Modular Miracles

Over the last 20 years, the modular function has become widely known through its miraculous intervention in two great mathematical achievements: the proof of Fermat\u27s last theorem and the moonshine of the monster simple group. In both cases, the modular function appears where no one expected it, and it bridges a chasm between seemingly unrelated fields. It is probably fair to say that, in these two cases, we do not yet fully understand how the modular magic works. However, it can at least be said that these are not the first modular miracles. Ever since its discovery, in the early 19th century, the modular function has been an engine for spectacular and unexpected results. Now that things modular are back in the news, it is a good time to recall some of the modular miracles of the 19th century. They help us see the recent results in some perspective, and encourage us to believe that there is a lot more to be learned

Mirror Symmetry and Other Miracles in Superstring Theory

The dominance of string theory in the research landscape of quantum gravity physics (despite any direct experimental evidence) can, I think, be justified in a variety of ways. Here I focus on an argument from mathematical fertility, broadly similar to Hilary Putnam's 'no miracles argument' that, I argue, many string theorists in fact espouse. String theory leads to many surprising, useful, and well-confirmed mathematical 'predictions' - here I focus on mirror symmetry. These predictions are made on the basis of general physical principles entering into string theory. The success of the mathematical predictions are then seen as evidence for framework that generated them. I attempt to defend this argument, but there are nonetheless some serious objections to be faced. These objections can only be evaded at a high (philosophical) price.Comment: For submission to a Foundations of Physics special issue on "Forty Years Of String Theory: Reflecting On the Foundations" (edited by G. `t Hooft, E. Verlinde, D. Dieks and S. de Haro)

Macdonald Polynomials from Sklyanin Algebras: A Conceptual Basis for the pp-Adics-Quantum Group Connection

We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate'' between the zonal spherical functions of related real and $p$\--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$\--Baxter models. We show that in the n \air \infty limit, the Jost function for the scattering of {\em first} level excitations in the $Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--function constructed from the Macdonald polynomials associated to the root system $A_1$. The partition function of the $Z_2$\--Baxter model itself is also expressed in terms of this Macdonald\--Harish\--Chandra\ $c$\--function, albeit in a less simple way. We relate the two parameters $q$ and $t$ of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the $p$\--adic ``regimes'' in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of ``$q$\--deforming'' Euler products.Comment: 25 page

Modularity in action.GNU/Linux and free/Open source sotfware development model unleashed.

Organizational and managerial theories of modularity applied to the design and production of complex artifacts are used to interpret the rise and success of Free/Open Source Software methodologies and practices in software engineeringmodularity; software project management; free/open source software; division of labor; coordination; information hiding

Modular Theory and Eyvind Wichmann's Contributions to modern Particle Physics Theory

Some of the consequences of Eyvind Wichmann's contributions to modular theory and the QFT phase-space structure are presented. In order to show the power of those ideas in contemporary problems, I selected the issue of algebraic holography as well as a new nonperturbative constructive approach (based on the modular structur of wedge-localized algebras and modular inclusions) and show that these ideas are recent consequences of the pathbreaking work which Wichmann together with his collaborator Bisognano initiated in the mid 70$^{ies}.Comment: A rogue address which entered chapter 2 has since been omitted. 21 pages, tcilatex, to be published in a special Festschrift volume, dedicated to Prof. E. Wichmann on the occasion of his seventieth birthda

Worldsheet instantons and coupling selection rules in heterotic orbifolds

We review recent results on string coupling selection rules for heterotic orbifolds, derived using conformal field theory. Such rules are the first step towards understanding the viability of the recently obtained compactifications with potentially realistic particle spectra. They arise from the properties of the worldsheet instantons that mediate the couplings, and include stringy effects that would seem 'miraculous' to an effective field theory observer.Comment: 4 pages, talk presented at SQS'13, JINR, Dubna, Russia, 29 July - 03 August, 201

Surprises in Open-String Perturbation Theory

The perturbative analysis of models of open and closed superstrings presents a number of surprises. For instance, variable numbers of antisymmetric tensors ensure their consistency via generalized Green-Schwarz cancellations and a novel type of singularity occurs in their moduli spaces. All these features are related, in one way or another, to the presence of boundaries on the world sheet or, equivalently, of extended objects (branes) interacting with the bulk theory in space time. String dualities have largely widened the interest in these models, that exhibit a wealth of generic non-perturbative features of String Theory.Comment: 13pages, LATEX with espcrc2, 3 eps figures Contribution to the Proceedings of the XXX Ahrenshoop Symposium, Buckow (Berlin), August 1996 references adde

The Five-Phases of Economic Development and Institutional Evolution in China and Japan

Based on the variable rate of gross domestic product per capita growth and its sources, this paper first identifies five phases of economic development that are common to China, Japan, and Korea: M (Malthusian), G (government-led), K (à la Kuznets), H (human capital based) and PD (post demographic-transition). But there are also marked differences in the onset, duration, and institutional forms of these phases across these economies. In order to understand these differences, this paper explores the agrarian origins of institutions in Qing China and Tokugawa Japan (and briefly Chosŏn Korea) and their path-dependent transformations over those phases. In doing so, the paper employs game-theoretic reasoning and interpretations of divergent institutional evolution between China and Japan, which also clarifies the simplicity of prevailing arguments that identify East Asian developmental and institutional features with authoritarianism, collectivism, kinship-dominance, Confucianism and the like. Finally, the paper examines the relevance of the foregoing developmental discussions to the institutional agendas faced by the People’s Republic of China (PRC) and Japan in their respective emergent phase-transitions. In what way can the PRC avoid the “middle income trap”? What institutional shortcomings become evident from the Fukushima catastrophe and how can they be overcome in an aging Japan?development phases; institutional evolution; agrarian origin; prc economy; middle income trap; post demographic transition; east asia; norm