SP Theory Research Articles

S-duality and the prepotential of N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ theories (II): the non-simply laced algebras

We derive a modular anomaly equation satisfied by the prepotential of the $$ \mathcal{N}={2}^{\star } $$ supersymmetric theories with non-simply laced gauge algebras, including the classical B r and C r infinite series and the exceptional F 4 and G 2 cases. This equation determines the exact prepotential recursively in an expansion for small mass in terms of quasi-modular forms of the S-duality group. We also discuss the behaviour of these theories under S-duality and show that the prepotential of the SO(2r + 1) theory is mapped to that of the Sp(2r) theory and viceversa, while the exceptional F 4 and G 2 theories are mapped into themselves (up to a rotation of the roots) in analogy with what happens for the $$ \mathcal{N}=4 $$ supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper.

Symplectic critical models in 6 + ϵ dimensions

We consider nontrivial critical models in d=6+ϵ spacetime dimensions with anticommuting scalars transforming under the symplectic group Sp(N). These models are nonunitary, but the couplings are real and all operator dimensions are positive. At large N we can take ϵ→1 consistently with the loop expansion and thus provide evidence that these theories may be used to define critical models in d=7. The relation of these theories to critical Sp(N) theories, defined similarly to the well-known critical O(N) theories, is examined, and some similarities are pointed out.

Critical Sp(N ) models in 6 − ϵ dimensions and higher spin dS/CFT

Theories of anti-commuting scalar fields are non-unitary, but they are of interest both in statistical mechanics and in studies of the higher spin de Sitter/Conformal Field Theory correspondence. We consider an Sp(N ) invariant theory of N anti-commuting scalars and one commuting scalar, which has cubic interactions and is renormalizable in 6 dimensions. For any even N we find an IR stable fixed point in 6 − ϵ dimensions at imaginary values of coupling constants. Using calculations up to three loop order, we develop ϵ expansions for several operator dimensions and for the sphere free energy F . The conjectured F -theorem is obeyed in spite of the non-unitarity of the theory. The 1/N expansion in the Sp(N ) theory is related to that in the corresponding O(N ) symmetric theory by the change of sign of N . Our results point to the existence of interacting non-unitary 5-dimensional CFTs with Sp(N ) symmetry, where operator dimensions are real. We conjecture that these CFTs are dual to the minimal higher spin theory in 6-dimensional de Sitter space with Neumann future boundary conditions on the scalar field. For N = 2 we show that the IR fixed point possesses an enhanced global symmetry given by the super-group OSp(1|2). This suggests the existence of OSp(1|2) symmetric CFTs in dimensions smaller than 6. We show that the 6 − ϵ expansions of the scaling dimensions and sphere free energy in our OSp(1|2) model are the same as in the q → 0 limit of the q-state Potts model.

General instanton counting and 5d SCFT

Instanton partition functions of 5d $$ \mathcal{N}=1 $$ gauge theories are Witten indices for the ADHM gauged quantum mechanics with (0, 4) SUSY. We derive the integral contour prescriptions for these indices using the Jeffrey-Kirwan method, for gauge theories with hypermultiplets in various representations. The results can be used to study various 4d/5d/6d QFTs. In this paper, we study 5d SCFTs which are at the UV fixed points of 5d SYM theories. In particular, we focus on the Sp(N ) theories with N f ≤ 7 fundamental and 1 antisymmetric hypermultiplets, living on the D4-D8-O8 systems. Their superconformal indices calculated from instantons all show $$ {E_N}_{{}_f+1} $$ symmetry enhancements. We also discuss some aspects of the 6d SCFTs living on the M5-M9 system. It is crucial to understand the UV incompleteness of the 5d SYM, coming from small instantons in our problem. We explain in our examples how to fix them. As an aside, we derive the index for general gauged quantum mechanics with (0, 2) SUSY.

Application of the SP theory of intelligence to the understanding of natural vision and the development of computer vision.

The SP theory of intelligence aims to simplify and integrate concepts in computing and cognition, with information compression as a unifying theme. This article is about how the SP theory may, with advantage, be applied to the understanding of natural vision and the development of computer vision. Potential benefits include an overall simplification of concepts in a universal framework for knowledge and seamless integration of vision with other sensory modalities and other aspects of intelligence. Low level perceptual features such as edges or corners may be identified by the extraction of redundancy in uniform areas in the manner of the run-length encoding technique for information compression. The concept of multiple alignment in the SP theory may be applied to the recognition of objects, and to scene analysis, with a hierarchy of parts and sub-parts, at multiple levels of abstraction, and with family-resemblance or polythetic categories. The theory has potential for the unsupervised learning of visual objects and classes of objects, and suggests how coherent concepts may be derived from fragments. As in natural vision, both recognition and learning in the SP system are robust in the face of errors of omission, commission and substitution. The theory suggests how, via vision, we may piece together a knowledge of the three-dimensional structure of objects and of our environment, it provides an account of how we may see things that are not objectively present in an image, how we may recognise something despite variations in the size of its retinal image, and how raster graphics and vector graphics may be unified. And it has things to say about the phenomena of lightness constancy and colour constancy, the role of context in recognition, ambiguities in visual perception, and the integration of vision with other senses and other aspects of intelligence.

Big Data and the SP Theory of Intelligence

This paper is about how the SP theory of intelligence and its realization in the SP machine may, with advantage, be applied to the management and analysis of big data. The SP system-introduced in this paper and fully described elsewhere-may help to overcome the problem of variety in big data; it has potential as a universal framework for the representation and processing of diverse kinds of knowledge, helping to reduce the diversity of formalisms and formats for knowledge, and the different ways in which they are processed. It has strengths in the unsupervised learning or discovery of structure in data, in pattern recognition, in the parsing and production of natural language, in several kinds of reasoning, and more. It lends itself to the analysis of streaming data, helping to overcome the problem of velocity in big data. Central in the workings of the system is lossless compression of information: making big data smaller and reducing problems of storage and management. There is potential for substantial economies in the transmission of data, for big cuts in the use of energy in computing, for faster processing, and for smaller and lighter computers. The system provides a handle on the problem of veracity in big data, with potential to assist in the management of errors and uncertainties in data. It lends itself to the visualization of knowledge structures and inferential processes. A high-parallel, open-source version of the SP machine would provide a means for researchers everywhere to explore what can be done with the system and to create new versions of it.

Autonomous Robots and the SP Theory of Intelligence

This article is about how the "SP theory of intelligence" and its realisation in the "SP machine" (both outlined in the article) may help to solve computer-related problems in the design of autonomous robots, meaning robots that do not depend on external intelligence or power supplies, are mobile, and are designed to exhibit as much human-like intelligence as possible. The article is about: how to increase the computational and energy efficiency of computers and reduce their bulk; how to achieve human-like versatility in intelligence; and likewise for human-like adaptability in intelligence. The SP system has potential for substantial gains in computational and energy efficiency and reductions in the bulkiness of computers: by reducing the size of data to be processed; by exploiting statistical information that the system gathers; and via an updated version of Donald Hebb's concept of a "cell assembly". Towards human-like versatility in intelligence, the SP system has strengths in unsupervised learning, natural language processing, pattern recognition, information retrieval, several kinds of reasoning, planning, problem solving, and more, with seamless integration amongst structures and functions. The SP system's strengths in unsupervised learning and other aspects of intelligence may help to achieve human-like adaptability in intelligence via: the learning of natural language; learning to see; building 3D models of objects and of a robot's surroundings; learning regularities in the workings of a robot and in the robot's environment; exploration and play; learning major skills; and secondary forms of learning. Also discussed are: how the SP system may process parallel streams of information; generalisation of knowledge, correction of over-generalisations, and learning from dirty data; how to cut the cost of learning; and reinforcements, motivations, goals, and demonstration.

The SP Theory of Intelligence: Benefits and Applications

This article describes existing and expected benefits of the SP theory ofintelligence, and some potential applications. The theory aims to simplify and integrate ideasacross artificial intelligence, mainstream computing, and human perception and cognition,with information compression as a unifying theme. It combines conceptual simplicitywith descriptive and explanatory power across several areas of computing and cognition.In the SP machine—an expression of the SP theory which is currently realized in theform of a computer model—there is potential for an overall simplification of computingsystems, including software. The SP theory promises deeper insights and better solutions inseveral areas of application including, most notably, unsupervised learning, natural languageprocessing, autonomous robots, computer vision, intelligent databases, software engineering,information compression, medical diagnosis and big data. There is also potential inareas such as the semantic web, bioinformatics, structuring of documents, the detection ofcomputer viruses, data fusion, new kinds of computer, and the development of scientifictheories. The theory promises seamless integration of structures and functions within andbetween different areas of application. The potential value, worldwide, of these benefits andapplications is at least $190 billion each year. Further development would be facilitatedby the creation of a high-parallel, open-source version of the SP machine, available toresearchers everywhere.

An Assessment Method for Individual Credit Risk Based on SP Theory

An Assessment Method for Individual Credit Risk Based on SP Theory

The SP Theory of Intelligence: An Overview

This article is an overview of the SP theory of intelligence, which aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme. It is conceived of as a brain-like system that receives "New" information and stores some or all of it in compressed form as "Old" information; and it is realised in the form of a computer model, a first version of the SP machine. The matching and unification of patterns and the concept of multiple alignment are central ideas. Using heuristic techniques, the system builds multiple alignments that are "good" in terms of information compression. For each multiple alignment, probabilities may be calculated for associated inferences. Unsupervised learning is done by deriving new structures from partial matches between patterns and via heuristic search for sets of structures that are "good" in terms of information compression. These are normally ones that people judge to be "natural", in accordance with the "DONSVIC" principle—the discovery of natural structures via information compression. The SP theory provides an interpretation for concepts and phenomena in several other areas, including "computing", aspects of mathematics and logic, the representation of knowledge, natural language processing, pattern recognition, several kinds of reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience and human perception and cognition. Examples include the parsing and production of language with discontinuous dependencies in syntax, pattern recognition at multiple levels of abstraction and its integration with part-whole relations, nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks, including "explaining away", causal diagnosis, and the solving of a geometric analogy problem.

F-theory, Seiberg-Witten curves and $ \mathcal{N} = {2} $ dualities

F-theoretic constructions can alternatively be understood as consequences of certain N = 2 Seiberg-Witten theories via type IIB r D3s probing the quantum corrected orientifold backgrounds. We present four models that come out from such consideration. In Model 1, the 7-branes wrap the flat R^4 directions, leading to the well known Sp(2r) theories. We study singularity structure of moduli space of Seiberg-Witten curve, such as maximal Argyres-Douglas loci, in order to construct 1-1 map between moduli spaces. In Model 2, the 7-branes are wrapped on Taub-NUT and multi Taub-NUT spaces instead of R^4. These configurations may explain many of the Gaiotto-type constructions including possible extensions to non-conformal models with cascading behaviors. In this model the UV is described by the probe D3s decomposed into D5-anti D5 pairs wrapped on multi Taub-NUT space, while the IR remains a 4d theory. For certain arrangements of the 7-branes, this model may be dualized to the brane networks of Benini, Benvenuti and Tachikawa. The Gaiotto dualities in Model 2 are explained by chiral anomaly cancellations, anti-GSO projections and brane transmutations. Model 3 is described by seven-branes wrapped on a K3 manifold and D3 and anti-D3 probes whose number may differ at most by 24. These constructions could lead to new N = 2 models with possible dualities to both type IIB and heterotic theories on non-Kahler manifolds. In the limit where the number of probes becomes very large, the physics is captured by M(atrix) theory on K3 x K3 manifold. Finally, Model 4 is described by k D3s probing intersecting seven-brane backgrounds with Sp(2k) x Sp(2k) gauge group. These constructions could produce new N =1 heterotic dual on a non-Kahler K3 manifold that is no longer conformally Calabi-Yau. We discuss possible constraints on these models coming from global charge and anomaly cancellations in F-theory.

Superconformal Indices of $${{\mathcal N}=4}$$ SYM Field Theories

Superconformal indices (SCIs) of 4d ${\mathcal N}=4$ SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For $F_4, E_6, E_7, E_8$ gauge groups this yields first examples of integrals of such type. S-duality transformation for G_2 and F_4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N+1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d $\mathcal{N}=2$ SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.

From SO/Sp instantons to W-algebra blocks

We study instanton partition functions for N=2 superconformal Sp(1) and SO(4) gauge theories. We find that they agree with the corresponding U(2) instanton partitions functions only after a non-trivial mapping of the microscopic gauge couplings, since the instanton counting involves different renormalization schemes. Geometrically, this mapping relates the Gaiotto curves of the different realizations as double coverings. We then formulate an AGT-type correspondence between Sp(1)/SO(4) instanton partition functions and chiral blocks with an underlying W(2,2)-algebra symmetry. This form of the correspondence eliminates the need to divide out extra U(1) factors. Finally, to check this correspondence for linear quivers, we compute expressions for the Sp(1)-SO(4) half-bifundamental.

Conformal windows of SP(2N) and SO(N) gauge theories from topological excitations on ℝ3×S1

We derive an estimate of the lower boundary of the conformal window of SP(2N) and SO(N) gauge theories with fermionic matter in several different representations. We calculate the index of topological excitations for these groups on the manifold 3 × S1, from which we deduce the scale of the generation of the mass gap of the theory. This is then used to approximate the critical value of the number of species nf* for the onset of conformality on 4. We also provide a detailed comparison with other estimates of the conformal window.

The dimensions of [formula omitted] codes

A family of LDPC codes, called LU ( 3 , q ) codes, has been constructed from q-regular bipartite graphs. Recently, P. Sin and Q. Xiang determined the dimensions of these codes in the case that q is a power of an odd prime. They also obtained a lower bound for the dimension of an LU ( 3 , q ) code when q is a power of 2. In this paper we prove that this lower bound is the exact dimension of the LU ( 3 , q ) code. The proof involves the geometry of symplectic generalized quadrangles, the representation theory of Sp ( 4 , q ) , and the ring of polynomials.

A note on S-duality for the Script N = 1* Sp(2n) and SO(2n+1) super-Yang-Mills theories

We study the N=1* supersymmetric gauge theories with gauge groups Sp(2n) and SO(2n+1). These theories are obtained from the corresponding N=4 supersymmetric Yang-Mills theories via a mass deformation. We show that the number of quantum vacua in the Sp(2n) theory is equal to the number of quantum vacua in the SO(2n+1) theory. This constitutes non-trivial support for S-duality between these theories. The verification of the equality of the number of quantum vacua involves a rather esoteric identity due to Ramanujan.

From the Hamiltonian to the Lagrangean formalism for 1-reducible theories. The Freedman-Townsend model

Abstract The paper presents a possible path to the sp(3) BRST Lagrangean formalism for a 1-reducible gauge field theory starting from the Hamiltonian one. This appears to be not at all a trivial attempt and will allow explanation of the structure of generators and the form of the master equations in the Lagrangean sp(3) theories. The Freedman-Townsend model, for which a Lagrangean (covariant) sp(3) theory is important, is presented.

The Simplicity and Power model for inductive inference

With this paper we wish to present a simplicity (informally `simple explanations are the best') formalism that is easily and directly applicable to modeling problems in cognitive science. While simplicity has been extensively advocated as a psychologically relevant principle, a general modeling formalism has been lacking. The Simplicity and Power model (SP) is a particular simplicity-based framework, that has been supported in machine learning (Wolff, Unifying computing and cognition: the SP theory and its applications, 2006). We propose its utility in cognitive modeling. For illustration, we provide SP demonstrations of the trade-off between encoding with whole exemplars versus parts of stimuli in learning and the effect of wide versus narrow distributions in categorization. In both cases, SP computations show how simplicity can account for these contrasts, in terms of how the frequency of individual exemplars in training compares to the frequency of their constituent parts.

Towards an intelligent database system founded on the SP theory of computing and cognition

The SP theory of computing and cognition, described in previous publications, is an attractive model for intelligent databases because it provides a simple but versatile format for different kinds of knowledge, it has capabilities in artificial intelligence, it can function effectively in the face of errors in its input data, and it can function like established database models when that is required. This paper first describes the SP theory in outline and the computer models in which it is expressed. The main sections of the paper describe, with examples from the SP62 computer model, how the SP framework can emulate other abstract models used in database applications: the relational model (including retrieval of information in the manner of query-by-example, creating a join between two or more tables, and aggregation), object-oriented models (including class-inclusion hierarchies, part-whole hierarchies and their integration, inheritance of attributes, cross-classification and multiple inheritance), and hierarchical and network models (including discrimination networks). Comparisons are made between the SP model and those other models. The artificial intelligence capabilities of the SP model are briefly reviewed: representation and integration of diverse kinds of knowledge in one versatile format; fuzzy pattern recognition and recognition at multiple levels of abstraction; best-match and semantic forms of information retrieval; various kinds of exact reasoning and probabilistic reasoning; analysis and production of natural language; planning; problem solving; and unsupervised learning. Also considered are ways in which current prototypes may be translated into an ‘industrial strength’ working system.

Medical diagnosis as pattern recognition in a framework of information compression by multiple alignment, unification and search

This paper describes a novel approach to medical diagnosis based on the SP theory of computing and cognition. The main attractions of this approach are: a format for representing diseases that is simple and intuitive; an ability to cope with errors and uncertainties in diagnostic information; the simplicity of storing statistical information as frequencies of occurrence of diseases; a method for evaluating alternative diagnostic hypotheses that yields true probabilities; and a framework that should facilitate unsupervised learning of medical knowledge and the integration of medical diagnosis with other AI applications.