Lax pairs for string Newton Cartan geometry

In this paper, based on a systematic formulation of Lax pairs, we show \textit{classical} integrability for nonrelativistic strings propagating over \textit{stringy} Newton-Cartan (NC) geometry. We further construct the corresponding \textit{monodromy} matrix which leads to an infinite tower of \textit{non-local} conserved charges over stringy NC background.


Overview and Motivation
During last one decade, nonrelativistic strings coupled to background fields has started to unveil a new fascinating sector of string theory whose dynamics is governed by a 2D relativistic non-linear sigma model [1]- [16] that enjoys a deformed global string Galilei invariance on the world-sheet. The corresponding target space geometry 1 is obtained by gauging the centrally extended string Galilei algebra which eventually turns out to be an extension from point particle Newton-Cartan (NC) background [17]- [22] to strings. As a natural consequence of this gauging procedure, the corresponding target space geometry is naturally attributed to both longitudinal (τ i M ) as well as transverse (e q M ) vielbein fields which together with the U (1) connection (m i M ) define what is known as the stringy NC background [4], [8]. From the perspective of a 2D world-sheet theory, these gauge fields on the target space have a natural interpretation of being the coupling constants for the non-linear sigma model in the so called NC limit [4].
Given the above structure, the purpose of the present analysis is to look ahead one step further and address the following fundamental question namely whether nonrelativistic strings propagating over string NC geometry are integrable 2 . In other words, whether the non linear sigma model (characterizing the string propagation over stringy NC geometry) allows infinite symmetry generators associated to a one parameter family of flat connections those are constructed by means of symmetry currents corresponding to generators of extended string Galilei algebra [4]. Technically speaking, the present analysis is therefore the nonrelativistic counterpart of the earlier observations made in the context of AdS 5 × S 5 superstrings 3 where following similar spirits we are finally able to construct the so called Lax pairs using a suitable combination of global symmetry currents corresponding to generators of extended string Galilei algebra. Based on the notion of spacetime foliation [8], we split these currents into longitudinal, as well as transverse components where H i and P q are respectively the generators corresponding to longitudinal as well as transverse translations. Here, Z i is the central extension [4] associated to U (1) gauge connection (m i M ). As a first step towards proving integrability, we show that the non-linear sigma model under consideration could be expressed as a bilinear in the currents (1) and (2) whose equations of motion could be reproduced by means of a one parameter family of flat connections. These are precisely the Lax connections for the sigma model under consideration that could be expanded in a basis of string Galilei generators {H i , Z i , P q }. Finally, using Lax connection(s), we construct the so called monodromy matrix that leads to an infinite tower of non-local charges thereby proving integrability of the non-linear sigma model.
The rest of the paper is organized as follows. In Section 2, we provide a detailed analysis on the construction of Lax pairs for the nonrelativistic strings propagating over stringy NC background and construct the corresponding tower of non-local charges. Finally, we conclude in Section 3 mentioning some of its possible future extensions.

Road to integrability 2.1 Lax pairs
We start our analysis with a formal introduction to the Nambu-Goto (NG) strings in NC background. The corresponding NG action associated with the bosonic sector could be formally expressed as 4 [8], where, we read off the pull-back of the target space metric as, together with, X M (M = 0, .., d) as embedding coordinates and α = σ 0 , σ 1 being the coordinates on the world-sheet. Here, B (2) is the background NS-NS two form field which the string is coupled to. Following [4], we express the metric field as, where, i, j = 0, 1 are the longitudinal directions together with p, q = 2, .., d as being that of the transverse directions associated with the tangent space T p at a point p in the string NC manifold (M). Here, E a M is a d + 1 dimensional relativistic vielbein that could be splited into following two parts [4], where, τ i M and e p M are respectively the longitudinal as well as transverse vielbein fields associated with the corresponding generators of translation [4], [8]. Here, ω is the so called NC parameter that needs to be set to infinity for a consistent non relativistic limit [4]. Finally, we introduce m i M as U (1) connection corresponding to the generator associated with the non central extension of the string Galilei algebra [8].
Using (5), it is now straightforward to show, where, we introduce the so called transverse metric [8], together with, Considering the NS-NS field strength to be of the following form [5], it is now straightforward to obtain the finite world-sheet action in the NC (ω → ∞) limit, where we introduce 2 × 2 matrices, a αβ = τ i α τ j β η ij together with the inverse a αβ a βγ = δ γ α [4]. In order to proceed further, as a first step, we define currents associated with the longitudinal translation, transverse translation as well as the non central extension of string Galilean algebra. These currents could be formally expressed as 5 , where, we identify H i (i = 0, 1) and P q (q = 2, .., d) respectively as generators corresponding to longitudinal as well as transverse translations [8] and Z i as the generator corresponding to the non central extension of string Galilei algebra [8], where, G ip is the generator associated with string Galilei boosts. We denote, j as the longitudinal current and identify j α as that of the current associated with transverse isometries. Here, g ≡ g(ξ α ) is a field with values in the Lie group. From, (12), it immediately follows, dj + j ∧ j = 0 (14) which yields, where, we identify the current in its most generic form, Here, T a (= H i , Z i , P q ) could be identified with one of the translation generators (12) that satisfies [4], With the above terminology in place, it is now quite straightforward to re-express the NG action (11) in terms of currents (12) defined above, where, we introduce the following entities, Next, we note down the equation of motion corresponding to the current j (xa)a , ∂ α ( −|a| a αβ βa ) = 0 (20) where, we introduce new variables, Finally, we note down the equation of motion corresponding to the current j (z b )b . A similar calculation reveals, Based on (15), (20) and (22), we propose the following flat (Lax) connection associated with string NC geometry, where, we express the individual coefficients as, where Λ is the so called spectral parameter together with, ε σ 0 σ 1 = −ε σ 1 σ 0 = 1. Using (15), (20) and (22) it is now trivial to show, On the other hand, a careful computation reveals, where, by virtue of extended string Galilei algebra [3]- [4], we identify each of the above commutators equal to zero. This finally allows us to write down the zero curvature condition for Lax pairs in two dimensions [28],

Conserved charges
The flat connection (24) provides us a canonical way of estimating conserved charges associated with the 2D sigma model. The first step is to write down the corresponding monodromy matrix [28], where, the exponential is subjected to a path ordered form.
Using (24), one could further simplify (28) as 6 , where, we assume that all the dynamical variables associated with the world-sheet theory are periodic functions of σ.
Using (29), it is now straightforward to construct the corresponding transfer matrix, A straightforward computation reveals [28], 6 See Appendix for the details.
which eventually leads to an infinite tower of conserved charges Q n (∀n ≥ 0), thereby making the theory integrable. By Taylor expanding the transfer matrix (30) near the origin we finally read off the family of non-local conserved charges, associated with the 2D non-linear sigma model corresponding to stringy NC background.

Summary and final remarks
The present paper was an attempt to explore classical integrability for non-relativistic strings propagating over stringy Newton-Cartan (NC) background. We address this issue considering the bosonic sector of the full superstring spectra. The analysis of this paper suggests that the non-linear sigma model under consideration indeed allows an underlying integrable structure that further opens up a number of possible future directions. The first and foremost question that one should clarify is whether the integrable structure could be extended for the full superstring spectrum. The other question that remains to be addressed is what are the natural consequences of the above integrable structure in the context of AdS/CFT duality. In order to address the second question, one must have a detailed understanding of the stringy dynamics in the presence of a negative cosmological constant [4]. We leave these issues for the purpose of future investigations.
Acknowledgements : It is indeed a great pleasure to thank J. Hartong, E. Bergshoeff and A. A. Tseytlin for their valuable comments on the draft. The author is indebted to the authorities of IIT Roorkee for their unconditional support towards researches in basic sciences.
Appendix: Expression for Γ b (α, Λ) Here, in the Appendix, we provide the detail expression for the function Γ b (α, Λ) appearing in the expression for the monodromy matrix (28), −|a|a 00 (j)j where we use short-hand notation for the derivative, ∂ ∂σ α ≡ ∂ α .