2004/04/30 by Boris Pioline, Andrew Waldron · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Computation #Invariant (physics) #Membrane #Modular design #Modular invariance #Quantum #Quantum Chromodynamics and Particle Interactions #Toroid #Winding number #hep-th
paper · pdf · doi:10.1088/1126-6708/2004/06/009
published as JHEP 0406 (2004) 009 · 43 pages, 1 figure, JHEP3.cls style; v2: misprints corrected, differs from JHEP version
openalex publication_date 2004/06/07 · arxiv created 2010/01/15 · arxiv updated 2010/02/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We present a 1-loop toroidal membrane winding sum reproducing the conjectured M-theory, four-graviton, eight derivative, R4 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,\Zint)× Sl(2,\Zint) U-duality and Sl(3,\Zint) world-volume modular groups are embedded in E6(6)(\Zint). 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 R4 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 E6 theta series also describes five-branes wrapped on T6 in a manifestly U-duality invariant way. In addition we uncover a new action of E6 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.