68 problems

All-loop BPS symmetry conjecture. The algebra $\mathfrak{g}$ is a symmetry of BPS amplitudes at all loop orders.

Framed no-exotics conjecture. Framed BPS states form the trivial representation under $SU(2)_R$; consequently, ${\overline{\underline\Omega}}(\mathbb{L},u,\gamma,q)$ is a character…

Let $X^5$ be a generic Sasaki–Einstein manifold, and consider the states obtained from D3-branes wrapping an $S^3$ inside $AdS_5$ and moving along a trajectory in $X^5$. BPS-geodes…

Conjectured BPS-invariant relation. The BPS invariants for other dimension vectors should work out so that the general relation is satisfied.

No-exotics conjecture. The Hodge diamond of $\mathcal{M}_\gamma^Q$ is diagonal for every physical quiver describing a four-dimensional supersymmetric quantum field theory, so that

Zero-angular-momentum conjecture. All microstates of single-centered $\tfrac{1}{8}$-BPS black holes carry vanishing intrinsic angular momentum once the universal goldstino modes ar…

Non-graviton absence conjecture. There are no non-graviton operators in the Macdonald (or Schur) sector of the $\mathcal{N}=4$ $SU(2)$ theory.

One-loop exactness conjecture. The BPS spectrum is one-loop exact: states that remain BPS at one-loop do not lift at higher perturbative or non-perturbative orders.

Let $\widehat{M}_\beta$ be the moduli space associated with curve class $\beta$, and choose a Calabi–Yau orientation such that the resulting perverse sheaf of vanishing cycles…

A gauge theory coupled to gravity is assumed to contain charged matter in every representation of its gauge group. Completeness Hypothesis. Any gauge theory coupled to gravity shou…

Consider the phase portrait with stability parameters $\zeta^{\circ}$ and $\zeta^{\bullet}$. In the region where $\zeta^{\circ}\geq 0$ and $\zeta^{\bullet}\leq 0$, both $R$-dominat…

BPS states in supersymmetric theories may carry algebraic structures, such as quiver Yangian structures, whose representations depend on stability parameters and their chambers. Wa…

Topological-string resurgence conjecture. The resurgent structure of the topological string contains information about the spectrum of BPS states.

In a four-dimensional supersymmetric theory, consider the spectrum of BPS states describing a black hole, in both the macroscopic and microscopic regimes, and let the moduli lie at…

BPS Cardy conjecture. In every three-dimensional superconformal field theory with $\mathcal{N}=2$ supersymmetry, the spectrum of supersymmetric (BPS) states obeys this Cardy asympt…

Let refined BPS counting denote the refined enumeration of BPS states encoded by the refined partition functions, and let the refined topological string denote the corresponding re…

Let twisted quiver Yangians be constructed as associative subalgebras of quiver Yangians, and let their truncations be the algebras obtained by imposing the corresponding level con…

BPS interpretation conjecture. The Stokes coefficients $\mathsf{S}^{\mathcal A}_{\omega\omega'}$ appearing in this resurgent structure are interesting invariants counting BPS state…

Let $\mathcal{T}$ be a four-dimensional $\mathcal{N}=2$ field theory with Coulomb branch $B$, discriminant locus $\Delta$, Seiberg–Witten integrable system $(M,\Omega_I)$, holomorp…

A four-dimensional $\mathcal{N}=2$ quantum field theory has a corresponding Seiberg–Witten integrable system, and its holomorphic Floer theory is defined for that system. BPS spect…

Consider a four-dimensional $\mathcal{N}=2$ supersymmetric theory with Schur index … Here the trace is over the supersymmetric states contributing to the Schur index, $F$ is fermio…

Let a general Calabi–Yau threefold be given, and let a chamber of its BPS moduli space be fixed. A crystal model is a crystal whose melting configurations encode the BPS states in…

Let $d^f_N(\ell)$ denote the index of BPS states of charge $\ell$ in the four-dimensional gauge theory, equivalently in its dual gravitational theory, and let … be the generating f…

Quiver Yangian–BPS equivalence conjecture. With this prescription, the equivalence between the quiver Yangian and BPS algebra descriptions should extend to all representations, inc…

Let $\mathcal K$ be a hyperbolic knot, and consider the Stokes matrices associated with the perturbative expansions of its complex Chern–Simons theory. The Dimofte–Gaiotto–Gukov 3D…