Young's double slit interference pattern from a twisted beam

The interference pattern of a Laguerre Gaussian beam in a double slit experiment is reported. Whereas a typical laser beam phase front is planar, a Laguerre Gaussian beam exhibits a wave front that is twisting along the direction of propagation. This leads to a distorted interference pattern. The topological charge also called the order of the twisted beam can be then readily and simply determined. More precisely, the naked eye resolution of the distortion shift of the interference pattern directly informs about the number of twists made as well as on the sign of the twist. These results are in very good agreement with theoretical calculations that offer a general description of the double slit interference with twisted beams.Comment: 11 pages, 4 figure

Spherical Hecke algebra in the Nekrasov-Shatashvili limit

The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of $\mathcal{N}=2$ gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter $\epsilon_2$. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.Comment: 29 pages, 3 figures (v3: redaction of section 4 improved, results unchanged

Beyond Title VII: Rethinking Race, Ex-Offender Status, and Employment Discrimination in the Information Age

More than sixty-five million people in the United States—more than one in four adults—have had some involvement with the criminal justice system that will appear on a criminal history report. A rapidly expanding, for-profit industry has developed to collect these records and compile them into electronic databases, offering employers an inexpensive and readily accessible means of screening prospective employees. Nine out of ten employers now inquire into the criminal history of job candidates, systematically denying individuals with a criminal record any opportunity to gain work experience or build their job qualifications. This is so despite the fact that many individuals with criminal records have never been convicted of a crime, as one-third of felony arrests never result in conviction. And criminal records databases routinely contain significant errors, including false positive identifications and sealed or expunged information. The negative impact of employers’ reliance on criminal records databases falls most heavily on Black and Latino populations, as studies show that the stigma of having a criminal record is significantly more damaging for racial minorities than for Whites. This criminal record “penalty” limits profoundly the chance of achieving gainful employment, creating new and vexing problems for regulators, employers, and minorities with criminal records. Our existing regulatory apparatus, which is grounded in Title VII of the Civil Rights Act of 1964 and the Fair Credit Reporting Act, is ill-equipped to resolve this emerging dilemma because it fails to address systematic information failures and the problem of stigma. This Article, therefore, proposes a new framework drawn from core aspects of anti-discrimination laws that govern health law, notably the Americans with Disabilities Act, and the Genetic Information Nondiscrimination Act. These laws were designed to regulate the flow of information that may form the basis of an adverse employment decision, seeking to prevent discrimination preemptively. More fundamentally, they conceptualize discrimination through the lens of social stigma, which is critical to understanding and prophylactically curbing the particular discrimination that results from dual criminal record and minority status. This health law framework attends to the interests of minorities with criminal records, allows for more robust enforcement of existing laws, and enables employers to make appropriate and equitable hiring decisions, without engaging in invidious discrimination or contributing to the establishment of a new, and potentially enduring, underclass

Fiber-base duality from the algebraic perspective

Quiver 5D $\mathcal{N}=1$ gauge theories describe the low-energy dynamics on webs of $(p,q)$-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping $(p,q)$-branes to branes of charge $(-q,p)$, and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator $\mathcal{T}$ constructed by gluing the algebra's intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and $\mathcal{T}$-operators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki's automorphism.Comment: 49 pages, 7 figures (v3: statement on universal R-matrix corrected

Formal Geometric Quantization III, Functoriality in the spin-c setting

In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds

Non-negative Principal Component Analysis: Message Passing Algorithms and Sharp Asymptotics

Principal component analysis (PCA) aims at estimating the direction of maximal variability of a high-dimensional dataset. A natural question is: does this task become easier, and estimation more accurate, when we exploit additional knowledge on the principal vector? We study the case in which the principal vector is known to lie in the positive orthant. Similar constraints arise in a number of applications, ranging from analysis of gene expression data to spike sorting in neural signal processing. In the unconstrained case, the estimation performances of PCA has been precisely characterized using random matrix theory, under a statistical model known as the `spiked model.' It is known that the estimation error undergoes a phase transition as the signal-to-noise ratio crosses a certain threshold. Unfortunately, tools from random matrix theory have no bearing on the constrained problem. Despite this challenge, we develop an analogous characterization in the constrained case, within a one-spike model. In particular: $(i)$~We prove that the estimation error undergoes a similar phase transition, albeit at a different threshold in signal-to-noise ratio that we determine exactly; $(ii)$~We prove that --unlike in the unconstrained case-- estimation error depends on the spike vector, and characterize the least favorable vectors; $(iii)$~We show that a non-negative principal component can be approximately computed --under the spiked model-- in nearly linear time. This despite the fact that the problem is non-convex and, in general, NP-hard to solve exactly.Comment: 51 pages, 7 pdf figure

Quantum integrability of N=2\mathcal{N}=2 4d gauge theories

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories. In this regime of the background, -- we shall show --, the instanton partition function is characterised by the solution of a TQ-equation. Exploiting a symmetry of the contour integrals expressing the partition function, we derive a 'dual' TQ-equation, sharing the same T-polynomial with the former. This fact allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for $Q$) and, then, to reproduce the NS TBA-like equation. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the 'dual' Bethe Ansatz equations.Comment: 21 page