scholarly journals COORDINATE-INVARIANT PATH INTEGRAL METHODS IN CONFORMAL FIELD THEORY

2006 ◽  
Vol 21 (17) ◽  
pp. 3525-3563 ◽  
Author(s):  
ANDRÉ VAN TONDER
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Path Integral ◽  
Direct Calculation ◽  
Conformal Anomaly ◽  
Operator Formalism ◽  
Conformal Field ◽  
Integral Methods ◽  
Change Of Variables ◽  
Definition Of

We present a coordinate-invariant approach, based on a Pauli–Villars measure, to the definition of the path integral in two-dimensional conformal field theory. We discuss some advantages of this approach compared to the operator formalism and alternative path integral approaches. We show that our path integral measure is invariant under conformal transformations and field reparametrizations, in contrast to the measure used in the Fujikawa calculation, and we show the agreement, despite different origins, of the conformal anomaly in the two approaches. The natural energy–momentum in the Pauli–Villars approach is a true coordinate-invariant tensor quantity, and we discuss its nontrivial relationship to the corresponding nontensor object arising in the operator formalism, thus providing a novel explanation within a path integral context for the anomalous Ward identities of the latter. We provide a direct calculation of the nontrivial contact terms arising in expectation values of certain energy–momentum products, and we use these to perform a simple consistency check confirming the validity of the change of variables formula for the path integral. Finally, we review the relationship between the conformal anomaly and the energy–momentum two-point functions in our formalism.

scholarly journals SU(2)k×SU(2)l/SU(2)k+l COSET CONFORMAL FIELD THEORY AND TOPOLOGICAL MINIMAL MODEL ON HIGHER GENUS RIEMANN SURFACE

1993 ◽  
Vol 08 (31) ◽  
pp. 5537-5561 ◽  
Author(s):  
HITOSHI KONNO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Riemann Surface ◽  
Vertex Operator ◽  
Correlation Functions ◽  
Minimal Model ◽  
The Other ◽  
Operator Formalism ◽  
Conformal Field ◽  
Higher Genus

We consider the Feigin-Fuchs-Felder formalism of the SU (2)k× SU (2)l/ SU (2)k+l coset minimal conformal field theory and extend it to higher genus. We investigate a double BRST complex with respect to two compatible BRST charges, one associated with the parafermion sector and the other associated with the minimal sector in the theory. The usual screened vertex operator is extended to the BRST-invariant screened three-string vertex. We carry out a sewing operation of these vertices and derive the BRST-invariant screened g-loop operator. The latter operator characterizes the higher genus structure of the theory. An analogous operator formalism for the topological minimal model is obtained as the limit l=0 of the coset theory. We give some calculations of correlation functions on higher genus.

scholarly journals Correlator correspondences for subregular $$ \mathcal{W} $$-algebras and principal $$ \mathcal{W} $$-superalgebras

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Thomas Creutzig ◽  
Yasuaki Hikida ◽  
Devon Stockal
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Path Integral ◽  
Vertex Operator ◽  
Operator Algebras ◽  
Natural Generalization ◽  
Weak Duality ◽  
Two Dimensional ◽  
Conformal Field ◽  
Corner Vertex

Abstract We examine a strong/weak duality between a Heisenberg coset of a theory with $$ \mathfrak{sl} $$ sl n subregular $$ \mathcal{W} $$ W -algebra symmetry and a theory with a $$ \mathfrak{sl} $$ sl n|1-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rapčák and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal $$ \mathcal{W} $$ W -superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts.

The group of local biholomorphisms of ℂ1 and conformal field theory in the operator formalism

1989 ◽  
Vol 120 (3) ◽  
pp. 481-499 ◽  
Author(s):  
Robert J. Budzyński ◽  
Sławomir Klimek ◽  
Paweł Sadowski
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Operator Formalism ◽  
Conformal Field

scholarly journals INTRODUCTION TO VERTEX ALGEBRAS, BORCHERDS ALGEBRAS AND THE MONSTER LIE ALGEBRA

1993 ◽  
Vol 08 (31) ◽  
pp. 5441-5503 ◽  
Author(s):  
REINHOLD W. GEBERT
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Point Of View ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Conformal Field ◽  
Formal Calculus ◽  
Developing Area ◽  
Definition Of

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory In this context Borcherds algebras arise as certain “physical” subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction to this rapidly developing area of mathematics. Based on the machinery of formal calculus, we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analyzed from the point of view of symmetry in quantum theory and the construction of the monster Lie algebra is sketched.

A MATHEMATICA™ PACKAGE FOR COMPUTING OPERATOR PRODUCT EXPANSIONS

1991 ◽  
Vol 02 (03) ◽  
pp. 787-798 ◽  
Author(s):  
K. THIELEMANS
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Free Field ◽  
Standard Form ◽  
General Purpose ◽  
Operator Product ◽  
Conformal Anomaly ◽  
Moody Algebra ◽  
Conformal Field ◽  
Free Field Realization

A general purpose Mathematica™ package for computing Operator Product Expansions of composite operators in meromorphic conformal field theory is described. Given the OPEs for a set of “basic” fields, OPEs of arbitrarily complicated composites can be computed automatically. Normal ordered products are always reduced to a standard form. Two explicit examples are presented: the conformal anomaly for superstrings and a free field realization for the [Formula: see text] Kač-Moody-algebra.

scholarly journals The entanglement entropy of typical pure states and replica wormholes

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Erez Y. Urbach
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Path Integral ◽  
Entanglement Entropy ◽  
Singular Boundary ◽  
Von Neumann ◽  
Short Intervals ◽  
Conformal Field ◽  
Circle Size ◽  
Pure States

Abstract In a 1+1 dimensional QFT on a circle, we consider the von Neumann entanglement entropy of an interval for typical pure states. As a function of the interval size, we expect a Page curve in the entropy. We employ a specific ensemble average of pure states, and show how to write the ensemble-averaged Rényi entropy as a path integral on a singular replicated geometry. Assuming that the QFT is a conformal field theory with a gravitational dual, we then use the holographic dictionary to obtain the Page curve. For short intervals the thermal saddle is dominant. For large intervals (larger than half of the circle size), the dominant saddle connects the replicas in a non-trivial way using the singular boundary geometry. The result extends the ‘island conjecture’ to a non-evaporating setting.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

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