Non-negativity of BMN two-point functions with three string modes

Published for SISSA by Springer Received: May 6, 2021 Revised: June 22, 2021 Accepted: July 13, 2021 Published: August 2, 2021 Bao-ning Du and Min-xin Huang Interdisciplinary Center for Theoretical Study, Department of Modern Physics, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, China Peng Huanwu Center for Fundamental Theory, 96 Jinzhai Road, Hefei, Anhui 230026, China E-mail: , Abstract: Recently, we proposed a novel entry of the pp-wave holographic dictionary, which equated the Berenstein-Maldacena-Nastase (BMN) two-point functions in free N = 4 super-Yang-Mills theory with the norm squares of the quantum unitary transition amplitudes between the corresponding tensionless strings in the infinite curvature limit, for the cases with no more than three string modes in different transverse directions. A seemingly highly non-trivial conjectural consequence, particularly in the case of three string modes, is the non-negativity of the BMN two-point functions at any higher genus for any mode numbers. In this paper, we further perform the detailed calculations of the BMN two-point functions with three string modes at genus two, and explicitly verify that they are always non-negative through mostly extensive numerical tests. Keywords: 1/N Expansion, AdS-CFT Correspondence, Long strings ArXiv ePrint: 2104.12502 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2021)006 JHEP08(2021)006 Non-negativity of BMN two-point functions with three string modes Contents 1 2 More on the physical motivation 4 3 Genus one 5 4 Genus two 4.1 Some useful symmetries 4.2 The result 7 8 13 5 Conclusion 18 A Some standard integrals 19 1 Introduction We continue the studies of free BMN correlators in our recent paper [1]. The general motivations and the physical settings are explained in the previous paper. We shall provide a recapitulation with some new comments more relevant to the current context. The AdS/CFT correspondence [2–4] is a conceptual breakthrough in our understandings of quantum gravity, in particular provides a non-perturbative definition of string theory in AdS background in terms of N = 4 SU(N ) super-Yang-Mills theory. We consider the Penrose limit [5], which gives rise to another maximally supersymmetric background [6], known as the pp-wave or plane wave geometry ds2 = −4dx+ dx− − µ2 (~r 2 + ~y 2 )(dx+ )2 + d~r 2 + d~y 2 , (1.1) where x+ , x− are light cone coordinates, ~r, ~y are 4-vectors, and the parameter µ measures the spacetime curvature as well as the Ramond-Ramond flux F+1234 = F+5678 ∼ µ. This appears to be a promising ground for quantitative explorations of the holographic duality in stringy regimes, as the dual theories on both sides can be either free or weakly coupled. In the groundbreaking paper [7], Berenstein, Maldacena and Nastase (BMN) proposed a type of near-BPS operators, which correspond to the type IIB closed strings on the ppwave background. The free string spectrum is correctly reproduced by gauge interactions as the planar conformal dimensions of BMN operators. The BMN scaling limit with large √ R-charge J ∼ N ∼ ∞ appears to be the right Goldilocks limit in this situation, since a smaller R-charge would not provide finite string interactions in the strict N ∼ ∞ limit, while a larger R-charge may blow up the strings into D-branes, known as giant gravitons, studied in early papers e.g. [8–13]. Some recent studies relating to the large R-charge –1– JHEP08(2021)006 1 Introduction J hŌ(m OJ i = δm1 ,n1 · · · δmk ,nk , 1 ,m2 ,··· ,mk ) (n1 ,n2 ,··· ,nk ) 0 J hŌ(m OJ i ∼ g 2h . 1 ,m2 ,··· ,mk ) (n1 ,n2 ,··· ,nk ) h (1.2) As discussed in [1], the BMN two-point functions are real and symmetric, and there is a nice normalization relation summing over one set of mode numbers X Pk i=1 J hŌ(m OJ i = 1 ,m2 ,··· ,mk ) (n1 ,n2 ,··· ,nk ) h g 2h . 22h (2h + 1)! (1.3) nk =0 We may define a matrix element, summing up all genus contributions with a proper nor- –2– JHEP08(2021)006 limit or Penrose limit, as well as applications in more general theories can be found in e.g. [14–20]. As in our previous papers [1, 21–25], we focus on free gauge theory and study the BMN correlation functions. This corresponds to the pp-wave background with infinite curvature and infinite Ramond-Ramond flux as µ ∼ ∞ in the geometry (1.1), where the strings are tensionless with completely degenerate spectrum. There are still interesting string 2 interactions as we identify the finite genus-counting parameter g := JN as the effective string coupling constant in this case. Some non-planar BMN correlators are first computed in [26, 27]. The celebrated standard AdS holographic dictionary [4] seems not directly applicable for string interactions in the pp-wave background, as the geometries are quite different. So in some cases, certain guessworks may be required to identify the correct entries of the “ppwave holographic dictionary”. In this paper we focus on a probability interpretation of BMN two-point functions [1, 24]. There are other interesting entries of the pp-wave holographic dictionary, namely the comparisons of free planar BMN three-point functions with GreenSchwarz light-cone string field cubic vertices [21, 28–31], the factorization formulas [22, 23], which are most recently explored in the recent paper [1] in the context of many string modes. Higher point correlators, including the planar three-point functions, actually always vanish in the strict BMN limit, and are now perceived by us as a kind of virtual processes. Let us introduce some notations. The BMN vacuum operator is simply proportional Tr(Z J ) where Z is a complex scalar in the N = 4 SU(N ) super-Yang-Mills theory. One can insert the four remaining real scalars into the trace with phases, corresponding to string modes in four of the eight transverse dimensions. The BMN operators are then denoted as J O(m , where the positive and negative integer modes represent the left and right 1 ,m2 ,··· ,mk ) moving stringy excited modes, while the zero modes are supergravity modes representing discretized momenta in the corresponding traverse direction. We will consider the case of string modes in different transverse directions, otherwise the BMN operators are no longer near-BPS and there may be some potential issues as discussed in [1]. Due to the closed P string level matching condition i mi = 0, the excited stringy states have at least two string modes. The BMN operators are properly normalized to be orthonormal at planar level, and the genus h two-point functions are proportional to g 2h as malization by the all-genera formula of vacuum correlator p(m1 ,m2 ,··· ,mk ),(n1 ,n2 ,··· ,nk ) = ∞ X g hŌJ OJ ih , g 2 sinh( 2 ) h=0 (m1 ,m2 ,··· ,mk ) (n1 ,n2 ,··· ,nk ) (1.4) so that it looks like a probability distribution X Pk i=1 p(m1 ,m2 ,··· ,mk ),(n1 ,n2 ,··· ,nk ) = 1. (1.5) nk =0 p(m1 ,··· ,mk ),(n1 ,··· ,nk ) (...truncated)