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Register a new userCorrelators, Probabilities and Topologies in N=4 SYM
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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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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 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.
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.
We compute two-point functions of lowest weight operators at the next-to-leading order in the couplings for the beta-deformed N=4 SYM. In particular we focus on the CPO Tr(Phi_1^2) and the operator Tr(Phi_1 Phi_2) not presently listed as BPS. We find that for both operators no anomalous dimension is generated at this order, then confirming the results recently obtained in hep-th/0506128. However, in both cases a finite correction to the two-point function appears.
In this paper we study the form factors for the half-BPS operators $mathcal{O}^{(n)}_I$ and the $mathcal{N}=4$ stress tensor supermultiplet current $W^{AB}$ up to the second order of perturbation theory and for the Konishi operator $mathcal{K}$ at first order of perturbation theory in $mathcal{N}=4$ SYM theory at weak coupling. For all the objects we observe the exponentiation of the IR divergences with two anomalous dimensions: the cusp anomalous dimension and the collinear anomalous dimension. For the IR finite parts we obtain a similar situation as for the gluon scattering amplitudes, namely, apart from the case of $W^{AB}$ and $mathcal{K}$ the finite part has some remainder function which we calculate up to the second order. It involves the generalized Goncharov polylogarithms of several variables. All the answers are expressed through the integrals related to the dual conformal invariant ones which might be a signal of integrable structure standing behind the form factors.
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