Complex Critical Points Research Articles

Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity

This paper focuses on the semiclassical behavior of the spinfoam quantum gravity in four dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries that contribute dominantly to the large-$j$ Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude. The dominant contribution from the curved geometry to the spinfoam amplitude is proportional to ${e}^{i\mathcal{I}}$, where $\mathcal{I}$ is the Regge action of the geometry plus corrections of higher order in curvature. As a result, in the semiclassical regime, the spinfoam amplitude reduces to an integral over Regge geometries weighted by ${e}^{i\mathcal{I}}$, where $\mathcal{I}$ is the Regge action plus corrections of higher order in curvature. As a by-product, our result also provides a mechanism to relax the cosine problem in the spinfoam model. Our results provide important evidence supporting the semiclassical consistency of the spinfoam quantum gravity.

Analytic continuation of spinfoam models

The Lorentzian Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK) spinfoam model and the Conrady-Hnybida (CH) timelike-surface extension can be expressed in the integral form $\int e^S$. This work studies the analytic continuation of the spinfoam action $S$ to the complexification of the integration domain. Our work extends our knowledge from the real critical points well-studied in the spinfoam large-$j$ asymptotics to general complex critical points of $S$ analytic continued to the complexified domain. The complex critical points satisfying critical equations of the analytic continued $S$. In the large-$j$ regime, the complex critical points give subdominant contributions to the spinfoam amplitude when the real critical points are present. But the contributions from the complex critical points can become dominant when the real critical point are absent. Moreover the contributions from the complex critical points cannot be neglected when the spins $j$ are not large. In this paper, we classify the complex critical points of the spinfoam amplitude, and find a subclass of complex critical points that can be interpreted as 4-dimensional simplicial geometries. In particular, we identify the complex critical points corresponding to the Riemannian simplicial geometries although we start with the Lorentzian spinfoam model. The contribution from these complex critical points of Riemannian geometry to the spinfoam amplitude give $e^{-S_{Regge}}$ in analogy with the Euclidean path integral, where $S_{Regge}$ is the Riemannian Regge action on simplicial complex.

Towards Lefschetz Thimbles in Sigma Models, I

We study two dimensional path integral Lefschetz thimbles, i.e. the possible path integration contours. Specifically, in the examples of the $O(N)$ and ${\bf CP}^{N-1}$ models, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, with various chemical potentials corresponding to the symmetries of the models. In this paper we discuss the case of the $O(2m)$ and the ${\bf CP}^{N-1}$ models in the sector of zero instanton charge, as well as some solutions of the $O(2m+1)$ model. The ${\bf CP}^{N-1}$-model for all instanton charges and a more general class of solutions of the $O(N)$-model with odd $N$ will be discussed in the forthcoming paper.

Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of $n\times n$ Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is $2n-3$.

Double bifurcation for a cubic Kolmogorov model

Our work focuses on the study of a kind of new bifurcation behavior about limit cycles (namely double bifurcation) for a class of cubic Kolmogorov model with a degenerate critical point (1, 1). The positive critical point (1, 1) is a 3-multiple nilpotent fine focus which can become a fine focus of 5th order. By analyzing the change in stability, we show (1, 1) can yield 4 limit cycles. Moreover, the high-order critical point (1, 1) can be broken into three lower-order critical points including a real critical points and two complex critical points under small parameters’ perturbation. Under the condition letting the multiplicity be same one, obtained real critical point (1, 1) can become a fine focus of 2nd order and yield two small limit cycles. In sum, 6 limit cycle can bifurcate near point (1, 1). This is a kind of new bifurcation behavior via double perturbation; in addition, in terms of the Hilbert number of cubic Kolmogorov model, our results are also new.

Critical points via monodromy and local methods

A fundamental problem in many areas of applied mathematics and statistics is to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the function restricted to the model.We compile ideas arising from numerical algebraic geometry to compute these critical points. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.

Bound the number of limit cycles bifurcating from center of polynomial Hamiltonian system via interval analysis

The algebraic criterion for Abelian integral was posed in (Grau et al. Trans Amer Math Soc 2011) and (Mañosas et al. J Differ Equat 2011) to bound the number of limit cycles bifurcating from the center of polynomial Hamiltonian system. Thisapproach reduces the estimation to the number of the limit cycle bifurcating from the center to solve the associated semi-algebraic systems (the system consists of polynomial equations, inequations and polynomial inequalities). In this paper, a systematic procedure with interval analysis has been explored to solve the SASs. In this application, we proved a hyperelliptic Hamiltonian system of degree five with a pair of conjugate complex critical points that could give rise to at most six limit cycles at finite plane under perturbations ɛ(a+bx+cx3+x4)y∂∂x. Moreover we comment the results of some related works that are not reliable by using numerical approximation.

Suppression of quantum tunneling for all spins for easy-axis systems

The semi-classical limit of quantum spin systems corresponds to a dynamical Lagrangian which contains the usual kinetic energy, the couplings and interactions of the spins and an additional, first order kinematical term which corresponds to the Wess-Zumino-Novikov-Witten (WZNW) term for the spin degree of freedom \cite{og}. It was shown that in the case of the kinetic dynamics determined only by the WZNW term, half odd integer spin systems show a lack of tunneling phenomena whereas integer spin systems are subject to it \cite{l} in the case of potentials with easy-plane easy-axis symmetry. Here we prove, for the theory with a normal quadratic kinetic term of arbitrary strength or the first order theory with azimuthal symmetry (which is equivalently the so-called easy-axis situation), that the tunneling is in fact suppressed for all non-zero values of spin. This model exemplifies the concept that in the presence of complex Euclidean action, it is necessary to use the ensuing complex critical points in order to define the quantum (perturbation) theory \cite{ampr}. In the present example, if we do not do so, exactly the opposite, erroneous conclusion, that the tunneling is unsuppressed for all spins, is reached.

Density of Complex Critical Points of a Real Random SO(m+1) Polynomial

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper [Mac09], the author used the Poincare- Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac- Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.

Algebraic Degree of Polynomial Optimization

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP), and pth order cone programming (POCP), in analogy to the algebraic degree of semidefinite programming [J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming, Math. Programm., to appear].

Problems with complex actions

We consider Euclidean functional integrals involving actions that are not exclusively real. This situation arises, for example, when there are t-odd terms in the the Minkowski action. Writing the action in terms of only real fields (which is always possible), such terms appear as explicitly imaginary terms in the Euclidean action. The usual quanization procedure that involves finding the critical points of the action and then quantizing the spectrum of fluctuations about these critical points fails. In the case of complex actions, there do not exist, in general, any critical points of the action on the space of real fields, the critical points are in general complex. The proper definition of the function integral then requires the analytic continuation of the functional integration into the space of complex fields so as to pass through the complex critical points according to the method of steepest descent. We show a simple example where this procedure can be carried out explicitly. The procedure of finding the critical points of the real part of the action and quantizing the corresponding fluctuations, treating the (exponential of the) complex part of the action as a bounded integrable function is shown to fail in our explicit example, at least perturbatively. PACS Nos.: 11.10.–z, 03.70.+k

Partition function zeros for the two-dimensional Ising model VII

The density of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices is investigated near real and complex critical points. The non-uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically, and numerically for a lattice with interactions in the ratios 3:2:1. The limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane. In the neighbourhood of interior critical points one finds only a single layer of zeros. But generally there are two layers or superposed sets of zeros with different distributions, and different limiting densities at boundary critical points. On anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical, by symmetry of the partition function. The divergence of the (two- dimensional) density of zeros at real critical points is discussed briefly in relation to scaling theory.

Extrema of Landau polynomials

For Landau polynomials which are invariant under some representation of a finite group and which satisfy a non-degeneracy condition two methods of calculating the number of orbits of complex critical points with a given isotropy subgroup are given. Neither method uses the explicit form of the polynomial.