Newton-Cartan D0 branes from D1 branes and integrability

We explore analytic integrability criteria for D1 branes probing 4D relativistic background with a null isometry direction. We use both the Kovacic’s algorithm of classical (non)integrability as well as the standard formulation of Lax connections to show the analytic integrability of the associated dynamical configuration. We further use the notion of double null reduction and obtain the world-volume action corresponding to a torsional Newton-Cartan (TNC) D0 brane probing a 3D torsional Newton-Cartan geometry. Moreover, following Kovacic’s method, we show the classical integrability of the TNC D0 brane configuration thus obtained. Finally, considering a trivial field redefinition for the D1 brane world-volume fields, we show the equivalence between two configurations in the presence of vanishing NS fluxes.


Overview and motivation
The extension of non relativistic (NR) string sigma models [1]- [3] to arbitrary backgrounds and understanding two of its primary aspects namely, (i) the UV completion and (ii) the underlying integrable structure (if any) stands extremely important in its own right. The target space geometry corresponding to NR propagating strings could be classified into two different categories. One of these goes under the name of string Newton-Cartan geometry obtained via gauging the centrally extended string Galilean algebra [4]- [9]. The other is obtained via null reduction of (relativistic) Lorentzian manifolds giving rise to what is known as torsional Newton-Cartan (TNC) geometry [10]- [18]. A recent analysis of [16] reveals that under certain specific assumptions, these two seemingly different string theories could in principle be mapped into each other in a consistent manner. The exciting evidence behind the existing integrable structure at the tree level of the Newton-Cartan (closed string) sigma models [8,15] has opened up a tremendous possibility of analyzing the NR stringy dynamics using the standard techniques of integrable models. This is therefore quite similar in spirit to that of its relativistic counterpart [19,20]. However, the understanding of similar questions in the corresponding open string sector still JHEP06(2020)120 remains as a challange. The present article therefore aims to fill up some of these gaps and widen our current understanding beyond the closed string sector by taking into account the dynamics of extended objects like Dp branes 1 [21,22] those probing the Galilean invariant manifolds. The corresponding target space geometry that we choose to work with happens to be a 2 + 1 dimensional TNC spacetime (those are obtained via null reduction of 3 + 1 dimensional Lorentzian manifolds [12]) with R × S 2 topology.
We start our analysis considering D1 branes propagating over 4D relativistic manifolds with a null isometry direction. Given the D1 brane configuration, we address the above issue of classical integrability following two traditional paths. One of these approaches goes under the name of Kovacic's algorithm [23,24] of classical (non)integrability which has been applied with remarkable success in various examples of relativistic sigma models [25]- [31] with or without supersymmetries. The other approach is based on the systematic formulation of Lax connection [20] and thereby establishing its flatness following the equations of motion. This further allows us to compute the infinite tower of conserved charges associated with the 2D world-volume theory and thereby proving the integrability.
In the second part of the analysis, we use a double null reduction of the D1 brane worldvolume action and obtain a world-volume description for torsional Newton-Cartan (TNC) D0 branes propagating over 3D torsional Newton-Cartan geometry with R × S 2 topology. We further show the classical integrability of the configuration following Kovacic's method. On top of it, we show that following a trivial field redefinition, the D1 brane dynamics could be mapped to that of TNC D0 brane dynamics in the presence of vanishing NS fluxes. Finally, we draw our conclusion in section 3.

Kovacic's method: a review
For the sake of comprehensiveness, we briefly outline the essentials of Kovacic's algorithm that was proposed originally in [23]. The algorithm essentially provides road to explore the classical (non)integrability criteria associated with dynamical phase space configurations. The steps are in fact quite straightforward to follow: (1) choose an invariant plane in the dynamical phase space and (2) consider fluctuations normal to this plane. These fluctuations generally obey differential equations, known as Normal Variational Equations (NVEs) [26]. Here, a , b and c are in general complex rational functions. The associated phase space configuration is said to be classically integrable if there exists simple algebraic/logarithmic/exponential solutions to (2.1) known as Liouvillian solutions [23][24][25][26]. In summary, the algorithm sets rules to check whether NVE (2.1) admits Liouvillian solutions or not.
Interestingly enough, we discover that for extended objects like D1 branes (those probing 4D relativistic backgrounds) as well as nonrelativistic D0 branes (those probing 3D TNC geometries) it is indeed possible to find a very special form of NVEs (2.1) with a = 0 together with b = c = 0 which therefore uniquely sets the potential V (τ ) = 0 as well as the rational polynomial w(τ ) ∼ 1 τ with degree 1. The most general expression for these Polynomials goes under the name of Mobius transformations that generate the group of automorphisms of the Riemann sphere.

Relativistic D1 branes
We consider D1 brane dynamics over 4D relativistic backgrounds with a null isometry direction. For technical simplicity, we set the dilaton as well as the background RR fluxes to zero and take into account only the background NS-NS fluxes (B M N ).
The resulting DBI action [21] is given by, where we identify, Here, T 1 = l −2 s stands for the D1 brane tension together with ξ α (α = 0, 1) as world-volume directions. Moreover, we identify F αβ = ∂ α a β − ∂ β a α as being the world-volume field strength tensor where a α is the corresponding U(1) gauge field.
To proceed further, we consider the following 4D geometry, where we identify individual metric functions [12,15] JHEP06(2020)120 Notice that, here X u ≡ u is the so called null isometry direction associated with the target space manifold. Using (2.6), it is therefore trivial to show where we choose to work with NS-NS two form B θϕ = B sin θ [18] that corresponds to some specific values of the page charge, Q D ∼ S 2 B 2 which takes quantized values on the world-volume of the D1 brane. This is related to the underlying mechanism known as flux stabilization which states that the background NS fluxes sort of prevents D1 branes (wrapping S 2 ) from shrinking it to zero size.

The world-volume theory
To start with, we consider that the D1 brane is placed at a point X u = constant, along the axis of null isometry. The ansatz that we choose to work with is that of a D1 brane wrapping the azimuthal direction of S 2 , where κ is the corresponding winding number. The resulting matrix elements A αβ are given by, where we define,ψ = ψ 2 .

Equations of motion
The corresponding Lagrangian density is given by, where we choose, The resulting equations of motion could be formally expressed as, JHEP06(2020)120
Substituting the above ansatz into (2.20) we find, which thereby yields, ψ(τ ) ∼ τ + c. Substituting this back into (2.19) and considering fluctuations δθ ∼ η(τ ) at leading order, we arrive at the following NVË which admits Loiuvillian solution of the form, Therefore, following our discussion in the previous section, we conclude that the associated dynamical phase space configuration is classically integrable.

Lax pairs and integrability
In this section, we look forward towards identifying the D1 brane dynamics in terms of a proper formulation of Lax connections [20] over X u = constant sub-manifold of the full relativistic/Lorentz invariant 3 + 1 dimensional manifold (2.6). The corresponding worldvolume theory turns out to be, where we identify, together with [12], v = ψ 2 + t which we collectively identify as time. JHEP06(2020)120

The 2D world-volume current
Our starting point is the consideration of the D1 brane dynamics over group manifold G ∼ S 2 with SO(3) isometries. The Killing generators that span the so(3) ∼ su(2) Lie algebra could be schematically expressed as [12], where e a M (X M ; M = v, θ, ϕ) are the expansion coefficients that could be fit into the following 3 × 3 matrix as, In the following, we introduce 2D world-volume currents as

Equations of motion
With the above set-up in hand, the D1 world-volume theory (2.24) could be formally expressed as, where we introduce the notation, J α · J β ≡ J a α J βa = J a α J b β Ω ab . Before we proceed further, it is customary first to note down the following identity where we use the fact, δJ a =ẽ a M δX M .

JHEP06(2020)120
Using (2.47), the equation of motion corresponding to world-volume currents could be formally expressed as, where we identify, g αβ = A αβ + A βα = g βα as a symmetric tensor filed on the world-volume of the D1 brane. Moreover, here A αβ has been introduced as an inverse of the induced world-volume metric namely, A αβ A βγ = δ α γ . On the other hand, the dynamics associated to world-volume gauge fields reveals, The above set of equations (2.49)-(2.50) could be combined into a single equation, is the antisymmetric two form on the world-volume. Furthermore, as a natural consequence of (2.49)-(2.50) it is in fact trivial to see, τ σ = Π = constant.

The flat connection
Given the above dynamics, we propose the Lax connection of the following form where, ℘ 1 , ℘ 2 and ℘ 3 are arbitrary constats that will be fixed from the flatness condition [20] of the Lax connection. Moreover, here ε τ σ = −ε στ = 1 is the 2D Levi-Civita symbol together with its inverse ε αβ (= −ε βα ) which is defined through the relation, ε αβ ε αγ = δ β γ . Using (2.32), (2.48), (2.49) and (2.50) it is now quite straightforward to show, On the other hand, after some trivial algebra we find Combing (2.53) and (2.54) together, we arrive at the following relation

JHEP06(2020)120
The flatness condition [20] of Lax implies that the r.h.s. of (2.55) must vanish identically. This is naturally achieved by setting the following constraint, A non trivial choice that solves (2.56) could be of the form, (2.57) Using (2.57), the flat connection (2.52) could be formally expressed as, where, we identify Π as the spectral parameter associated with the 2D integrable model.

Conserved charges
The flat connection (2.58) estimated above could be used to define tower of conserved charges associated with the 2D world-volume theory. The first step is to introduce the so called monodromy matrix [19,20], where we presume that one of the world-volume directions of D1 brane is compact where the world-volume fields obey periodic boundary conditions. Using (2.59), it is in fact straightforward to show, (2.60) The next step would be to introduce the transfer matrix, which by means of (2.60) yields, an infinite tower of conserved charges, associated with the D1 brane dynamics over 4D relativistic backgrounds. JHEP06(2020)120

Equations of motion
Next, we note down equations of corresponding to different world-volume fields. To proceed further, we set t = τ = ξ 0 . Below we enumerate equations of motion for world volume fields,φȧ  [25,26]. To start with, we choose to work with the dynamical phase space configuration with F τ σ =ȧ σ = 0 (= constant) together with ψ = constant. Notice that, both the invariant planes that we choose below belong to the ψ = constant and Π ψ = constant subspace of the full dynamical phase space configuration. This further simplifies (2.73)-(2.76), ϕθ cos θ +φ sin θ − 2l 2 s Λθ sin θ − 2l 2 s Λθ 2 cos θ = 0 (2.77) sin θθφ − cos θφ = 0 (2.78) where the second equation (2.78) follows from setting dΛ dτ = 0. In order to implement Kovacic's algorithm, we further choose to work with the following invariant plane θ =θ =θ = 0 which identically satisfies (2.77). The invariant plane one might wish to think of as a submanifold with, θ = 0 , Π θ = 0 and Π ϕ = constant within the subspace of the full dynamical phase space configuration.
Upon substitution into (2.78) this further yields, which ensures the integrability of the associated phase space configuration. The second phase space configuration that one might choose to work with is to set the invariant plane as, ϕ =φ =φ = 0 which trivially solves (2.78). This is a submanifold that satisfies, ϕ = 0 and Π ϕ = 0. Substituting this into (2.77) yields, The above analysis confirms that the second phase space configuration is also classically integrable in the sense of Kovacic.