Monopoles, Abelian projection, and gauge invariance

Abstract

A direct connection is proved between the Non-Abelian Bianchi Identities(NABI), and the abelian Bianchi identities for the 't Hooft tensor. As a consequence the existence of a non-zero magnetic current is related to the violation of the NABI's and is a gauge-invariant property. The construction allows to show that not all abelian projections can be used to expose monopoles in lattice configurations: each field configuration with non-zero magnetic charge identifies its natural projection, up to gauge transformations which tend to unity at large distances. It is shown that the so-called maximal-abelian gauge is a legitimate choice. It is also proved, starting from the NABI, that monopole condensation is a physical gauge invariant phenomenon, independent of the choice of the abelian projection.

QCD Monopoles on the Lattice and Gauge Invariance

“…One can then project on a complete set of independent diagonal matrices, which, by definition of rank, are as many as the rank r of the gauge group. A convenient choice for it are the fundamental weights [11] φ a 0 (a = 1, ..r), one for each simple root of the algebra, which are defined by their algebraic relations to the commuting operators of the Cartan algebra H i , (i = 1, ..r), and to the non diagonal operators E ± α corresponding to the…”

“…That direction coincides [11] with that of the residual U(1) symmetry in the maximal abelian gauge [Section 3]. Only in that projection, modulo gauge transformations which tend to the identity at large distances, the magnetic charge obeys the Dirac quantization condition, which is at the basis of the detection procedure of Ref.…”

“…[11]. By use of the non abelian Bianchi identities it was shown that the violations of abelian Bianchi identities in any abelian projection are projections on the corresponding fundamental weights of the gauge covariant non-abelian Bianchi identities.…”

“…3] All this is based on the results of Ref. [11] and of a paper in preparation [13]. In Sect.4 we discuss gauge invariance of dual superconductivity.…”

Topology and θ dependence in finite temperature G 2 lattice gauge theory

“…For this reason, together with the fact that G 2 has the low rank value 2, the G 2 gauge theory was often used as a testbed for possible confinement mechanisms (see e.g. [6][7][8][9][10]). The second of the aforementioned properties 1 makes the G 2 pure gauge theory quite similar to a theory coupled to 1 Which follows from the fact that the Clebsch-Gordan series of the product 7 ⊗ 14 ⊗ 14 ⊗ 14 contains a singlet, see e.g.…”