Ghost constraints and the covariant quantization of the superparticle in ten dimensions

Abstract

We present a modification of the Berkovits superparticle. This is firstly in order to covariantly quantize the pure spinor ghosts, and secondly to covariantly calculate matrix elements of a generic operator between two states. We proceed by lifting the pure spinor ghost constraints and regaining them through a BRST cohomology. We are then able to perform a BRST quantization of the system in the usual way, except for some interesting subtleties. Since the pure spinor constraints are reducible, ghosts for ghosts terms are needed, which have so far been calculated up to level 4. Even without a completion of these terms, we are still able to calculate arbitrary matrix elements of a physical operator between two physical states. * M.J.Chesterman@qmul.ac.uk 1

On the cohomology and inner products of the Berkovits superparticle and superstring

“…One is by canonical gauge-fixing using U(5) co-ordinates, in a manner analogous to choosing light-cone gauge for the bosonic particle [8,9]. The second is a BRST approach, in which a second BRST operator Q gc = C a λΓ a λ + .…”

“…For a discussion, see [8,9,12]. Roughly speaking, a basis of BRSTclosed but not exact states {ψ A g } can be chosen, such that…”

“…There is also however non-zero cohomology at ghost numbers −8, −9, −10, −11. In my previous work [8,9], the cohomology at ghost numbers ±9 was used to calculate amplitudes for the free superparticle with the standard inner product. However, the cohomology as a whole was not studied.…”

Partition functions of pure spinors

“…In papers [10,11,12] the authors reproduce the massless spectrum of open and closed superstrings using some additional ghost fields and imposing a grading constraint on the functional space to retrieve the correct constraints. Several studies [13,14,15,16,17] followed the original papers extending the analysis in different directions.…”

The quantum superstring as a WZNW model with N=2 superconformal symmetry

“…These grading conditions were shown to be equivalent to equivariant cohomology [3]. Homological perturbation theory [17] leads to the same results, at least at the classical level (i.e., with only Poisson brackets, or with single contractions): if one removes (co)homology classes by adding new ghosts, one needs in general an infinite set of such ghosts [18], but one may again truncate this series by introducing the b, c z system. The grading number turned out to be the antifield number (as defined in homological perturbation theory) minus the ghost number [2,6].…”