The protected chiral algebra of the T_n SCFT for n at least 3

Let $T_n$ be the four-dimensional SCFT associated to a three-punctured sphere of type $A_{n-1}$, and let its protected chiral algebra be the two-dimensional chiral algebra assigned by the four-dimensional chiral algebra construction. Write $\bigwedge^{\ell}$ for the $\ell$th exterior-power representation of $\mathfrak{su}(n)$.

The T_n chiral algebra conjecture. The protected chiral algebra of the $T_n$ SCFT for any $n\geqslant3$ is a $\mathcal{W}$-algebra with the following generators: three sets of $\mathfrak{su}(n)^3$ affine currents at critical level $k=-n$; one current of dimension $\frac12\ell(n-\ell)$ transforming in $(\bigwedge^{\ell},\bigwedge^{\ell},\bigwedge^{\ell})$ for each $\ell=1,\ldots,n-1$; and $\mathfrak{su}(n)^3$-singlet operators $W_i$, $i=1,\ldots,n-1$, of dimension $i+1$. The dimension-two operator is a stress tensor, $W_1(z)\equiv T(z)$, with Virasoro central charge

$$c_{2d}=-2n^3+3n^2+n-2.$$

In special cases some of these operators may be redundant.

For $n=3$, the stated characterization is realized by the $\mathfrak{e}_6$ current algebra at the appropriate level; uniqueness is under investigation for $T_4$, while the general existence and structure of the proposed $\mathcal{W}$-algebras remain open.