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Abstract. First, we establish the relation between the associated varieties of modules over Kac-Moody algebras g and those over affine W -algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated varieties of G-integrable admissible representations are irreducible Ad G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C 2 -cofiniteness of a large number of simple W -algebras, including all the minimal series principal W -algebras [FKW] and the exceptional W -algebras recently discovered by .

Interacting quantum many-body systems constitute a fascinating research field because they form quantum liquids with remarkable properties and universal behaviour The idea is that they behave as an ensemble of non-interacting 'quasi-particles'. However, non-equilibrium properties have still to be established and remain a key issue of many-body physics. Here, we show a precise experimental demonstration of Landau Fermi liquid theory extended to the non-equilibrium regime in a zero-dimensional system. Combining transport and ultra-sensitive current noise measurements, we have unambiguously identified the SU(2) (ref. The Kondo effect19 is a typical example of a quantum manybody effect, where a localized spin is screened by the surrounding conduction electrons at low temperature to form a unique correlated ground state. The Kondo state is described well by the Fermi liquid theory at equilibrium 1,9 , which makes it an ideal testbed to go beyond equilibrium. To unveil the universal behaviour of nonequilibrium Fermi liquid 11 , we have used the current fluctuations or shot noise in a Kondo-correlated nanotube quantum dot 18 . When electrons are transmitted through this system, the scattering induces the shot noise, which sensitively reflects the nature of the quasi-particles 20 , as shown in the upper panel of Fig. 1a. A remarkable prediction of the non-equilibrium Fermi liquid theory is that the residual interaction between quasi-particles creates an additional scattering of two quasi-particles which enhances the noise (see the lower panel of Fig. 1a) 10,12-15 . This two-particle scattering is characterized by an effective charge e * larger than e (electron charge). This value, closely related to the Wilson ratio, is universal for the Fermi liquid in the Kondo regime as it depends only on the symmetry group of the system [13][14][15] . Although some aspects of Kondo-associated noise have been reported 8,16,17 , a rigorous, selfconsistent treatment in a regime where universal results apply is at the core of the present work. Actually, by investigating the same nanotube quantum dot in the spin degenerate SU(2) Kondo regime and in the spin-orbit degenerate SU(4) regime, the noise is proved to contain distinct signatures of these two symmetries, confirming theoretical developments of Fermi liquid theory out of equilibrium.In our experiment, we measured the conductance and current noise through a carbon nanotube quantum dot grown by chemical vapour deposition 21 . Iron catalyst was deposited on an oxidized undoped silicon wafer and exposed to 10 mbar of acetylene for 9 s at 900• C. The nanotube was connected with a Pd(6 nm)/Al(70 nm) bilayer deposited by e-gun evaporation. The distance between the contacts is 400 nm and a side gate electrode is deposited to tune the potential of the quantum dot (see Fig. 1b). A magnetic field of 0.08 T is applied to suppress superconductivity of the contacts. To measure accurately the shot noise, our sample is connected to a resonant (2.58 MHz) LC circuit thermalized at the mixing chambe...

We study the representation theory of the W -algebra W k (ḡ) associated with a simple Lie algebra ḡ at level k. We show that the "−" reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k ∈ C. Moreover, we show that the character of each irreducible highest weight representation of W k (ḡ) is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra g of ḡ. As a consequence we complete (for the "−" reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of W -algebras.