Conformal correlators as simplex integrals in momentum space

Abstract

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

Nonperturbative dynamics of (2+1)d ϕ4-theory from Hamiltonian truncation

“…It would also be useful to push the LCT basis to higher values of ∆ max , in order to extract critical exponents and confirm that the critical point is described by the 3d Ising CFT. One useful tool for reaching higher values of ∆ max would be to compute the general expression for CFT three-point functions of spinning operators in momentum space, building on recent results [29,[52][53][54][55][56][57], which would greatly improve the efficiency for computing Hamiltonian matrix elements. Though in practice our Dirichlet basis states do not individually correspond to primary operators, one can construct a map between the two bases, in order to more efficiently construct matrix elements from CFT data.…”

The conformal anomaly action to fourth order (4T) in $$d=4$$ in momentum space

“…Examples are the classification of the minimal number of form factors present in their expansions in the external momenta, the identification of the arbitrary functions which appear in the solution of the corresponding CWIs for n > 3, [11][12][13][14], or the search for exact solutions in the presence of dual conformal symmetry [15]. Among these correlators, an important role is played by those involving the stress energy tensor, due to the appearance of a conformal (trace) anomaly in even spacetime dimensions (see [16] for a general discussion).…”

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“…In the massless limit, the resulting constraints are precisely those that have recently been studied in the context of the momentum space conformal bootstrap (see e.g. [23][24][25][26][27][28][29][30][31][32][33]) with applications in cosmology or condensed matter physics.…”