Note about Hamiltonian formalism for Newton–Cartan string and p-brane

Abstract

We construct non-relativistic string and p-brane actions in Newton-Cartan background using the limiting procedure from the relativistic string and p-brane action in general background. We also find their Hamiltonian formulations when however we restrict ourselves to the case of the vanishing gauge field m μ .

Non-relativistic M2-brane

“…Observe that this Hamiltonian constraint has the same structure as the Hamiltonian constraint for non-relativistic string that was derived in [13]. Further, this Hamiltonian constraint is the generalization of the Hamiltonian constraint found in [14] to the case of non-zero gauge field m a M which is again nice consistency check. In the next section we will discuss another important aspect of non-relativistic M2-brane which is its dimensional reduction.…”

“…Then in order to cancel this divergence we should introduce an appropriate form of the background three form C M N P . We studied this problem in more details in case of non-relativistic string [13,14] when we proposed such a form of NSNS two form that leads to finite action for non-relativistic string in stringy NC background. Clearly this approach can be generalized to higher dimensional objects and hence it is natural to presume that C M N P has the form…”

Remark about non-relativistic string in Newton-Cartan background and null reduction

“…Non-relativistic strings were firstly introduced from different point of view in [11,12], for further analysis in the context of string theory, see for example [13,14,15,16,17]. Non-relativistic strings and p-branes gained renewed interest recently when these objects were studied in the context of Newton-Cartan gravity and Carroll gravity [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33].…”

Hamiltonian for a string in a Newton-Cartan background

“…For example, in was argued in [23] that in order to define correctly an action for non-relativistic string in Newton-Cartan background two longitudinal directions have to be selected and hence we obtain more general form of the Newton-Cartan geometry. The canonical analysis of this string was performed recently in [25]. During this analysis we met an obstacle which was an impossibility to derive Hamiltonian constraint for the string with non-zero gauge field m a µ that will be defined in the next section.…”