22 problems

Let $Q$ and $D$ be the basic affine coherent-state operators, and let ${\cal H}$ be a self-adjoint Hamiltonian operator. For $0<\beta\leq 1/2$, suppose the propagator and the assoc…

The paper compares the trivial and Poisson structures for $SU(2)$, where the symplectic leaves are spheres and the transformation to Darboux coordinates makes the path integrals ha…

Localization conjecture. Developing the localization method might, in the long run, increase our understanding of evolution operators of integrable models.

UV-completion conjecture. The Euclidean GPI will only truly be justified in a UV-complete theory of quantum gravity.

Let the initial state be the sharply peaked state $\Psi_{T=0}(z)=\delta(z-z_\mathrm{i})$, and consider its time evolution in a false-vacuum region (FV region), with the desired exc…

Consider the canonical real-time quantization, the discretized lattice formulation, and the formal continuous formulation of JT gravity. Formulation-equivalence conjecture. All thr…

Let $Dg$ denote the formal ultralocal DeWitt measure on the space of metrics, and let the lattice formulation define a continuum measure on constant-curvature metrics. Lattice meas…

Let ${\bf M}$ be a complete Kähler manifold satisfying the condition of Proposition as a Riemannian manifold, and let its Kähler form be the symplectic form. Assume that ${\bf M}$…

The Berezin–Simon quantization is considered for bounded functions, so that the resulting operators are bounded. Let a geometric path integral quantization mean a formulation exten…

Padé determinant-limit conjecture. The propagation amplitude is the infinite-size limit of the quotient of two amplitudes, each expressed as a determinant; specifically,

Let the Schwarzian path integral be defined by the formal Duistermaat–Heckman calculation around the fixed point, whose quadratic action has indefinite directions. Alekseev–Shatash…

Let $\mathrm{G}$ be a finite-dimensional non-compact Lie group with an invariant Riemannian metric $\mathrm{g}$, so that the standard Brownian motion on $(\mathrm{G},\mathrm{g})$ e…

Let $Z(\beta)$ be a generic finite-temperature path integral defined using a thermal circle $S^1_{\beta}$, and let $N_0$ denote the number of zero modes along that circle. Write…

Let $Z(\beta)$ be a finite-temperature path integral, with thermal circle $S^1_{\beta}$, and let ${\rm R.}$ denote a regularized sum of its Kaluza–Klein zero modes on that circle.…

Path-integral connection conjecture. The path integral is a connection, in the sense of differential geometry, on a $hbar$-family of vacua. Its content for states in the vacuum sec…

One-to-one correspondence conjecture. A similar argument should show a one-to-one correspondence between $\Sigma_{x_i,x_f}$ and an appropriate subset of $\mathbb{Z}^2/\sim$ determi…

Consider the four-dimensional supersymmetric Yang–Mills matrix model, with $\tilde{\beta}$ the bosonic mass parameter and $\tilde{\xi}$ the scalar-curvature parameter, equivalently…

Consider the supersymmetric Yang–Mills matrix model in dimension $D=4$, with scalar-curvature parameter $\tilde{\xi}$ representing the mass of the fermionic matrices. Generic-fermi…

One-term asymptotic conjecture. This asymptotic form is valid for sufficiently large $|z|$ and for $-\pi/2<\arg z<\pi/2$.

Let $Y$ be a target space-time, and let $Z_{D2}(Y)$ and $Z_{M2}(Y)$ denote the path integrals of perturbative D2-brane and M2-brane theories, respectively. The relevant worldvolume…

Let $Y$ be a target space-time and let $Z_{D3}(Y)$ denote the path integral of a perturbative D3-brane theory, with contributions summed over fundamental closed orientable $4$-mani…

The discussion concerns the limit of products of factors … where the parameters $\varepsilon_{\sigma^3}$ tend to zero, potentially in different orders for different $3$-simplices…