On membrane interactions and a three-dimensional analog of Riemann surfaces

Published for SISSA by Springer Received: November 2, 2015 Accepted: January 20, 2016 Published: February 8, 2016 On membrane interactions and a three-dimensional analog of Riemann surfaces a Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland b ICTP South American Institute for Fundamental Research, IFT-UNESP, São Paulo, SP 01440-070, Brazil c National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwartersrand, Wits 2050, South Africa d Okayama Institute for Quantum Physics, Okayama, Japan E-mail: , , Abstract: Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of R3 connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions. Keywords: Penrose limit and pp-wave background, Solitons Monopoles and Instantons, M(atrix) Theories, M-Theory ArXiv ePrint: 1508.03367 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP02(2016)050 JHEP02(2016)050 Stefano Kovacs,a,b Yuki Satoc and Hidehiko Shimadad Contents 2 2 BPS instantons 2.1 Instanton equations 2.2 Some properties of BPS instantons and their moduli spaces 4 5 8 3 Mapping to three-dimensional Laplace equation 3.1 Continuum approximation 3.2 Classical membrane perspective 10 10 12 4 Solutions 4.1 Stable sphere 4.2 Single membrane splitting and Riemann space 4.2.1 Qualitative picture: introduction of the concept of Riemann space 4.2.2 Analytic solution 4.3 General Riemann spaces 4.3.1 More solutions 4.3.2 Comments on the moduli space of solutions 13 14 15 15 20 28 31 33 5 Conclusions and discussion 42 A Geometric proof of the equivalence between Laplace and continuum Nahm equations 48 B Behaviour at the splitting point 50 C Other solutions C.1 BHP solution C.2 Solution with positive and negative point charges 54 54 55 D Relation between Hobson’s and Sommerfeld’s solutions 56 E Boundary conditions at branch disks and reformulation in terms of integral equations 58 F Equations for general pp-wave matrix model –1– 61 JHEP02(2016)050 1 Introduction 1 Introduction –2– JHEP02(2016)050 The best candidate for a microscopic description of M-theory is the matrix model originally proposed in [1] as a regularisation of the supermembrane theory and subsequently conjectured in [2] to describe the full dynamics of the theory in light-front quantisation when the size of the matrices is sent to infinity. The matrix model provides a formulation of M-theory which is capable of describing multi-membrane configurations, arising as block-diagonal matrices. Membranes in M-theory should interact via splitting and joining processes and therefore the matrix model should capture such effects. However, no concrete proposal for the description of these splitting/joining interactions in the matrix model has been formulated. More generally there is no quantitative prescription for the study of this type of membrane interactions, which would make it possible to evaluate the associated transition amplitudes. This makes it difficult to test any results for these amplitudes that may be obtained from the matrix model. This is of course a general difficulty with all predictions of the matrix model, which so far have been mostly tested at the level of the low energy supergravity approximation. The AdS/CFT correspondence — and specifically the duality proposed in [3], relating M-theory in AdS4 × S 7 /Zk to an N = 6 Chern-Simons theory — provides in principle new means of testing the matrix model predictions by comparing them to a dual CFT. In the context of the duality of [3] it was recently shown in [4] that the matrix model description of M-theory in AdS4 ×S 7 /Zk can be quantitatively compared to the dual gauge theory without relying on the supergravity approximation or compactification to type IIA string theory in ten dimensions. The crucial observation of [4] was that, focussing on large angular momentum states in M-theory and the dual CFT sector involving monopole operators, natural approximation schemes arise on the two sides of the duality, so that a systematic, quantitative comparison is possible. On the gravity side, M-theory states with large angular momentum, J, along a great circle in S 7 can be studied using the pp-wave approximation. The associated matrix model uses J × J matrices and its action was constructed in [5]. We will use basic properties of the model which were further studied in [6]. Multi-membrane states in the pp-wave matrix model consist of concentric membranes and their fluctuations. More precisely the vacua of the theory consist of spherical membranes which extend in AdS4 directions and are point-like in S 7 . They are classified by a set of integers, Ji , i = 1, . . . , n, corresponding to a partition of the total angular momentum among n membranes. The fluctuations of the spherical membranes described by the pp-wave matrix model are associated, in the dual gauge theory, with certain monopole operators. The latter are characterised by their integer GNO charges, which are in one-to-one correspondence with the angular momenta, Ji , of the membranes. The sector of monopole operators with large GNO charges can be reliably studied using a weakly-coupled effective low-energy approximation. The AdS4 /CFT3 duality in this M-theoretic regime relates correlation functions of monopole operators in the ABJM theory to processes on the gravity side in which the dual states interact in the bulk and propagate to the boundary. The simplest such process involves a single membrane splitting into two — with the associated three states propagating (...truncated)