27 problems

Let $\widehat Y^{i\infty}_{\vec{\eta}}(\sigma|\beta)$ denote the degeneration at the cusp $\tau\to i\infty$ of the integrals $Y^\tau_{\vec{\eta}}(\sigma|\beta)$, depending on Mande…

Let $s_{ij}$ be the Mandelstam variables in genus-zero closed-string amplitudes, and expand these amplitudes in a Taylor series at $s_{ij}=0$. The coefficients are numbers in the r…

Huang’s universality conjecture. The leading transcendental terms of tree-level string amplitudes are universal.

D’Hoker–Green conjecture. The general form of the non-analytic part involves only numbers of the type $Z^{\mathrm{DG}}_m-Z^{\mathrm{DG}}_n$, the combinations…

Gerasimov–Shatashvili amplitude conjecture. One has

Let $A_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$ be the $p$-adic Koba–Nielsen amplitudes, and let $A_{\mathbb{C((}T\mathbb{))}}^{(N)}(\boldsymbol{L})$ denote the motivic string amplitu…

Let $A_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$ denote the tree-level $p$-adic Koba–Nielsen string amplitudes, and let $A_{\mathrm{DL}}^{(N)}(\boldsymbol{s})$ denote their Denef–Loese…

Let $\widehat Y^{i\infty}_{\vec\eta}(\sigma|\rho)$ be the cusp degeneration of the four-point generating series, with parameters $\eta_i$, their combinations $\eta_{ij\cdots}$, con…

Let $Y^\tau_{\vec\eta}$ be the generating series constructed from the genus-one integrals, and let modular graph forms (MGFs) denote the modular graph objects generated by their…

Smallest-number-of-solutions conjecture. The number $(n-3)!$ is the smallest possible number of saddle points among all stringy canonical forms reproducing the ABHY associahedron.

Minimal saddle-point conjecture. Among all stringy canonical forms for the ABHY associahedron, ${\bf I}_n$ has the smallest number of saddle points, namely

Let the coefficients be those in the Taylor expansions around $\alpha'=0$ of closed-string integrals, and let multiple zeta values (MZVs) denote the corresponding periods. Single-v…

Let the low-energy expansion of the genus-one four-graviton amplitude be organized by derivative order, and assign transcendental weights to the constants and functions appearing i…

Let $\rho,\sigma$ be permutations in the symmetric group $\mathfrak S_k$, and let $J_{\rho,\sigma}(\mathbf a,\mathbf b,\mathbf c)$ be the corresponding multiple complex integral. A…

Genus-one closed string amplitudes have low-energy expansions involving modular graph functions and forms. The single-valued map is defined on iterated integrals of modular forms.…

The low-energy expansion of open string amplitudes is expressed in terms of multiple zeta values, while closed string amplitudes at tree level involve single-valued multiple zeta v…

Let $\widehat{\cal I}^{(4,0)}_{1234}(s_{ij},\tau)$ be the closed integral form introduced for the modular-weight $(4,0)$ quantity associated with the four-point heterotic-string am…

Brödel–Schlotterer–Zerbini's esv conjecture. For all graphs $\Gamma$,

Consider the asymptotic expansions of genus-zero open- and closed-string amplitudes in the Mandelstam variables for any number of punctures. Let a single-valued map be the map … ex…

Let $\Gamma$ be a graph with $l$ edges, and write its modular graph function as … where $\tau\in\mathbb{H}$, $q=\exp(2\pi i\tau)$, and $y=\pi\,\operatorname{Im}(\tau)$. The coeffic…

Let the non-planar part of the one-loop open-string amplitude be expanded at low energy in the inverse string tension $\alpha'$. The relevant iterated integrals are twisted ellipti…

Obers' conjecture. The renormalized integral should be proportional to the sum

The differential relation conjecture for $D_5$.

Let $j^{(1,1)}(\tau)$ be the modular function occurring in the weight-five low-energy expansion, and let $D_{2,1,1,1}(\tau)=C_{3,1,1}(\tau)$ be the corresponding modular graph func…

KP decomposition conjecture. There exists such a KP decomposition for which, as $\alpha'\to0$, the following hold: (i) integrating over the outer domain gives only a subleading con…