Ambitwistor superstring in the Green–Schwarz formulation
Eur. Phys. J. C
Ambitwistor superstring in the Green-Schwarz formulation
Osvaldo Chandía 1
Brenno Carlini Vallilo 0
0 Departamento de Ciencias Físicas, Universidad Andres Bello , Sazié 2212, Santiago , Chile
1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez , Diagonal Las Torres 2640, Peñalolén, Santiago , Chile
In this paper we construct the ambitwistor superstring in the Green-Schwarz formulation. The model is obtained from the related pure spinor version. We show that the spectrum contains only ten-dimensional supergravity and that kappa symmetry in a curved background implies some of the standard constraints.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . .
2 Flat superspace formulation . . . . . . . . . . . . .
3 Relation to the pure spinor formalism . . . . . . . .
4 Light-cone gauge quantization and spectrum . . . .
5 Curved background . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Perturbative superstring theory determines not only physical
states but also their interactions. These include supergravity
interactions, gauge interactions and a tower of very massive
states in ten dimensions. The presence of this infinite set
of states complicates calculations of scattering amplitudes
but it also provides a control of the quantum aspects of the
theory. One could eliminate the massive states in the limit
α → 0. In this way, the spectrum obtained in
perturbation quantization is the massless states only. The world-sheet
description of these class of strings has been recently
introduced in [
1
]. This new string can be seen as the α → 0
limit of the usual string. Note that they provide the scattering
amplitudes proposed by Cachazo, He and Ye (CHY) [
2–4
]
in gauge and gravity theories. A manifest space-time
supersymmetric extension was constructed in [
5
] using the pure
spinor formalism where the CHY scattering amplitudes were
generalized to ten-dimensional superspace. Since
supergravity theories are not exact quantum theories, it is expected
that world-sheet loops contributions to scattering amplitudes
should modify them. In our case, the modification should be
done on type II supergravity. A one-loop analysis was done
in [
6
] for the NS–NS part of the spectrum. It would be
interesting to determine the RR contributions by using our pure
spinor formulation.
A natural next step is to consider the string moving in a
curved space. In the RNS case it was done in [
7
] where the
superdiffemorphism algebra is satisfied if the background
NS–NS fields satisfy the supergravity equations of motion
in ten dimensions. Their result is exact for all orders in
world-sheet perturbation theory. The RR contributions can
be naturally obtained using the pure spinor formalism. The
coupling to a curved background was studied in [
8
], where
the BRST invariance of the particle-like hamiltonian of the
system helps to put the background superfields on-shell. It
was noted that the system has an additional symmetry whose
BRST invariance implies the so-called nilpotency constraints
in [
9
] involving H = d B, where B is the super two-form of
type II supergravity in ten dimensions. It is natural to consider
the generator of this symmetry as part of the BRST charge.
This was done in [
10
] where the nilpotency of the new BRST
charge and the BRST invariance of the particle-like
hamiltonian determine the type II supergravity constraints in ten
dimensions of [
9
]. In the present paper, we obtain the Green–
Schwarz superstring for the pure spinor string of [
10
].
This paper is organized as follows. In Sect. 2 we define
the GS ambitwistor superstring in flat superspace. We show
that the action is invariant under space-time
supersymmetry transformations as well as under kappa symmetry. In
this case, the necessary Virasoro-like constraints are
determined. In Sect. 3, we relate the GS superstring to the pure
spinor ambitwistor string in flat space. The relation allows
one to obtain the particle-like hamiltonian of the pure spinor
ambitwistor superstring from the Virasoro-like constraints
of the GS ambitwistor superstring. In Sect. 4 the spectrum of
the GS ambitwistor superstring is computed using the
lightcone gauge quantization. The spectrum is described by the
linearization of the type II supergravity fields, as expected.
It would be interesting to apply the well known light-cone
methods to compute scattering amplitudes in this string and
see if they match with the ones obtained from the Mason–
Skinner string and the pure spinor version. We conclude in
Sect. 5 by studying the GS action in a curved superspace. The
action turns out to be invariant under kappa symmetries if the
constraints of type II supergravity are imposed.
2 Flat superspace formulation
We construct an action for the ambitwistor in Green–Schwarz
formalism. We find the κ symmetry of the theory and the
Virasoro-like constraints. (...truncated)