Exact Three-Point Functions of Determinant Operators in Planar N=4 Supersymmetric Yang-Mills Theory
Yunfeng Jiang Shota Komatsu Edoardo Vescovi
hep-th math-ph math.MP nlin.SI





















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    ^1Theoretical Physics Department, CERN,  1211 Geneva 23, Switzerland
    ^2School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA
    ^3The Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
    
    We introduce a nonperturbative approach to correlation functions of two determinant operators and one non-protected single-trace operator in planar 𝒩=4 supersymmetric Yang-Mills theory. Based on the gauge/string duality, we propose that they correspond to overlaps on the string worldsheet between an integrable boundary state and a state dual to the single-trace operator. We determine the boundary state using symmetry and integrability of the dual superstring sigma model, and write down expressions for the correlators at finite coupling, which we conjecture to be valid for operators of arbitrary size. The proposal is put to test at weak coupling.
    Exact Three-Point Functions of Determinant Operators in Planar 𝒩=4 Supersymmetric Yang-Mills Theory
        Yunfeng Jiang^1, Shota Komatsu^2, Edoardo Vescovi^3
        Received ⋯ / Accepted ⋯
    ===================================================================================================
    
    
    
    § INTRODUCTION
    
    To advance our understanding of nonperturbative dynamics in gauge theories, it is useful to study simple models with rich enough structures. 𝒩=4 supersymmetric Yang-Mills theory in four dimensions (𝒩=4 SYM) is one of the leading candidates for the following reasons: First it admits the planar large N_c limit, which makes it amenable to analytical studies. Second it is a conformal field theory (CFT) and all the correlation functions can be decomposed into two- and three-point functions. Third it can be described alternatively in terms of two-dimensional string worldsheets which we can analyze exactly using integrability.
    The application of integrability led to a complete determination of two-point functions of local operators <cit.>.
    It was applied also to the three-point function <cit.>, but the result is still unsatisfactory since it is given by a series expansion which one needs to resum.
    
    In this letter, we present the first fully nonperturbative result for the three-point function valid for a large class of operators Some three-point functions are already known at finite coupling. For instance, the structure constant of a Lagrangian operator and two identical operators is given by a derivative of the conformal dimension _λΔ<cit.>. However, it is determined solely by the conformal dimension and does not provide truly new conformal data.. Specifically, we study the correlator of two determinant operators and one non-protected single-trace operator. By interpreting this correlator as an overlap on the string worldsheet between a boundary state and a state dual to the single-trace operator, we write down nonperturbative expressions using the framework of thermodynamic Bethe ansatz (TBA) <cit.>.
    
    
    
    § SET UP AND BASIC STRATEGY
    
    The main subject is the three-point function of a non-protected single-trace operator 𝒪 and two determinant operators 𝒟_1,2≡ℨ(a_1,2)[We chose a configuration suitable for analyzing the symmetry. More general configurations can be obtained by performing the conformal and the R-symmetry transformations. The structure constant 𝔇_𝒪 is not affected by such transformations.] with
    
        ℨ(a)≡.(1+a^2)Φ^1+i(1-a^2)Φ^2+2i aΦ^4/√(2)|_x^μ=  
        (0,a,0,0) ,
    
    where Φ^1,2,4 are real scalar fields in 𝒩=4 SYM. Owing to the superconformal symmetry, its spacetime dependence is fixed to be  <cit.>
        ⟨𝒟_1 𝒟_2 𝒪(0)⟩=(a_1-a_2/a_1 a_2)^Δ-J𝔇_𝒪 ,
    
    where 𝔇_𝒪 is the structure constant while Δ and J are the conformal dimension and the R-charge of 𝒪.
    
    The goal of this letter is to compute  𝔇_𝒪 nonperturbatively using the gauge/string duality.
    As discussed in <cit.>, the duality maps (<ref>) to a closed string in AdS_5×S^5 which ends on a geodesic of a D-brane dual to determinant operators (see Figure <ref>).
    
    On the string worldsheet, this corresponds to an overlap between a boundary state ⟨𝒢| and a state dual to 𝒪. To evaluate such an overlap, we first consider a partition function Z(J,R) of a cylinder worldsheet whose ends are capped off by the boundary states (see Figure <ref>).
    
    In the limit R→∞, the expansion of Z(J,R) in the closed string channel is dominated by the ground state |Ω⟩
        Z(J,R)   =∑_ψ_ c⟨𝒢|ψ_ c⟩ e^-E_ψ_ cR⟨ψ_ c |𝒢⟩
           R→∞→ |⟨𝒢|Ω⟩|^2 e^-E_ΩR
    
    By contrast, in the open string channel, Z(J,R) can be viewed as the thermal free energy, and the limit corresponds to the thermodynamic limit in which the volume of the space becomes infinite. This allows us to compute ⟨𝒢|Ω⟩ using TBA. The result for excited states can be obtained from ⟨𝒢|Ω⟩ by analytic continuation <cit.>.
    
    
    
    § CONSTRAINTS ON BOUNDARY STATES
    
    To apply the aforementioned strategy, we first determine the boundary state ⟨𝒢| in the infinite volume (J→∞) limit. For this, we assume that ⟨𝒢| is an  integrable boundary state, namely a state corresponding to a boundary condition which preserves infinite many conserved charges <cit.>. The assumption is justified a posteriori by agreement with weak-coupling computations as we see later. For integrable boundary states, the overlap in the J→∞ limit  can be factorized into two-particle overlaps
    
        F_ A B(u)≡.⟨𝒢|𝒳_ A(u)𝒳_ B(u̅)⟩|_J→∞ ,
    
    where 𝒳's are magnons in the 𝒩=4 SYM spin chain, and A=AȦ and B=BḂ are in the bifundamental representation of the 𝔭𝔰𝔲(2|2)^2 symmetry <cit.>. The rapidities u and u̅ are parity-conjugate to each other, and satisfy
    x^±(u̅)=-x^∓(u)
    where f^± (u)≡ f(u±i/2) and the Zhukovsky variable x(u) is defined by
    x(u)≡u+√(u^2-4g^2)/2g with g≡√(λ)/4π and λ being the 't Hooft coupling constant.
    
    
    
      
    Boundary Yang-Baxter equation
    The integrable boundary state satisfies the so-called boundary Yang-Baxter equation (bYBE), which reads (see Figure <ref>)
    
        .⟨𝒢|𝕊_24𝕊_34|𝒳_1(u)𝒳_2(v)𝒳_3(v̅)𝒳_4(u̅)⟩|_J→∞
        
        =   .⟨𝒢|𝕊_13𝕊_12|𝒳_1(u)𝒳_2(v)𝒳_3(v̅)𝒳_4(u̅)⟩|_J→∞
    
    where 𝕊_kl is the bulk S-matrix <cit.> between 𝒳_k and 𝒳_l.
    
    
      
    Watson's equation
    The second constraint is Watson's equation, which states that an exchange of particles is equivalent to a multiplication of the S-matrix. Explicitly, it reads (see Figure <ref>)
    
        F_𝐀𝐁(u)=𝕊_𝐀𝐁^𝐂𝐃(u,u̅)F_𝐂𝐃(u̅) .
    
    
    
      
    Decoupling equation
    The last condition is the decoupling condition, which is equivalent to the boundary unitarity in <cit.>. It states that a pair of particle-antiparticle pairs must decouple from the rest of the overlap (see Figure <ref>) and reads
    
        F_ A B(u)
        ℭ^ B B^'F_ B^' C^'(u̅^2γ)ℭ^ C^' C=δ_ A^ C
    
    where ℭ is the charge conjugation matrix <cit.>, and u^2γ is the crossing transformation defined by
    x^±(u^2γ)=1/x^±(u).
    
    
      
    Solution Solving these constraints, the two-particle overlap is fixed to be
    
    F_ A B(u)=x^+/x^-u-i/2/uσ_B (u)/σ (u,u̅)(-1)^|Ȧ||B|M_ A, B
    where |∙| denotes the grading of the index ∙ and σ (u,v) is the bulk dressing phase <cit.>. As shown in Appendix <ref>, there are two choices for the matrix part M_ A, B and we conjecture that the true boundary state is given by a sum of the two. See also discussions below (<ref>). σ_B(u) is the  boundary dressing phase satisfying σ_B (u̅)=σ_B(u)    σ_B (u)σ_B (u^2γ)=u^2/u^2+1/4
    A solution is given by σ_B (u)=4^1+x^+x^-/1-x^+x^-G(x^+)/G(x^-)
    with
    
        log G(x)=    ∮dz/2π ilog𝔊(g(z+z^-1))/x-z
    
    and 𝔊(u)≡ Γ(1/2-iu)Γ(1+iu)/Γ(1/2+iu)Γ(1-iu).
    
    
    § G-FUNCTION FOR GROUND STATE
    
    We now discuss the ground-state overlap ⟨𝒢|Ω⟩ for finite J.
    For this we consider Z(J,R) in the open string channel (also known as the  mirror channel) and take the limit R→∞:
    
    Z(J,R)=∑_ψ_oe^-Ẽ_ψ_oJ R→∞→ 𝒩∫𝒟ρ e^-RS_ eff[ρ]
    As shown above, in the limit R→∞ one can replace the sum over ψ_o with a path integral of densities ρ.
    
    
      
    Bethe equation in the mirror channel The crucial input for writing down S_ eff is the boundary asymptotic Bethe equation (bABA), which constrains the rapidities of magnons. Schematically, it reads (see Figure <ref>)
     1=ℝ_L(u_j)∏_k≠ j𝕊(u_j,u_k)  ℝ_R (u_j)  ∏_k≠ j𝕊(u_j,u̅_k)
    where ℝ_L,R are the left/right reflection matrices. The reflection matrices are related to the infinite-volume overlap (<ref>) by [ℝ_L]_ A^ B(u)=[ℝ_R]_ A^ B(u̅)=F_ A  C(u^γ)ℭ^ C B with u^γ being the mirror transformation defined by
    x^+(u^γ)=1/x^+(u) and x^-(u^γ)=x^-(u).
    As a result, we find
    
        [ℝ_L]^BḂ_AȦ(u)=u-i/2/uσ_B (u^γ)/σ (u̅^γ,u^γ)𝒮_AȦ^BḂ(u̅^γ,u^γ)
    
    where 𝒮 is a single copy of the 𝔭𝔰𝔲(2|2) S-matrix <cit.>. The structure of ℝ_L,R allows for the  unfolding interpretation; the bABA (<ref>) can be mapped to an ABA of a closed string with a single 𝔭𝔰𝔲(2|2) symmetry (see Figure <ref>).
    
    
    
      
    TBA equationS_ eff can be derived from ABA following the standard derivation of TBA. In the R→∞ limit, (<ref>) can be approximated by the saddle point δ S_ eff/δρ=0. Due to the unfolding structure, the saddle-point equations coincide with the standard TBA for the spectrum <cit.> with the identification Y_a,s(u̅)=Y_a,-s(u). They take a form of log Y_a,s=φ^+_a,s. For instance, φ^+_a,0 reads (the full equations are given in <cit.>)
    
        φ^+_a,0≡ -JẼ_a+log(1+Y_b,0)∗ (𝒦_+)^∙∙_b,a
           +log(1+Y_m,1)⋆(𝒦_+)^▹∙_m-1,a+log(1+1/Y_2,2)  ⋆  (𝒦_+)^y∙_+a
           +log(1+Y_1,1)  ⋆  (𝒦_+)^y∙_-a .
    
    Here we follow the notations in <cit.>, and ⋆, ∗, and ⋆ denote the convolutions along [-∞,∞], [0,∞] and [-2g,2g] respectively. 𝒦_+ is a symmetrized kernel defined by K(u,v)+K(u,v̅) with K being the standard TBA kernel.
    
    
      
    g-function The saddle point value of S_ eff only gives the exponentially decaying piece in (<ref>), e^-E_ΩR. To read off the overlap, we need to consider the one-loop fluctuation around the saddle point and the O(1) normalization factor 𝒩 in (<ref>). Such analysis was performed in the literature <cit.>, and the application to our problem leads to an expression for the ground states, which corresponds to BPS operators in 𝒩=4 SYM,
    
    
        ⟨𝒢|Ω⟩   =e^∑_a∫_0^∞du/2πΘ_a log (1+Y_a,0)√( Det(1-Ĝ))/ Det (1-Ĝ_+) .
    
    Here a runs from 1 to ∞ and Θ_a (u)=i_u log r_a (u̅)-πδ (u)+i/2_u log S_aa^∙∙ (u,u̅) where S_ab^∙∙ is the bound-state S-matrix and r_a is the bound-state reflection factor given in Appendix <ref>. Det denotes the Fredholm determinant [In what follows, Det denotes the Fredholm determinant while det denotes a standard determinant of a finite-dimensional matrix.] and Ĝ_+ is an integral kernel defined by [Ĝ_+]^(a,s)_(b,t)(u,v)≡δφ^+_a,s(u)/δlog Y_b,t(v). Similarly, Ĝ is given by δφ_a,s(u)/δlog Y_b,t(v) where φ_a,s are the “right hand sides” of the standard TBA, log Y_a,s=φ_a,s, without identification of Y-functions.
    
    
    § CONJECTURE FOR SL(2) SECTOR
    
    We now generalize (<ref>) to excited states in the SL(2) sector using the analytic continuation trick <cit.> following the standard TBA analysis.
    
    
      
    g-function for excited states
    After the analytic continuation, poles of 1/(1+1/Y_1,0) cross the integration contour and modify the overlap (<ref>).
    As a result, we find that the structure constant 𝔇_𝒪 is given by
    𝔇_𝒪   =-i^J+(-i)^J/√(J) e^∑_a∫_0^∞du/2πΘ_a log (1+Y_a,0)
       ×√(∏_1≤ s≤M/2u_s^2+1/4/u_s^2σ_B^2(u_s))√( Det (1-Ĝ^∙))/ Det (1-Ĝ^∙_+)
    This is the main result of this letter which we conjecture to be valid for any length J and at finite λ.
    Here the J-dependent prefactor reflects the fact that the true boundary state is a sum of two boundary states as mentioned below (<ref>). Ĝ^∙ and Ĝ_+^∙ are given by [The kernels in this letter are related to the ones in <cit.> by the transposition.]
        Ĝ^∙· f=∑_k=1^MiK^∙, X_1,x(u_k,u)/∂_ulog Y_1,0(u_k)f(u_k)
        +Ĝ· f
        Ĝ_+^∙· f=∑_k=1^M/2i(𝒦_+)^∙, X_1,x(u_k,u)/∂_ulog Y_1,0(u_k)f(u_k)
        +Ĝ_+· f
    
    Here and below, x and X take various indices and symbols which represent different bound states. The sum in (<ref>) come from the poles crossing the contours, and u_k's are the magnon rapidities satisfying the parity condition,
    
    u_M/2+k=u̅_k      (1≤ k≤ M/2)
    They are the solutions to the  exact Bethe equations<cit.>,
    ϕ (u_j)=2π i(n_j+1/2)     n_j ∈ℤ
    with
    ϕ (u)≡    -JẼ_1+∑_k=1^Mlog S^∙∙_11 (u,u_k)
       +∑_x,Xlog (1+Y_X,x)⋆ K^X,∙_x,1
    
      
    Exact Gaudin determinants The result (<ref>) can be rewritten into a form similar to the so-called Gaudin determinants. For this, we first split the kernel Ĝ^∙ into a sum S and an integral I (see (<ref>)), and rewrite Det(1-Ĝ^∙) as Det(1- S- I)= Det(1-S̃)× Det(1- I) with S̃≡ S/(1- I). Similarly, Det(1-Ĝ^∙_+) can be split into a sum S_+ and an integral I_+, and can be re-expressed as Det(1-Ĝ^∙_+)= Det(1-S̃_+)× Det(1- I_+) with S̃_+≡ S_+/(1- I_+).
    
    Next we consider _u_kϕ (u_j)(j,k=1,…, M). The derivative _u_k can act either on p(u_k), log S^∙∙_11 or Y_X,x in (<ref>). We then eliminate _u_k Y_X,x by considering the  excited state TBA (see <cit.> for the full set of equations)
    log Y_a,0=   -J Ẽ_a+∑_k=1^Mlog S^∙∙_a,1(u,u_k)
       +∑_x,Xlog (1+Y_X,x)⋆ K_x,a^X,∙
    taking a derivative _u_k of both sides, and solving for _u_kY_X,x. The parity condition (<ref>) is imposed only at the end of the computation. As a result of these manipulations, we find that (_u_kϕ(u_j))∝ Det(1-S̃) up to some constant of proportionality. The relation in particular shows that Det(1-S̃) is actually a finite-dimensional determinant although it was initially defined as the Fredholm determinant. Details of the rewriting are explained in a toy example in Appendix <ref>.
    
    On the other hand, if we first impose the parity condition (<ref>) and compute the derivatives _u_kϕ(u_j), we find that (_u_kϕ(u_j))(j,k=1,…M/2) is now proportional to Det(1-S̃_+). Upon taking the ratio, the constants of proportionality cancel out and we obtain
    √( Det(1-S̃))/ Det(1-S̃_+)=√( (_u_kϕ(u_j)))/ (_u_kϕ^+(u_j))
    where ϕ^+ denotes that we are imposing the parity condition before computing derivatives. These determinants can be viewed as the finite-volume version of the Gaudin determinants for the norm of the spin chain. They also resemble the finite-volume one-point functions in sin(h)-Gordon model <cit.>.
    
    Using this rewriting, we obtain an alternative representation for the Fredholm determinants in (<ref>),
    √( Det (1-Ĝ^∙))/ Det (1-Ĝ^∙_+)=√( (_u_kϕ(u_j)))/ (_u_kϕ^+(u_j))√( Det(1-Ĩ))/ Det(1-Ĩ_+)
    
      
    Asymptotic formula Using the representation (<ref>), one can take the  asymptotic limit of (<ref>),  in which the size of the operator becomes large J≫ 1. In this limit, the middle-node Y-functions are exponentially suppressed, Y_a,0→ 0, and one can show that both
    
    e^∑_a∫_0^∞du/2πΘ_a log (1+Y_a,0)  and  √( Det(1-Ĩ))/ Det(1-Ĩ_+)
    tend to unity. We thus obtain the following expression for the structure constant in the asymptotic limit:
    𝔇_𝒪^ asym=   -i^J+(-i)^J/√(J)√(∏_1≤ s≤M/2u_s^2+1/4/u_s^2σ_B^2(u_s))
       ×√( (_u_kϕ(u_j)))/ (_u_kϕ^+(u_j))
    Note that the determinants on the second line are the standard Gaudin-like determinants since all the finite-size corrections can be dropped.
    For generalization of (<ref>) to operators outside the SL(2) sector, see <cit.>. A similar formula was found at weak coupling for the defect one-point functions <cit.>.
    
    
    
    § WEAK COUPLING TEST
    
    To test our formula (<ref>), we computed the four-point function of 𝒟_1,2 and two 20^' operators 𝒪_ 20^' up to O(λ^2). We then performed the operator product expansion to read off the conformal data of the spin-S twist-2 operators 𝒪_S. The details are given in <cit.>.
    
    The results of the computation are summarized in Table <ref>. We compared them against the integrability prediction (<ref>) and observed a perfect match. This is quite a nontrivial test of our formalism since the results contain the transcendental number ζ_3 and include the contributions from the boundary dressing phase σ_B(u). Further tests at weak and strong couplings are provided in <cit.>.
    
    
    
    We also found that the structure constants exhibit a simple large spin behavior up to two loops,
    log[𝔇_𝒪_S/.𝔇_𝒪_S|_tree level]=   f_1 log S^'+f_2+O(1/S^')
    
    with log S^'≡log S +γ_E, where γ_E is the Euler-Mascheroni constant, and
    
    f_1    =-4 g^2 log 2+8g^4[ζ_2log 2+9/2ζ_3]+O(g^6)
    
    f_2   =-2g^2ζ_2+8g^4[4/5(ζ_2)^2+3/2ζ_3log 2]+O(g^6)
    
    § CONCLUSION
    
    In this letter, we applied the TBA formalism to write down a nonperturbative expression for the structure constant of two determinant operators and a single-trace operator in the SL(2) sector of arbitrary size. Our result would provide a foundation for future developments, such as the reformulation in terms of the Quantum Spectral Curve <cit.>, as was the case with the TBA for the spectrum. It would also be worth trying to extract various interesting physics from our formula. We also hope that our approach gives useful insights into the three-point functions of single-trace operators <cit.>.
    
    
      
    *Acknowledgement
    We thank Amit Sever for helpful discussions. SK is supported by DOE
    grant number DE-SC0009988. EV is funded by the FAPESP grants 2014/18634-9 and 2016/09266-1, and by
    the STFC grant ST/P000762/1.
    
    
    § OVERLAP AND REFLECTION MATRIX
    
    The matrix part of the two-particle overlap is given by
    
    M_aȧ,bḃ   =1+B/2ϵ^aḃϵ^bȧ+1-B/2ϵ^aȧϵ^bḃ
    
    M_αα̇,ββ̇   =-1+E/2ϵ^αβ̇ϵ^βα̇-1+E/2ϵ^αα̇ϵ^ββ̇
    
    M_aȧ,ββ̇   =Gϵ^aȧϵ^ββ̇  
    M_αα̇,bḃ=Gϵ^αα̇ϵ^bḃ
    
    M_aα̇,bβ̇   =Cϵ^abϵ^α̇β̇  
    M_aα̇,βḃ=Hϵ^aḃϵ^βα̇
    
    M_αȧ,bβ̇   =Hϵ^bȧϵ^αβ̇  
    M_αȧ,βḃ=Cϵ^ȧḃϵ^αβ
    where
       
    B=-x^-+z^4(x^+)^3/ z^2x^+(1+x^+x^-)   E=z^4x^++(x^-)^3/z^2 x^-(1+x^+x^-)
       C=-i[z^4(x^+)^2-(x^-)^2]/2z^2√(x^+x^-)(1+x^+x^-)
       G=x^-+z^2x^+/2 z√(x^+x^-)   H=x^--z^2x^+/2 z√(x^+x^-)
    There are two allowed solutions depending on the value of z, z=± i.
    
    The bound-state reflection factor r_a is given by
    
    r_a(u)=(∏_k=-a-1/2^a-1/2u+ik+i/2/u+ik)σ_B,a(u̅^γ)/σ_aa(u^γ,u̅^γ)σ_B,a(u)≡ 2^-E_a(u)G(x^[+a])/G(x^[-a])
    with f^[± a]≡ f(u±i a/2) and
    
    E_a(u)=a/2+g/i(1/x^[-a]-1/x^[+a])
    
    § FREDHOLM DETERMINANT IN TOY MODEL
    
    Here we write down an explicit expression for the Fredholm determinant in a simple toy model which only contains a single species of particles without bound states. We also elucidate the relation between the Fredholm determinants and the exact Gaudin determinants (<ref>).
    
    The excited state TBA is given by log Y =ϕ with
    ϕ (u)≡ -J Ẽ+∑_k=1^Mlog S (u,u_k) +log (1+Y)⋆ K
    where S is the S-matrix and K(u,v)≡ -i_ulog S(u,v). The rapidities u_k's satisfy the exact Bethe equation
    ϕ (u_j)=2π i (n_j+1/2)     n_j ∈ℤ
    Following the discussion in the main text, one obtains the deformed Fredholm kernel Ĝ^∙· f= S· f + I· f, with
     S· f≡∑_k=1^Mi K(u_k,u)/_u ϕ(u_k)f(u_k)   I· f≡f/1+1/Y⋆ K
    
    Let us now consider the exact Gaudin norm (_u_kϕ(u_j)). Using (<ref>), we obtain
    _u_kϕ(u_j)=_u ϕ(u_k)δ_jk-iK(u_k,u_j)+ I· (_u_kϵ) (u_j)
    with ϵ(u)≡log Y (u). To eliminate _u_kϵ, we differentiate the excited state TBA,
    _u_kϵ(u)=-iK(u_k,u)+ I· (_u_kϵ)(u)
    and invert it to get
    _u_kϵ (u)=1/1- I·[-iK(u_k,u)]
    Plugging it back to (<ref>), we obtain
    _u_kϕ(u_j)=_u ϕ (u_k)δ_jk-.1/1- I·[iK(u_k,u)]|_u=u_j
    
    Comparing (<ref>) and (<ref>), we find
     (_u_kϕ(u_j))=(∏_k=1^M_uϕ (u_k)) Det (1-S̃)
    with S̃≡ S/(1- I). This relation allows us to relate the Fredholm determinant and the exact Gaudin determinant. The derivation can be readily generalized to Ĝ_+^∙ and to theories with bound states such as 𝒩=4 SYM, leading to the identity (<ref>).
