M5-brane in three-form flux and multiple M2-branes

Abstract

Abstract:We investigate the Bagger-Lambert-Gustavsson model associated with the Nambu-Poisson algebra as a theory describing a single M5-brane. We argue that the model is a gauge theory associated with the volume-preserving diffeomorphism in the threedimensional internal space. We derive gauge transformations, actions, supersymmetry transformations, and equations of motions in terms of six-dimensional fields. The equations of motions are written in gauge-covariant form, and the equations for tensor fields have manifest self-dual structure. We demonstrate that the double dimensional reduction of the model reproduces the non-commutative U(1) gauge theory on a D4-brane with a small noncommutativity parameter. We establish relations between parameters in the BLG model and those in M-theory. This shows that the model describes an M5-brane in a large C-field background.

Lorentzian Lie (3-)algebra and toroidal compactification of M/string theory

“…In section 5, we construct the BLG model (or super Yang-Mills theory) based on loop algebras which are the simplest nontrivial examples of generalized Lorentzian Lie (3-)algebra. Finally we comment that the description of M5-brane [8,9] can be also regarded as the typical example of the compactification through the Lorentzian 3-algebra.…”

“…Therefore, at least when the 3-algebra is finite dimensional, it is impossible to construct Lorentzian algebra with nontrivial F ijkl = 0. The situation is totally different if the 3-algebra is infinite dimensional [8,9] which is related to the description of M5-brane (for the various aspects of M5-brane in BLG context, see also [22] for example). The realization of the three-algebra was given as follows.…”

Effective action for Dp-brane in large RR (p − 1)-form background

“…In the large B-field background, the noncommutative structure can be approximated by the Poisson bracket. In fact, the low energy effective theory of M5-brane in large C-field background has already been shown [6,8] to agree with both D4-brane in NS-NS B-field background and D4-brane in RR C-field background, depending on whether the direction of compactification is xμ =5 or x 2 . Since the D4-brane theories for the two different backgrounds reduce to D3-branes in NS-NS B-field or RR field background through dimensional reduction on x 2 or xμ =5 , respectively, it is more or less obvious that, to the leading order (Poisson approximation), the noncommutative U(1) gauge theory is dual (equivalent) to the effective theory defined by (48) for p = 3, as they are both dimensional reduction of the same M5-brane theory on a torus in the directions of x 2 and xμ =5 .…”

“…Starting with the D4-brane case [8] which is derived from the M5-brane theory in Cfield background [5,6,7], we can straightforwardly go down to D3-brane and D2-brane theories via the T-duality transformation rules listed above. To go up to D5-brane, we need to look for Lagrangians that would reduce to the D4-brane theory through the replacements (39)-(47).…”

“…When there is a large longitudinal NS-NS B-field background, the effective theory of a D-brane is deformed to a non-commutative gauge theory [3,4], as a result of the fact that the B-field background changes the zero modes of open strings, so that higher derivative terms are turned on. Similarly, when there is a large longitudinal C-field background in M theory, the effective theory of M5-brane is a Nambu-Poisson gauge theory [5,6,7]. The interaction mediated by open membranes under the effect of the C-field is characterized by the gauge symmetry of volume-preserving diffeomorphism, where the volume-form is defined by the C-field.…”