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SUMMARY:Hyper-Kähler and HKT geometry through supersymmetric glasses
DTSTART;VALUE=DATE-TIME:20181126T133000Z
DTEND;VALUE=DATE-TIME:20181126T143000Z
DTSTAMP;VALUE=DATE-TIME:20200920T205307Z
UID:indico-event-4188@indico.math.cnrs.fr
DESCRIPTION:In the first introductory part of the talk (that may be skippe
d by the request of the audience)\, we remind the classical Witten's resul
t that the de Rham complex is isomorphic to the Hilbert space of wave func
tions in a certain supersymmetric quantum mechanical system. There is an `
`industrial'' way to construct supersymmetric systems which is based on th
e superspace formalism\, which we also describe. We then discuss Kahler
manifolds and show how the classical result that any Kahler metric can be
derived from the Kahler potential\, $h_{m \\bar n} = \\partial_m \\bar \\p
artial_n \\\, K(z^p\, \\bar z^p)$ can be easily derived in the {\\it exten
ded} superspace formalism.\n\nThe hyper-Kahler models enjoy ${\\cal N} = 8
$ supersymmetry. We do not know how to fully implement the latter in the c
onventional superspace approach\, it is only possible in the harmonic supe
rspace formalism that we briefly outline. Any hyper-Kahler metric can be
derived from a certain\nfunction (prepotential) depending on the harmonic
superfields and harmonics. In contrast to the Kahler case\, the relation
of the metric to the prepotential is not so simple\, one has to solve a sy
stem of certain differential equations.\n\n Finally we go over to the
so-called HKT manifolds. These are the manifolds admitting three quaterni
onic complex structures that are covariantly constant with respect to a ce
rtain torsionful connection. (HK manifolds is a subclass of HKT manifolds
where this connection is torsionless). We describe them in the harmonic su
perspace framework. In constrast to the HK metric\, a general HKT metric
needs {\\it two} different functions for its description. We show that
the space of all HKT metrics is divided in families such that the Obata cu
rvatures of all members of one family coincide. \n\nhttps://indico.math
.cnrs.fr/event/4188/
LOCATION:I.H.E.S. Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/4188/
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