29 problems

For supersymmetric quantum mechanics with a generic quiver $\mathfrak{Q}$ as target space, let $Y(\mathfrak{Q})$ denote the respective quiver Yangian algebra. Quiver Yangian conjec…

Let an ${\cal N}=2$ supersymmetric quantum mechanical system be a quantum mechanical model with two supersymmetries. A model is tractable for Hodge theory when its physical symmetr…

GAL partner-potential conjecture. The following relationships hold:

Finite-band-gap conjecture. Every GAL potential whose four parameters $a,b,f,g$ are integers has a finite number of band gaps.

Let $L$ be an $N$th-order mild differential transformation, where “mild” means that it leaves most of the original spectrum invariant, with at most finitely many spectral points ch…

Let a potential and its eigenfunctions satisfy shape invariance, meaning that the corresponding Darboux-transformed eigenfunctions and potentials occur in pairwise shape-invariant…

Let $a$ be an integer, and consider the periodic potential $[(a-1)a,2,(a-2)(a-1),2]$. A potential is finite-gap if it has only finitely many spectral band gaps. Finite-gap conjectu…

Shape-invariance necessity conjecture. Shape invariance is perhaps a necessary condition for the lowest-order SWKB quantization condition to reproduce the exact bound-state spectru…

Let $M$ be the moduli space of instantons, let $S(TM)$ denote its chiral spin bundle, and let ${\cal L}^{(I)}$ be a line-bundle object over $M$ for which only…

Let the SWKB condition be the supersymmetric WKB quantization condition for a potential, and let a conventional shape-invariant (SI) potential be one for which this condition is ex…

Let $I_{\mathrm{SWKB}}$ be the leading-order supersymmetric WKB quantization integral for a quantum system, and let the energy eigenvalues describe its level structure. Nasuda's co…

Let $V(x)$ be a supersymmetric quantum-mechanical potential with groundstate eigenfunction $\phi_0(x)$, and let the SWKB quantization condition be … where $a$ and $b$ are the roots…

Consider the mixed vacuum branches obtained by solving the mass, F-term, and D-term equations of the bubbling supersymmetric quantum mechanics, for FI parameters with either…

Consider the bubbling supersymmetric quantum mechanics for the $SU(2)$ ${\rm N}_f=4$ theory, including its mixed vacuum branches specified by the mass, F-term, and D-term equations…

Let $Z_{mono}(P,\mathrm{v})$ denote the monopole-bubbling contribution associated with magnetic charge $P$ and gauge fugacity $\mathrm{v}$, and let $\beta$ be the inverse-temperatu…

Zero-mode conjecture. Only one linear combination of $\psi_0^B$ and $\psi_0^F$ is a zero mode of $A$ for each choice of self-adjoint extension of $Q$.

Let $G=U(1)^\alpha$ be the total gauge group of the quiver with a decoupled overall $U(1)$, so that $\widetilde\alpha=\alpha-1$. Let $\Sigma^F$ be the fundamental domain of the mod…

Consider an $\mathcal N=4$ quiver with total gauge group $G\cong U(1)^\alpha$ and a decoupled overall $U(1)$ factor. Let $J$ and $K$ be the index sets of thimble contributions, let…

Let $G$ be a quiver gauge group of rank $\text{rank}(G)=\alpha$, and let $\widetilde{F}_{\infty}$ and $\widetilde{F}_{0}$ be the real-part restrictions of the asymptotic moduli-spa…

The refined Witten index is a function of the fugacity $y$, and the moduli space of the quiver theory has singularities whose crossings determine intersection coefficients. Interse…

First-excited-level conjecture. For large $n$, the energy $\Lambda_n^{(-)}$ is the first excited level. The calculation establishes this value as an upper bound for the first excit…

Let $M$ and $N$ be the ranks of the two vertices of the $k$-arrow Kronecker quiver, and let $\Omega(M,N,k)$ denote its ground-state degeneracy. Fix $k$, a limiting ratio $N/M\to r$…

Single-intersection conjecture. For general $m$, the index receives a contribution only from the intersection

In the single-channel case, starting from the zero potential, a single non-conservative supersymmetric transformation can be equivalent to a particular pair of conservative transfo…

The paper considers supersymmetric transformations of a zero potential and compares conservative and non-conservative transformations. A potential family rests on the zero potentia…