Off-critical local height probabilities on a plane and critical partition functions on a cylinder

We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a $4 N$-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of $N$, the winding number of the spiral, and $\tau$, the departure from criticality of the model, and observe that the result depends only on the product $N \, \tau$. In the limit $N \rightarrow 1$, $\tau \rightarrow \tau_0$, such that $\tau_0$ is finite, we recover the off-critical local height probability on a plane, $\tau_0$-away from criticality. In the limit $N \rightarrow \infty$, $\tau \rightarrow 0$, such that $N \, \tau = \tau_0$ is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio $\tau_0$. We conclude that the off-critical local height probability on a plane, $\tau_0$-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio $\tau_0$, in agreement with a result of Saleur and Bauer.Comment: 28 page

Macdonald topological vertices and brane condensates

We show, in a number of simple examples, that Macdonald-type $qt$-deformations of topological string partition functions are equivalent to topological string partition functions that are without $qt$-deformations but with brane condensates, and that these brane condensates lead to geometric transitions.Comment: 23 pages, 5 figures. v2: minor changes, published versio

Polynomial identities of the Rogers--Ramanujan type

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions.Comment: 27 page

Virasoro character identities from the Andrews--Bailey construction

We prove $q$-series identities between bosonic and fermionic representations of certain Virasoro characters. These identities include some of the conjectures made by the Stony Brook group as special cases. Our method is a direct application of Andrews' extensions of Bailey's lemma to recently obtained polynomial identities.Comment: 22 pages. Expanded version with new result

AGT, N-Burge partitions and W_N minimal models

Let ${\mathcal B}^{\, p, \, p^{\prime}, \, {\mathcal H}}_{N, n}$ be a conformal block, with $n$ consecutive channels \chi_{\i}, \i = 1, \cdots, n, in the conformal field theory $\mathcal{M}^{\, p, \, p^{\prime}}_N \! \times \! \mathcal{M}^{\mathcal{H}}$, where $\mathcal{M}^{\, p, \, p^{\prime}}_N$ is a $\mathcal{W}_N$ minimal model, generated by chiral fields of spin $1, \cdots, N$, and labeled by two co-prime integers $p$ and $p^{\prime}$, $1 < p < p^{\prime}$, while $\mathcal{M}^{\mathcal{H}}$ is a free boson conformal field theory. $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$ is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra $A_{N-1}$, spanned by weight vectors $\omega_1, \cdots, \omega_{N-1}$. We restrict our attention to conformal blocks with vertex operators whose charge vectors point along $\omega_1$. The charge vectors that label the initial and final states can point in any direction. Following the $\mathcal{W}_N$ AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute $\mathcal{B}^{\, p, \, p^{\prime}, \mathcal{H}}_{N, n}$, leads to ill-defined expressions. We show that restricting the states that flow in the channels \chi_{\i}, \i = 1, \cdots, n, to states labeled by $N$ partitions that satisfy conditions that we call $N$-Burge partitions, leads to well-defined expressions that we identify with $\mathcal{B}^{\, p, \, p^{\prime}, \, \mathcal{H}}_{N, n}$. We check our identification by showing that a specific non-trivial conformal block that we compute, using the $N$-Burge conditions satisfies the expected differential equation.Comment: 34 pages. More references, same conten

Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models

We study Virasoro minimal-model 4-point conformal blocks on the sphere and 0-point conformal blocks on the torus (the Virasoro characters), as solutions of Zamolodchikov-type recursion relations. In particular, we study the singularities due to resonances of the dimensions of conformal fields in minimal-model representations, that appear in the intermediate steps of solving the recursion relations, but cancel in the final results.Comment: 26 pages, 1 figure minor modification

OPE in planar QCD from integrability

We consider the operator product expansion of local single-trace operators composed of the self-dual components of the field strength tensor in planar QCD. Using the integrability of the one-loop matrix of anomalous dimensions of such operators, we obtain a determinant expression for certain tree-level structure constants in the OPE.Comment: 20 pages, 5 figures; v2: added refs and discussion of general structure constant

Grand Plans in Glass Bottles: The Economic, Social, and Technological History of Beer in Egypt 1880-1970

Contrary to common perceptions, the history of beer (and indeed of other alcoholic beverages) in the Muslim-majority context of Egypt has not been a history of government officials desperately seeking to extirpate the evil of alcohol as rumrunners, backyard brewers, and moonshiners stayed one step ahead. Rather it was a history of a commercially-marketed product that enjoyed relatively wide popularity and robust growth from 1880 to 1980, and sat at the cutting edge of technological innovation in Egypt in that same period. Its success was not only evident from the profitability of the companies that sold it, but also from its increasing appearances in all popular forms of art and media. The title of my dissertation is Grand Plans in Glass Bottles: An Economic, Social, and Technological history of Beer in Egypt, 1880-1970 . My dissertation studies Egypt during an exciting period, when the country was transitioning from being a quasi-colonial state, under British Occupation after 1882 and, until 1914, under Ottoman influence as well, to being an independent country within a highly competitive global economy. Using American, Dutch, and Egyptian archival sources, as well as Arabic literary sources, I focus on two closely linked companies, Crown and Pyramid Breweries. Originally founded by Belgian expatriates in Egypt, these two firms in their various incarnations developed the Egyptian beer industry and cultivated a wide customer base. I take the story past the 1950s, when the Egyptian government under Gamal Abdel Nasser nationalized the beer industry (which was by then led by Stella Beer and owned primarily by Heineken) much as it nationalized the Suez Canal. Through the study of this beverage, my research connects the history of Egypt to Belgium, Netherlands, Britain, and elsewhere; the history of a business to developments in technology, politics, and consumer culture; and the history of the people - of everyday Egyptians - to business elites. Viewed through a mug of beer, we can tell the economic, political, and cultural history of Egypt at large