The Automorphic Membrane

Abstract

We present a 1-loop toroidal membrane winding sum reproducing the conjectured M -theory, four-graviton, eight derivative, R 4 amplitude. The U -duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for M -theory compactified on a 3-torus, where the target-space SL(3, Z) × SL(2, Z) U -duality and SL(3, Z) world-volume modular groups are embedded in E 6(6) (Z). Unlike previous semi-classical expansions, U -duality is built in manifestly and realized at the quantum level thanks to Fourier invariance of cubic characters. In addition to winding modes, a pair of new discrete, flux-like, quantum numbers are necessary to ensure invariance under the larger group. The action for these modes is of Born-Infeld type, interpolating between standard Polyakov and Nambu-Goto membrane actions. After integration over the membrane moduli, we recover the known R 4 amplitude, including membrane instantons. Divergences are disposed of by trading the non-compact volume integration for a compact integral over the two variables conjugate to the fluxes -a constant term computation in mathematical parlance. As byproducts, we suggest that, in line with membrane/fivebrane duality, the E 6 theta series also describes five-branes wrapped on T 6 in a manifestly U-duality invariant way. In addition we uncover a new action of E 6 on ten dimensional pure spinors, which may have implications for ten dimensional super Yang-Mills theory. An extensive review of SL(3) automorphic forms is included in an Appendix.

D3-instantons, mock theta series and twistors

“…NS5-brane corrections were constructed at linear order in [31] (see also [32][33][34]), but an understanding of these effects at non-linear level is still missing.…”

Heterotic-type II duality in twistor space

“…Unfortunately, the twistorial description of these scalar-valued automorphic forms is in terms of H 1 (Z, O(−2)), rather than H 1 (Z, O(2)) as required for worldsheet instanton corrections to the QK metric. It would be very interesting to identify a class of heterotic/type II duals where a discrete subgroup SO(4, n, Z) of the isometry group of the moduli space M H could arguably stay unbroken at the non-perturbative level in the spirit of [22,27,28]. Determining the automorphic form in H 1 (Z, O(2)) governing these quantum corrections could then provide an alternative way to arrive at the afore-mentioned topological invariants.…”

“…On the type II side, the HM moduli space M H receives D-brane and NS5-brane instanton corrections [21]. Although much progress has been achieved recently in describing these instanton corrections [22][23][24][25][26][27][28][29][30][31][32][33][34][35] (see [36] for a review), a complete understanding is still lacking. In particular, the computation of the generalized Donaldson-Thomas (DT) invariants which govern Dinstanton contributions appears to be out of reach in general.…”

“…The twocycles in Pic ′ (S) are associated to ADE singularities of S, and the corresponding periods γ x I are frozen due to the gauge bundle. The metric and B-field moduli then arise from the remaining 2-cycles in 27) where Trans(S) is the orthogonal complement of Pic(S) inside H 2 (S, Z), known as the transcendental lattice. We refer to Trans Q (S) as the 'quantum transcendental lattice'.…”

M-theory and the string genus expansion

“…After world volume dualisation of one the scalar fields, the membrane action can be identified with the D2 action [5]. The relation of the BPS sector of the membrane to dualities in string theory has been studied in [6] and important work on M-theory loops and higher derivative corrections to the string effective action corrections has been done in [8].…”