Noncommutativity and Duality through the Symplectic Embedding Formalism

Abstract

Abstract. This work is devoted to review the gauge embedding of either commutative and noncommutative (NC) theories using the symplectic formalism framework. To sum up the main features of the method, during the process of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and brings some light on the so-called "arbitrariness problem". This alternative embedding formalism also presents a way to obtain a set of dynamically dual equivalent embedded Lagrangian densities which is obtained after a finite number of steps in the iterative symplectic process, oppositely to the result proposed using the BFFT formalism. On the other hand, we will see precisely that the symplectic embedding formalism can be seen as an alternative and an efficient procedure to the standard introduction of the Moyal product in order to produce in a natural way a NC theory. In order to construct a pedagogical explanation of the method to the nonspecialist we exemplify the formalism showing that the massive NC U (1) theory is embedded in a gauge theory using this alternative systematic path based on the symplectic framework. Further, as other applications of the method, we describe exactly how to obtain a Lagrangian description for the NC version of some systems reproducing well known theories. Naming some of them, we use the procedure in the Proca model, the irrotational fluid model and the noncommutative self-dual model in order to obtain dual equivalent actions for these theories. To illustrate the process of noncommutativity introduction we use the chiral oscillator and the nondegenerate mechanics.

Noncommutative mapping from the symplectic formalism

“…The symplectic formalism 25 is a powerful tool in the field theory and it was extended to also deal with constrained systems [26][27][28] , to induce symmetries into non-invariant systems [29][30][31][32] , and as well as to give an alternative way to introduce the Clebsch parameters 33 into some models. Further, it was also extended to induce noncommutativity into commutative systems 24,34,35 . There are a lot of ways to introduce NC 5,15,[36][37][38][39][40][41][42][43] into a system, however, a brief presentation of the NC symplectic induction formalism 24,34,35 will be necessary, since the Boop's shifts will be mathematically generalized through the symplectic framework.…”

“…Further, it was also extended to induce noncommutativity into commutative systems 24,34,35 . There are a lot of ways to introduce NC 5,15,[36][37][38][39][40][41][42][43] into a system, however, a brief presentation of the NC symplectic induction formalism 24,34,35 will be necessary, since the Boop's shifts will be mathematically generalized through the symplectic framework.…”

An alternative way to explain how non-commutativity arises in the bosonic string theory

“…In order to propose a NC version of the string theory, we apply the NC symplectic induction method [20][21] [22] or the general Boop's shifts matrix method [23], also presented in section 2, into the two-dimensional commutative harmonic oscillator of unit mass and frequency, which it will be mapped into the bosonic open string theory with string ends attached to the D3-brane. The two-dimensional harmonic oscillator has its dynamic governed by the following Lagrangian,…”

“…In [17,18] different aspects of non-commutative Yang-Mills theory within string theory and field theory setups were studied and in [19] the quantization of open strings ending on a D-branes in the presence of a B µν -field was reexamined and also quoted that a sigma model, with a specific boundary interaction and gauge fixing terms, it is a special case of the deformed quantization theory used by Kontsevich [15], which was well elucidated in [16]. We purchase this idea and, in this paper, we will obtain the open bosonic string attached to a D3-brane in a presence of a B µν -field from the non-commutative version of two-dimensional harmonic oscillator, which it is obtained applying the general Bopp's shift matrix method [23] (this result could be obtained using the non-commutative symplectic induction formalism [20][21][22]) and through the connection between self-dual theory and twodimensional harmonic oscillator [24,25]. In order to become the present work self-contained, it was organized as follow.…”

Noncommutative approach to diagnose degenerate Higgs bosons at 125 GeV

“…Noncommutativity has been extensively investigated in different contexts: quantization procedure [1][2][3][4][5][6][7][8][9][10][11], the Yang-Mills theory on a NC torus [12,13], matrix model of M-theory [14][15][16][17], string theory [18][19][20][21][22][23][24][25] and D-brane [26][27][28][29][30][31], SSB and Higgs-Kibble mechanisms [32][33][34][35][36]. At last scenario, it was investigated how noncommutativity affects the IR-UV mixing and the appearance of massless excitations [32,33], the relation between symmetry breaking in NC cut-off field theories [34,35] and the role played by the noncommutativity in the masses generation of new bosons [36].…”