Abstract
We construct an action for four-dimensional extended string Newton-Cartan gravity which is an extension of the string Newton-Cartan gravity that underlies nonrelativistic string theory. The action can be obtained as a nonrelativistic limit of the EinsteinHilbert action in General Relativity augmented with a term that contains an auxiliary two-form and one-form gauge field that both have zero flux on-shell. The four-dimensional extended string Newton-Cartan gravity is based on a central extension of the algebra that underlies string Newton-Cartan gravity.The construction is similar to the earlier construction of a three-dimensional Chern-Simons action for extended Newton-Cartan gravity, which is based on a central extension of the algebra that underlies Newton-Cartan gravity. We show that this three-dimensional action is naturally obtained from the four-dimensional action by a reduction over the spatial isometry direction longitudinal to the string followed by a truncation of the extended string Newton-Cartan gravity fields. Our construction can be seen as a special case of the construction of an action for extended p-brane Newton-Cartan gravity in p + 3 dimensions.
Highlights
JHEP01(2019)178 string theory [1].3 Taking the nonrelativistic limit of the off-shell Einstein-Hilbert term, one again obtains divergent terms
We find that all divergent terms proportional to ω2 cancel and that the remaining expression, after taking the limit ω → ∞, is given by the following nonrelativistic action describing the Extended String Newton-Cartan (ESNC) gravity theory: 1 S = 2κ2
In this work we showed how an action can be constructed for the 4D string Newton-Cartan (NC) gravity theory underlying nonrelativistic string theory provided an extra gauge field is introduced that signals a central extension in the string NC algebra
Summary
We start by writing down the symmetries and generators of the 4D extended string NewtonCartan algebra. 2. String Newton-Cartan algebra is a noncentral extension of string Galilei algebra that includes the generators ZA and Z. String Newton-Cartan algebra is a noncentral extension of string Galilei algebra that includes the generators ZA and Z 3. Extended String Newton-Cartan (ESNC) algebra is a central extension of the 4D string NC algebra in four dimensions that includes the generator S. Extended String Newton-Cartan (ESNC) algebra is a central extension of the 4D string NC algebra in four dimensions that includes the generator S Rμν (S) = 2∂[μsν] − A B Ω[μAA Ων]AB At this stage one usually imposes a maximal set of so-called conventional curvature constraints in order to obtain an irreducible gauge theory. For completeness and to compare, we will give the details of the irreducible gauging (without an action) in appendix A
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