Towards the quantization of the non-relativistic D2-brane in the pure spinor formalism

An attempt is made to apply the pure spinor formalism to the non-relativistic IIA D2-brane. The fermionic constraints corresponding to the rescaled fermionic coordinates are given. Two commuting spinor fields are introduced, each one corresponding to a fermionic constraint. A BRST charge is constructed via the ansatz proposed by Berkovits in (JHEP 04:018, arXiv:hep-th/0001035, 2000, JHEP 09:046, arXiv:hep-th/0006003, 2000, JHEP 09:016, arXiv:hep-th/0105050, 2001, ICTP Lect. Notes Ser. 13:57, arXiv:hep-th/0209059, 2003). The nilpotency of the BRST charge leads to a set of constraints for the two spinor fields including pure spinor constraints. A novel non-trivial solution is given for one of the spinor fields which can be written as a sum of two pure spinors.


Introduction
The pure spinor formalism introduced by Berkovits [1][2][3][4] is a successful attempt to solve the longstanding problem of finding a manifestly supersymmetric and covariant superstring formalism. The basic ingredient is the BRST-like operator Q = dzλ α d α where d α is the fermionic constraint that appears in the conventional Green-Schwarz formalism and λ α is a bosonic chiral spinor that plays the role of the associated "ghost". For Q to be regarded as a BRST operator must be nilpotent and this leads to the relation λ α γ m αβ λ β = 0. This in 10 dimensions is the condition for λ α to be characterized as a pure spinor.
An important property of Q is that its cohomology correctly reproduces the spectrum of the superstring. The pure spinor formalism has been used as well for the covariant a e-mail: aherrera@ifuap.buap.mx b e-mail: paschalis@physics.auth.gr quantization of the superparticle [3] and also to study several aspects of string theory, for example the propagation of strings in curved backgrounds [5][6][7][8].
Another important application of the pure spinor formalism is the calculation of scattering amplitudes within the framework of superstring theory [9][10][11][12][13]. The manifest Lorentz covariance and spacetime supersymmetry make the calculation much easier than in other formalisms. Thus, pure spinors play a crucial role within string perturbation theory. However, within the context of D-branes, there are no nontrivial solutions reported in the literature. Moreover, so far it is not known whether this formalism can be consistently extended or generalized to be applied to non-relativistic systems with kappa symmetry, a fact that could allow us to draw some conclusions for the general case too, especially if there are problems in solving the pure spinor constraints in that case.
In this paper we try to extend the pure spinor formalism to the case of the non-relativistic IIA D2-brane. The non-relativistic limit of string theories [14,15] give us a deeper understanding of string theories themselves. The nonrelativistic limit of Dp branes has been studied in [16,17]. It is important to note that in this limit the kappa symmetry is maintained and this allows us to treat non-relativistic Dp branes in the framework of the pure spinor formalism.
Here we present a novel non-trivial solution for the nonrelativistic D2-brane within the pure spinor formalism. This fact could lead to the quantization of branes with interesting and relevant results.
Our starting point is the action of a IIA D2-brane in a flat 10d background. The fields consist of the 10d superspace coordinates (x m , θ) and an Abelian gauge field A μ [18][19][20]: where T is the string tension and the Wess-Zumino action reads and m, n = 0, ..., 9/μ, ν = 0, 1, 2. The action (1.1) has a global supersymmetry and also a local supersymmetry (kappa symmetry).
The conjugate momenta of the variables x μ , x a , θ + , θ − , w i are given by: where l = 1, ..., 32; and (1.31) The fermionic constraints are given by where the last derivative does not include differentiation with respect to u a 0 , F 0i , and where the last derivative does not include differentiation with respect to u a 0i . We introduce now two commuting spinor fields λ + , λ − corresponding to the fermionic constraints F + , F − and we write down a BRST charge according to the ansatz proposed in [1][2][3][4]: A set of BRST transformations is the following: where again μ, ν = 0, 1, 2/i, j = 1, 2/a = 3, ..., 9/l = 1, ... 32; from which we obtain and where we used the ansatz The full set of constraints required for the nilpotency of the BRST charge is obtained by studying the equal time Poisson bracket {Q(σ ), Q(σ )} where Q(σ ) is given in (1.34). A basis for expanding a 32 × 32 matrix M is given in Appendix A.

Conclusions
This paper is the first attempt to extend the application of the pure spinor formalism to non-relativistic systems with kappa symmetry expecting to draw some conclusions for the general case as well. This can very useful especially in cases where it is difficult to solve the pure spinor constraints in the general case. We treated the non-relativistic IIA D2 brane in the framework of this formalism [23] and we derived the fermionic constraints corresponding to the rescaled fermionic coordinates. We introduced two commuting spinor fields each one corresponding to a fermionic coordinate. The nilpotency of the BRST charge leads to a set of constraints for the two spinor fields including pure spinor constraints. Nontrivial solutions are found for the spinor field λ + which corresponds to the fermionic coordinate θ + . It is interesting to note that this solution can be written as the sum of two pure spinors λ 1+ = P + λ + and λ 2+ = P − λ + where P ± = 1 2 (1 ± 0 1 2 ). The solution for the spinor field λ − corresponding to θ − which, according to the proof given in [16] constitutes a gauge degree of freedom, is the trivial one. So for example the expression of the BRST charge for θ − = 0 in the Schrödinger representation is given by . Cohomological issues concerning the BRST charge are currently under investigation [26][27][28][29]. This study can also be performed for more general manifolds and for general dimensions as well. We would like to finally mention that the treatment of the relativistic Dp-brane in the framework of the pure spinor formalism has been reported in [24]. grant, he thanks as well SNI and PRODEP for partial financial support, while JEP is grateful to the Instituto de Física y Matemáticas, at UMSNH, in Morelia, for the warm hospitality and the stimulating atmosphere generated during his visit at the Institute.

Data Availability Statement
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