Quiver Varieties, Category O for Rational Cherednik Algebras, and Hecke Algebras

We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Z-algebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the c-function) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the a-function) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZ-functor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties

Worldsheet Instantons and the amplitude for string pair production in an external field as a WKB exact functional integral

We revisit the problem of charged string pair creation in a constant external electric field. The string states are massive and creation of pairs from the vacuum is a tunnelling process, analogous to the Schwinger process where charged particle-anti-particle pairs are created by an electric field. We find the instantons in the worldsheet sigma model which are responsible for the tunnelling events. We evaluate the sigma model partition function in the multi-instanton sector in the WKB approximation which keeps the classical action and integrates the quadratic fluctuations about the solution. We find that the summation of the result over all multi-instanton sectors reproduces the known amplitude. This suggests that corrections to the WKB limit must cancel. To show that they indeed cancel, we identify a fermionic symmetry of the sigma model which occurs in the instanton sectors and which is associated with collective coordinates. We demonstrate that the action is symmetric and that the interaction action is an exact form. These conditions are sufficient for localization of the worldsheet functional integral onto its WKB limit.Comment: 40 pages; Expanded discussion section, added reference

Universe creation on a computer

The purpose of this paper is to provide an account of the epistemology and metaphysics of universe creation on a computer

Immigration as Commerce: A New Look at the Federal Immigration Power and the Constitution

The relationship of immigration law to the Constitution has long been incoherent. One result is that there is little clarity on the appropriate standard of review for constitutional violations when aspects of immigration law and policy are challenged in the federal courts. This Article advances the Commerce Clause as the anchor of a new understanding of the link between the government\u27s immigration power and the Constitution. Despite the extensive early history of the Foreign Commerce Clause as the presumed source of the immigration power, it plays almost no role in immigration jurisprudence today, and few scholars have seriously considered its suitability for that role. More strikingly, none have explored the Interstate Commerce Clause as an appropriate source of the immigration power, one that could open the door to a normalization of constitutional analysis in the immigration context. The Article argues that both the Foreign and the Interstate Commerce Clauses should be understood to undergird the contemporary immigration power, and suggests that acknowledging immigration’s relationship to the Commerce Clause clears a path to more routine constitutional review of immigration law and policy