The exact expressions for integrated maximal $U(1)_Y$ violating (MUV) $n$-point correlators in $SU(N)$ ${\mathcal N}=4$ supersymmetric Yang--Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of $N$ and $τ=θ/(2π)+4πi/g_{_{YM}}^2$, and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights $(w,-w)$ where $w=n-4$. The correlators satisfy Laplace-difference equations that relate the $SU(N+1)$, $SU(N)$ and $SU(N-1)$ expressions and generalise the equations previously found in the $w=0$ case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight $(w,-w)$. For any fixed value of $N$ the perturbation expansion of this correlator is found to start at order $( g_{_{YM}}^2 N)^w$. The contributions of Yang--Mills instantons of charge $k>0$ are of the form $q^k\, f(g_{_{YM}})$, where $q=e^{2πi τ}$ and $f(g_{_{YM}})= O(g_{_{YM}}^{-2w})$ when $g_{_{YM}}^2 \ll 1$. Anti-instanton contributions have charge $k<0$ and are of the form $\bar q^{|k|} \, \hat f(g_{_{YM}})$, where $\hat f(g_{_{YM}}) = O(g_{_{YM}}^{2w})$ when $g_{_{YM}}^2 \ll 1$. Properties of the large-$N$ expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of $n$-point free-field MUV correlators with the integrands of $(n-4)$-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of $SL(2, \mathbb{Z})$-covariance in the construction.
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 i…
This paper concerns a special class of $n$-point correlation functions of operators in the stress tensor supermultiplet of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory. These are "maximal $U(1)_Y$-violating" correlators that violate the bonus $U(1)_Y$ charge by a maximum of $2(n-4)$ units. We will demonstrate that such correlators satisfy $SL(2,\mathbb{Z})$-covariant recursion relations that relate $n$-point correlators to $(n-1)$-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory.…
In this paper we use conformal field theory techniques to constrain the form of the correlations functions of an inflationary scalar-vector model described by the interaction term $f_1(ϕ)F_{μν}F^{μν} + f_2(ϕ)\tilde{F}_{μν}F^{μν}$. We use the fact that the conformal group is the relevant symmetry group acting on super horizon scales in an inflationary de Sitter background. As a result, we find that super horizon conformal symmetry, constraints the form of the coupling functions $f_1, f_2$ to be homogeneous functions of the same degree. We derive the general form of the correlators involving sca…
We present a constructive derivation of all four-point tree-level holographic correlators for eleven dimensional supergravity on $AdS_7 \times S^4$. These correlators correspond to four-point functions of arbitrary one-half BPS operators in the six-dimensional $(2,0)$ theory at large central charge. The crucial observation is that the polar part of the correlators in Mellin space is fully captured by a drastically simpler Maximally R-symmetry Violating (MRV) amplitude, while the contact part is fully fixed by superconformal Ward identities and the flat space limit.
Recently four-point holographic correlators with arbitrary external BPS operators were constructively derived in [1,2] at tree-level for maximally superconformal theories. In this paper, we capitalize on these theoretical data, and perform a detailed study of their analytic properties. We point out that these maximally supersymmetric holographic correlators exhibit a hidden dimensional reduction structure à la Parisi and Sourlas. This emergent structure allows the correlators to be compactly expressed in terms of only scalar exchange diagrams in a dimensionally reduced spacetime, where formall…
This thesis describes progresses made by the author and collaborators in the positive geometry description of superamplitudes and supercorrelators in planar N = 4 SYM.
We point out that an infinite class of Witten diagrams is invariant under a Yangian symmetry. These diagrams are building blocks of holographic correlators and are related by a web of differential recursion relations. We show that Yangian invariance is equivalent to the consistency conditions of the recursion relations.
We consider four-point correlation functions of protected single-trace scalar operators in planar N = 4 supersymmetric Yang-Mills (SYM). We conjecture that all loop corrections derive from an integrand which enjoys a ten-dimensional symmetry. This symmetry combines spacetime and R-charge transformations. By considering a 10D light-like limit, we extend the correlator/amplitude duality by equating large R-charge octagons with Coulomb branch scattering amplitudes. Using results from integrability, this predicts new finite amplitudes as well as some Feynman integrals.
N-point correlation functions of conserved currents and weight-two scalar operators of the three-dimensional free fermion vector model are found as invariants of the higher-spin symmetry in four-dimensional AdS. These are the correlators of the unbroken Vasiliev higher-spin theory. The results extend the recent work arXiv:1210.7963 and are complementary to arXiv:1301.3123 where the correlators were computed entirely on the boundary.
We study one and two point functions of conformal field theories on spaces of maximal symmetry with and without boundaries and investigate their spectral representations. Integral transforms are found, relating the spectral decomposition to renormalized position space correlators. Several applications are presented, including the holographic boundary CFTs as well as spacelike boundary CFTs, which provide realizations of the pseudo-conformal universe.
We consider the correlator of two concentric circular Wilson loops with equal radii for arbitrary spatial and internal separation at strong coupling within a defect version of $\mathcal{N}=4$ SYM. Compared to the standard Gross-Ooguri phase transition between connected and disconnected minimal surfaces, a more complicated pattern of saddle-points contributes to the two-circles correlator due to the defect's presence. We analyze the transitions between different kinds of minimal surfaces and their dependence on the setting's numerous parameters.
The generating functions for the Wilson loops in the symmetric and antisymmetric representations of the gauge group $U(N)$ are expressed in terms of the connected correlators of multiply-wound Wilson loops, using ingredients from the representation theory of the symmetric group. This provides a proof of a recent observation by Okuyama. As a by-product, we present a new calculation of the connected 2-point correlator of multiply-wound Wilson loops at leading order in $1/N$.
We study the connected correlator of 1/2 BPS winding Wilson loops in $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory, where those Wilson loops are on top of each other along the same circle. We find the exact finite $N$ expression of the connected correlator of such Wilson loops. We show that the $1/N$ expansion of this exact result is reproduced from the topological recursion of Gaussian matrix model. We also study the exact finite $N$ expression of the generating function of 1/2 BPS Wilson loops in the symmetric representation.
We obtain the planar correlation function of four half-BPS operators of arbitrary weights, up to three loops. Our method exploits only elementary properties of the integrand of the planar correlator, such as its symmetries and singularity structure. This allows us to write down a general ansatz for the integrand. The coefficients in the ansatz are fixed by means of a powerful light-cone OPE relation between correlators with different weights. Our result is formulated in terms of a limited number of functions built from known one-, two- and three-loop conformal integrals. These results are usef…
We present calculation of the correlation function of three twist-2 operators in the BFKL limit. The calculation is performed in N = 4 SYM but the result is valid in other gauge theories such as QCD. The obtained leading order structure constant is exact for any number of colors.
We calculate the four-point correlation function of half-BPS operators with weights 2, 3, 3, 4 in N=4 SYM to two-loop order. The OPE of this correlation function provides a nontrivial check of the integrability conjecture for a class of three-point functions formulated in arXiv:1311.6404. Our perturbative calculation exploits the supergraph formalism in N=2 harmonic superspace.
An integrated correlator of four superconformal stress-tensor primaries of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory (SYM), originally obtained by localisation, is re-expressed as a two-dimensional lattice sum that is manifestly invariant under $SL(2,\mathbb{Z})$ S-duality. This expression is shown to satisfy a novel Laplace equation in the complex coupling constant $τ$ that relates the $SU(N)$ integrated correlator to those of the $SU(N+1)$ and $SU(N-1)$ theories. The lattice sum is shown to precisely reproduce known perturbative and non-perturbative properties of $\mathcal{N}=…
How much spectral information is needed to determine the correlation functions of a conformal theory? We study this question in the context of planar supersymmetric Yang-Mills theory, where integrability techniques accurately determine the single-trace spectrum at finite 't Hooft coupling. Corresponding OPE coefficients are constrained by dispersive sum rules, which implement crossing symmetry. Focusing on correlators of four stress-tensor multiplets, we construct combinations of sum rules which determine one-loop correlators, and we study a numerical bootstrap problem that nonperturbatively b…
A non-trivial consequence of the super-correlator/super-amplitude duality is that the integrand of the four-point correlation function of stress-tensor multiplets in planar N=4 super Yang-Mills contains a certain combination of n-point amplitude integrands for any n. This combination is the sum of products of all helicity super-amplitudes with their corresponding helicity conjugates. The four-point correlator itself is described by a single scalar function whose loop level integrands possess a hidden permutation symmetry facilitating its computation up to ten loops. We discover that assuming Y…
We examine the cosmological correlators induced by the simultaneous breaking of parity and of statistical isotropy, e.g., in presence of the coupling ${\cal L} = f(ϕ) ( - \frac{1}{4} F^2 + \fracγ{4} F \tilde{F} )$ between the inflaton $ϕ$ and a vector field with vacuum expectation value ${\bf A}$. For a suitably chosen function $f$, the energy in the vector field $ρ_{\rm A}$ does not decay during inflation. This results in nearly scale-invariant signatures of broken statistical isotropy and parity. Specifically, we find that the scalar-scalar correlator of primordial curvature perturbations in…
Non-Renormalizability Properties of Quarter-BPS Correlators in $\mathcal{N}$=4 SYM
ICTP, International Centre for Theoretical Physics（国际理论物理中心）；INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文系）；Department of Physics, Boston University（波士顿大学物理系）；Fields and Strings Laboratory, Institute of Physics, École Polytechnique Fédéral de Lausanne (EPFL)（洛桑联邦理工学院物理学院弦场实验室）；Department of Mathematical Sciences, Durham University（杜伦大学数学科学系）
Three-point correlators of twist-2 operators in N=4 SYM at Born approximation
本文在弱耦合近似下计算了两种不同类型的三点关联函数，它们涉及扭2算符，在N=4 SYM理论中，它们在N_c中的所有阶。第一个相关器中的三个算符中的每一个都可以是扭转2超多重态的任何分量，尽管显式计算是针对SU（4）单重态的特定分量进行的。本文在玻恩近似下对任意自旋j_1，j_2，j_3进行了计算。当至少有一个自旋大于或等于零，并且坐标被限制在由两条光线跨越的2d平面时，结果显着简化。第二相关器涉及两个扭转2算子Tr X \nabla^{j_1} X +. Tr Z \nabla^{j_2} Z+.和一个小西算子Tr[ \bar Z，\bar X]^2。它在最低的g^0阶为零，在g^2阶近似下计算。
We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for $SU(N)$ in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formall…
Recently Dorigoni, Green and Wen conjectured a remarkable exact formula for an integrated correlator of four superconformal primary operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory. In this work, we investigate its large $N$ limit in detail. We show that the formula of Dorigoni, Green and Wen can be recast into the sum over the contributions of $(p,q)$-strings. Due to the $SL(2,\mathbb{Z})$ duality, all the contributions are governed by a single function, typically appearing as the fundamental string contribution. The large order behavior for the perturbative genus expansion of th…
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 fi…
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 …
