scholarly journals Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Ilija Burić ◽  
Sylvain Lacroix ◽  
Jeremy Mann ◽  
Lorenzo Quintavalle ◽  
Volker Schomerus
Keyword(s):  
Differential Operators ◽  
Arbitrary Dimension ◽  
Tensor Fields ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Higher Dimensional ◽  
Sutherland Model ◽  
Joint Eigenfunctions ◽  
Space Formalism ◽  
Gaudin Models

Abstract It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

scholarly journals Gaudin models and multipoint cnformal blocks: general theory

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Ilija Burić ◽  
Sylvain Lacroix ◽  
Jeremy A. Mann ◽  
Lorenzo Quintavalle ◽  
Volker Schomerus
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Differential Operators ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Commuting Differential Operators ◽  
Higher Dimensional ◽  
Important Open Problem ◽  
Complete Set ◽  
Gaudin Models

Abstract The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d.

4D Einstein equations in a general gauge Kaluza–Klein space

2015 ◽  
Vol 12 (03) ◽  
pp. 1550036
Author(s):  
Aurel Bejancu ◽  
Constantin Călin
Keyword(s):  
Einstein Equations ◽  
Tensor Fields ◽  
New Approach ◽  
Higher Dimensional ◽  
High Level ◽  
General Gauge ◽  
Kaluza Klein

Using the new approach on higher-dimensional Kaluza–Klein theories developed by the first author, we obtain the 4D Einstein equations on a (4 + n)D relativistic gauge Kaluza–Klein space. Adapted frame and coframe fields, adapted tensor fields, and the Riemannian adapted connection, have a fundamental role in the study. The high level of generality of the study, enables us to recover several results from earlier papers on this matter.

TETRAHEDRA AND POLYNOMIAL EQUATIONS IN TOPOLOGICAL FIELD THEORY

1991 ◽  
Vol 06 (20) ◽  
pp. 3571-3598 ◽  
Author(s):  
NOUREDDINE CHAIR ◽  
CHUAN-JIE ZHU
Keyword(s):  
Field Theory ◽  
Topological Field Theory ◽  
Fundamental Representation ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Witten Theory ◽  
Topological Field ◽  
General Program ◽  
The One ◽  
Chern Simons

Some tetrahedra in SUk(2) Chern-Simons-Witten theory are computed. The results can be used to compute an arbitrary tetrahedron inductively by fusing with the fundamental representation. The results obtained are in agreement with those of quantum groups. By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT, we also derive an explicit expression for the modular transformation matrix S(k) of the one-point conformal blocks on a torus in RCFT and the equations satisfied by S(k), in agreement with those required in RCFT. The implication of our results on the general program of classifying RCFT is also discussed.

scholarly journals On 2d CFTs that interpolate between minimal models

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Exactly Solvable ◽  
Structure Constants

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

scholarly journals PROJECTIVE AND CONFORMAL SCHWARZIAN DERIVATIVES AND COHOMOLOGY OF LIE ALGEBRAS VECTOR FIELDS RELATED TO DIFFERENTIAL OPERATORS

2006 ◽  
Vol 03 (04) ◽  
pp. 667-696 ◽  
Author(s):  
SOFIANE BOUARROUDJ
Keyword(s):  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Projective Manifold ◽  
Conformal Class ◽  
Tensor Fields ◽  
Relative Cohomology ◽  
Schwarzian Derivatives ◽  
Cohomology Of Lie Algebras

Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class of the metric [g], respectively. Furthermore, we compute the first cohomology group of Vect(M) with coefficients in the space of symmetric contravariant tensor fields valued in the space of δ-densities, and we compute the corresponding sl(n + 1, ℝ)-relative cohomology group.

scholarly journals Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

2019 ◽  
Vol 29 (06) ◽  
pp. 951-1007
Author(s):  
Nithi Rungtanapirom ◽  
Jakob Stix ◽  
Alina Vdovina
Keyword(s):  
Arbitrary Dimension ◽  
Congruence Subgroups ◽  
Quaternion Algebras ◽  
Ramanujan Graphs ◽  
Dimension Formula ◽  
Cubical Sets ◽  
Higher Dimensional ◽  
Residually Finite Groups ◽  
Doubling Construction ◽  
Effective Use

We construct vertex transitive lattices on products of trees of arbitrary dimension [Formula: see text] based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher-dimensional cubical version of Ramanujan graphs (optimal expanders).

scholarly journals Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

2012 ◽  
Vol 860 (3) ◽  
pp. 377-420 ◽  
Author(s):  
Benoit Estienne ◽  
Vincent Pasquier ◽  
Raoul Santachiara ◽  
Didina Serban
Keyword(s):  
Conformal Blocks ◽  
Sutherland Model
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