169 problems

Let $Z_N$ denote the Kontsevich–Penner matrix model for $N\geq 1$, and let the refinement be obtained by replacing the complex integral in the refined open partition-function model…

Large-$N$ critical energy conjecture. In the large-$N$ limit,

In the infinite momentum frame, let $N$ point-like D0-branes be described by quantum mechanics, with their dynamics governed by a supersymmetric matrix model. BFSS M-theory conject…

The Kontsevich matrix model is a matrix-model generating function, and a BKP hypergeometric tau function is a tau function of the BKP hierarchy with hypergeometric coefficients. Al…

Large-$D$ scaling conjecture. The function satisfies

Kontsevich–Penner conjecture. The function $\tau^{o,\mathrm{ext}}_N$ coincides with the tau-function $\tau_N$ given by the Kontsevich–Penner matrix integral.

Pole structure conjecture. All $\omega_{g,n}$ are meromorphic differentials on the spectral curve and have poles of finite degree only at the branch point $z_j=0$.

The IKKT matrix model is a dimensionally reduced ten-dimensional super Yang–Mills theory whose degrees of freedom are described by ten Lie-algebra-valued matrices and their sixteen…

Let $tilde{\alpha}$ denote the inverse temperature (inverse gauge coupling) and let $N$ be the matrix size, in the mass-deformed supersymmetric Yang–Mills matrix model with…

Matrix-integral conjecture. The generating function of Gromov–Witten invariants should have a matrix integral expression.

Convergent matrix-integral decomposition conjecture. The convergent matrix integral $Z_{\rm conv}$ is a finite linear combination of convergent formal solutions of the loop equatio…

Dijkgraaf–Vafa conjecture. The superpotential $W(S)$ is the sum of the zero-momentum planar diagrams of the $\mathcal{N}=1$ theory under consideration.

Let $F^{(1)}$ denote the coefficient of order $1/N^2$ in the free energy of the two-matrix model in the genus-one case, and let the model determine an associated algebraic curve. L…

Let the superalgebra be defined by the relations in Eqs. (5.2.5), (5.2.6), and (5.2.7), with $\tilde{f}^I_J$ and $\gamma^I_J$ treated as independent generators, and let…

Let $\hat{M}'_{\Lambda}$ be the ideal of the heterix algebra described in the source, and consider the quotient algebra whose elements are the cosets…

Let $N$ denote the number of D0-branes, and let the low-energy dynamics of D0-branes be described by dimensionally reduced supersymmetric Yang–Mills quantum mechanics. Large-$N$ M-…

Consider the model of two-dimensional quantum gravity with $\Theta=\pi/2$, whose partition function is formally expanded as … Let $\lambda_1(\epsilon)$ and $\lambda_2(\epsilon)$ be…

Let $W(x)$ and $f(x)$ be polynomials of degrees $d+1$ and $d-1$, respectively, and consider the local Calabi–Yau background … Its associated hyperelliptic spectral curve is … which…

Let a noncompact Calabi–Yau manifold be a local background whose geometry reduces to a complex curve, and let $F^{(g)}(t)$ denote the holomorphic genus-$g$ free energies of the ass…

Let the IKKT matrix model be the matrix model whose partition function is denoted by $Z$ and whose finite-$N$ formulation uses matrices of size $N$. Its large-$N$ limit is obtained…

The scalar field is represented by a hermitian finite matrix, with an action consisting of a kinetic term and a potential term whose trace has pure matrix-model form and whose…

The authors study Witten's open string field theory (SFT), whose Neumann coefficients for surface states satisfy the Hirota equations of the dispersionless KP hierarchy, while thos…

Let the $osp(1|32;\mathbf{R})$ matrix model be the cubic matrix model whose matrices are built from the super Lie algebra $osp(1|32;\mathbf{R})$. Consider expansions of this model…

Let the fermionic matrix integral be defined by the matrix model in the source, with matrix size $N$, and consider its genus expansion in the large $N$ limit. Borel summability con…

Let $k_{\max}$ denote the level at which the finite-$N$ difference of the counting functions, such as $\Delta {\cal Z}(N,d,k)$ or $\Delta Z(N,d,k)$, attains its maximum, and let th…