We study the effective gravitational couplings of four-dimensional gauge theories with eight supercharges. The class of theories we analyse are arrived at via Kaluza-Klein compactification of five-dimensional gauge theories. We consider both pure SU(N) Yang-Mills theories with Chern-Simons couplings and the conformal gauge theories with 2N fundamental flavours. The resolvent of the gauge theory plays a crucial role in the calculation of these gravitational couplings. The results obtained from the Seiberg-Witten geometry are matched against independent computations using localisation.

India

Sujay Ashok

Renjan John

T. Layon

Sujoy Mahato

Madhusudhan Raman

Effective gravitational couplings of Kaluza-Klein gauge theories

When placed on four-manifolds,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauge theories couple to topological invariants of the background via two functions A and B. General considerations allow for these functions to be fixed in terms of the Coulomb moduli and other parameters in the theory, but only up to multiplicative factors about which little is known. We extend earlier work on the microscopic study of these functions in the Ω-background to   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 *  gauge theories with higher-rank U(N) gauge groups. We complement this analysis by carrying out a perturbative study of these functions. This allows us to determine the manner in which these multiplicative factors scale with the rank of the gauge group and the mass of the adjoint hypermultiplet.

Brazil

Effective gravitational couplings of higher-rank supersymmetric gauge theories

Consistency with position space OPE limit requires three-point momentum space CFT correlators to have only total energy singularity. We show that this requirement gives a simple proof of the known result that in three dimensions the parity-odd structure cannot exist for three-point correlators of exactly conserved currents with spins s  i  , s  j  , s  k , when triangle inequality s  i  ≤ s  j  + s  k  is violated. We also show that even for parity even correlation functions the properties are different inside and outside the triangle. It was previously shown that if we allow for weakly broken higher spin symmetry, parity-odd correlators can exist even when triangle inequality is violated. In this paper we establish a relation between non-conservation Ward-Takahashi (WT) identities for weakly broken currents at large N and the WT identities for exactly conserved currents with the help of a few examples. This allows us to calculate the parity violating results outside the triangle using parity-even free bosonic and free fermionic results.

Sachin Jain

Abhishek Mehta

K. Dhruva

Constraining momentum space CFT correlators with consistent position space OPE limit and the collider bound

In this paper we relate the parity-odd part of two and three point correlation functions in theories with exactly conserved or weakly broken higher spin symmetries to the parity-even part which can be computed from free theories. We also comment on higher point functions.

Relation between parity-even and parity-odd CFT correlation functions in three dimensions

In this paper we use the spinor-helicity formalism to calculate 3-point functions involving scalar operators and spin-s conserved currents in general 3d CFTs. In spinor-helicity variables we notice that the parity-even and the parity-odd parts of a correlator are related. Upon converting spinor-helicity answers to momentum space, we show that correlators involving spin-s currents can be expressed in terms of some simple conformally invariant conserved structures. This in particular allows us to understand and separate out contact terms systematically, especially for the parity-odd case. We also reproduce some of the correlators using weight-shifting operators.

Amin Nizami

Adithya Suresh

Higher spin 3-point functions in 3d CFT using spinor-helicity variables

We study the parity-odd sector of 3-point functions comprising scalar operators and conserved currents in conformal field theories in momentum space. We use momentum space conformal Ward identities as well as spin-raising and weight-shifting operators to fix the form of some of these correlators. Wherever divergences appear we discuss their regularisation and renormalisation using appropriate counter-terms.

Momentum space parity-odd CFT 3-point functions

We show that general parity-violating 3d conformal field theories show a double copy structure for momentum space 3-point functions of conserved currents, stress tensor and marginal scalar operators. Splitting up the CFT correlator into two parts — called homogeneous and non-homogeneous — we show that double copy relations exist for each part separately. We arrive at similar conclusions regarding double copy structures using tree-level correlators of massless fields in dS 4. We also discuss the flat space limit of these correlators. We further extend the double copy analysis to correlators involving higher-spin conserved currents, which suggests that the spin-s current correlator can be thought of as s copies of the spin one current correlator.

Double copy structure of parity-violating CFT correlators

In this work, building up on [1] we present momentum space Ward identities related to broken higher spin symmetry as an alternate approach to computing correlators of spinning operators in interacting theories such as the quasi-fermionic and quasi-bosonic theories. The direct Feynman diagram approach to computing correlation functions is intricate and in general has been performed only in specific kinematic regimes. We use higher spin equations to obtain the parity even and parity odd contributions to two-, three- and four-point correlators involving spinning and scalar operators in a general kinematic regime, and match our results with existing results in the literature for cases where they are available.

Italy

Vinay Malvimat

Constraining momentum space correlators using slightly broken higher spin symmetry

In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator   <math> <mover> <mi>ψ</mi> <mo>¯</mo> </mover> <mi>ψ</mi> </math>  $$ \overline{\psi}\psi $$  is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of   <math> <mover> <mi>ψ</mi> <mo>¯</mo> </mover> <mi>ψ</mi> </math>  $$ \overline{\psi}\psi $$  operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.

Momentum space spinning correlators and higher spin equations in three dimensions

The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint ϕ 3 theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra [2]. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [3], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations introduced in [4, 5] to these theories. We then give a detailed analysis of how the recursion relations in ϕ  p  theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra. Following the very recent development [6] we also extend our analysis to one-loop integrands in the quartic theory.