Abstract
We revisit the leading irrelevant deformation of mathcal{N} = 4 Super Yang-Mills theory that preserves sixteen supercharges. We consider the deformed theory on S3× ℝ. We are able to write a closed form expression of the classical action thanks to a formalism that realizes eight supercharges off shell. We then investigate integrability of the spectral problem, by studying the spin-chain Hamiltonian in planar perturbation theory. While there are some structural indications that a suitably defined deformation might preserve integrability, we are unable to settle this question by our two-loop calculation; indeed up to this order we recover the integrable Hamiltonian of undeformed mathcal{N} = 4 SYM due to accidental symmetry enhancement. We also comment on the holographic interpretation of the theory.
Highlights
It has been a long-standing speculation that the canonical AdS/CFT duality might extend beyond the low energy/near horizon limit — that the full D3 brane effective field theory might be dual to closed string theory in the full asymptotically flat D3 brane geometry
We have shown that the su(2|2) R symmetry uniquely fixes the form of the planar time-translation generator to be the same as the integrable planar N = 4 SYM dilatation operator up to two loops, apart from the definition of the coupling constant
Thanks to an off-shell formalism well suited for the symmetries of our problem, we have managed to write down the full deformed classical action in closed form
Summary
Before analyzing the deformed action of (2.16), let us provide a complementary point of view on its construction. It is clear that it is given by a,α (σ0)αα S+a αa ̇ ,α (σ0)αα S+αa Evaluating this expression is straightforward: the action of the Poincaré/special conformal supercharges will move us up/down in the 105 supermultiplet. Recalling that the adjoint representation of SU(4) decomposes as 15 → (3, 1)0 ⊕ (1, 3)0 ⊕ (2, 2)+2 ⊕ (2, 2)−2 under the subgroup SU(2)a × SU(2)a × U(1)J , it is clear that the nonzero anticommutators of the Poincaré and special conformal supercharges appearing in (2.18) result in R-symmetry generators in the (2, 2)+2. The special conformal variation of O7 is non-zero, demanding the addition of O6 and so forth This process keeps going until we reach an operator that is annihilated by all special conformal supercharges appearing in the flat-space limit of the supercharges. Convince oneself that it is not annihilated by all S appearing in the flat-space limit of the supercharges.
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