scholarly journals Smooth functorial field theories from B-fields and D-branes

2021 ◽  
Vol 16 (1) ◽  
pp. 75-153
Author(s):  
Severin Bunk ◽  
Konrad Waldorf
Keyword(s):  
Field Theory ◽  
Sigma Models ◽  
Lagrangian Approach ◽  
Field Theories ◽  
Open Strings ◽  
Homotopy Invariant ◽  
Target Manifold ◽  
Definition Of ◽  
Smooth Map

AbstractIn the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.

Integrability in 2D fields theory/sigma-models

2019 ◽  
pp. 248-318
Author(s):  
Sergei L. Lukyanov ◽  
Alexander B. Zamolodchikov
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Sigma Models ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field ◽  
Zero Curvature ◽  
Basic Facts ◽  
Linear Sigma Models ◽  
General Aspects

This is a two-part course about the integrability of two-dimensional non-linear sigma models (2D NLSM). In the first part general aspects of classical integrability are discussed, based on the O(3) and O(4) sigma-models and the field theories related to them. The second part is devoted to the quantum 2D NLSM. Among the topics considered are: basic facts of conformal field theory, zero-curvature representations, integrals of motion, one-loop renormalizability of 2D NLSM, integrable structures in the so-called cigar and sausage models, and their RG flows. The text contains a large number of exercises of varying levels of difficulty.

The Principles of Renormalisation

2021 ◽  
pp. 304-328
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Power Counting ◽  
Field Theories ◽  
Asymptotic Freedom ◽  
Green Functions ◽  
Quantum Field Theories ◽  
Quantum Field

Loop diagrams often yield ultraviolet divergent integrals. We introduce the concept of one-particle irreducible diagrams and develop the power counting argument which makes possible the classification of quantum field theories into non-renormalisable, renormalisable and super-renormalisable. We describe some regularisation schemes with particular emphasis on dimensional regularisation. The renormalisation programme is described at one loop order for φ‎4 and QED. We argue, without presenting the detailed proof, that the programme can be extended to any finite order in the perturbation expansion for every renormalisable (or super-renormalisable) quantum field theory. We derive the equation of the renormalisation group and explain how it can be used in order to study the asymptotic behaviour of Green functions. This makes it possible to introduce the concept of asymptotic freedom.

scholarly journals TOWARDS A CLASSIFICATION OF FINITE FIELD THEORIES

1994 ◽  
Vol 09 (27) ◽  
pp. 2555-2567
Author(s):  
PETER GRANDITS
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Finite Field ◽  
Field Theories ◽  
Quantum Field ◽  
Particle Content ◽  
Finiteness Conditions ◽  
Gauge Groups ◽  
Yukawa Couplings

We consider the finiteness conditions on the Yukawa couplings of a general quantum field theory for gauge groups SU (n)(n>6) and a rather general particle content. It is shown that in the class of theories considered (149 different particle contents), only two models are able to fulfill the finiteness conditions. Only one of these is supersymmetric. For the nonsupersymmetric one the appropriate Yukawa couplings are constructed explicitly.

STATISTICS OF 2D SOLITONS

1992 ◽  
Vol 07 (11) ◽  
pp. 2589-2600 ◽  
Author(s):  
LEE BREKKE ◽  
TOM D. IMBO
Keyword(s):  
Sigma Models ◽  
Gauge Theories ◽  
Higher Dimensions ◽  
Fractional Statistics ◽  
Topological Solitons ◽  
Multiply Connected ◽  
Target Manifold ◽  
Definition Of ◽  
Spin And Statistics ◽  
Nonlinear Sigma Models

We study the inequivalent quantizations of (1 + 1)-dimensional nonlinear sigma models with space manifold S1 and target manifold X. If X is multiply connected, these models possess topological solitons. After providing a definition of "spin" and "statistics" for these solitons and demonstrating a spin-statistics correlation, we give various exmples where the solitons can have exotic statistics. In some of these models, the solitons may obey a generalized version of fractional statistics called ambistatistics. The relevance of these 2D models to the statistics of vortices in (2 + 1)-dimensional spontaneously broken gauge theories is also discussed. We close with a discussion concerning the extension of our results to higher dimensions.

scholarly journals SYMMETRIES IN CLASSICAL FIELD THEORY

2004 ◽  
Vol 01 (05) ◽  
pp. 651-710 ◽  
Author(s):  
MANUEL DE LEÓN ◽  
DAVID MARTÍN DE DIEGO ◽  
AITOR SANTAMARÍA-MERINO
Keyword(s):  
Field Theory ◽  
Conservation Laws ◽  
Classical Field Theory ◽  
Cauchy Data ◽  
Classical Field ◽  
Field Theories ◽  
Classical Field Theories

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.

CLASSIFICATION OF 2- AND 3-FIELD RATIONAL CONFORMAL FIELD THEORIES

1990 ◽  
Vol 05 (25) ◽  
pp. 2063-2070 ◽  
Author(s):  
GIL RIVLIS
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Fusion Rule ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Algebraic Equations ◽  
Conformal Field ◽  
Modular Transformation ◽  
Fusion Algebra

Using the fact that the fusion algebra of a rational conformal field theory is specified in terms of integers that are related to modular transformation properties, we completely classify 2-field chiral RCFT's. We show that the only possibilities for the non-trivial fusion rule are ϕ × ϕ = 1 or ϕ × ϕ = 1 + ϕ. We reduce the 3-field classification to a set of algebraic equations and solve them in a few cases.

scholarly journals NEW SPIN-2 GAUGED SIGMA MODELS AND GENERAL CONFORMAL FIELD THEORY

1998 ◽  
Vol 13 (26) ◽  
pp. 4487-4512 ◽  
Author(s):  
J. DE BOER ◽  
M. B. HALPERN
Keyword(s):  
Field Theory ◽  
Sigma Models ◽  
Large Class ◽  
Sigma Model ◽  
Field Theories ◽  
Nonlinear Sigma Model ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Local Gauge Symmetry ◽  
Action Formulation

Recently, we have studied the general Virasoro construction at one loop in the background of the general nonlinear sigma model. Here, we find the action formulation of these new conformal field theories when the background sigma model is itself conformal. In this case, the new conformal field theories are described by a large class of new spin-2 gauged sigma models. As examples of the new actions, we discuss the spin-2 gauged WZW actions, which describe the conformal field theories of the generic affine-Virasoro construction, and the spin-2 gauged g/h coset constructions. We are able to identify the latter as the actions of the local Lie h-invariant conformal field theories, a large class of generically irrational conformal field theories with a local gauge symmetry.

scholarly journals Gerbes in Geometry, Field Theory, and Quantisation

2021 ◽  
Vol 8 (1) ◽  
pp. 150-182
Author(s):  
Severin Bunk
Keyword(s):  
Field Theory ◽  
Line Bundles ◽  
Differential Cohomology ◽  
Symplectic Forms ◽  
Survey Article ◽  
Geometric Quantisation ◽  
Symplectic Groupoids ◽  
Bundle Gerbes ◽  
Definition Of

Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.

Development and Evaluation of Fuzzy Criteria for the Diagnosis of Rheumatoid Arthritis

1996 ◽  
Vol 35 (04/05) ◽  
pp. 334-342 ◽  
Author(s):  
K.-P. Adlassnig ◽  
G. Kolarz ◽  
H. Leitich
Keyword(s):  
Rheumatoid Arthritis ◽  
Reference Model ◽  
American Rheumatism Association ◽  
Uniform Definition ◽  
Sensitivity Rate ◽  
Fuzzy Techniques ◽  
Definition Of ◽  
Different Levels ◽  
Specificity Rate

Abstract:In 1987, the American Rheumatism Association issued a set of criteria for the classification of rheumatoid arthritis (RA) to provide a uniform definition of RA patients. Fuzzy set theory and fuzzy logic were used to transform this set of criteria into a diagnostic tool that offers diagnoses at different levels of confidence: a definite level, which was consistent with the original criteria definition, as well as several possible and superdefinite levels. Two fuzzy models and a reference model which provided results at a definite level only were applied to 292 clinical cases from a hospital for rheumatic diseases. At the definite level, all models yielded a sensitivity rate of 72.6% and a specificity rate of 87.0%. Sensitivity and specificity rates at the possible levels ranged from 73.3% to 85.6% and from 83.6% to 87.0%. At the superdefinite levels, sensitivity rates ranged from 39.0% to 63.7% and specificity rates from 90.4% to 95.2%. Fuzzy techniques were helpful to add flexibility to preexisting diagnostic criteria in order to obtain diagnoses at the desired level of confidence.

ABOUT THE PROBLEM OF CLASSIFICATION OF THE CONCEPTS OF INFORMATICS STUDIED AT SECONDARY SCHOOL

2018 ◽  
pp. 4-7
Author(s):  
S. I. Zenko
Keyword(s):  
Secondary School ◽  
Foreign Language ◽  
Computer Science ◽  
Parts Of Speech ◽  
The Third ◽  
The Cross ◽  
Definition Of

The article raises the problem of classification of the concepts of computer science and informatics studied at secondary school. The efficiency of creation of techniques of training of pupils in these concepts depends on its solution. The author proposes to consider classifications of the concepts of school informatics from four positions: on the cross-subject basis, the content lines of the educational subject "Informatics", the logical and structural interrelations and interactions of the studied concepts, the etymology of foreign-language and translated words in the definition of the concepts of informatics. As a result of the first classification general and special concepts are allocated; the second classification — inter-content and intra-content concepts; the third classification — stable (steady), expanding, key and auxiliary concepts; the fourth classification — concepts-nouns, conceptsverbs, concepts-adjectives and concepts — combinations of parts of speech.

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