Exact properties of an integrated correlator in $$ \mathcal{N} $$ = 4 SU(N) SYM
Daniele Dorigoni, Michael B. Green, Congkao Wen
Abstract
A bstract We present a novel expression for an integrated correlation function of four superconformal primaries in SU( N ) $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 supersymmetric Yang-Mills ( $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper 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 $$ \tau =\theta /2\pi +4\pi i/{g}_{\mathrm{YM}}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>πi</mml:mi> <mml:mo>/</mml:mo> <mml:msubsup> <mml:mi>g</mml:mi> <mml:mi>YM</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> . In this form it is manifestly invariant under SL(2 , ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU( N ) correlator to the SU( N + 1) and SU( N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, $$ E\left(s;\tau, \overline{\tau}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mfenced> <mml:mi>s</mml:mi> <mml:mi>τ</mml:mi> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math> with s ∈ ℤ, and rational coefficients that depend on the values of N and s . The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n -loop coefficient of order ( g YM / π ) 2 n is a rational multiple of ζ (2 n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM field theory. Likewise, the charge- k instanton contributions (| k | = 1 , 2 , . . . ) have an asymptotic, but Borel summable, series of perturbative corrections. The large- N expansion of the correlator with fixed τ is a series in powers of $$ {N}^{\frac{1}{2}-\mathrm{\ell}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>−</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msup> </mml:math> ( ℓ ∈ ℤ) with coefficients that are rational sums of $$ E\left(s;\tau, \overline{\tau}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mfenced> <mml:mi>s</mml:mi> <mml:mi>τ</mml:mi> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math> with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large- N Yang-Mills theory in which $$ \lambda ={g}_{\mathrm{YM}}^2N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>g</mml:mi> <mml:mi>YM</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>N</mml:mi> </mml:math> is fixed. The coefficient of each order in the 1 /N expansion can be expanded as a series of powers of λ that converges for |λ| < π 2 . For large λ this becomes an asymptotic series when expanded in powers of $$ 1/\sqrt{\lambda } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>λ</mml:mi> </mml:msqrt> </mml:math> with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large- λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is $$ O\left(\exp \left(-2\sqrt{\lambda}\right)\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mi>λ</mml:mi> </mml:msqrt> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> .