15 problems

For positive integers $N$, $K$, and $r$ with $r\mid N$, consider the three-dimensional $(\operatorname{SU}(N)/\mathbb Z_r)^0_K$ theory and its naive Witten index … Here … where…

The charges of a single-centered $\frac{1}{4}$-BPS black hole are encoded by integers $(m,n,\ell)$, and let $d_*(m,n,\ell)$ denote the immortal dyon index, namely the indexed degen…

Let $N,M\in\mathbb{Z}_{>0}$, let $k=-N-M+\frac12$, and consider the three-dimensional $\mathcal{N}=2$ $SU(N)$ theory with one fundamental Neumann chiral and $M$ fundamental Fermi m…

The $SU(N)_{-N-M}$ half-index conjecture. This integral equals

Let $N\in\mathbb{Z}$, and let $\langle W_1\rangle^{SU(N)_{-N-1}}(x;q)$ denote the one-point function of the fundamental Wilson line in the three-dimensional $\mathcal{N}=2$ $SU(N)$…

Let $N\in\mathbb{Z}$, and consider the three-dimensional $\mathcal{N}=2$ $SU(N)$ pure Chern Simons theory at level $-N-1$. Let $\mathbb{II}_{\mathcal{N}}^{SU(N)_{-N-1}}(x;q)$ denot…

Let $N\in\mathbb{Z}$, and consider the three-dimensional $\mathcal{N}=2$ $SU(N)$ Chern Simons theory at level $-N-\frac12$ with the boundary conditions described in the source. Let…

Let $C_0$ denote the coefficient appearing in the large-genus expansion of the index for a genus $g$ compactification with no flux and no punctures, so that … The index counts, wit…

Let $I_{3,m}$ be an irregular singularity with $\gcd(3,m)=0$. Write $m=3j+k$ with $k=1,2$, let $q$ and $t$ be the index parameters, and let $\delta_{k,1}$ be the Kronecker delta. C…

Let $I_{2,2m}$ be an irregular singularity, let $\lambda$ label a representation, let $a$ be a flavor fugacity, and let $t$ be the Macdonald parameter. Song's scaling conjecture. T…

Let $I_{2,2m-1}$ be an irregular singularity with $m\geq 1$, let $\lambda$ label the wavefunction, and let $t$ be the Macdonald parameter. Song's conjecture. The wavefunction is ……

Let $\mathcal{I}(p,q,y)$ be a superconformal index and suppose that the corresponding partition function can be written in the form of Eq., with the transformed index and exponenti…

Let $\mathbf{z}_{E_8}$ be an $E_8$ fugacity, and let $[k,0,0,0,0,0,0,0]_{\mathbf z}$ denote the character of the irreducible $E_8$ representation with Dynkin labels…

The $T_3$ index conjecture. The index of $T_3$ is

For a generalized quiver, let $f^\alpha(\mathbf{a})$ be functions of the flavor fugacities that diagonalize the structure constants of the associated index. Let…