Abstract
We study 4D systems in which parameters of the theory have position dependence in one spatial direction. In the limit where these parameters jump, this can lead to 3D interfaces supporting localized degrees of freedom. A priori, this sort of position dependence can occur at either weak or strong coupling. Demanding time-reversal invariance for $U(1)$ gauge theories with a duality group $\Gamma \subset SL(2,\mathbb{Z})$ leads to interfaces at strong coupling which are characterized by the real component of a modular curve specified by $\Gamma$. This provides a geometric method for extracting the electric and magnetic charges of possible localized states. We illustrate these general considerations by analyzing some 4D $\mathcal{N} = 2$ theories with 3D interfaces. These 4D systems can also be interpreted as descending from a six-dimensional theory compactified on a three-manifold generated by a family of Riemann surfaces fibered over the real line. We show more generally that 6D superconformal field theories compactified on such spaces also produce trapped matter by using the known structure of anomalies in the resulting 4D bulk theories.
Highlights
Insights from geometry and topology provide a nontrivial handle on many quantum systems, even at strong coupling
Since we have already explained the significance of the time-reversal invariant components of these modular curves, we review the graphical rules developed in [45] which enumerate which (Γ-equivalence classes of) cusps are on a given real component
We have investigated 3D interfaces generated from 4D theories at strong coupling
Summary
Insights from geometry and topology provide a nontrivial handle on many quantum systems, even at strong coupling. Passing through such cusps is inevitable and means that singularities in the family of elliptic curves are dictated purely by topological considerations For each such cusp, we can fix the associated electric and magnetic charges, indicating the corresponding charge of states localized on an interface. This raises the question as to whether more singular transitions such as a change from a genus zero to a genus one curve could arise, and if so, what this would mean in terms of the 4D effective field theory Along these lines, we consider a more general way to construct 3D interfaces from compactifying six-dimensional superconformal field theories on a three-manifold with boundaries. We consider a more general way to construct 3D interfaces from compactifying six-dimensional superconformal field theories on a three-manifold with boundaries In this setting, we present explicit examples where the genus jumps as a function of x⊥. Some additional details and examples are presented in the Appendices
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
