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Register a new userExact expressions for $n$-point maximal $U(1)_Y$-violating integrated correlators in $SU(N)$ $mathcal{N}=4$ SYM
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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 $tau=theta/(2pi)+4pi i/g_{_{YM}}^2$, 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 $q=e^{2pi i tau}$ 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 relation of $n$-point MUV correlators to $(n-4)$-loop contributions to the four-point correlator. In particular, we argue that it is important to ensure the $SL(2, mathbb{Z})$-covariance even in the construction of perturbative loop integrands.
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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. These recursion relations are used to determine terms in the large-$N$ expansion of $n$-point maximal $U(1)_Y$-violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and $(n-4)$ chiral Lagrangian operators, starting from known properties of the $n=4$ case. We concentrate on the first three orders in $1/N$ beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in $AdS_5 times S^5$ at the same orders as $R^4, d^4R^4$ and $d^6R^4$. The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights $(n-4,4-n)$ that are $SL(2, mathbb{Z})$-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to $U(1)_Y$-violating $n$-particle interactions ($n>4$) in the low-energy expansion of type IIB superstring amplitudes in $AdS_5times S^5$.
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 $O(exp(-2sqrt{lambda}))$.
We study supersymmetric sectors at half-BPS boundaries and interfaces in the 4d $mathcal{N}=4$ super Yang-Mills with the gauge group $G$, which are described by associative algebras equipped with twisted traces. Such data are in one-to-one correspondence with an infinite set of defect correlation functions. We identify algebras and traces for known boundary conditions. Ward identities expressing the (twisted) periodicity of the trace highly constrain its structure, in many cases allowing for the complete solution. Our main examples in this paper are: the universal enveloping algebra $U(mathfrak{g})$ with the trace describing the Dirichlet boundary conditions; and the finite W-algebra $mathcal{W}(mathfrak{g},t_+)$ with the trace describing the Nahm pole boundary conditions.
We compute the one-loop non-holomorphic effective potential for the N=4 SU(n) supersymmetric Yang-Mills theory with the gauge symmetry broken down to the maximal torus. Our approach remains powerful for arbitrary gauge groups and is based on the use of N=2 harmonic superspace formulation for general N=2 Yang-Mills theories along with the superfield background field method.
We calculate transition probabilities for various processes involving giant gravitons and small gravitons in AdS space, using the dual N=4 SYM theory. The normalization factors for these probabilities involve, in general, correlators for manifolds of non-trivial topology which are obtained by gluing simpler four-manifolds. This follows from the factorization properties which relate CFT correlators for different topologies. These points are illustrated, in the first instance, in the simpler example of a two dimensional Matrix CFT. We give the bulk five dimensional interpretation, involving neighborhoods of Witten graphs, of these gluing properties of the four dimensional boundary CFT. As a corollary we give a simple description, based on Witten graphs, of a multiplicity of bulk topologies corresponding to a fixed boundary topology. We also propose to interpret the correlators as topology-changing transition amplitudes between LLM geometries.
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