Free Quantum Fields in 4D and Calabi-Yau Spaces

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation).We introduce a partition function, termed the Hilbert series, to enumerate the operator basis -correspondingly, the S-matrix -and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries.In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes.Open Access, c The Authors. Article funded by SCOAP 3 .https://doi.org/10.1007/JHEP10(2017)199 JHEP10(2017)199We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the Smatrix in the form of soft limits. The most naïve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.Although primarily discussed in the language of EFT, some of our results -conceptual and quantitative -may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.

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“…Note added: During the finalization of this manuscript, the preprints [87,88] appeared, with some overlap with the ideas of this paper.…”

Section: Discussionmentioning

confidence: 99%

Operator bases, S-matrices, and their partition functions

Henning

Melia

et al. 2017

J. High Energ. Phys.

161

331

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“…In this paper we extend the study of [7,8] by carrying out a systematic study of primaries in free fermion field theories in four dimensions. In section 2 we obtain formulae for the counting of primary fields constructed from n copies of a left handed Weyl fermion, using the characters of representations of so (4,2).…”

Section: Introductionmentioning

confidence: 99%

From spinning primaries to permutation orbifolds

Koch

Rabambi

Zyl

2018

J. High Energ. Phys.

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We carry out a systematic study of primary operators in the conformal field theory of a free Weyl fermion. Using SO(4, 2) characters we develop counting formulas for primaries constructed using a fixed number of fermion fields. By specializing to particular classes of primaries, we derive very explicit formulas giving the generating functions for the number of primaries in these classes. We present a duality map between primary operators in the fermion field theory and polynomial functions. This allows us to construct the primaries that were counted. Next we show that these classes of primary fields correspond to polynomial functions on certain permutation orbifolds. These orbifolds have palindromic Hilbert series. 1 robert@neo.phys.wits.ac.za 2 457990@students.wits.ac.za 3

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“…We have considered the problem of constructing primary fields in free scalar CFTs in general dimensions, combining insights from [4,1,2] and [3]. This has been a fruitful avenue, with the key results described in the introduction and developed in the bulk of the paper.…”

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confidence: 99%

“…This ring should be the ring of polynomials on C 2 subject to the condition µ Z(1) µ Z (2) µ = 0 (B.5)…”

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confidence: 99%

“…The counting and construction of primary fields in free scalar field theories was found to have surprisingly simple and elegant geometrical structures in [1,2]. General primary fields in scalar field theory in d dimensions, which are composites of n elementary fields, are in 1-1 correspondence with polynomials in nd variables, x I µ where 1 ≤ µ ≤ d, 1 ≤ I ≤ n which solve a system of linear first and second order partial differential differential equations, and obey an invariance condition under S n , the symmetric group of permutations of n distinct objects.…”

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confidence: 99%

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Free field primaries in general dimensions: counting and construction with rings and modules

Koch

Ramgoolam

2018

J. High Energ. Phys.

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We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under S n correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d×(n−1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.

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Free Quantum Fields in 4D and Calabi-Yau Spaces

Cited by 10 publications

References 33 publications

Operator bases, S-matrices, and their partition functions

Operator bases, S-matrices, and their partition functions

From spinning primaries to permutation orbifolds

Free field primaries in general dimensions: counting and construction with rings and modules

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