Descent Equations Research Articles

Closed string vertex operators with various ghost number

We construct closed string vertex operators with various ghost numbers in addition to the conventional ones, using the Faddeev-Popov procedure for the gauge fixing of the conformal Killing group, from matter primary fields. We find that these operators give solutions to the descent equations in the framework of the BRST formalism. Similarly, we also construct solutions to the descent equations for the dilaton vertex operator with the Lorentz covariant form. Using the unintegrated vertex operator of the dilaton with the ghost number three, we obtain the correct result of the tadpole amplitude on the disk, including a non-zero contribution from a BRST exact term which comes from a conformal transformation.

Renormalization of the topological charge density in QCD with dimensional regularization

To all orders of perturbation theory, the renormalization of the topological charge density in dimensionally regularized QCD is shown to require no more than an additive renormalization proportional to the divergence of the flavour-singlet axial current. The proof is based on the standard BRS analysis of the QCD vertex functional in the background gauge and exploits the special algebraic properties of the charge density through the Stora–Zumino chain of descent equations.

Improving Convergence in Therapy Scheduling Optimization: A Simulation Study

The infusion times and drug quantities are two primary variables to optimize when designing a therapeutic schedule. In this work, we test and analyze several extensions to the gradient descent equations in an optimal control algorithm conceived for therapy scheduling optimization. The goal is to provide insights into the best strategies to follow in terms of convergence speed when implementing our method in models for dendritic cell immunotherapy. The method gives a pulsed-like control that models a series of bolus injections and aims to minimize a cost a function, which minimizes tumor size and to keep the tumor under a threshold. Additionally, we introduce a stochastic iteration step in the algorithm, which serves to reduce the number of gradient computations, similar to a stochastic gradient descent scheme in machine learning. Finally, we employ the algorithm to two therapy schedule optimization problems in dendritic cell immunotherapy and contrast our method’s stochastic and non-stochastic optimizations.

Notes on the 11D pure spinor wordline vertex operators

The construction of the ghost number zero and one vertex operators for the 11D pure spinor superparticle will be revisited. In this sense, an alternative way of defining the ghost number one vertex operator will be given after introducing a ghost number -2 operator made out of physical operators defined on the 11D non-minimal pure spinor superspace. This procedure will make explicit and transparent the relation between the ghost number three and one vertex operators. In addition, using a non-Lorentz covariant b-ghost, ghost number zero and two vertex operators satisfying standard descent equations will be presented in full form.

Towards Holography in the BV-BFV Setting

We show how the BV-BFV formalism provides natural solutions to descent equations, and discuss how it relates to the emergence of holographic counterparts of given gauge theories. Furthermore, by means of an AKSZ-type construction we reproduce the Chern-Simons to Wess-Zumino-Witten correspondence from infinitesimal local data, and show an analogous correspondence for BF theory. We discuss how holographic correspondences relate to choices of polarisation relevant for quantisation, proposing a semi-classical interpretation of the quantum holographic principle.

Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.

We further develop a new framework, called PDE acceleration, by applying it to calculus of variation problems defined for general functions on , obtaining efficient numerical algorithms to solve the resulting class of optimization problems based on simple discretizations of their corresponding accelerated PDEs. While the resulting family of PDEs and numerical schemes are quite general, we give special attention to their application for regularized inversion problems, with particular illustrative examples on some popular image processing applications. The method is a generalization of momentum, or accelerated, gradient descent to the PDE setting. For elliptic problems, the descent equations are a nonlinear damped wave equation, instead of a diffusion equation, and the acceleration is realized as an improvement in the CFL condition from Δt ~ Δx 2 (for diffusion) to Δt ~ Δx (for wave equations). We work out several explicit as well as a semi-implicit numerical scheme, together with their necessary stability constraints, and include recursive update formulations which allow minimal-effort adaptation of existing gradient descent PDE codes into the accelerated PDE framework. We explore these schemes more carefully for a broad class of regularized inversion applications, with special attention to quadratic, Beltrami, and total variation regularization, where the accelerated PDE takes the form of a nonlinear wave equation. Experimental examples demonstrate the application of these schemes for image denoising, deblurring, and inpainting, including comparisons against primal-dual, split Bregman, and ADMM algorithms.

Descent Equations Starting from High Rank Chern-Simons**Supported by National Natural Science Foundation of China under Grant Nos. 11475116, 11401400

In this paper a set of generalized descent equations are proposed. The solutions to those descent equations labeled by r for any r (r ≥ 2, r ϵ ℕ) are forms of degrees varying from 0 to (2r − 1). And the case of r = 2 is mainly discussed.

Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.

Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g. in the Chern-Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as $p= \langle F, F\rangle$ where $F$ is the curvature 2-form and $\langle \cdot, \cdot\rangle$ is an invariant scalar product on the corresponding Lie algebra $\mathfrak{g}$. The descent for $p$ gives rise to an element $\omega=\omega_3 + \omega_2 + \omega_1 + \omega_0$ of mixed degree. The 3-form part $\omega_3$ is the Chern-Simons form. The 2-form part $\omega_2$ is known as the Wess-Zumino action in physics. The 1-form component $\omega_1$ is related to the canonical central extension of the loop group $LG$. In this paper, we give a new interpretation of the low degree components $\omega_1$ and $\omega_0$. Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara-Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class $p$. In more detail, we define a 1-cocycle $C$ which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara-Vergne equation $F$ is mapped to $\omega_1=C(F)$. Furthermore, the component $\omega_0$ is related to the associator corresponding to $F$. It is surprising that while $F$ and $\Phi$ satisfy the highly non-linear twist and pentagon equations, the elements $\omega_1$ and $\omega_0$ solve the linear descent equation.

Six-dimensional regularization of chiral gauge theories

We propose a regularization of four dimensional chiral gauge theories using six-dimensional Dirac fermions. In our formulation, we consider two different mass terms having domain-wall profiles in the fifth and the sixth directions, respectively. A Weyl fermion appears as a localized mode at the junction of two different domain-walls. One domain-wall naturally exhibits the Stora-Zumino chain of the anomaly descent equations, starting from the axial U(1) anomaly in six-dimensions to the gauge anomaly in four-dimensions. Another domain-wall implies a similar inflow of the global anomalies. The anomaly free condition is equivalent to requiring that the axial U(1) anomaly and the parity anomaly are canceled among the six-dimensional Dirac fermions. Since our formulation is based on a massive vector-like fermion determinant, a non-perturbative regularization will be possible on a lattice. Putting the gauge field at the four-dimensional junction and extending it to the bulk using the Yang-Mills gradient flow, as recently proposed by Grabowska and Kaplan, we define the four-dimensional path integral of the target chiral gauge theory.

Axial symmetry, anti-BRST invariance, and modified anomalies

It is shown that, anti-BRST symmetry is the quantized counterpart of local axial symmetry in gauge theories. An extended form of descent equations is worked out, which yields a set of modified consistent anomalies.

Algebraic description of chiral anomalies and superspace geometry

The supersymmetry of the non abelian chiral anomaly is exhibited, up to trivial terms, using descent equations and triangle formulas in the framework of N=1 superspace geometry in four dimensions.

Quantification of the Effect of Water Temperature on the Fall Rate of Expendable Bathythermographs

AbstractA very large portion of the historical information on ocean temperatures has been measured using expendable bathythermograph (XBT) devices. For decades, these devices provided the majority of global information. It is, therefore, important to quantify their accuracy and identify biases in this important historical dataset. Here, calculations are made of the influence of water temperature on the rate of descent of the XBT devices into the ocean waters. In colder regions, the larger viscosity of the water is expected to cause a greater drag force on the device, which would slow the descent. It was found through computational fluid dynamic models that the impact of temperature and viscosity on the probe descent is approximately 2.2% for water temperatures that range from 0° to 27°C. Probe-specific temperature-dependent fall rate equations were applied to 269 collocated XBT/conductivity–temperature–depth (CTD) measurements from two different research cruises. It was found that the probe-specific descent equations were an improvement over the uncorrected method. Next, in an effort to automate the process, the fall rate coefficients were related to the topmost measured temperature in the water column. With this relationship, comparisons were made between the probe-specific descent calculations and 2937 high-resolution XBT–CTD pairs. It was found that again, the new methodology outperformed the standard fall rate equation. The new method was also compared with an independent correction method that was previously published. It was found that both new methods were improvements upon the industry-standard fall rate calculation. Subsequent calculations using the top-100-m water temperature were performed and were found to be statistically insignificant compared to the proposed simplified method.

Explicit computation of the Chern character forms

We propose a method for explicit computation of the Chern character form of a holomorphic Hermitian vector bundle (E, h) over a complex manifold X in a local holomorphic frame. First, we use the descent equations arising in the double complex of (p, q)-forms on X and find the explicit degree decomposition of the Chern–Simons form \({\mathrm {cs}}_{k}\) associated to the Chern character form \({\mathrm {ch}}_{k}\) of (E, h). Second, we introduce the so-called ascent equations that start from the \((2k-1,0)\) component of \({\mathrm {cs}}_{k}\), and use the Cholesky decomposition of the Hermitian metric h to represent the Chern–Simons form, modulo d-exact forms, as a \(\partial \)-exact form. This yields a formula for the Bott–Chern form \({\mathrm {bc}}_{k}\) of type \((k-1,k-1)\) such that \(\displaystyle {{\mathrm {ch}}_{k}=\frac{\sqrt{-1}}{2\pi }\bar{\partial }\partial {\mathrm {bc}}_{k}}\). Explicit computation is presented for the cases \(k=2\) and 3.

Matching-constrained active contours with affine-invariant shape prior

In the object segmentation by active contours, an initial contour provided by user is often required. This paper extends the conventional active contour model by incorporating feature matching in the formulation for automatic object segmentation, yielding a novel matching-constrained active contour. The key to our formulation is a mathematical model of the relationship between interior feature points and object shape, called the interior-points-to-shape relation. According to this interior-points-to-shape relation, we are able to achieve the automatic object segmentation in two steps. Specifically, we are able to estimate the object boundary position given the matched interior feature points. Afterwards, we are able to further optimize the boundary position in the active contour framework. To obtain a unified optimization model for this task, we additionally formulate the matching score as a constraint to active contour model, resulting in our matching-constrained active contour. We also derive the projected-gradient descent equations to solve the constrained optimization. In the experiments, we show that our method achieves automatic object segmentation, and it clearly outperforms the related methods.

Topological insulators and the QCD vacuum: The theta parameter as a Berry phase

There is considerable evidence, based on large ${N}_{c}$ chiral dynamics, holographic QCD, and Monte Carlo studies, that the QCD vacuum is permeated by discrete quasivacua separated by domain walls across which the local value of the topological $\ensuremath{\theta}$ parameter jumps by $\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}$. This scenario is realized in a 2D U(1) gauge theory, the $C{P}^{N\ensuremath{-}1}$ sigma model, where a pointlike charge is a domain wall, and $\ensuremath{\theta}$ describes the background electric flux and the polarization of charged pairs in the vacuum. The transition between discrete $\ensuremath{\theta}$ vacua occurs via the transport of integer units of charge between the two spatial boundaries of the domain. We show that this screening process, and the role of $\ensuremath{\theta}$ as an order parameter describing electric polarization, are naturally formulated in terms of Bloch wave eigenstates of the Dirac Hamiltonian in the background gauge field. This formulation is similar to the Berry phase description of electric polarization and quantized charge transport in topological insulators. The Bloch waves are quasiperiodic superpositions of localized Dirac zero modes and the charge transport takes place coherently via topological charge-induced spectral flow. The adiabatic spectral parameter becomes the Bloch wave momentum, which defines a Berry connection around the Brillouin zone of the zero mode band. It describes the local polarization of vacuum pairs, analogous to its role in topological insulator theory. In 4D Yang-Mills theory, the $\ensuremath{\theta}$ domain walls are $2+1$-dimensional Chern-Simons membranes, and the $\ensuremath{\theta}$ parameter describes the local polarization of brane-antibrane pairs. The topological description of polarization in 2D U(1) gauge theory generalizes to membrane polarization in 4D QCD by exploiting a relationship between the Berry connection and the gauge cohomology structure encoded in the descent equations of 4D Yang-Mills theory.

Topological insulators and the QCD vacuum: the theta parameter as a Berry phase

There is considerable evidence, based on large $N_c$ chiral dynamics, holographic QCD, and Monte Carlo studies, that the QCD vacuum is permeated by discrete quasivacua separated by domain walls across which the local value of the topological $\theta$ parameter jumps by $\pm2\pi$. In the 2-dimensional $CP^{N-1}$ sigma model, a pointlike charge is a domain wall, and $\theta$ describes the background electric flux and the polarization of charged pairs in the vacuum. We show that the screening process, and the role of $\theta$ as an order parameter describing electric polarization, are naturally formulated in terms of Bloch wave eigenstates of the Dirac Hamiltonian in the background gauge field. This formulation is similar to the Berry phase description of electric polarization and quantized charge transport in topological insulators. The Bloch waves are quasiperiodic superpositions of localized Dirac zero modes. They define a Berry connection around the Brillouin zone of the zero mode band which describes the local polarization of vacuum pairs, analogous to its role in topological insulator theory. In 4D Yang-Mills theory, the $\theta$ domain walls are 2+1-dimensional Chern-Simons membranes. The interpretation of the $\theta$ parameter as a Berry phase describing the local polarization of brane-antibrane pairs follows from the descent equations of 4D Yang-Mills theory.

A first look at Weyl anomalies in shape dynamics

One of the more popular objections towards shape dynamics is the suspicion that anomalies in the spatial Weyl symmetry will arise upon quantization. The purpose of this short paper is to establish the tools required for an investigation of the sort of anomalies that can possibly arise. The first step is to adapt to our setting Barnich and Henneaux's formulation of gauge cohomology in the Hamiltonian setting, which serve to decompose the anomaly into a spatial component and time component. The spatial part of the anomaly, i.e., the anomaly in the symmetry algebra itself ([Ω, Ω] ∝ ℏ instead of vanishing) is given by a projection of the second ghost cohomology of the Hamiltonian BRST differential associated to Ω, modulo spatial derivatives. The temporal part, [Ω, H] ∝ ℏ is given by a different projection of the first ghost cohomology and an extra piece arising from a solution to a functional differential equation. Assuming locality of the gauge cohomology groups involved, this part is always local. Assuming locality for the gauge cohomology groups, using Barnich and Henneaux's results, the classification of Weyl cohomology for higher ghost numbers performed by Boulanger, and following the descent equations, we find a complete characterizations of anomalies in 3+1 dimensions. The spatial part of the anomaly and the first component of the temporal anomaly are always local given these assumptions even in shape dynamics. The part emerging from the solution of the functional differential equations explicitly involves the shape dynamics Hamiltonian, and thus might be non-local. If one restricts this extra piece of the temporal anomaly to be also local, then overall no Weyl anomalies, either temporal or spatial, emerge in the 3+1 case.

Guidance and Control for Planetary Landing: Flatness-Based Approach

The guidance, navigation, and control problems for autonomous planetary entry descent and landing pose a number of interesting challenges. In this paper, the guidance and control problems for the descent phase are considered, and an approach based on the flatness property of the descent equations is proposed. A flat reformulation of the 2-D descent equations is presented and flatness-based analytical solutions and numerical schemes based on polynomial and pseudo-spectral approximations for the problem of guidance law design are compared. Subsequently, the problem of designing trajectory tracking control laws is considered and two design approaches are developed and compared in a simulation study. The first approach is based on feedforward linearization and leads to a nonlinear controller, while the second one is based on linear time-varying feedback computed by linearizing the descent equations along the computed trajectory. Finally, a simulation study is carried out, the results of which show that the nonlinear flatness-based controller provides a more satisfactory solution in terms of robustness.

Anomaly inflow and membrane dynamics in the QCD vacuum

Large ${N}_{c}$ and holographic arguments, as well as Monte Carlo results, suggest that the topological structure of the QCD vacuum is dominated by codimension-one membranes which appear as thin dipole layers of topological charge. Such membranes arise naturally as $D6$-branes in the holographic formulation of QCD based on IIA string theory. The polarizability of these membranes leads to a vacuum energy $\ensuremath{\propto}{\ensuremath{\theta}}^{2}$, providing the origin of nonzero topological susceptibility. Here we show that the axial U(1) anomaly can be formulated as anomaly inflow on the brane surfaces. A 4D gauge transformation at the brane surface separates into a 3D gauge transformation of components within the brane and the transformation of the transverse component. The in-brane gauge transformation induces currents of an effective Chern-Simons theory on the brane surface, while the transformation of the transverse component describes the transverse motion of the brane and is related to the Ramond-Ramond closed string field in the holographic formulation of QCD. The relation between the surface currents and the transverse motion of the brane is dictated by the descent equations of Yang-Mills theory.