Conformal Field Theory, Quantum Group and Berry Phase

Abstract

From an analysis of the relationship between the conformal field theory in 1 + 1 dimension, the Chern-Simons theory in 2+ 1 dimension and chiral anomaly in 3+ 1 dimension, we study here the relationship between the central charge c, Chern-Simons coupling k and the Berry phase factor µ where the phase is given by e i2πµ which is associated with the chiral anomaly and their association with the quantum symmetry in the framework of the Landau problem of 2D electron gas in a magnetic field. A possible link between the deformation parameter q of the deformed algebra Uq(SL(2)) with these parameters is suggested. It is also shown how a Zp spin system is realized for a quantum space and the parameter p is linked up with the deformation parameter q and; in turn with the central charge c and the Berry phase factor µ. Finally some implications of this result in a statistical system are discussed.

Scaling of entanglement entropy in a quantum phase transition in the transverse Ising model induced by a quench

“…It has been shown in an earlier work [20] that the central charge associated with conformal anomaly in conformal field theory has a correspondence with the Berry phase associated with chiral anomaly in quantum field theory. It has been shown that the factor m in Eq.…”

Noncommutative Geometry, Geometrical Aspects of Skyrmions and String Theory

“…In this space the angular momentum is given by the relation (3). The fact that in such an anisotropic space the angular momentum can take the value 1/2 is found to be analogous to the result that a monopole-charged particle composite representing a dyon satisfying the condition eµ = 1/2 have their angular momentum shifted by 1 2 unit and their statistics shift accordingly. 16 Evidently a fermion can be viewed as a scalar particle moving with = 1/2 in an anisotropic space.…”

“…6 it has been shown that in the noncommutative space where the coordinate is given by z µ = x µ + iξ µ , where ξ µ is a "direction vector" attached to the space-time point helps us to consider the decomposition of a conformal spinor in a pseudo-Euclidean manifold R 4,2 into two Cartan semispinors defined in Minkowski space R 3,1 where these two semispinors are characterized by the fact that space, time as well as conformal reflection interchanges one into the other. Indeed, the wave function of the form φ(x µ , ξ µ ) where ξ is an attached vector extends the Lorentz group SO (3,1) to the De Sitter group SO(4, 1). The irreducible representations of SO(4), the maximal compact subgroup of SO(4, 1) are characterized by two members (k, n) where k is an integer or half integer and n is a natural number.…”

Geometric phase and fractional orbital-angular-momentum states in electron vortex beams

“…It is observed that the specific case for the quantized fractional value µ = 1 2 , corresponds to the non-paraxial beam, where the vortex is orthogonal to the wave-front propagation direction. In analogy with the central charge of conformal field theory, the monopole charge undergoes the renormalization group (RG) flow [16,17]. When µ depends on a certain parameter λ, for certain fixed points λ * , µ takes quantized values.…”