A derivation of an off-shell 2D, supergravity chiral projection operator

S James GatesJr and Akin Morrison

Center for String and Particle Theory Department of Physics, University of Maryland College Park, MD 20742-4111, USA

E-mail: gatess@wam.umd.edu

Received 28 August 2009
Published 8 October 2009

Abstract. Utilizing the known off-shell formulation of 2D,  =  (2, 2) supergravity containing a finite number of auxiliary fields, there is shown to exist a simple form for a `chiral projection operator' and an explicit expression for it is given. PACS number: 04.65. + e

1. Introduction

Some years ago, the first discussion of a 2D,  =  (2, 2) supergravity theory was given [1] and elaborated upon in a subsequent work [2]. In both of these works, no discussion of the auxiliary fields required to close the local supersymmetry algebra without resorting to a set of equations of motion was undertaken. Finally, 13 years after the first such discussion, a formulation of 2D,  =  4 supergravity theory that included a finite number of auxiliary fields was presented [3]. The relation between these on-shell versus off-shell supergravity theories is not as direct as one might imagine. The reason for this is the existence of a great plethora of 2D,  =  4 scalar multiplets [4]. Previous experience has shown that when this situation exists, the diverse scalar multiplets foreshadow distinct formulations of correspondingly diverse off-shell supergravity theories.

The existence of a relatively simple off-shell formulation of a 2D,  =  4 supergravity theory implies that there should exist a straightforward way to completely develop an efficient local integration theory for the associated local Salam–Strathdee superspace. In this work, we will take the first major step in this direction by providing the initial discussion of a local 2D,  =  4 chiral projection operator.

2. A review of an off-shell 2D, superspace supergravity geometry

Let us begin by reviewing the results in [3]. This work showed that there exist component fields (e_a^m, ψ_a^αi, A_ai^j, B, G, H) which describe an off-shell 2D, supergravity theory. These are the components that remain after imposing the following constraints on the 2D, N  =  4 superspace supergravity covariant derivative, (with _αβ ≡ –i[C_αβ G  +  i(γ³)_αβ H])

The consistency of the Bianchi identities constructed from the commutator algebra above required the conditions,

The component gauge fields occur in the above supertensors in the following manner.

where is the usual two-dimensional curvature in terms of e_a^m and .

3. 2D, local superspace integration and chiral projector

In a rigid superspace (i.e. in a `flat supergravity background'), the derivation of component results follows most efficiently [5] from replacing the integration of fermionic coordinates by a process using first application of the superspace covariant derivative followed by taking a limit in which all Grassmann coordinates are taken to zero. For the discussion of this communication, this amounts to the validity of the following equation,

In the presence of a supergravity background the integration over the superspace measure is modified by the insertion of the supergravity vielbein

and this in turn implies a modification of (5) to the form [6]

which is written in terms of two differential operators, and . The first of these is called the `density projection operator' and the second is called `chiral projection operator'. The appearance of these two distinct operators is characteristic of any superspace in which chirality has a well defined meaning.

As outlined in [6], the density projection operator must be of the form

in terms of some field-dependent coefficients b_(4–i) and powers of the spinorial superspace supergravity covariant derivative ∇_αi. In the work of [5], a `handicraft' method for finding these coefficients was described. The works of [6] describe more powerful methods for deriving this operator (which will be a topic of future efforts). In a similar manner, the chiral projection operator must be of the form

in terms of some field-dependent coefficients a_(4–i) (that are distinct from the b_(4–i) coefficients) and powers of the spinorial superspace supergravity covariant derivative . Since the two sets of coefficients are distinct it follows that is not the conjugate of .

Here we give our main result for the local 2D,  =  (2, 2) superspace described by (1)

where on the first line E^–1 is the density factor (i.e. the superdeterminant of the vielbein of the full 2D,  =  (2, 2) superspace) and on the second line of this equation is the chiral density factor. The superdifferential operator

is the 2D,  =  (2, 2) chiral projection operator. The main new result we have to report is its explicit form which satisfies

for any general scalar superfield Ψ.

Upon comparing (8) with (10), we note that the coefficients for the former can be read from the latter and imply that

So that explicitly we have

Even without knowing the explicit form of the density projection operator, (9) implies

i.e. the superspace described by (1) has a vanishing supervolume. The derivation of these results is described in the appendix.

4. Conclusion

With this present work, we have completed half of the task of developing an efficient local superspace integration theory for two-dimensional theories that possess eight supercharges. The crux of this presentation was the unveiling of the explicit form of the chiral projection operator given in (11).

Acknowledgments

This research was supported in part by the endowment of the John S Toll Professorship, the University of Maryland Center for String & Particle Theory, National Science Foundation Grant PHY-0354401. AM wishes to acknowledge the hospitality of the University of Maryland and in particular of the Center for String and Particle Theory. Additionally he wishes to recognize his participation in the 2008 SSTPRS (Student Summer Theoretical Physics Research Session).

Appendix A. Definitions and conventions

For two-dimensional superspaces, we use the following conventions for the quantities associated with spinors.

The last one of these relations imply

Some useful Fierz identities are as follows:

In terms of an explicit representation, we can define the 2D γ-matrices in terms of the usual Pauli matrices according to

As can be seen, these satisfy the second line in (A.1). The spinor metric C_αβ and its inverse C^αβ can be identified as

Using this explicit representation, it is easy to show the following symmetry properties

In a similar manner the following complex conjugation properties can be derived

Due to the first relation in (A.8) we see that this choice of gamma matrices is in a Majorana representation and thus the simplest spinors such as ψ^α(x), may be chosen to be real, i.e.

and we can raise and lower spinor indices according to

It can be seen as a consequence that

Of course, it is always possible to introduce complex spinors also.

The two generators that define the holonomy group of the 2D,  =  (2, 2) superspace are and , respectively for the 2D Lorentz group (SL(2,R)) and an internal SU(2) group. These are defined to act as

The rules for manipulating the SU(2) spinors are very much similar to the ones used for the SL(2,R) spinor indices. The SU(2) metric C_ij and its inverse C^ij can be identified as

so that

We raise and lower SU(2) indices according to

that are directly the analogs for raising and lowering indices on SL(2,R) tensors.

Appendix B. Derivation of the chiral projector

In this appendix, we give a presentation that leads to the form of the chiral projector.

The expressions in (1) give the commutator algebra of the 4D, (2, 2) supergravity derivative. However, in order to derive the chiral projection formula we need to go beyond that algebra. For this purpose, we introduce new second-order differential operators denoted by ∇⁽²⁾_αβ and ∇⁽²⁾_ij and defined by the equations

Using these definitions implies

Next we use (1), (B.1) and (B.2) to derive

Using more manipulations there are found additional identities involving ∇⁽²⁾_αβ.

With the identities of (B.4)–(B.6) in hand, the proof of (11) follows from the steps described below.

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