31 problems

AGT conjecture. One has

Let $\Phi^{\vec v}_{\vec u}(w):\mathcal{F}_{\vec u}\to\mathcal{F}_{\vec v}$ be the normalized intertwining operator, and let $|K_{\vec\mu}\rangle$ and $\langle K_{\vec\lambda}|$ be…

Let $q$ and $t$ be indeterminates, let $M(h)$ be the Verma module of the deformed Virasoro algebra, and let $w$ be its Whittaker vector. Denote by $\langle w,w\rangle$ its norm wit…

Alday–Tachikawa conjecture. The “fully-ramified” Nekrasov instanton partition function should be related to Hitchin's integrable system.

Five-dimensional AGT conjecture. The $K$-theoretic Nekrasov partition function coincides with the norm of the Gaiotto state:

Let $\mathcal{C}$ be the curve underlying the class $S$ construction, and consider an (asymptotically) conformal four-dimensional $\mathcal{N}=2$ $SU(r)$ gauge theory on the…

Let $\mathcal{W}_{M|N\times\infty}$ denote the relevant rectangular $\mathcal{W}$-algebra, let $\mathtt{T}$ be a toroidal BPS algebra associated with a generalized conifold, and le…

Let $M_i$ be representations of $\mathcal{W}^{\kappa_i}(\mathfrak{gl}_{N_i},f_i)$ for $i=1,2$, and let $\Phi_l$ be the map from the affine Yangian associated with the parabolic-ind…

The AGT correspondence relates Nekrasov partition functions of supersymmetric gauge theories to conformal blocks of two-dimensional conformal field theories with infinite-dimension…

Let $cmathcal{U}$ be the DingIoharaMiki algebra, let $cmathcal{A}(N)$ be the algebra generated by the currents $X^{(i)}_n$, and let $Ket{X_{\vec{\lambda}}}$ be the PBW-type vectors…

The Ding–Iohara–Miki algebra is the symmetry algebra of the network matrix models, a class that includes Dotsenko–Fateev and gauge-theory matrix integrals and admits a topological-…

AGT limit conjecture. The limit of $\U^{(1)}_{q_1,q_2,q_3}$ as $q_1\rightarrow \omega_n$ and $q_2\rightarrow 1$ should be related to the affine Yangian of $\mathfrak{sl}_n$. This c…

Let $\widetilde\Phi(z):\mathcal F_{\vec u}\to\mathcal F_{\vec v}$ be the crystal-limit vertex operator, with $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2)$, and let…

Let $N$ be a positive integer, let $\vec{u}=(u_1,\ldots,u_N)$, and let $\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$ be an $N$-tuple of partitions. Write…

Let $r$ and $p$ be parameters, let $U(r)$ denote the gauge group, and let $\mathcal{A}(r,p)$ be the chiral algebra associated with the corresponding coset model. The quotient…

Let $\mathcal H(q,t)$ be the deformed Heisenberg algebra generated by $a_n$ for $n\in\mathbb Z\setminus\{0\}$, with … Let a level-$r$ representation of the Ding–Iohara algebra be g…

AGT coset conjecture. Conformal blocks of the coset construction $\mathcal A(r,p)$ correspond to the instanton partition function of the $U(r)$ gauge theory on…

AGT root-of-unity conjecture. The AGT correspondence between the $U(r)$ instanton counting on $\mathbb R^4/\mathbb Z_p$ and the two-dimensional field theories with conformal symmet…

Torus conformal-block prefactor conjecture.

The W-algebra symmetry conjecture. The two-dimensional symmetry associated with the surface operator is $W(\widehat{\mathfrak{sl}}(N),\vec{N})$.

The AGT correspondence for gauge theories on ALE spaces involves the moduli spaces of framed torsion-free sheaves and a torus action whose equivariant parameters determine a parame…

Let $G$ be an ADE group, let $m$ be a positive integer, and consider the AGT correspondence on $\mathbb R^4/\mathbb Z_m$. On the two-dimensional side, consider the $m$-th para-…

Let $G$ be an arbitrary gauge group, consider the pure four-dimensional theory with a surface operator of arbitrary type, and let the two-dimensional conformal field theory side ca…

In the relevant orbifold vortex and super Liouville correspondence, let $q_1^{(2)}$ and $q_2^{(2)}$ denote the two instanton charges associated with the second configuration. Parit…

The partition function of certain $\mathcal{N}=2$ gauge theories is related, after an appropriate translation of parameters, to correlation functions in two-dimensional Liouville o…