Topological Supersymmetry Research Articles

Criticality or Supersymmetry Breaking?

In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1/f-type power spectra and/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found approximation-free supersymmetric theory of stochastics (STS) identifies this phase as the noise-induced chaos (N-phase), i.e., the phase where the topological supersymmetry pertaining to all stochastic dynamical systems is broken spontaneously by the condensation of the noise-induced (anti)instantons. Here, we support this picture in the context of neurodynamics. We study a 1D chain of neuron-like elements and find that the dynamics in the N-phase is indeed featured by positive stochastic Lyapunov exponents and dominated by (anti)instantonic processes of (creation) annihilation of kinks and antikinks, which can be viewed as predecessors of boundaries of neuroavalanches. We also construct the phase diagram of emulated stochastic neurodynamics on Spikey neuromorphic hardware and demonstrate that the width of the N-phase vanishes in the deterministic limit in accordance with STS. As a first result of the application of STS to neurodynamics comes the conclusion that a conscious brain can reside only in the N-phase.

Chaos as a symmetry-breaking phenomenon

Chaos is an ubiquitous and fundamental phenomenon with a wide range of features pointing to a similar phenomenology. Although apparently distinct, it is natural to ask if all these features emerge from a unifying principle. Recently, it was realized that all continuous-time stochastic dynamical systems (DSs) — the most relevant in physics because natural DSs are always subject to noise and time is continuous — possess a topological supersymmetry (TS). It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. This conclusion stems from the fact that such phenomenon encompasses features that are traditionally associated with chaotic dynamics such as non-integrability, positive topological entropy, sensitivity to initial conditions and the Poincaré–Bendixson theorem. Here, we strengthen and complete this picture by showing that the remaining hallmarks of deterministic chaos — topological transitivity/mixing and dense periodic orbits — while being consistent with the TS breaking, do not actually admit a stochastic generalization. This is a major limitation, since all physical systems are always noisy to some extent. We, therefore, conclude that spontaneous TS breaking can be considered as the most general definition of continuous-time dynamical chaos. Contrary to the common perception and semantics of the word “chaos,” this phenomenon should then be truly interpreted as the low-symmetry, or ordered phase of the DSs that manifest it. We also argue that the concept of chaos in low-dimensional, discrete-time DSs that do not obey the Poincaré–Bendixson theorem, is related to the explicit TS breaking.

Supersymmetric theory of stochastic ABC model

In this paper, we investigate numerically the stochastic ABC model, a toy model in the theory of astrophysical kinematic dynamos, within the recently proposed supersymmetric theory of stochastics (STS). STS characterizes stochastic differential equations (SDEs) by the spectrum of the stochastic evolution operator (SEO) on elements of the exterior algebra or differential forms over the system’s phase space, X. STS can thereby classify SDEs as chaotic or non-chaotic by identifying the phenomenon of stochastic chaos with the spontaneously broken topological supersymmetry that SEOs of all SDEs possess. We demonstrate the following three properties of the SEO, deduced previously analytically and from physical arguments: the SEO spectra for zeroth and top degree forms never break topological supersymmetry, all SDEs possess pseudo-time-reversal symmetry, and each de Rham cohomology class provides one supersymmetric eigenstate. Our results also suggest that the SEO spectra for forms of complementary degrees, i.e., k and dim X −k, may be isospectral.

Topological Field Theory and Computing with Instantons

AbstractIt is well known that dynamical systems may be employed as computing machines. However, not all dynamical systems offer particular advantages compared to the standard paradigm of computation, in regard to efficiency and scalability. Recently, it was suggested that a new type of machines, named digital –hence scalable– memcomputing machines (DMMs), that employ non‐linear dynamical systems with memory, can solve complex Boolean problems efficiently. This result was derived using functional analysis without, however, providing a clear understanding of which physical features make DMMs such an efficient computational tool. Here, we show, using recently proposed topological field theory of dynamical systems, that the solution search by DMMs is a composite instanton. This process effectively breaks the topological supersymmetry common to all dynamical systems, including DMMs. The emergent long‐range order – a collective dynamical behavior– allows logic gates of the machines to correlate arbitrarily far away from each other, despite their non‐quantum character. We exemplify these results with the solution of prime factorization, but the conclusions generalize to DMMs applied to any other Boolean problem.

Kinematic dynamo, supersymmetry breaking, and chaos

The kinematic dynamo (KD) describes the growth of magnetic fields generated by the flow of a conducting medium in the limit of vanishing backaction of the fields onto the flow. The KD is therefore an important model system for understanding astrophysical magnetism. Here, the mathematical correspondence between the KD and a specific stochastic differential equation (SDE) viewed from the perspective of the supersymmetric theory of stochastics (STS) is discussed. The STS is a novel, approximation-free framework to investigate SDEs. The correspondence reported here permits insights from the STS to be applied to the theory of KD and vice versa. It was previously known that the fast KD in the idealistic limit of no magnetic diffusion requires chaotic flows. The KD-STS correspondence shows that this is also true for the diffusive KD. From the STS perspective, the KD possesses a topological supersymmetry and the dynamo effect can be viewed as its spontaneous breakdown. This supersymmetry breaking can be regarded as the stochastic generalization of the concept of dynamical chaos. As this supersymmetry breaking happens in both the diffusive and the non-diffusive case, the necessity of the underlying SDE being chaotic is given in either case. The observed exponentially growing and oscillating KD modes prove physically that dynamical spectra of the STS evolution operator that break the topological supersymmetry exist with both, real and complex ground state eigenvalues. Finally, we comment on the non-existence of dynamos for scalar quantities.

Topological sigma models & dissipative hydrodynamics

We outline a universal Schwinger-Keldysh effective theory which describes macroscopic thermal fluctuations of a relativistic field theory. The basic ingredients of our construction are three: a doubling of degrees of freedom, an emergent abelian symmetry associated with entropy, and a topological (BRST) supersymmetry imposing fluctuationdissipation theorem. We illustrate these ideas for a non-linear viscous fluid, and demonstrate that the resulting effective action obeys a generalized fluctuation-dissipation theorem, which guarantees a local form of the second law.

Topological supersymmetry breaking: The definition and stochastic generalization of chaos and the limit of applicability of statistics

The concept of deterministic dynamical chaos has a long history and is well established by now. Nevertheless, its field theoretic essence and its stochastic generalization have been revealed only very recently. Within the newly found supersymmetric theory of stochastics (STS), all stochastic differential equations (SDEs) possess topological or de Rahm supersymmetry and stochastic chaos is the phenomenon of its spontaneous breakdown. Even though the STS is free of approximations and thus is technically solid, it is still missing a firm interpretational basis in order to be physically sound. Here, we make a few important steps toward the construction of the interpretational foundation for the STS. In particular, we discuss that one way to understand why the ground states of chaotic SDEs are conditional (not total) probability distributions, is that some of the variables have infinite memory of initial conditions and thus are not “thermalized”, i.e., cannot be described by the initial-conditions-independent probability distributions. As a result, the definitive assumption of physical statistics that the ground state is a steady-state total probability distribution is not valid for chaotic SDEs.

Topological field theory of dynamical systems

Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively stable due to what is generically known as non-renormalization theorems. As a result, all (equilibrium) dynamical models are divided into three major categories: Markovian models with unbroken Q-symmetry, chaotic models with Q-symmetry spontaneously broken on the mean-field level by, e.g., fractal invariant sets (e.g., strange attractors), and intermittent or self-organized critical (SOC) models with Q-symmetry dynamically broken by the condensation of instanton-antiinstanton configurations (earthquakes, avalanches, etc.) SOC is a full-dimensional phase separating chaos and Markovian dynamics. In the deterministic limit, however, antiinstantons disappear and SOC collapses into the "edge of chaos." Goldstone theorem stands behind spatio-temporal self-similarity of Q-broken phases known under such names as algebraic statistics of avalanches, 1/f noise, sensitivity to initial conditions, etc. Other fundamental differences of Q-broken phases is that they can be effectively viewed as quantum dynamics and that they must also have time-reversal symmetry spontaneously broken. Q-symmetry breaking in non-equilibrium situations (quenches, Barkhausen effect, etc.) is also briefly discussed.

The topological B-model with antisymmetric tensor matter fields

A new type of topological matter interactions involving second-rank antisymmetric tensor matter fields with an underlying N T≧1 topological supersymmetry are proposed. The construction of the 4-dimensional, N T =1 super-BF model and the N T =2 topological B-model with tensor matter are explicitly worked out.

Basic and equivariant cohomology in balanced topological field theory

We present a detailed algebraic study of the N=2 cohomological set-up describing the balanced topological field theory of Dijkgraaf and Moore. We emphasize the role of N=2 topological supersymmetry and sl(2, R) internal symmetry by a systematic use of superfield techniques and of an sl(2, R) covariant formalism. We provide a definition of N=2 basic and equivariant cohomology, generalizing Dijkgraaf’s and Moore’s, and of N=2 connection. For a general manifold with a group action, we show that: (i) the N=2 basic cohomology is isomorphic to the tensor product of the ordinary N=1 basic cohomology and a universal sl(2, R) group theoretic factor; (ii) the affine spaces of N=2 and N=1 connections are isomorphic.

Perturbative Chern-Simons theory on noncommutative Bbb R3

A U(N) Chern-Simons theory on noncommutative $\mathbb{R}^{3}$ is constructed as a $\q$-deformed field theory. The model is characterized by two symmetries: the BRST-symmetry and the topological linear vector supersymmetry. It is shown that the theory is finite and $\q_{\m\n}$-independent at the one loop level and that the calculations respect the restriction of the topological supersymmetry. Thus the topological $\q$-deformed Chern-Simons theory is an example of a model which is non-singular in the limit $\q \to 0$.

A short comment on the supersymmetric structure of Chern-Simons theory in the axial gauge

The topological supersymmetry of the pure Chern-Simons model in three dimensions is established in the case where the theory is defined in the axial gauge.

Topological solitons and zero modes in high-T c superconductivity

The realization of topological solitons, zero modes, and supersymmetry in theCP 1 nonlinear sigma model, a theory of high-T c superconductivity, is presented.

Topological singularities and supersymmetry breaking

The influence of topological singularities (fluxons) on two- and three-dimensional supersymmetric quantum mechanics is investigated. It is proved that the Aharonov-Bohm effect is incompatible with supersymmetry. The violation is a quantum effect and its quantitative amount is determined by the topology.

Bootstrap theory of particle

Abstract Recent developments in S-matrix theory are re-viewed, which have resulted in a topological bootstrap theory of particles. Each component of an S-matrix topological expansion is associated with a pair of intersecting surfaces: a “classical” surface de-scribing space-time aspects of particle interactions, and a “quantum” surface carrying the internal quantum numbers. The theory implies a definite and restricted set of elementary particles, all of which are composite structures built from two kinds of topological constituents that are not themselves particles. One of the two constituents is identified with topological quarks, which appear in 3 colors, 8 flavors, and with integral charges. The lowest level of the topological expansion exhibits a topological supersymmetry involving a single dimensionless coupling constant that has been identified tentatively with the fine-structure constant.

Dual Feynman rules—Topological asymptotic freedom

Feynman-graph rules are formulated for the strong—interaction components of the topological expansion—defined as those graphs all of whose vertices are zero—entropy connected parts. These rules imply a “topological asymptotic freedom” and admit a corresponding perturbative evaluation where the zeroth order exhibits topological supersymmetry.

Topological supersymmetric structure of hadron cross sections

A way of fully implementing unitarity and confinement in the framework of a dual-topological-unitarization theory, including not only mesons but also baryons, has recently been found. This theory consists in the topological description of hadron interactions in terms of two two-dimensional surfaces (a closed "quantum" surface and a bounded "classical" surface). We show that this description directly leads, at the zeroth order of the topological expansion, to certain relations between hadron cross sections, in nice agreement with experimental data. A new topological suppression mechanism is shown to play an important dynamical role. We also point out a new topological supersymmetry property, which leads to realistic experimental consequences. A possible topological origin of the $\ensuremath{\rho}$ and $\ensuremath{\omega}$ universality relations emerges as a by-product of our study.

Zero-Entropy Bootstrap and the Fine-Structure Constant

"Topological supersymmetry," a newly recognized feature of the zero-entropy $S$-matrix bootstrap, leads to a zero-entropy coupling constant ${g}_{0}$ whose order of magnitude is controlled by the relation $8{N}_{g}(8{N}_{g}\ensuremath{-}1)\frac{{{g}_{0}}^{2}}{16{\ensuremath{\pi}}^{2}}\ensuremath{\sim}1$, where ${N}_{g}=4$ is the number of quark generations. A unification principle, suggested by electromagnetic extension of topological theory and encouraged by this relation, is ${g}_{0}=e$. The posture of this unifying $S$-matrix condition with respect to field-theoretical grand unification is discussed.

Topological Theory of Elementary-Hadron Coupling Constants

Topological particle theory is applied to the computation of elementary-hadron coupling constants. All are determined through topological supersymmetry by a single dimensionless "zero-entropy" constnat ${g}_{0}$. The predicted coupling-constant ratios encompass and justify those of Mandelstam. The universality conjecture, ${g}_{0}=e$, is supported (to 6%) by the accurately measured value of the pion-nucleon coupling constant. An explanation emerges for the experimental failure to find baryonium.