37 problems

Let $V$ be a graded vector space with differential $Q$, and let $\mu_n:V^{\otimes n}\to V$ be multilinear operations of degree $2-n$, with $\mu_1=Q$. An $A_{\infty}$-algebra is req…

The Lian–Zuckerman homotopy algebra is the homotopy algebra constructed from the operations and differential associated with a topological vertex operator algebra. A $BV_{\infty}$-…

Let $V^\bullet[\cdot]$ be a topological vertex operator algebra (TVOA), with dot product and bracket product defined by the vertex-operator operations above. The homotopy Gerstenha…

Let $A$ range over the homotopy or deformation algebras mentioned in the paper, including the various $( - )_{\infty}$-algebras, and let $\operatorname{Mod}(A)$ denote the category…

Extension property converse. If every $(S,\underline D)$-diagram of ${\cal P}$-algebras can be extended to an $S$-diagram of ${\cal P}$-algebras, then the couple $(S,D)$ has the…

Lian–Zuckerman's lifting conjecture. The product and bracket extend to a $G_\infty$ structure on $V$. The conjecture is presented as an open problem in the source, with no resoluti…

Let the homotopy Batalin–Vilkovisky maps of Lian and Zuckerman be given on a topological vertex operator algebra. Kimura–Voronov–Zuckerman's conjecture. The antisymmetrization of t…

Let $X$ be a homotopy differential graded non-unital algebra, and let $X(1)$ be the underlying chain complex of the associated homotopy semigroup. The construction described in the…

Let $X$ be a homotopy semigroup in the category of based topological spaces or chain complexes, with underlying object $X(1)$. In the topological case, the associated structure is…

Let the strict $\mathrm{BV}^{\square}$ algebra identified in the paper be given, and let the full $\mathrm{BV}_{\infty}^{\square}$ algebra of Yang–Mills theory be given. A suitable…

Trial projection conjecture. The solution of the first equation of the master system is the map $\pi_{\mathrm{trial}}$ defined by

Open-closed Deligne conjecture. For any open-closed homotopy algebra $(\mathfrak l,\mathfrak q)$, there is a $G_\infty$ algebra structure on the space…

Let $X$ be an oriented smooth manifold. Let $(\operatorname{coHoch}_\bullet(\mathcal{C}(X)),d,P)$ be the associated mixed complex, and consider its relative cyclic chains. An…

Consider the canonical supermultiplet in the $D=11$ supergravity example, whose nonzero higher structure includes the $4$-bracket $[D_a,D_b,D_c,D_d]=H_{abcd}$. Absence of higher br…

Let $s$ be the original differential, let $\omega$ be the constructed $3$-bracket deformation, and let $d_{CE}$ denote the Chevalley–Eilenberg differential. Total deformation conje…

Jordan triple-system conjecture. There is an isomorphism

Let $\mathcal{P}$ be an operad and let $\mathcal{A}_\infty\to\mathcal{P}$ be a morphism that lifts to the category of non-$\Sigma$ cyclic operads. Suppose the associated deformatio…

Let $\mathcal{P}$ be an operad and let $\mathcal{A}_\infty\to\mathcal{P}$ be a morphism. Its associated deformation complex carries the operadic Lie bracket. Deligne's conjecture.…

Let ${\mathcal T}_{poly}({\mathbb R}^n)$ be the space of polyvector fields, with the standard Schouten bracket $[\,\ ]_S$, and let $[\,\ldots,\ ]_{2p}$ for $p\geq 1$ denote the hig…

Let ${\mathcal T}_{poly}({\mathbb R}^n)$ denote the space of polyvector fields on ${\mathbb R}^n$, equipped with the standard Schouten bracket $[\,\ ]_S$. A universal homotopy defo…

Let $\Lambda=(\xi,f)\in\mathbb{F}^1_-$ generate an infinitesimal symmetry of the generalized Maurer-Cartan equation, and let $\mathbb{M}$ be the corresponding component of a soluti…

Let $\Psi=(\mathbb{M},\eta,\phi,\bar{\phi})$ be a solution of the generalized Maurer-Cartan equation for the $L_{\infty}$-algebra on ${\bf F}^{\cdot}_-$, with … Assume that…

Let $(\mathcal{F}^{\cdot},\mathcal{Q})$ be a complex, with jet version $(\mathcal{F}^{\cdot}_{\infty},\mathcal{Q})$ and complex conjugate…

Let $\mathbb{M}$ be the metric and $B$-field component of a solution of the generalized Maurer–Cartan equation, and let $\Lambda=(\xi,f)\in\mathbb{F}^1_-$ generate its infinitesima…

Let ${\bf F}^{\cdot}_-$ be the invariant subcomplex from Conjecture 4.1a, and let its degree-two Maurer–Cartan elements be written as $\Psi=(\mathbb{M},\eta,\phi,\bar{\phi})$. Set…