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Register a new userStrict Almost Non-Abelian Deformations
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A. Much
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An almost non-abelian extension of the Rieffel deformation is presented in this work. The non-abelicity comes into play by the introduction of unitary groups which are dependent of the infinitesimal generators of $SU(n)$. This extension is applied to quantum mechanics and quantum field theory.
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In this contribution to the study of one dimensional point potentials, we prove that if we take the limit $qto 0$ on a potential of the type $v_0delta({y})+{2}v_1delta({y})+w_0delta({y}-q)+ {2} w_1delta({y}-q)$, we obtain a new point potential of the type ${u_0} delta({y})+{2 u_1} delta({y})$, when $ u_0$ and $ u_1$ are related to $v_0$, $v_1$, $w_0$ and $w_1$ by a law having the structure of a group. This is the Borel subgroup of $SL_2({mathbb R})$. We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the exceptional cases emerging in the study are also described in full detail. It is shown that for the $v_1=pm 1$, $w_1=pm 1$ values of the $delta^prime$ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.
We construct the general solution of a class of Fuchsian systems of rank $N$ as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the $W_N$-algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for $c=N-1$.
In contrast to Hamiltonian perturbation theory which changes the time evolution, spacelike deformations proceed by changing the translations (momentum operators). The free Maxwell theory is only the first member of an infinite family of spacelike deformations of the complex massless Klein-Gordon quantum field into fields of higher helicity. A similar but simpler instance of spacelike deformation allows to increase the mass of scalar fields.
The U(1) BF Quantum Field Theory is revisited in the light of Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition function is related to the BF one and how the latter on its turn coincides with an abelian Turaev-Viro invariant. Significant differences compared to the non-abelian case are highlighted.
We consider antibracket superalgebras realized on the smooth Grassmann-valued functions with compact supports in n-dimensional space and with the grading inverse to Grassmanian parity. The deformations with even and odd deformation parameters of these superalgebras are presented for arbitrary n.
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