# Can we gauge away curvature in supergravity?

In locally supersymmetric theories we can make a supertransformation which rotates one field into another. With enough supersymmetries could one not make a supertransformation at each point which made the graviton field zero at each point? Thereby turning it into a simpler non-gravitational theory?

i.e. Could we gauge away the gravitational field alltogether?

Could we use this to show renormalizability?

• Can you write down the transformation you're thinking of? – ACuriousMind Oct 11 '18 at 22:12

Can't happen.

You are proposing to gauge away Grassmann "body" degrees of freedom in favor of "soul" ones―nothing doing. Rather than plunging into superfields, worthy of somebody else's answer, if it hasn't already been covered in someone's thesis, let me give you the primitive, thuggish reassurance of mugs in the street.

Canonical fields and local (gauge) supersymmetry aside, the particle content of SG is a 2-state massless spin 3/2 particle, and a 2-state massless spin 2 particle, and that is without redundancy in your description. So, by theorem, the graviton is there, and hence nonlinear couplings and curvature in the QFT you might construct out of them.

Your Dadaist vision implicitly asks "what happened to the SG gauge (redundant) degrees of freedom?" Indeed, being fermionic, $$\partial_\mu \epsilon (x) +...$$, they could only be absorbed by a fermion field $$\psi_\mu$$ to neutralize some of its d.o.f. redundancy; but not by a boson field like the Vierbein, or the spin connection, or the metric, or, certainly, the gravitons inside them; or rather, they cannot suck gravitons into them and hence nothingness.

In practice, this local supersymmetry is crucial in yielding the celebrated gravitino physical degrees of freedom in D dimensions, so $$2^{\lfloor D/2\rfloor -1} (D-3)$$, hence 2 in 4 dimensions. That is, the canonical field $$\psi_\mu$$ starts with 8 (in general, $$2^{\lfloor D/2\rfloor -1} D$$) degrees of freedom (spins (1,0) ⊗ 1/2 = 3/2⊕ 1/2 ⊕1/2), of which 4 are pruned out by removal of spin 1/2 constraints, $$\partial\cdot \psi=0=\gamma\cdot \psi$$ and 2 by pulling out the above gauge $$\partial_\mu \epsilon (x) +...$$, i.e. fixing the gauge and eliminating redundancy of description. Does it remind you of the Lorentz gauge in QED?

In extended SG, there are more gauge parameters, but, ineluctably, an equal number of gravitini, likewise soaking up their respective gauge freedom to produce real particles.

As for finiteness and renormalizability, the ranting gaggles have been attempting magic bullet nonlinear field redefinitions in search of superior survey handles thereof, but if something trenchant has turned up, I have missed it.

• Good answer but does this apply to more complicated supergravities such as N=8. Which would have 8 times as many gravitinos for example? – zooby Oct 12 '18 at 19:45
• Yes, as covered in the penultimate paragraph: 8 gravitini, 8 fermionic parameters to soak up their superflous +- 1/2 spin polarization states! – Cosmas Zachos Oct 12 '18 at 19:48