locations of various known models in the component of M associated to K3 surfaces. Among them are the Z 2 and Z 4 orbifold theories obtained from the torus component of M. Here, SO(4, 4) triality is found to play a dominant role. We obtain the B-field values in direction of the exceptional divisors which arise from orbifolding. We prove T-duality for the Z 2 orbifolds and use it to derive the form of M purely within conformal field theory. For the Gepner model (2) 4 and some of its orbifolds we find the locations in M and prove isomorphisms to nonlinear σ models. In particular we prove that the Gepner model (2) 4 has a geometric interpretation with Fermat quartic target space.

A family of SCFTs hosting all ``very attractive'' relatives of the (2)4Gepner model

“…These joint forces have already led to the appropriate description of all orbifold SCFTs obtained from toroidal models within the moduli space [30,31], which the present work is built on. As a final ingredient Witten's results on the phase structure of the parameter space of supersymmetric gauge theories [33] comes to aid.…”

“…by the results of [27] each of these theories is a SCFT associated to K3 according to Definition 3.3. The results of [30,67] allow me to describe the four-plane x α,β,β ,γ ⊂ H even (X, R) which specifies this theory in terms of the lattice…”

“…Recall from [68,30,67] the description of the lattice H 2 (X, Z) in terms of π * H 2 (A, Z) Z 4 and the exceptional divisors coming from the resolution of A α,β,γ /Z 4 : Consider the action of the subgroup Z 2 of Z 4 on A α,β,γ . It has 16 fixed points, labeled by an affine F 4 2 over the field F 2 with two elements 0, 1, where…”

“…The claims about the lattice π * H 2 (A, R) Z 4 ∩ H 2 (X, Z) follow from [69], where Ω 0 X gives the complex structure of X α,β,γ , see also [30,67]. Moreover, in [30], [67,Theorem 3.3] it is proved that C α,β,β ,γ has a geometric interpretation…”

Crystallographic orbifolds: towards a classification of unitary conformal field theories with central charge c = 2

“…. , d} (see [1,16]). Moreover, by the above restrictions on the scalar products between the p i , these vectors generate the root lattice of a simply laced Lie group.…”