The particle interpretation of N=1 supersymmetric spin foams

Whilst supergravity theories go a certain way to tame the infinities of their non-supersymmetric cousins, they also generalise these theories by coupling fermionic degrees of freedom to the original gravity theory. One can easily see this at the continuum classical level. We want to investigate this issue in the discrete quantum case. Thus, in this paper, we shall analyze a rather simplified model: Riemannian supergravity in three dimensions. As is well known, the conventional metric-dependent action may be recast as particular gauge theory action, which is dependent on the triad and the spin connection: BF theory. Extensive work on the discretisation and quantisation of this theory with SU(2) chosen as the gauge group arises in the literature (see [1, 2] and references therein). One can see that the theory maintains its topological nature once quantised, and the discrete quantum model is the Ponzano-Regge model. Several approaches to coupling matter within spin foams were embarked upon [2–7]. The most tractable and indeed most successful of these procedures embedded the Feynman diagrams of the field theory into the spin foam. Remarkably, summing over the gravitational degrees of freedom, the effective matter amplitude was seen to arise as the Feynman diagram of a non-commutative field theory [8]. To add to this position, it was shown that an explicit 2nd quantised theory of this gravity matter theory could be provided by group field theory, while later the non-commutative field theory was seen to arise as a phase around a classical solution of a related group field theory [9]. Of course, one may approach the subject with the view that one should discretise the field directly on the spin foam, since in the continuum theory, we expect that the field has a non-trivial energy-momentum tensor, and should affect the state sum globally. This method has yielded to a succinct initial quantisation for Yang-Mills and fermionic theories [4–6], but due to the non-topological nature of the resulting amplitudes, further calculations proved unwieldy. Now, it was not our intention that this work would or should settle this debate, but we find that this theory is more in line with the arguments of the former way. The path we follow in our analysis is to start from continuum BF theory with gauge group UOSP(1|2), discretise and quantise. Once this lattice gauge theory quantisation has been completed, we Fourier transform to uosp(1|2) representation space. Owing to its algebraic structure, there is an su(2) structure embedded within uosp(1|2) [10–12]. We may rewrite the amplitudes to make this dependence explicit. The aim of the game is then to give an accurate interpretation of these amplitudes in terms of matter coupled to gravity. To do so, we Fourier transform again, but this time to functions on the group SU(2). In this form, we can identify within the state sum Feynman diagrams of a massless spin12 fermionic field. Therefore, we arrive at