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Résumé |
Witten index of d=1 N=2,4 supersymmetric gauged quantum
mechanics is considered. A canonical set of examples is
the N=4 quiver dynamics, well-known to capture wall-crossing
of BPS states of d=4 N=2 theories. For N=4 quivers, index were
traditionally computed either by Coulomb or Higgs approximations.
However, this is neither the most efficient nor the most complete
computation. The Coulomb one in particular is known to miss
a large number of wall-crossing-safe states, known as quiver
invariants. After outlining recent progress on this front, we turn
to honest and complete Witten index computation via localization.
While the naive localization argument seems to conflict with
wall-crossing, this is merely due to neglecting important
subtleties with new flat or runaway asymptotic direction that
emerges when some of FI constant vanishes. After taking
account of this subtlety, one arrives at universal index
formula, via JK residue, determined entirely by field contents
and R-charge assignment, which reproduces most up-to-date
results in literature, also reproducing many explicit wall-crossing
formulae such as predicted by Kontsevich-Soibelman.
Finally, we comment on the wall-crossing-safe subsector of
d=1 GLSM, exemplified by the quiver invariants, and propose
how this invariant subsector might be counted again as a path
integral.
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