Bianca L. Cerchiai

A cohomological approach to the non-Abelian Seiberg-Witten map

We present a cohomological method for obtaining the non-abelian Seiberg-Witten map for any gauge group and to any order in θ. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator.

Euler angles for G2

We provide a simple parameterization for the group G2, which is analogous to the Euler parameterization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H.

Geometrical Tools for Quantum Euclidean Spaces

Abstract: We apply one of the formalisms of noncommutative geometry to ℝNq, the quantum space covariant under the quantum group SOq(N). Over ℝNq there are two SOq(N)-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of ℝ3q. As in the case N=3, one has to slightly enlarge the algebra ℝNq; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over ℝNq. While in our previous article the frame was found “by hand”, here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of ℝNq with Uqso(N) into ℝNq, an interesting result in itself.

From qubits to E7

There is an intriguing relation between quantum information theory and super gravity, discovered by M. J. Duff and S. Ferrara. It relates entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant on the 56-dimensional representation of the Lie group E7. In this paper we recall the relatively straightforward manner in which three-qubits lead to E7, or at least to the Weyl group of E7. We also show how the Fano plane emerges in this context.

On the Euler angles for SU(N)

In this paper we reconsider the problem of the Euler parametrization for the unitary groups. After constructing the generic group element in terms of generalized angles, we compute the invariant measure on SU(N) and then we determine the full range of the parameters, using both topological and geometrical methods. In particular, we show that the given parametrization realizes the group SU(N+1) as a fibration of U(N) over the complex projective space CPn. This justifies the interpretation of the parameters as generalized Euler angles.

The Seiberg-Witten map for a time-dependent background

In this paper the Seiberg-Witten map for a time-dependent background related to a null-brane orbifold is studied. The commutation relations of the coordinates are linear, i.e. it is an example of the Lie algebra type. The equivalence map between the Kontsevich star product for this background and the Weyl-Moyal star product for a background with constant noncommutativity parameter is also studied.

The Seiberg-Witten map for noncommutative gauge theories

The Seiberg-Witten map for noncommutative Yang-Mills theories is studied and methods for its explicit construction are discussed which are valid for any gauge group. In particular the use of the evolution equation is described in some detail and its relation to the cohomological approach is elucidated. Cohomological methods which are applicable to gauge theories requiring the Batalin-Vilkoviskii antifield formalism are briefly mentioned. Also, the analogy of the Weyl-Moyal star product with the star product of opestring field theory and possible ramifications of this analogy are briefly mentioned.

Iwasawa N=8 attractors

Starting from the symplectic construction of the Lie algebra e7(7) due to Adams, we consider an Iwasawa parametrization of the coset E7(7)SU(8), which is the scalar manifold of N=8, d=4 supergravity. Our approach, and the manifest off-shell symmetry of the resulting symplectic frame, is determined by a noncompact Cartan subalgebra of the maximal subgroup SL(8,R) of E7(7). In the absence of gauging, we utilize the explicit expression of the Lie algebra to study the origin of E7(7)SU(8) as scalar configuration of a 18-BPS extremal black hole attractor. In such a framework, we highlight the action of a U(1) symmetry spanning the dyonic 18-BPS attractors. Within a suitable supersymmetry truncation allowing for the embedding of the Reissner–Nordstrom black hole, this U(1) action is interpreted as nothing but the global R-symmetry of pure N=2 supergravity. Moreover, we find that the above mentioned U(1) symmetry is broken down to a discrete subgroup Z4, implying that all 18-BPS Iwasawa attractors are nondyonic ...

On quantum groups in the Hubbard model with phonons

The correct Hamiltonian for an extended Hubbard model with quantum group symmetry as introduced by Montorsi and Rasetti is derived for a D-dimensional lattice. It is shown that the superconducting holds as a true quantum symmetry only for D = 1 and that terms of higher order in the fermionic operators are needed in addition to phonons. A discussion of quantum symmetries in general is given in a formalism that should be readily accessible to non-Hopf algebraists.

Properties of Perturbative Solutions of Unilateral Matrix Equations

A left-unilateral matrix equation is an algebraic equation of the form a0+a1x+a2x2+·+anxn=0 where the coefficients ar and the unknown x are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra.