23 research outputs found
Dolbeault Complex on S^4\{.} and S^6\{.} through Supersymmetric Glasses
S^4 is not a complex manifold, but it is sufficient to remove one point to
make it complex. Using supersymmetry methods, we show that the Dolbeault
complex (involving the holomorphic exterior derivative and its Hermitian
conjugate) can be perfectly well defined in this case. We calculate the
spectrum of the Dolbeault Laplacian. It involves 3 bosonic zero modes such that
the Dolbeault index on S^4\{.} is equal to 3
Self-duality and supersymmetry
We observe that the Hamiltonian H = D^2, where D is the flat 4d Dirac
operator in a self-dual gauge background, is supersymmetric, admitting 4
different real supercharges. A generalization of this model to the motion on a
curved conformally flat 4d manifold exists. For an Abelian self-dual
background, the corresponding Lagrangian can be derived from known harmonic
superspace expressions.Comment: 14 page
Supersymmetric Proof of the Hirzebruch-Riemann-Roch Theorem for Non-K\"ahler Manifolds
We present the proof of the HRR theorem for a generic complex compact
manifold by evaluating the functional integral for the Witten index of the
appropriate supersymmetric quantum mechanical system
Born--Oppenheimer corrections to the effective zero-mode Hamiltonian in SYM theory
We calculate the subleading terms in the Born--Oppenheimer expansion for the
effective zero-mode Hamiltonian of N = 1, d=4 supersymmetric Yang--Mills theory
with any gauge group. The Hamiltonian depends on 3r abelian gauge potentials
A_i, lying in the Cartan subalgebra, and their superpartners (r being the rank
of the group). The Hamiltonian belongs to the class of N = 2 supersymmetric QM
Hamiltonia constructed earlier by Ivanov and I. Its bosonic part describes the
motion over the 3r--dimensional manifold with a special metric. The corrections
explode when the root forms \alpha_j(A_i) vanish and the Born--Oppenheimer
approximation breaks down.Comment: typos correcte
Screening vs. Confinement in 1+1 Dimensions
We show that, in 1+1 dimensional gauge theories, a heavy probe charge is
screened by dynamical massless fermions both in the case when the source and
the dynamical fermions belong to the same representation of the gauge group
and, unexpectedly, in the case when the representation of the probe charge is
smaller than the representation of the massless fermions. Thus, a fractionally
charged heavy probe is screened by dynamical fermions of integer charge in the
massless Schwinger model, and a colored probe in the fundamental representation
is screened in with adjoint massless Majorana fermions. The screening
disappears and confinement is restored as soon as the dynamical fermions are
given a non-zero mass. For small masses, the string tension is given by the
product of the light fermion mass and the fermion condensate with a known
numerical coefficient. Parallels with 3+1 dimensional and supersymmetric
gauge theories are discussed.Comment: 29 pages, latex, no figures. slight change in the wording on page 2,
references adde
Quantum entanglement via nilpotent polynomials
We propose a general method for introducing extensive characteristics of quantum entanglement. The
method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference
vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a
special canonical form and expressed via polynomials of nilpotent variables, we show how this description
provides a simple criterion for entanglement as well as a universal method for constructing the invariants
characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging
from our approach. We derive the equation of motion for the tanglemeter and, in representative examples
of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We
extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced
idea of generalized entanglement. Possible future developments and applications of the method are discussed