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C-deformation of Supergravity

A four-dimensional supergravity toy model in an arbitrary self-dual gravi-photon background is constructed in Euclidean space, by freezing out the gravi-photon field strength in the standard N=(1,1) extended supergravity with two non-chiral gravitini. Our model has local N=(1/2,0) supersymmetry. Consistency of the model requires the background gravi-photon field strength to be equal to the self-dual (bilinear) anti-chiral gravitino condensate.


Introduction
As was shown by Ooguri and Vafa in ref. [1], the superworldvolume of a supersymmetric D-brane in a constant Ramond-Ramond type flux gives rise to the remarkable new structure in the corresponding superspace, which is now called Non-AntiCommutativity (NAC). The non-anticommutativity means that the fermionic superspace coordinates are no longer Grassmann (i.e. they no longer anti-commute), but satisfy a Clifford algebra. In other words, the impact of the RR flux on the D-brane dynamics can be simply described by the non-anticommutativity in the Dbrane superworldvolume. In its turn, the non-anticommutativity in superspace can be easily described by the (Moyal-Weyl type) non-anticommutative star product among superfields, which gives rise to the NAC deformed supersymmetric field theories with partially broken supersymmetry [2,3,4]. When gluino background is added, one can deform the anticommutation relation of the spinors in the D-brane worldvolume in order to recover full supersymmetry [1].
As regards a D3-brane with its four-dimensional worldvolume, a ten-dimensional (self-dual) five-form flux upon compactification to four dimensions gives rise to the (self-dual) gravi-photon flux [1]. All recent studies of the NAC supersymmetric field theories after the pioneering papers [2,3,4] were limited to rigid supersymmetry, i.e. without gravity. In this paper we would like to investigate the impact of a self-dual gravi-photon flux on supergravity.
The simplest supergravity model with a gravi-photon is the pure N = (1, 1) (or N = 2 in the Lorentz case) supergravity unifying gravity with electromagnetism. Therefore, the easiest thing to do is to 'freeze out' the gravi-photon field in that supergravity model to some self-dual value of its field strength. Of course, such a condition would break N = (1, 1) supersymmetry, so we should like to investigate the residual supersymmetry, if any, and then cut the theory properly, in our search for a supergravity model with lower local supersymmetry but with a non-vanishing self-dual gravi-photon background. It is the task that we pursue in this paper. The Euclidean signature appears to be crucial here, similarly to the rigid NAC supersymmetric field theory [3,4]. We use the component formulation of supergravity [5,6], as regards rigid N=1/2 supersymmetric field theories in superspace, see e.g., refs. [2,3,4,7].
Our paper is organized as follows. In sect. 2 we briefly introduce our notation. In sect. 3 we formulate the standard (Euclidean) four-dimensional N = (1, 1) supergravity in our notation. Sect. 4 is devoted to our results. Our conclusion is sect. 5.
Raising and lowering of spinor indices is performed with the help of two-dimensional Levi-Civita symbols, while we have Chiral and anti-chiral (with bars) spinors are independent in Euclidean space with the signature (4, 0), as well as in Atiyah-Ward space with the signature (2, 2) [3,9]. 4 As a rule, we omit contracted spinor indices for simplicity in our equations, by using the notation As regards, the (anti)self-dual parts of an antisymmetric tensor K µν , we define The sigma-matrices in our notation are given by while their triple (totally antisymmetric) product is given by where σ are Pauli matrices and I is unit matrix.
Flat and curved vector indices are related by a vierbein e a µ and its inverse e µ a , as usual, e.g., σ µ = e a µ σ a and σ a = e µ a σ µ , etc.

N = (1, 1) supergravity
Our starting point is the pure N = (1, 1) extended supergravity in four Euclidean dimensions. The N = (1, 1) supergravity multiplet unifies a graviton field e a µ , two non-chiral gravitino fields ψ i µ , and a gravi-photon gauge field A µ . The N = (1, 1) supergravity action was first constructed in ref. [6] by Noether procedure. When using our notation with chiral and antichiral gravitini, and making the O(2) internal symmetry manifest, the N = (1, 1) supergravity Lagrangian of ref. [6] reads where we have used the standard definitions [5], of the gravi-photon field strength F µν and its supercovariant extensionF µν . The dimensional parameter κ is the gravitational coupling constant.
The spinor covariant derivative D µ (ω) ≡ D µ contains the independent spin connection ω ab µ that is supposed to be fixed as a function of the vierbein and gravitini by solving its algebraic equation of motion, δS/δω = 0 (it is known as the 1.5 order or Palatini formalism), as usual [5].
By construction, the action S = d 4 x L of eq. (7) is invariant under the following transformation rules of local N = (1, 1) supersymmetry: where ǫ i andǭ i stand for the infinitesimal chiral and anti-chiral anticommuting spinor parameters of local N = (1, 1) supersymmetry, respectively.

C-deformation
We now impose the condition whereF µν is the covariantized gravi-photon field strength (8), and C µν (x) = C + µν (x) is an arbitrary given function. We have chosenF µν instead of F µν in eq. (11) becauseF µν is a tensor under the local supersymmetry (sect. 3), though it turns to be unimportant (see the end of this section). Also, we do not require the background C µν (x) to be constant, because it turns out to be unessential too (cf. a coordinate-dependent deformation of rigid N = 1/2 supersymmetry in superspace [8]).
Of course, the condition (11) is not compatible with the full N = (1, 1) local supersymmetry (9). The consistency condition has, however, the residual N = (1/2, 0) local supersymmetry with the infinitesimal spinor parameter ǫ 1 (x), when choosing Note that then δψ 2 µ = 0 is automatically satisfied, while the Lagrangian (7) takes the form The fieldψ 2 γḂ enters the Lagrangian (14) as a Lagrange multiplier, so that it gives rise to a constraint (after varying the action with respect to that field), whose solution is In the simplified notation we thus arrive at the Lagrangian where By our construction, the supergravity action S = d 4 x L with the Lagrangian (18) is invariant under the N = 1/2 local supersymmetry with the transformation laws Note that eqs. (8) and (16) also imply The result (21) may prompt us to choose another constraint, from the very beginning of this section, instead of eq. (11). Then its consistency with local supersymmetry, under the transformations (9) again has a solution (13), with an arbitrary infinitesimal parameter ǫ 1 (x). The Lagrangian (7) then takes the form The algebraic equation of motion of the fieldψ 2 γḂ , now has a solution When using the notation (17), we arrive at the N = 1/2 supergravity Lagrangian subject to the constraint which are both invariant under the same N = 1/2 local supersymmetry transformations (20). In this case we haveF instead of eq. (21).

Conclusion
The new model we constructed is, of course, a toy model with local N = 1/2 supersymmetry. Nevertheless, we found that an N = 1/2 supergravity is possible, while it can be very simple, like the C-deformed N = 1/2 supersymmetric gauge theory constructed in the NAC-deformed superspace [4]. Perhaps, the most remarkable feature of our construction is the very simple relation it implies between the expectation value of C + µν and that of the self-dual product of two anti-chiral gravitini -see eqs. (19) and (28). Another approach to a construction of a C-deformed N = 1/2 supergravity in four Euclidean dimensions can be based on the NAC deformed N = (1/2, 1/2) superspace, by imposing the NAC relation on the chiral superspace coordinates, {θ A , θ B } * = C AB , with C µν = C AB ǫ BC (σ µν ) A C . When using the Moyal-Weyl-type star product ( * ) for the supergravity superfields [4] and applying the explicit superfield star-product summation formulae of ref. [7], it may be possible to get another C-deformed N = 1/2 supergravity action in components (in a Wess-Zumino gauge). However, the last procedure seems to be much more complicated [10], and we do not claim that the result is going to be equivalent to our model constructed above.
A non-vanishing gravitino condensate is the well known tool for spontaneous breaking of local supersymmetry, which may lead to a natural solution to the hierarchy problem in elementary particles physics (see e.g., the pioneering paper [11] for the dynamical mechanism due to gravitational instantons in quantum gravity, and ref. [12] for its possible realization in superstring theory). Because of the relations (19) and (28), the C-deformation may be the viable alternative mechanism of spontaneous local supersymmetry breaking. To explore its physical consequences, one has to go beyond our toy model of C-deformed pure supergravity by adding some matter content [10].