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Noncommutative Kn\"{o}rrer periodicity and noncommutative Kleinian singularities
We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of
noncommutative invariant theory. Namely, let be a left noetherian
AS-regular algebra, let be a normal and regular element of of positive
degree, and take . Then there exists a bijection between the set of
isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules
over and those over (a noncommutative analog of) its second double branched
cover . Our results use and extend the study of twisted matrix
factorizations, which was introduced by the first three authors with Cassidy.
These results are applied to the noncommutative Kleinian singularities studied
by the second and fourth authors with Chan and Zhang.Comment: Numerous typos fixed, removed unnecessary finite order hypothesi
index for four-dimensional field theories
In theories with supersymmetry on , BPS bound states can decay
across walls of marginal stability in the space of Coulomb branch parameters,
leading to discontinuities in the BPS indices . We consider a
supersymmetric index which receives contributions from 1/2-BPS states,
generalizing the familiar Witten index . We expect
to be smooth away from loci where massless particles appear, thanks to
contributions from the continuum of multi-particle states. Taking inspiration
from a similar phenomenon in the hypermultiplet moduli space of string
vacua, we conjecture a formula expressing in terms of the BPS indices
, which is continuous across the walls and exhibits the
expected contributions from single particle states at large . This gives
a universal prediction for the contributions of multi-particle states to the
index . This index is naturally a function on the moduli space after
reduction on a circle, closely related to the canonical hyperk\"ahler metric
and hyperholomorphic connection on this space.Comment: 7 pages; v2: introduction expanded, minor corrections, differs from
published version in PRL in that supplemental material is included as an
Appendi
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