From spinning primaries to permutation orbifolds

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The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of nonadditive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of N = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U , abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U ,2 , related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.

Section: 1)mentioning

confidence: 99%

Perturbative 4D conformal field theories and representation theory of diagram algebras

Koch

Ramgoolam

2020

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$$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries

Koch

Ramgoolam

2023

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The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring $$ \mathcal{R} $$ R (8|8) which admits a realisation of psu(2, 2|4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of $$ \mathcal{R} $$ R (8|8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2, 2|4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrate the construction using simple examples of representations of su(3) and su(4). These results lead to the definition of a notion of quantum mechanical emergence for oscillator realisations of symmetries, which is based on ideals in the ring of polynomials in the creation operators.

Holography of a single free matrix

de Mello Koch,

Roy,

Van Zyl

2024

In this paper we consider the collective field theory description of a single free massless scalar matrix theory in 2+1 dimensions. The collective fields are given by k-local operators obtained by tracing a product of k-matrices. For k = 2 and k = 3 we argue that the collective field packages the fields associated to a single and two Regge trajectories respectively. We also determine the coordinate transformation between the coordinates of the collective field theory and the bulk AdS space time. This is used to verify that the bulk equations of motion holds in the collective field theory description.