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Register a new userTwo-Point Correlator of Chiral Primary Operators with a Wilson Line Defect in $mathcal{N}=4$ SYM
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We study the two-point function of the stress-tensor multiplet of $mathcal{N}=4$ SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the $20$ irrep of the $mathfrak{so}(6)_text{R}$ R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expansions which capture defect and bulk data respectively. We present a closed-form formula for the defect CFT data, which allows to write an efficient Taylor series for the correlator in the limit when one of the operators is close to the line. The bulk channel is technically harder and closed-form formulae are particularly challenging to obtain, nevertheless we use our analysis to check against well-known data of $mathcal{N}=4$ SYM. In particular, we recover the correct anomalous dimensions of a famous tower of twist-two operators (which includes the Konishi multiplet), and successfully compare the one-point function of the stress-tensor multiplet with results obtained using matrix-model techniques.
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We generalize local operators of the leading twist-2 of N=4 SYM theory to the case of complex Lorentz spin j using principal series representation of sl(2,R). We give the direct computation of correlation function of two such non-local operators in the BFKL regime when j -> 1. The correlator appears to have the expected conformal coordinate dependence governed by the anomalous dimension of twist-2 operator in NLO BFKL approximation predicted by Kotikov and Lipatov.
We present a novel expression for an integrated correlation function of four superconformal primaries in $SU(N)$ $mathcal{N}=4$ SYM. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. The correlator is re-expressed as a sum over a two dimensional lattice that is valid for all $N$ and all values of the complex Yang-Mills coupling $tau$. In this form it is manifestly invariant under $SL(2,mathbb{Z})$ Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the $SU(N)$ to the $SU(N+1)$ and $SU(N-1)$ correlators. For any fixed value of $N$ the correlator is an infinite series of non-holomorphic Eisenstein series, $E(s;tau,bartau)$ with $sin mathbb{Z}$, and rational coefficients. The perturbative expansion of the integrated correlator is asymptotic and the $n$-loop coefficient is a rational multiple of $zeta(2n+1)$. The $n=1$ and $n=2$ terms agree precisely with results determined directly by integrating the expressions in one- and two-loop perturbative SYM. Likewise, the charge-$k$ instanton contributions have an asymptotic, but Borel summable, series of perturbative corrections. The large-$N$ expansion of the correlator with fixed $tau$ is a series in powers of $N^{1/2-ell}$ ($ellin mathbb{Z}$) with coefficients that are rational sums of $E_s$ with $sin mathbb{Z}+1/2$. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider t Hooft large-$N$ Yang-Mills theory. The coefficient of each order can be expanded as a convergent series in $lambda$. For large $lambda$ this becomes an asymptotic series with coefficients that are again rational multiples of odd zeta values. The large-$lambda$ series is not Borel summable, and its resurgent non-perturbative completion is $O(exp(-2sqrt{lambda}))$.
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 $tau$ 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 $mathcal{N}=4$ SYM for any finite $N$, as well as extending previously conjectured properties of the large-$N$ expansion.
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 propose a mechanism for calculating anomalous dimensions of higher-spin twist-two operators in N=4 SYM. We consider the ratio of the two-point functions of the operators and of their superconformal descendants or, alternatively, of the three-point functions of the operators and of the descendants with two protected half-BPS operators. These ratios are proportional to the anomalous dimension and can be evaluated at n-1 loop in order to determine the anomalous dimension at n loops. We illustrate the method by reproducing the well-known one-loop result by doing only tree-level calculations. We work out the complete form of the first-generation descendants of the twist-two operators and the scalar sector of the second-generation descendants.
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