Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds

171

We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s, t and u. We construct these modules for every value of the spacetime dimension D, and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by s 2 at fixed t. A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for D ≤ 6. For D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for D ≤ 6. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth bound, at least when D ≤ 6, even when the exchanged particles have low spin.

Section: Jhep02(2020)114mentioning

confidence: 97%

Classifying and constraining local four photon and four graviton S-matrices

Chowdhury

Gadde

Gopalka

et al. 2020

171

“…Previous studies [7] have explained how to map the algebraic problem of constructing primary fields in the quantum field theory of a free scalar field φ in four dimensions to one of finding polynomial functions on (R 4 ) n that are harmonic, translation invariant and which are in the trivial representation of S n . In this article, we have extended this construction to describe primary fields in the free quantum field theory of a single Weyl fermion.…”

Section: Discussionmentioning

confidence: 99%

“…The above formula produces an expression of the form in i P i (Z) wheren i are unit vectors inside the carrier space of V ⊗k H and P i (Z) are the polynomials that correspond to primary operators. To translate polynomials into momenta, the formula [7] z k ↔ (−1) k P k 2 k k! , (3.39) that we derived above, is very useful.…”

Section: )mentioning

confidence: 99%

“…This has the property Z ext n (q 1 , q 2 ) = Z ext n (q 2 , q 1 ). This follows because exchange of q 1 , q 2 amounts to the inversion of y, and by using the identity [7] Z SH (q −1 , Λ) = (−q) n−1 Z SH (q, Λ T ) .…”

Section: Geometrymentioning

confidence: 99%

“…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%

See 2 more Smart Citations

From spinning primaries to permutation orbifolds

Koch

Rabambi

Zyl

2018

Self Cite

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

Classification of four-point local gluon S-matrices

Chowdhury

Gadde

2021

In this paper, we classify four-point local gluon S-matrices in arbitrary dimensions. This is along the same lines as [1] where four-point local photon S-matrices and graviton S-matrices were classified. We do the classification explicitly for gauge groups SO(N) and SU(N) for all N but our method is easily generalizable to other Lie groups. The construction involves combining not-necessarily-permutation-symmetric four-point S-matrices of photons and those of adjoint scalars into permutation symmetric four-point gluon S-matrix. We explicitly list both the components of the construction, i.e permutation symmetric as well as non-symmetric four point S-matrices, for both the photons as well as the adjoint scalars for arbitrary dimensions and for gauge groups SO(N) and SU(N) for all N. In this paper, we explicitly list the local Lagrangians that generate the local gluon S-matrices for D ≥ 9 and present the relevant counting for lower dimensions. Local Lagrangians for gluon S-matrices in lower dimensions can be written down following the same method. We also express the Yang-Mills four gluon S-matrix with gluon exchange in terms of our basis structures.