Towards Covariant Quantization of the Supermembrane

By replacing ten-dimensional pure spinors with eleven-dimensional pure spinors, the formalism recently developed for covariantly quantizing the d=10 superparticle and superstring is extended to the d=11 superparticle and supermembrane. In this formalism, kappa symmetry is replaced by a BRST-like invariance using the nilpotent operator Q=int (lambda^alpha d_alpha) where d_alpha is the worldvolume variable corresponding to the d=11 spacetime supersymmetric derivative and lambda^alpha is an SO(10,1) pure spinor variable satisfying (lambda Gamma^c lambda)=0 for c=1 to 11. Super-Poincare covariant unintegrated and integrated supermembrane vertex operators are explicitly constructed which are in the cohomology of Q. After double-dimensional reduction of the eleventh dimension, these vertex operators are related to Type IIA superstring vertex operators where Q=Q_L+Q_R is the sum of the left and right-moving Type IIA BRST operators and the eleventh component of the pure spinor constraint, (lambda Gamma^{11} lambda)=0, replaces the (b_L^0 - b_R^0) constraint of the closed superstring. A conjecture is made for the computation of M-theory scattering amplitudes using these supermembrane vertex operators.Comment: Changed title since the hamiltonian still needs to be regularized. Added reference to Cederwall, Nilsson and Tsimpi

Free Field Representations and Screening Operators for the N=4N=4 Doubly Extended Superconformal Algebras

We present explicit free field representations for the $N=4$ doubly extended superconformal algebra, $\tilde{\cal{A}}_{\gamma}$. This algebra generalizes and contains all previous $N=4$ superconformal algebras. We have found $\tilde{\cal{A}}_{\gamma}$ to be obtained by hamiltonian reduction of the Lie superalgebra $D(2|1;\alpha)$. In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to present a natural conjecture for the Kac determinant generalizing a previous conjecture by Kent and Riggs for the singly extended case. The results support and illuminate several aspects of the characters of this algebra previously obtained by Taormina and one of us.Comment: 15 pages, Late

A Jost-Pais-type reduction of Fredholm determinants and some applications

We study the analog of semi-separable integral kernels in \cH of the type {equation*} K(x,x')={cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, {cases} {equation*} where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, F_j (x) \in \cB_2(\cH_j,\cH) and G_j(x) \in \cB_2(\cH,\cH_j) such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and {equation*} \|F_j(\cdot)\|_{\cB_2(\cH_j,\cH)} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\cB_2(\cH,\cH_j)} \in L^2((a,b)), \quad j=1,2, {equation*} with \cH and \cH_j, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a trace class operator \bsK in L^2((a,b);\cH), we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the Fredholm determinant {\det}_{L^2((a,b);\cH)}(\bsI - \alpha \bsK), \alpha \in \bbC, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \cH (and \cH_1 \oplus \cH_2). Explicit applications of this reduction theory are made to Schr\"odinger operators with suitable bounded operator-valued potentials. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.Comment: 50 pages; some typos remove

A Jost-Pais-type reduction of (modified) Fredholm determinants for semi-separable operators in infinite dimensions

We study the analog of semi-separable integral kernels in $\mathcal{H}$ of the type $$K(x,x')=\begin{cases} F_1(x)G_1(x'), & a<x'< x< b, \\ F_2(x)G_2(x'), & a<x<x'<b, \end{cases}$$ where $-\infty\leq a<b\leq \infty$, and for a.e.\ $x \in (a,b)$, $F_j (x) \in \mathcal{B}_2(\mathcal{H}_j,\mathcal{H})$ and $G_j(x) \in \mathcal{B}_2(\mathcal{H},\mathcal{H}_j)$ such that $F_j(\cdot)$ and $G_j(\cdot)$ are uniformly measurable, and $$\|F_j(\cdot)\|_{\mathcal{B}_2(\mathcal{H}_j,\mathcal{H})} \in L^2((a,b)), \; \|G_j (\cdot)\|_{\mathcal{B}_2(\mathcal{H},\mathcal{H}_j)} \in L^2((a,b)), \quad j=1,2,$$ with $\mathcal{H}$ and $\mathcal{H}_j$, $j=1,2$, complex, separable Hilbert spaces. Assuming that $K(\cdot, \cdot)$ generates a Hilbert-Schmidt operator $\mathbf{K}$ in $L^2((a,b);\mathcal{H})$, we derive the analog of the Jost-Pais reduction theory that succeeds in proving that the modified Fredholm determinant ${\det}_{2, L^2((a,b);\mathcal{H})}(\mathbf{I} - \alpha \mathbf{K})$, $\alpha \in \mathbb{C}$, naturally reduces to appropriate Fredholm determinants in the Hilbert spaces $\mathcal{H}$ (and $\mathcal{H} \oplus \mathcal{H}$). Some applications to Schr\"odinger operators with operator-valued potentials are provided.Comment: 25 pages; typos removed. arXiv admin note: substantial text overlap with arXiv:1404.073

Estimation of smooth functionals of covariance operators: jackknife bias reduction and bounds in terms of effective rank

Let $E$ be a separable Banach space and let $X, X_1,\dots, X_n, \dots$ be i.i.d. Gaussian random variables taking values in $E$ with mean zero and unknown covariance operator $\Sigma: E^{\ast}\mapsto E.$ The complexity of estimation of $\Sigma$ based on observations $X_1,\dots, X_n$ is naturally characterized by the so called effective rank of $\Sigma:$ ${\bf r}(\Sigma):= \frac{{\mathbb E}_{\Sigma}\|X\|^2}{\|\Sigma\|},$ where $\|\Sigma\|$ is the operator norm of $\Sigma.$ Given a smooth real valued functional $f$ defined on the space $L(E^{\ast},E)$ of symmetric linear operators from $E^{\ast}$ into $E$ (equipped with the operator norm), our goal is to study the problem of estimation of $f(\Sigma)$ based on $X_1,\dots, X_n.$ The estimators of $f(\Sigma)$ based on jackknife type bias reduction are considered and the dependence of their Orlicz norm error rates on effective rank ${\bf r}(\Sigma),$ the sample size $n$ and the degree of H\"older smoothness $s$ of functional $f$ are studied. In particular, it is shown that, if ${\bf r}(\Sigma)\lesssim n^{\alpha}$ for some $\alpha\in (0,1)$ and $s\geq \frac{1}{1-\alpha},$ then the classical $\sqrt{n}$-rate is attainable and, if $s> \frac{1}{1-\alpha},$ then asymptotic normality and asymptotic efficiency of the resulting estimators hold. Previously, the results of this type (for different estimators) were obtained only in the case of finite dimensional Euclidean space $E={\mathbb R}^d$ and for covariance operators $\Sigma$ whose spectrum is bounded away from zero (in which case, ${\bf r}(\Sigma)\asymp d$)

A perturbative study of two four-quark operators in finite volume renormalization schemes

Starting from the QCD Schroedinger functional (SF), we define a family of renormalization schemes for two four-quark operators, which are, in the chiral limit, protected against mixing with other operators. With the appropriate flavour assignments these operators can be interpreted as part of either the $\Delta F=1$ or $\Delta F=2$ effective weak Hamiltonians. In view of lattice QCD with Wilson-type quarks, we focus on the parity odd components of the operators, since these are multiplicatively renormalized both on the lattice and in continuum schemes. We consider 9 different SF schemes and relate them to commonly used continuum schemes at one-loop order of perturbation theory. In this way the two-loop anomalous dimensions in the SF schemes can be inferred. As a by-product of our calculation we also obtain the one-loop cutoff effects in the step-scaling functions of the respective renormalization constants, for both O(a) improved and unimproved Wilson quarks. Our results will be needed in a separate study of the non-perturbative scale evolution of these operators.Comment: 22 pages, 4 figure

Order a improved renormalization constants

We present non-perturbative results for the constants needed for on-shell $O(a)$ improvement of bilinear operators composed of Wilson fermions. We work at $\beta=6.0$ and 6.2 in the quenched approximation. The calculation is done by imposing axial and vector Ward identities on correlators similar to those used in standard hadron mass calculations. A crucial feature of the calculation is the use of non-degenerate quarks. We also obtain results for the constants needed for off-shell $O(a)$ improvement of bilinears, and for the scale and scheme independent renormalization constants, (Z_A), (Z_V) and (Z_S/Z_P). Several of the constants are determined using a variety of different Ward identities, and we compare their relative efficacies. In this way, we find a method for calculating $c_V$ that gives smaller errors than that used previously. Wherever possible, we compare our results with those of the ALPHA collaboration (who use the Schr\"odinger functional) and with 1-loop tadpole-improved perturbation theory.Comment: 48 pages. Modified "axis" source for figures also included. Typos corrected (version published in Phys. Rev. D