Anomaly resolution via decomposition

In this paper, we apply decomposition to orbifolds with quantum symmetries to resolve anomalies. Briefly, it has been argued by, e.g. Wang–Wen–Witten, Tachikawa that an anomalous orbifold can sometimes be resolved by enlarging the orbifold group so that the pullback of the anomaly to the larger orbifold group is trivial. For this procedure to resolve the anomaly, one must specify a set of phases in the larger orbifold, whose form is implicit in the extension construction. There are multiple choices of consistent phases, which give rise to physically distinct resolutions. We apply decomposition, and find that theories with enlarged orbifold groups are equivalent to (disjoint unions of copies of) orbifolds by nonanomalous subgroups of the original orbifold group. In effect, decomposition implies that enlarging the orbifold group is equivalent to making it smaller. We provide a general conjecture for such descriptions, which we check in a number of examples.

1846: Canada’s First Inquiries Act

Historical accounts of commissions of inquiry in Canada make only passing reference to the seminal 1846 Inquires Act. None explore the provenance of this legislation beyond a few sentences of the most general conjecture. This paper contends that Canada’s first Inquiries Act was a by-product of a political crisis that grew out of the politics and institutional processes integral to the resolution of claims for rebellion losses in Canada during the 1840s. As the events associated with the passage of the 1849 Rebellion Losses Bill would disclose, this crisis posed an existential threat to the viability of the Union. The passage of the Inquiries Act, precipitated by the immediate contingencies of the rebellion losses crisis, marked for Canada a fundamental shift in constitutional authority dating back to 1688. The Act embraced methods of inquiry denied to the Crown since the late seventeenth century. Though created by a democratic legislature, the Inquiries Act revived a Crown-driven inquisitional approach to public inquires long since inoperative in Great Britain. The Act thus marked a shift in the relationship between state and citizen, and opened a new terrain for the long struggle to protect the individual against the all-powerful state.

HECKE OPERATORS AND THE COHERENT COHOMOLOGY OF SHIMURA VARIETIES

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

(Mis-)matching type-B anomalies on the Higgs branch

Building on [1], we uncover new properties of type-B conformal anomalies for Coulomb-branch operators in continuous families of 4D $$ \mathcal{N} $$ N = 2 SCFTs. We study a large class of such anomalies on the Higgs branch, where conformal symmetry is spontaneously broken, and compare them with their counterpart in the CFT phase. In Lagrangian the- ories, the non-perturbative matching of the anomalies can be determined with a weak coupling Feynman diagram computation involving massive multi-loop banana integrals. We extract the part corresponding to the anomalies of interest. Our calculations support the general conjecture that the Coulomb-branch type-B conformal anomalies always match on the Higgs branch when the IR Coulomb-branch chiral ring is empty. In the opposite case, there are anomalies that do not match. An intriguing implication of the mismatch is the existence of a second covariantly constant metric on the conformal manifold (other than the Zamolodchikov metric), which imposes previously unknown restrictions on its holonomy group.

Improving Analytic Reasoning via Crowdsourcing and Structured Analytic Techniques

How might analytic reasoning in intelligence reports be substantially improved? One conjecture is that this can be achieved through a combination of crowdsourcing and structured analytic techniques (SATs). To explore this conjecture, we developed a new crowdsourcing platform supporting groups in collaborative reasoning and intelligence report drafting using a novel SAT we call “Contending Analyses.” In this paper we present findings from a large study designed to assess whether groups of professional analysts working on the platform produce better-reasoned reports than those analysts produce when using methods and tools normally used in their organizations. Secondary questions were whether professional analysts working on the platform produce better reasoning than the general public working on the platform; and how usable the platform is. Our main finding is a large effect size (Cohen’s d = 1.37) in favor of working on platform. This provides early support for the general conjecture. We discuss limitations of our study, implications for intelligence organizations, and future directions for the work as a whole.

Automorphic vector bundles with global sections on -schemes

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

Sequences, modular forms and cellular integrals

It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

-GRAPHS AND GYOJA’S -GRAPH ALGEBRA

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. These lead to a more general conjecture on monotonicity of ratios of Kummer hypergeometric functions, which remained open ever since. In this paper we prove some conjectures for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turán-type inequalities.

Families of weighted sum formulas for multiple zeta values

Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper, we prove a family of identities involving Bernoulli numbers and apply them to obtain infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.