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

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 R 3 connected via a generalisation of branch cuts.We discuss various general features of our proposal and we also present explicit analytic solutions.

Spherical transverse M5-branes from the plane wave matrix model

“…This corresponds to kink-like configurations in PWMM which connect two distinct vacua [18][19][20][21]. However, the instanton amplitudes are bounded from below by N 2 /λ times the difference of the quadratic Casimirs of the two vacua.…”

Giant graviton interactions and M2-branes ending on multiple M5-branes

“…In this section we show that the instanton equation (2.40) can be mapped to the 4d Laplace equation following the procedure laid out in [13]. As will be shown later, the key observation in [13] is the following relations, as illustrated in Fig. 3:…”

“…In the case of the BMN matrix model, the instanton equation analogous to (2.40) can be mapped to the 3d Laplace equation and various solutions, such as one membrane splitting into two membranes, have been found [13]. In this section we show that the instanton equation (2.40) can be mapped to the 4d Laplace equation following the procedure laid out in [13]. As will be shown later, the key observation in [13] is the following relations, as illustrated in Fig.…”

Towards a tensionless string field theory for the $$ \mathcal{N}=\left(2,0\right) $$ CFT in d = 6

“…Apart from the case of a few exceptional operators, 8 one can show that 20) where ·O denotes the action of the operator from the right on the complex conjugate of the field (as a bra-vector). For instance, in the present case, we have…”

“…Moreover, although the (2, 0) CFT is inherently non-perturbative, as implied by its M-theory origin, a Lagrangian description should make it possible to construct reliable weak-coupling approximation schemes valid in special sectors and/or for special observables, such as near-BPS quantities. These ideas were exploited in [19,20] in the case of the ABJM theory [21] -the maximally supersymmetric CFT in three dimensions, associated with coincident M2-branes -which is also intrinsically strongly coupled. In [19], using the AdS/CFT correspondence, a perturbative analysis of the spectrum in a special sector of the ABJM theory was successfully compared to the dual AdS description provided by the pp-wave matrix model [22].…”