Exponential Networks and Representations of Quivers

We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge theory to local Calabi-Yau threefolds and related models. The differential is multi-valued on the covering curve and features a new type of logarithmic singularity in order to account for D0-branes and non-compact D4-branes, respectively. We describe local rules for the three-way junctions of BPS trajectories relative to a particular framing of the curve. We reproduce BPS quivers of local geometries and illustrate the wall-crossing of finite-mass bound states in several new examples. We describe first steps toward understanding the spectrum of framed BPS states in terms of such "exponential networks."Comment: 82 pages, 60 figures, typos fixe

Theophany on the Shakespearean Stage

This thesis offers a reading of five of Shakespeare's late plays––Pericles, Cymbeline, The Winter's Tale, The Tempest, and The Two Noble Kinsmen––via the idea of theophany. Theophany takes a different form in each of these plays. In Pericles and Cymbeline, Diana and Jupiter appear, ostensibly in body, on the stage. In the other plays examined here, theophany might retire into the imaginative hinterland of the work, or be veiled in language or explicit artifice. The Two Noble Kinsmen does not divulge its cardinal deity openly; likewise The Winter's Tale offers a number of gods and divine suggestions, and burdens the reader or audience with deciding the contours of the play's implicit divine hierarchy. The Tempest presents nearly intractable difficulty and mystery, which the relevant chapter attempts to elucidate. Nevertheless, the thesis contends that each of these plays presents a moment, set of moments, or a general suffusion which is answerable to the term 'theophany'. In order to understand such peculiar moments in the Shakespearean corpus, the thesis draws on a number of considerations, such as 1) the various precedents in classical and contemporary literature and visual culture; 2) the importance of genre in understanding Shakespeare's theophanies and those on the early modern stage in general; and 3) the staging of these scenes. The thesis also enquires into Shakespeare's use of allegory and its importance for his thinking about the relationship between the gods and ideas. Owing to its focus on genre, the thesis also explores competing and coexisting concepts of Providence and Fortune in the plays, as well as other modes of thinking about destiny. Finally, the thesis finds that, instead of sidelining Shakespeare's theophanies as criticism has frequently done, placing them at the very centre of enquiry yields a rich and holistic reading of these complex plays

Hidden exceptional symmetry in the pure spinor superstring

The pure spinor formulation of superstring theory includes an interacting sector of central charge $c_{\lambda}=22$, which can be realized as a curved $\beta\gamma$ system on the cone over the orthogonal Grassmannian $\text{OG}^{+}(5,10)$. We find that the spectrum of the $\beta\gamma$ system organizes into representations of the $\mathfrak{g}=\mathfrak{e}_6$ affine algebra at level $-3$, whose $\mathfrak{so}(10)_{-3}\oplus {\mathfrak u}(1)_{-4}$ subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine $\mathfrak{e}_6$ characters. We interpret this as an instance of a more general pattern of enhancements in curved $\beta\gamma$ systems, which also includes the cases $\mathfrak{g}=\mathfrak{so}(8)$ and $\mathfrak{e}_7$, corresponding to target spaces that are cones over the complex Grassmannian $\text{Gr}(2,4)$ and the complex Cayley plane $\mathbb{OP}^2$. We identify these curved $\beta\gamma$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $\mathcal{N}=2$ SCFTs on $S^2$.Comment: 8 pages, 1 figure; v2: added references, minor update

Global Attracting Equilibria for Coupled Systems with Ceiling Density Dependence

In this paper, we present a system of two difference equations modeling the dynamics of a coupled population with two patches. Each patch can house only a limited number of individuals (called a carrying capacity) because resources like food and breeding sites are limited in each patch. We assume that the population in each patch is governed by a linear model until reaching a carrying capacity in each patch, resulting in map which is nonlinear and not sublinear. We analyze the global attractors of this model

Bounds on the dynamics of sink populations with noisy immigration

Sink populations are doomed to decline to extinction in the absence of immigration. The dynamics of sink populations are not easily modelled using the standard framework of per capita rates of immigration, because numbers of immigrants are determined by extrinsic sources (for example, source populations, or population managers). Here we appeal to a systems and control framework to place upper and lower bounds on both the transient and future dynamics of sink populations that are subject to noisy immigration. Immigration has a number of interpretations and can fit a wide variety of models found in the literature. We apply the results to case studies derived from published models for Chinook salmon (Oncorhynchus tshawytscha) and blowout penstemon (Penstemon haydenii)