We study parity symmetries and boundary conditions in the framework of gauged linear sigma models. This allows us to investigate the Kähler moduli dependence of the physics of D-branes as well as orientifolds in a Calabi–Yau compactification. We first determine the parity action on D-branes and define the set of orientifold-invariant D-branes in the linear sigma model. Using probe branes on top of orientifold planes, we derive a general formula for the type (SO versus Sp) of orientifold planes. As applications, we show how compactifications with and without vector structure arise naturally at different real slices of the Kähler moduli space of a Calabi–Yau compactification. We observe that orientifold planes located at certain components of the fixed-point locus can change type when navigating through the stringy regime.

We study the mixed Hodge theoretic aspects of the B-model side of local mirror symmetry. Our main objectives are to define an analogue of the Yukawa coupling in terms of the variations of the mixed Hodge structures and to study its properties. We also describe a local version of Bershadsky–Cecotti–Ooguri–Vafa’s holomorphic anomaly equation.

We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex." This gives open BPS invariants for a toric Calabi–Yau manifold without compact 4-cycles, where we have $D0/D2/D6$-branes wrapping holomorphic $0/2/6$-cycles, as well as $D2$- branes wrapping disks whose boundaries are on $D4$-branes wrapping noncompact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed by the ordinary topological vertex. We prove an identify expressing the non-commutative topological vertex of a toric Calabi–Yau manifold $X$ as a specialization of the closed BPS partition function of an orbifold of $X$, thus giving a closed expression for our vertex. We also clarify the action of the Weyl group of an affine $A_L$ Lie algebra on chambers, and comment on the generalization of our results to the case of refined BPS invariants.

We analyze the Cauchy problem for the Klein–Gordon equation on the type IIB supergravity backgrounds $\mathrm{AdS}^5 \times Y^{p,q}$, where $Y^{p,q}$ is any of the Sasaki–Einstein 5-manifolds introduced by Gaunlett et al. (Adv. Theor. Math. Phys. 8 (2004) 711–734). Although these spaces are not globally hyperbolic, we prove that there exists a unique admissible propagator and derive an integral representation thereof using spectral-theoretic techniques.

We describe an algebra $\mathcal{G}$ of diagrams that faithfully gives a diagrammatic representation of the structures of both the Heisenberg–Weyl algebra $\mathcal{H}$ – the associative algebra of the creation and annihilation operators of quantum mechanics – and $\mathcal{U}(\mathcal{L}_{\mathcal{H}})$, the enveloping algebra of the Heisenberg Lie algebra $\mathcal{L}_{\mathcal{H}}$. We show explicitly how $\mathcal{G}$ may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of $\mathcal{U}(\mathcal{L}_{\mathcal{H}})$. While both $\mathcal{H}$ and $\mathcal{U}(\mathcal{L}_{\mathcal{H}})$ are images of $\mathcal{G}$, the algebra $\mathcal{G}$ has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.