# N=4 SYM N.b. we are working in 4d. ## Action - schematically - $\mathcal{L}=\frac{1}{g^{2}} \operatorname{Tr}\left[-F_{\mu \nu} F^{\mu \nu}-D_{\mu} \phi_{I} D^{\mu} \phi_{I}+\bar{\lambda}^{i} \not D \lambda_{i}+\sum_{I, J}\left[\phi_{I}, \phi_{J}\right]^{2}+\bar{\lambda}^{i} \Gamma^{I} \phi_{I} \lambda_{i}+\theta \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma}\right]$ - schematically in $\mathcal{N}=1$ notation - $\mathcal{L}=\int d^{4} \theta \sum_{i} \bar{\Phi}_{i} e^{V} \Phi_{i}+\int d^{2} \theta \operatorname{Tr}\left[\tau \mathcal{W}^{\alpha} \mathcal{W}_{\alpha}+\Phi_{1}\left[\Phi_{2}, \Phi_{3}\right]\right]+\mathrm{h.c.}$ - where $\tau=\frac{\theta}{2 \pi}+i \frac{4 \pi}{g^{2}}$ is a complex gauge coupling ## Fields - $V_{\mathcal{N}=4}=\left(A_{\mu}, \lambda_{i=1, \ldots, 4}, \phi_{I=1, \ldots, 6}\right)$ - gauge fields - 4 fermions = fundamental of $SU(4)$ = Weyl spinor of $SO(6)$ - 6 scalar fields = fundamental of $SO(6)$ = antisymmetric of $SU(4)$ - in terms of $\mathcal{N}=1$ notation: $V_{\mathcal{N}=4}=V_{\mathcal{N}=1}$ plus three chiral $\Phi_{1,2,3}$ in the adjoint ## Remarks - the Lagrangian completely determined by SUSY - $\mathcal{N}=4$ is the maximum one can have in 4d flat space ## Symmetry - conformal SO(4,2) <-> isometry of AdS${}_5$ - global SO(6) (internal symmetry of 6 scalar fields) <-> isometry of $S^5$ - $SU(4)\cong SO(6)$, $SU(4)$ is the **R-symmetry** - SUSY 4 + 4 (conformal sym does not commute with the 4 SUSY generators so extra 4) Weyl spinors (two complex components) so 32 real charges <-> same number of large *local* SUSY in this string theory ## More symmetries - $\beta$ function for $g$ is 0 for all loop orders and non-perturbatively -> SCFT with superconformal group ==$SU(2,2|4)$== - **S-duality**: - EM duality: $\tau\rightarrow -\frac{1}{\tau}$ - shift of $\theta$ angle: $\tau\rightarrow\tau+1$ - => overall S-duality = $SL(2,\mathbb{Z})$