27 research outputs found
Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics
We consider a topological quantum mechanics described by a phase space path
integral and study the 1-dimensional analog for the path integral
representation of the Kontsevich formula. We see that the naive bosonic
integral possesses divergences, that it is even naively non-invariant and thus
is ill-defined. We then consider a super-extension of the theory which
eliminates the divergences and makes the theory naively invariant. This
super-extension is equivalent to the correct choice of measure and was
discussed in the literature. We then investigate the behavior of this extended
theory under diffeomorphisms of the extended phase space and despite of its
naive invariance find out that the theory possesses anomaly under nonlinear
diffeomorphisms. We localize the origin of the anomaly and calculate the lowest
nontrivial anomalous contribution.Comment: 36 page
Twisted characters and holomorphic symmetries
We consider holomorphic twists of arbitrary supersymmetric theories in four
dimensions. Working in the BV formalism, we rederive classical results
characterizing the holomorphic twist of chiral and vector supermultiplets,
computing the twist explicitly as a family over the space of nilpotent
supercharges in minimal supersymmetry. The BV formalism allows one to work with
or without auxiliary fields, according to preference; for chiral superfields,
we show that the result of the twist is an identical BV theory, the holomorphic
system with superpotential, independent of whether or not
auxiliary fields are included. We compute the character of local operators in
this holomorphic theory, demonstrating agreement of the free local operators
with the usual index of free fields. The local operators with superpotential
are computed via a spectral sequence, and are shown to agree with functions on
a formal mapping space into the derived critical locus of the superpotential.
We consider the holomorphic theory on various geometries, including Hopf
manifolds and products of arbitrary pairs of Riemann surfaces, and offer some
general remarks on dimensional reductions of holomorphic theories along the
-sphere to topological quantum mechanics. We also study an
infinite-dimensional enhancement of the flavor symmetry in this example, to a
recently-studied central extension of the derived holomorphic functions with
values in the original Lie algebra that generalizes the familiar Kac--Moody
enhancement in two-dimensional chiral theories