A novel representation of an integrated correlator in \mathcal{N}=4 SYM theory
aa r X i v : . [ h e p - t h ] F e b A novel representation of an integrated correlator in N = 4 SYM theory
Daniele Dorigoni, Michael B. Green,
2, 3 and Congkao Wen Centre for Particle Theory & Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge CB3 0WA, UK School of Physics and Astronomy, Queen Mary University of London, London, E1 4NS, UK
An integrated correlator of four superconformal stress-tensor primaries of 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 , Z ) S-duality. This expression isshown 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 shownto precisely reproduce known perturbative and non-perturbative properties of N = 4 SYM for anyfinite N , as well as extending previously conjectured properties of the large- N expansion. N = 4 supersymmetric Yang–Mills ( N = 4 SYM) the-ory [1] is a highly non-trivial four-dimensional conformalfield theory that is of exceptional interest for a varietyof reasons. It possesses maximal supersymmetry, whichis connected to the fact that it is integrable, which inturn enables many of its properties to be determined an-alytically. Furthermore, its relation to string theory in AdS × S via the AdS/CFT correspondence provides amodel for more general examples of holography.Of particular significance to this letter is the analy-sis of the integrated correlation function of four super-conformal primaries that was formulated in terms of a N -dimensional matrix model in [2], and further devel-oped in [3–6]. This integrated correlator was defined interms of the partition function of N = 2 ∗ SYM theory,which is a mass deformation of the superconformal N = 4SYM theory with mass parameter m . The suitably nor-malised N = 2 ∗ SU ( N ) partition function, on a round S , Z N ( m, τ, ¯ τ ), was determined by Pestun using super-symmetric localisation [8]. Our notation follows usualconventions where the complex Yang–Mills coupling con-stant is defined by τ = τ + iτ := θ/ π + i π/g Y M .In [2] the integrated correlator of four superconformalprimaries was identified with the m → Z N that has the form G N ( τ, ¯ τ ) := 14 ∆ τ ∂ m log Z N ( m, τ, ¯ τ ) (cid:12)(cid:12)(cid:12)(cid:12) m =0 = − π Z ∞ dr Z π dθ r sin ( θ ) U T N ( U, V ) , (1)where ∆ τ = 4 τ ∂ τ ∂ ¯ τ is the hyperbolic laplacian and thecross-ratios U, V are defined by U = x x x x , V = x x x x , (2) The normalisation of the integrated correlator differs from thatin [2] by a factor of c/ c = ( N − / and are related to r and θ by U = 1 + r − r cos( θ )and V = r . The function T N ( U, V ) is related to thefour-point correlator by hO ( x , Y ) . . . O ( x , Y ) i (3)= 1 x x [ T N, free ( U, V ; Y i ) + I ( U, V ; Y i ) T N ( U, V )] , where O ( x i , Y i ) is a superconformal primary in the ′ of the SU (4) R symmetry, which is encoded in the depen-dence on the null vectors Y i . T N, free ( U, V ; Y i ) is the freefield correlator and the pre-factor I ( U, V ; Y i ) is deter-mined by superconformal symmetry [9, 10]. So we onlyfocus on the non-trivial part, T N ( U, V ).As pointed out in [2], the relation (1) between the massderivatives of the localised partition function and the in-tegrated four-point correlator may lead to mixing withlong operators, such as the Konishi operator. Not onlydo such effects decouple in the large- N strong couplinglimit, as argued in [2], but they also do not appear atfinite N and finite coupling. We will see direct evidenceof this statement in our results later in this letter.The results in this letter follow from a reformulationof G N ( τ, ¯ τ ), as a two-dimensional lattice sum that makesmanifest many of its properties for all values of N and τ . These results, which are based on a wealth of ev-idence concerning the structure of G N ( τ, ¯ τ ) in variouslimits, take the form of a conjecture rather than a math-ematical theorem: Conjecture : The integrated correlation function offour superconformal primary operators in the stress ten-sor multiplet of N = 4 SU ( N ) supersymmetric Yang–Mills theory is given by the lattice sum G N ( τ, ¯ τ ) = X ( m,n ) ∈ Z Z ∞ e − tπ | m + nτ | τ B N ( t ) dt , (4) We thank Shai Chester, Silviu Pufu and Yifan Wang for clarifi-cations on this point. This letter presents our main results but details of the derivationand further results are contained in [7]. where B N ( t ) has the form B N ( t ) = Q N ( t )( t + 1) N +1 , (5) and where Q N ( t ) is a polynomial of degree N − definedby Q N ( t ) = − N ( N − − t ) N +1 − t ) (cid:26) (3 + (8 N + 3 t − t ) P (1 , − N (cid:18) t − t (cid:19) + 3 t − N t −
31 + t P (1 , − N (cid:18) t − t (cid:19)(cid:27) , (6) and P ( α,β ) N ( z ) is a Jacobi polynomial. The following general properties of B N ( t ) are of im-portance in the following, B N ( t ) = 1 t B N (cid:18) t (cid:19) , (7)and Z ∞ B N ( t ) dt = N ( N − , Z ∞ B N ( t ) 1 √ t dt = 0 . (8)Using relationships between derivatives of Jacobi poly-nomials leads to the recurrence relation t d dt ( t B N ( t )) = N ( N − B N +1 ( t ) − N − B N ( t )+ N ( N + 1) B N − ( t ) . (9)The lattice sum (4) is convergent for τ in the upperhalf plane τ = Im τ > SL (2 , Z ) transformations τ → γ · τ = aτ + bcτ + d , γ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ) , (10)which is in accord with the expectations of Montonen–Olive duality [11–13].An important consequence of (4) together with (9) isthat G N ( τ, ¯ τ ) satisfies the following corollary: Corollary : The integrated correlator satisfies aLaplace-difference equation of the form (∆ τ − G N ( τ, ¯ τ ) = N ( N − G N +1 ( τ, ¯ τ ) − N G N ( τ, ¯ τ ) + N ( N + 1) G N − ( τ, ¯ τ ) . (11)This follows by applying the laplacian ∆ τ to (4) andusing (9). Equation (11) provides powerful constraintson G N that relate the dependence on the coupling τ andthe dependence on N in a manner that will be discussedlater. For now we note that as N → ∞ , assuming G N is a differentiable function of N , (11) becomes a Laplaceequation in both τ and N , taking the form(∆ τ − G N ( τ, ¯ τ ) = N →∞ ( N ∂ N − N ∂ N ) G N ( τ, ¯ τ ) , (12)where terms of higher order in 1 /N have been suppressed. The structure of the integrated correlator
The N = 2 ∗ partition function appearing in (1) hasthe form [8] Z N ( m, τ, ¯ τ ) = Z d N a i δ ( X i a i ) (cid:16) Y i 12 + ∞ X s =2 ( s − s − Γ( s + 1) ζ (2 s ) ( − y ) s π s , (18)It will be useful to formally identify the k = 0 mode of G , in (17) with the average of the y s and y − s terms, G , ( y ) = 12 (cid:16) G ( i )2 , ( y ) + G ( ii )2 , ( y ) (cid:17) . (19)The non-zero modes corresponding to instanton andanti-instanton contributions can be extracted from the | ˆ Z inst | factor in (13) by extending the analysis in[5]. This involves a systematic decomposition of∆ τ ∂ m ˆ Z inst ( m, τ, a ij ) (cid:12)(cid:12) m =0 in terms of a sum over rect-angular Young diagrams with ˆ m rows and n columns,where k = ˆ m n is the instanton number. The resulting k -instanton contribution is G ,k ( τ, ¯ τ ) = (20) e πikτ X ( ˆ m,n ) =(0 , ˆ mn = k Z ∞ e − πτ ( ˆ m t + n t ) r τ t B ( t ) dt , with B ( t ) given in (5) for N = 2. This integral can beexpanded as an infinite sum of K -Bessel functions usingthe integral representation Z ∞ e − a t − b t t ν − dt = 2 (cid:16) ab (cid:17) ν K ν (2 ab ) , (21)with a = √ πτ ˆ m and b = p π/τ n .We now recognise that the total integrated correla-tor, G = G , + P k =0 G ,k is an infinite sum of non-holomorphic Eisenstein series with integer index and withrational coefficients G ( τ, ¯ τ ) = 14 + 12 ∞ X s =2 c (2) s E ( s ; τ, ¯ τ ) , (22)where c (2) s = ( − s s − − s ) Γ( s + 1) . (23)In making this identification we recall that a non-holomorphic Eisenstein series has a Fourier expansion ofthe form E ( s ; τ, ¯ τ ) := 1 π s X ( m,n ) =(0 , τ s | m + nτ | s = 2 ζ (2 s ) π s τ s + 2 √ π Γ( s − ) ζ (2 s − π s Γ( s ) τ − s (24)+ X k =0 e πikτ s ) | k | s − σ − s ( | k | ) √ τ K s − (2 π | k | τ ) . We further note that E ( s, τ, ¯ τ ) satisfies the Laplace equa-tion (∆ τ − s ( s − E ( s, τ, ¯ τ ) = 0 . (25)Upon substituting the integral representation E ( s ; τ, ¯ τ ) = X ( m,n ) =(0 , Z ∞ e − tπ | m + nτ | τ t s − Γ( s ) dt , (26)into (22) it takes the form given in (4) with N = 2. (ii) Gauge groups SU ( N ) with N > . Here the di-rect analysis of (1) is considerably more complicated andis presented in more detail in [7], where the expressionfor B N ( t ) in (5) is motivated. However, for the pur-poses of this letter it is more efficient to use the Laplace-difference equation (11) to generate the expression forthe integrated correlator when N > 2. Once we inputthe boundary conditions G = 0 and G given by (22),the correlators for theories with higher N are generatedrecursively. They may be expressed as G N ( τ, ¯ τ ) = N ( N − ∞ X s =2 c ( N ) s E ( s ; τ, ¯ τ ) , (27)where the coefficients c ( N ) s are rational numbers that de-pend on N and are generated by the expansion of B N ( t )in the form B N ( t ) = ∞ X s =2 c ( N ) s Γ( s ) t s − . (28)The coefficients c ( N ) s can also be determined up to anydesired order by substituting the series (27) into (11) andsolving iteratively in terms of the coefficients c (2) s givenin (23). Properties of the integrated correlator The integrated correlator has interesting behaviourwhen expanded in various domains of the parameters, N and τ . We will here discuss three of these domains. (a) Finite N , small λ = g Y M N . This is the domain ofstandard Yang–Mills perturbation theory. The expansionof the perturbative part of the expression (4) has the form G N, ( τ ) = ( N − (cid:20) ζ (3) a − ζ (5) a ζ (7) a − ζ (9) (cid:0) N − (cid:1) a 32 + 114345 ζ (11) (cid:0) N − (cid:1) a − ζ (13) (cid:0) N − + N − (cid:1) a O ( a ) , (29)where a = g Y M N/ (4 π ) and arbitrary N ≥ 2. When N = 2, this reduces to (16). Although the perturbativeexpansion of the unintegrated four-point correlator hasa very complicated dependence on the cross ratios U, V ,the above expression is remarkably simple, consisting ofa power series in a with coefficients that are rational mul-tiples of odd Riemann zeta values.This expansion is in rather impressive agreement withknown facts concerning the perturbative expansion ofthe four-point correlator of superconformal primaries of N = 4 SYM. The expressions for the unintegrated corre-lator up to three loops (up to order a ) are given in [15].In [7] we have verified the integrals of the one-loop andtwo-loop contributions agree with the coefficients propor-tional to a and to a in (29). In performing these integralswe make use of the all-order results for ladder diagrams[16]. The one-loop and two-loop contributions are spe-cial cases of such ladder diagrams. However, at higherloops the correlator contains more general diagrams thatwe have not evaluated.A further property that is apparent from the pertur-bative expansion (29) is the dependence on N . We seethat up to order a the coefficients do not depend on N , apart from the overall factor of ( N − N − at order a [17, 18]. From (29) we anticipate that an extra powerof N − will appear at every subsequent even power of a ,which is in agreement with the observations in [19]. Thisprecise agreement would be spoilt if there were any ad-ditional contribution such as mixing with a three-pointcorrelator involving the Konishi operator. (b) Large N , with fixed λ = g Y M N . In this limit in-stantons are of order e − π kN/λ and are therefore sup-pressed. The large- N expansion of the correlator is’t Hooft’s topological expansion, G N ( τ, ¯ τ ) ∼ ∞ X g =0 N − g G ( g ) ( λ ) , (30)where the leading term is of order N and is given bythe sum of planar Feynman diagrams in Yang–Mills per-turbation theory. Given our knowledge of the coefficients c ( N ) s in (28) we are able to determine the λ -dependenceof G ( g ) order by order in N . For small λ the leading termis given by the series G (0) ( λ ) = ∞ X n =1 − n +1 ζ (2 n + 1)Γ (cid:0) n + (cid:1) π n +1 Γ( n )Γ( n + 3) λ n , (31)which converges for | λ | < π . It can be resummed to give G (0) ( λ ) = λ Z ∞ dw w F (cid:16) ; 2 , − w λπ (cid:17) π sinh ( w ) , (32)which is well-defined for λ ≥ π and coincides with theresult of [2] after using an identity that relates F andBessel functions J α .However, following an analysis similar to [20], it is easyto see that the large- λ expansion of (32) is divergentand not Borel summable since the Borel integral, is ob-structed by a branch cut along the positive axis. This sig-nifies that in order to reproduce the exact result (32) oneneeds a resurgent non-perturbative completion ∆ G (0) ( λ ), which is determined in [7] to be of the form∆ G (0) ( λ ) = i (cid:16) ( e − √ λ ) + 18 Li ( e − √ λ ) λ / + 117 Li ( e − √ λ )4 λ + 489 Li ( e − √ λ )16 λ / + · · · (cid:17) . (33)This expression is a sum of ‘instantonic’ terms that arenon-perturbative in 1 / √ λ , with coefficients O ( e − √ λ )that is similar to those found in [21–23] for the cuspanomalous dimension and other quantities in N = 4 SYM[24]. Similar arguments lead to non-perturbative comple-tions of G ( g ) . For example, the expression for ∆ G (1) ( λ )is also determined in [7] and takes an analogous form as(33). We believe that such non-perturbative effects inthe large- λ expansion have a holographic interpretationin terms of string world-sheet instantons. (c) Large N , with fixed g Y M . This is the large- N limitin which Yang–Mills instantons contribute in a mannerthat ensures that SL (2 , Z ) S-duality is manifest. Theform of G N can be obtained (as in [7]) by a large- N ex-pansion of B N ( t ) (defined in (5)), which is an expansionin half-integer powers of N . It is easy to check that thisleads to a solution of (11) of the form G N ( τ, ¯ τ ) ∼ N ∞ X ℓ =0 N − ℓ ℓ + X s = d sℓ E ( s ; τ, ¯ τ ) , (34)which is a series of Eisenstein series with s ∈ Z + 1 / s = ℓ + satisfy the limiting large- N Laplace equation (12) but this does not determine theircoefficients, which have to be input from the expansionof B N ( t ), giving d ℓ + ℓ = ( ℓ + 1)Γ (cid:16) ℓ − (cid:17) Γ (cid:16) ℓ + (cid:17) Γ (cid:16) ℓ + (cid:17) ℓ +2 π / ℓ ! . (35)Once d ℓ + ℓ is input the Laplace-difference equation de-termines the rest of the solution. This reproduces andextends the results of [5], where the first few coefficientswere obtained. For example, terms with s = ℓ − > s = ℓ − > d ℓ − ℓ = − ( ℓ − (2 ℓ + 9)Γ (cid:0) ℓ − (cid:1) Γ (cid:0) ℓ + (cid:1) ℓ +3 π / ℓ ! ,d ℓ − ℓ = ( ℓ − (20 ℓ + 48 ℓ − (cid:0) ℓ − (cid:1) 45 2 ℓ +5 π / Γ( ℓ )Γ ( ℓ − ) Γ ( ℓ + ) . (36)Finally, we believe that the considerations of this let-ter generalise to a second integrated correlator that wasconsidered in [4] and further explored in [6]. This is ob-tained from the N = 2 ∗ partition function by apply-ing four derivatives with respective to mass, G ′ N ( τ, ¯ τ ) := ∂ m log Z ( N ) ( m, τ, ¯ τ ) (cid:12)(cid:12) m =0 , which again generates a super-symmetric integrated correlator of four superconformalprimaries, but with a different integration measure. ACKNOWLEDGMENTS We would like to thank Shai Chester, Lance Dixon, PaulHeslop, Silviu Pufu, Yifan Wang, and Gang Yang, for use-ful conversations and comments. DD would like to thank theAlbert Einstein Institute for the hospitality and support dur-ing the writing of this paper. MBG has been partially sup-ported by STFC consolidated grant ST/L000385/1. CW issupported by a Royal Society University Research FellowshipNo. UF160350.[1] L. Brink, J. H. Schwarz and J. Scherk, “SupersymmetricYang-Mills Theories,” Nucl. 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