Magnetic Force-Free Theory: Nonlinear Case

In this paper, a theory of force-free magnetic field useful for explaining the formation of convex closed sets, bounded by a magnetic separatrix in the plasma, is developed. This question is not new and has been addressed by many authors. Force-free magnetic fields appear in many laboratory and astrophysical plasmas. These fields are defined by the solution of the problem ∇×B=ΛB with some field conditions B∂Ω on the boundary ∂Ω of the plasma region. In many physical situations, it has been noticed that Λ is not constant but may vary in the domain Ω giving rise to many different interesting physical situations. We set Λ=Λ(ψ) with ψ being the poloidal magnetic flux function. Then, an analytic method, based on a first-order expansion of ψ with respect to a small parameter α, is developed. The Grad–Shafranov equation for ψ is solved by expanding the solution in the eigenfunctions of the zero-order operator. An analytic expression for the solution is obtained deriving results on the transition through resonances, the amplification with respect to the gun inflow. Thus, the formation of Spheromaks or Protosphera structure of the plasma is determined in the case of nonconstant Λ.

Overview of Machine Learning Process Modelling

Much research has been conducted in the area of machine learning algorithms; however, the question of a general description of an artificial learner’s (empirical) performance has mainly remained unanswered. A general, restrictions-free theory on its performance has not been developed yet. In this study, we investigate which function most appropriately describes learning curves produced by several machine learning algorithms, and how well these curves can predict the future performance of an algorithm. Decision trees, neural networks, Naïve Bayes, and Support Vector Machines were applied to 130 datasets from publicly available repositories. Three different functions (power, logarithmic, and exponential) were fit to the measured outputs. Using rigorous statistical methods and two measures for the goodness-of-fit, the power law model proved to be the most appropriate model for describing the learning curve produced by the algorithms in terms of goodness-of-fit and prediction capabilities. The presented study, first of its kind in scale and rigour, provides results (and methods) that can be used to assess the performance of novel or existing artificial learners and forecast their ‘capacity to learn’ based on the amount of available or desired data.

Complexity growth in integrable and chaotic models

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Logarithmic expansion of field theories: higher orders and resummations

A formal expansion for the Green’s functions of a quantum field theory in a parameter $$\delta $$ δ that encodes the “distance” between the interacting and the corresponding free theory was introduced in the late 1980s (and recently reconsidered in connection with non-hermitian theories), and the first order in $$\delta $$ δ was calculated. In this paper we study the $${\mathcal {O}}(\delta ^2)$$ O ( δ 2 ) systematically, and also push the analysis to higher orders. We find that at each finite order in $$\delta $$ δ the theory is non-interacting: sensible physical results are obtained only resorting to resummations. We then perform the resummation of UV leading and subleading diagrams, getting the $${\mathcal {O}}(g)$$ O ( g ) and $${\mathcal {O}}(g^2)$$ O ( g 2 ) weak-coupling results. In this manner we establish a bridge between the two expansions, provide a powerful and unique test of the logarithmic expansion, and pave the way for further studies.

Perturbative linearization of super-Yang-Mills theories in general gauges

Supersymmetric Yang-Mills theories can be characterized by a non-local and non-linear transformation of the bosonic fields (Nicolai map) mapping the interacting functional measure to that of a free theory, such that the Jacobi determinant of the transformation equals the product of the fermionic determinants obtained by integrating out the gauginos and ghosts at least on the gauge hypersurface. While this transformation has been known so far only for the Landau gauge and to third order in the Yang-Mills coupling, we here extend the construction to a large class of (possibly non-linear and non-local) gauges, and exhibit the conditions for all statements to remain valid off the gauge hypersurface. Finally, we present explicit results to second order in the axial gauge and to fourth order in the Landau gauge.

On Galilean conformal bootstrap

In this work, we develop conformal bootstrap for Galilean conformal field theory (GCFT). In a GCFT, the Hilbert space could be decomposed into quasiprimary states and its global descendants. Different from the usual conformal field theory, the quasiprimary states in a GCFT constitute multiplets, which are block-diagonized under the Galilean boost operator. More importantly the multiplets include the states of negative norms, indicating the theory is not unitary. We compute global blocks of the multiplets, and discuss the expansion of four-point functions in terms of the global blocks of the multiplets. Furthermore we do the harmonic analysis for the Galilean conformal symmetry and obtain an inversion formula. As the first step to apply the Galilean conformal bootstrap, we construct generalized Galilean free theory (GGFT) explicitly. We read the data of GGFT by using Taylor series expansion of four-point function and the inversion formula independently, and find exact agreement. We discuss some novel features in the Galilean conformal bootstrap, due to the non-semisimpleness of the Galilean conformal algebra and the non-unitarity of the GCFTs.

Large-momentum behaviour in quantum field theory (QFT)

Renormalization group (RG) equations are used to characterize the large momentum behaviour of renormalized quantum field theories (QFT), assuming implicitly that such a universal large momentum physics can be defined, something which, beyond perturbation theory is not obvious. Since the initial effective QFT is valid only up to an energy-momentum scale much smaller than some cut-off, large momentum means much larger than the renormalization scale, but still much smaller than the cut-off scale. The existence of this large momentum physics implies the existence of a crossover scale between low and large momentum physics. One theoretic reason for discussing the large momentum behaviour is the apparent connection between the existence of consistent interacting renormalized QFTs and the presence of ultraviolet (UV) fixed points. The absence of identified UV fixed points in infrared-free QFTs, like the φ4 field theory or quantum electrodynamics (QED), leads to the triviality issue. The physics reason is that in collisions it is observed that quarks, fundamental particles of the Standard Model (SM) of particle physics, behave like free particles at the shortest distances presently accessible (the property of asymptotic freedom). This property can be explained by RG arguments if the free theory is an attractive UV fixed point. Therefore, the identification of QFTs where the free theory is an UV fixed point is important, and this has led to examine the large momentum behaviour of all QFTs renormalizable in four dimensions. It is shown that only theories having a non-Abelian gauge symmetry can be asymptotically free. As an application, the total cross section of electron–positron annihilation into hadrons at large momentum is calculated.

Superconformal boundaries in 4 − ϵ dimensions

Boundaries in three-dimensional $$ \mathcal{N} $$ N = 2 superconformal theories may preserve one half of the original bulk supersymmetry. There are two possibilities which are characterized by the chirality of the leftover supercharges. Depending on the choice, the remaining 2d boundary algebra exhibits $$ \mathcal{N} $$ N = (0, 2) or $$ \mathcal{N} $$ N = (1) supersymmetry. In this work we focus on correlation functions of chiral fields for both types of supersymmetric boundaries. We study a host of correlators using superspace techniques and calculate superconformal blocks for two- and three-point functions. For $$ \mathcal{N} $$ N = (1) supersymmetry, some of our results can be analytically continued in the spacetime dimension while keeping the codimension fixed. This opens the door for a bootstrap analysis of the ϵ-expansion in supersymmetric BCFTs. Armed with our analytically-continued superblocks, we prove that in the free theory limit two-point functions of chiral (and antichiral) fields are unique. The first order correction, which already describes interactions, is universal up to two free parameters. As a check of our analysis, we study the Wess-Zumino model with a super-symmetric boundary using Feynman diagrams, and find perfect agreement between the perturbative and bootstrap results.

ABJ correlators with weakly broken higher spin symmetry

We consider four-point functions of operators in the stress tensor multiplet of the 3d $$ \mathcal{N} $$ N = 6 U(N)k× U(N + M)−k or SO(2)2k× USp(2 + 2M)−k ABJ theories in the limit where M and k are taken to infinity while N and λ ∼ M/k are held fixed. In this limit, these theories have weakly broken higher spin symmetry and are holographically dual to $$ \mathcal{N} $$ N = 6 higher spin gravity on AdS4, where λ is dual to the bulk parity breaking parameter. We use the weakly broken higher spin Ward identities, superconformal Ward identities, and the Lorentzian inversion formula to fully determine the tree level stress tensor multiplet four-point function up to two free parameters. We then use supersymmetric localization to fix both parameters for the ABJ theories in terms of λ, so that our result for the tree level correlator interpolates between the free theory at λ = 0 and a parity invariant interacting theory at λ = 1/2. We compare the CFT data extracted from this correlator to a recent numerical bootstrap conjecture for the exact spectrum of U(1)2M× U(1 + M)−2M ABJ theory (i.e. λ = 1/2 and N = 1), and find good agreement in the higher spin regime.

Fusion of conformal defects in four dimensions

We consider two conformal defects close to each other in a free theory, and study what happens as the distance between them goes to zero. This limit is the same as zooming out, and the two defects have fused to another defect. As we zoom in we find a non-conformal effective action for the fused defect. Among other things this means that we cannot in general decompose the two-point correlator of two defects in terms of other conformal defects. We prove the fusion using the path integral formalism by treating the defects as sources for a scalar in the bulk.