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Résumé |
Between all the magical properties of $\mathcal{N} = 4$ SU(N) super Yang-Mills
perhaps one of the most important is Montonen-Olive electric-magnetic $SL(2,Z)$
duality.In particular this leads to the constraint that observables must be
invariant under inversion of the complex YM coupling $\tau$, i.e. under
$\tau -> -1 / \tau$.
In this talk we will focus on one such physical quantity, namely an integrated
correlator of four super-conformal primaries of the stress-tensor multiplet.
I will firstly review how this correlator can be computed via supersymmetric
localisation on $S^4$, and then discuss how this quantity can be rewritten in a
manifestly $SL(2,Z)$ invariant way for any number of colours N, and any value of
the complex YM coupling \tau. Thanks to this novel expression we can explore
various different regimes: perturbative SYM, large-N supergravity approximation,
large-N 't Hooft expansion. All of these regimes are connected via a remarkable
Laplace-difference equation relating the SU(N) to the SU(N + 1) and SU(N −
1)
correlators. |
