We discuss $K$-theoretic matching of $D$-brane charges in the string duality between type I on ${\mathbb T}^4$ and type IIA on $K3$. This case is more complex than the familiar case of IIA/IIB duality, which is already well understood, but it turns out that replacing $K3$ by its orbifold blow-down seems to largely resolve the apparent problems with the theory.

There exist myriads of off-shell worldline supermultiplets for $(N \leq 32)$- extended supersymmetry in which every supercharge maps a component field to precisely one other component field or its derivative. A subset of these extends to off-shell worldsheet $(p, q)$-supersymmetry, and is characterized herein by evading an obstruction specified visually and computationally by the "bow-tie" and "spin sum rule" twin theorems. The evasion of this obstruction is proven to be both necessary and sufficient for a worldline supermultiplet to extend to worldsheet supersymmetry; it is also a necessary filter for dimensional extension to higher-dimensional spacetime. We show explicitly how to "re-engineer" an Adinkra—if permitted by the twin theorems—so as to depict an off-shell supermultiplet of worldsheet $(p, q)$-supersymmetry. This entails starting from an Adinkra depicting a specific type of supermultiplet of worldline $(p+q)$- supersymmetry, judiciously re-defining a subset of component fields and partitioning the worldline $(p, q)$-supersymmetry action into a proper worldsheet $(p, q)$-supersymmetry action.

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the $A$-polynomial $A(x, y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $ħ$; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a $q$-difference equation that annihilates the colored Jones polynomials and $SL(2, {\mathbb C})$ Chern–Simons partition functions. We propose refinements/categorifications of both conjectures that include an extra deformation parameter t and describe similar properties of homological knot invariants and refined BPS invariants. Much like their unrefined/decategorified predecessors, that correspond to $t = -1$, the new volume conjectures involve objects naturally defined on an algebraic curve $A^{ref} (x, y; t)$ obtained by a particular deformation of the A-polynomial, and its quantization $\widehat{A}^{ref} \widehat{x}, \widehat{y}; q, t)$. We compute both classical and quantum $t$-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

We present an explicit parameterization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves $\widetilde{C}_{\pm \varphi}$, which parameterize the lines. These curves are 125:1 covers of genus six curves $C_{\pm \varphi }$. The $C_{\pm \varphi}$ are first presented as curves in $\mathbb{P}1 \times \mathbb{P}1$ that have three nodes. It is natural to blow up $\mathbb{P}1 \times \mathbb{P}1$ in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up $\mathbb{P}1 \times \mathbb{P}1$ in three points is the quintic del Pezzo surface $dP_5$, whose automorphism group is the permutation group $S_5$, which is also a symmetry of the pair of curves $C_{\pm \varphi }$. The subgroup $A_5$, of even permutations, is an automorphism of each curve, whereas the odd permutations interchange $C_{\varphi}$ with $C_{- \varphi }$. The ten exceptional curves of $dP_5$ each intersect the $C_{\varphi}$ in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the curves $C_{\varphi}$ are the curves of the Wiman pencil. We consider the family of lines also for the cases that the manifolds of the Dwork pencil become singular. For the conifold, the curve $C_{\varphi}$ develops six nodes and may be resolved to a $\mathbb{P}1$. The group $A_5$ acts on this $\mathbb{P}1$ and we describe this action.