Construction of two-dimensional topological field theories with non-invertible symmetries

We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

The Virasoro-Shapiro amplitude in AdS5 × S5 and level splitting of 10d conformal symmetry

The genus zero contribution to the four-point correlator $$ \left\langle {\mathcal{O}}_{p_1}{\mathcal{O}}_{p_2}{\mathcal{O}}_{p_3}{\mathcal{O}}_{p_4}\right\rangle $$ O p 1 O p 2 O p 3 O p 4 of half-BPS single-particle operators $$ {\mathcal{O}}_p $$ O p in $$ \mathcal{N} $$ N = 4 super Yang-Mills, at strong coupling, computes the Virasoro-Shapiro amplitude of closed superstrings in AdS5× S5. Combining Mellin space techniques, the large p limit, and data about the spectrum of two-particle operators at tree level in supergravity, we design a bootstrap algorithm which heavily constrains its α′ expansion. We use crossing symmetry, polynomiality in the Mellin variables and the large p limit to stratify the Virasoro-Shapiro amplitude away from the ten-dimensional flat space limit. Then we analyse the spectrum of exchanged two-particle operators at fixed order in the α′ expansion. We impose that the ten-dimensional spin of the spectrum visible at that order is bounded above in the same way as in the flat space amplitude. This constraint determines the Virasoro-Shapiro amplitude in AdS5× S5 up to a small number of ambiguities at each order. We compute it explicitly for (α′)5,6,7,8,9. As the order of α′ grows, the ten dimensional spin grows, and the set of visible two-particle operators opens up. Operators illuminated for the first time receive a string correction to their anomalous dimensions which is uniquely determined and lifts the residual degeneracy of tree level supergravity, due to ten-dimensional conformal symmetry. We encode the lifting of the residual degeneracy in a characteristic polynomial. This object carries information about all orders in α′. It is analytic in the quantum numbers, symmetric under an AdS5 ↔ S5 exchange, and it enjoys intriguing properties, which we explain and detail in various cases.

Conformal blocks from celestial gluon amplitudes. Part II. Single-valued correlators

In a recent paper, here referred to as part I, we considered the celestial four-gluon amplitude with one gluon represented by the shadow transform of the corresponding primary field operator. This correlator is ill-defined because it contains branch points related to the presence of conformal blocks with complex spin. In this work, we adopt a procedure similar to minimal models and construct a single-valued completion of the shadow correlator, in the limit when the shadow is “soft.” By following the approach of Dotsenko and Fateev, we obtain an integral representation of such a single-valued correlator. This allows inverting the shadow transform and constructing a single-valued celestial four-gluon amplitude. This amplitude is drastically different from the original Mellin amplitude. It is defined over the entire complex plane and has correct crossing symmetry, OPE and bootstrap properties. It agrees with all known OPEs of celestial gluon operators. The conformal block spectrum consists of primary fields with dimensions ∆ = m + iλ, with integer m ≥ 1 and various, but always integer spin, in all group representations contained in the product of two adjoint representations.

Dispersion relations and exact bounds on CFT correlators

We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fields. Besides bounds on correlators, we apply this technology to determine new universal constraints on the Regge limit of arbitrary CFTs and obtain very simple and accurate representations of the 3d Ising spin correlator.

A new gauge-invariant double copy for heavy-mass effective theory

We propose a new form of the colour-kinematics/double-copy duality for heavy-mass effective field theories, which we apply to construct compact expressions for tree amplitudes with heavy matter particles in Yang-Mills and in gravity to leading order in the mass. In this set-up, the new BCJ numerators are fixed uniquely and directly written in terms of field strengths, making gauge invariance manifest. Furthermore, they are local and automatically satisfy the Jacobi relations and crossing symmetry. We construct these BCJ numerators explicitly up to six particles. We also discuss relations of the BCJ numerators in the heavy-mass effective theory with those in pure Yang-Mills amplitudes.

Local Zeta Functions and Koba–Nielsen String Amplitudes

This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

New positivity bounds from full crossing symmetry

Positivity bounds are powerful tools to constrain effective field theories. Utilizing the partial wave expansion in the dispersion relation and the full crossing symmetry of the scattering amplitude, we derive several sets of generically nonlinear positivity bounds for a generic scalar effective field theory: we refer to these as the P Q, Dsu, Dstu and $$ {\overline{D}}^{\mathrm{stu}} $$ D ¯ stu bounds. While the PQ bounds and Dsu bounds only make use of the s↔u dispersion relation, the Dstu and $$ {\overline{D}}^{\mathrm{stu}} $$ D ¯ stu bounds are obtained by further imposing the s↔t crossing symmetry. In contradistinction to the linear positivity for scalars, these inequalities can be applied to put upper and lower bounds on Wilson coefficients, and are much more constraining as shown in the lowest orders. In particular we are able to exclude theories with soft amplitude behaviour such as weakly broken Galileon theories from admitting a standard UV completion. We also apply these bounds to chiral perturbation theory and we find these bounds are stronger than the previous bounds in constraining its Wilson coefficients.