L

#### Lubos Motl

sci.physics.strings. Everyone is invited to answer the question "What was

the most interesting paper on hep-th, hep-ph, or gr-qc today?"

My answer for the night of April 4th is the paper by Pioline and Waldron

The Automorphic Membrane

http://www.arxiv.org/abs/hep-th/0404018

It is a part of their efforts to determine the identity of "M" - which

means the non-perturbative generalization of a string, relevant for

M-theory (much like a string is fundamental in perturbative string

theory).

There are various interesting terms in the effective action of M-theory

(on tori), namely the R^4 terms (R is the curvature tensor), and their

calculation is analogous to various calculations in string theory.

Perturbative terms as well as toroidal membrane instantons contribute much

like the worldsheet instantons in string theory would contribute to a

similar process perturbatively.

It has been possible to isolate the coefficient of the R^4 term for

M-theory on T^3 (by a combination of perturbative calculations and duality

arguments), and Pioline and Waldron study the modular forms - more

precisely the theta series and automorphic forms - that manifestly respect

the enhanced exceptional U-duality groups such as E_{6(6)} (Z) which

includes not only SL(3,Z) times SL(2,Z) (U-duality on T^3), but also

another copy of SL(3,Z) that generalizes the modular invariance SL(2,Z)

of a string to the case of membranes (of toroidal topology).

The math is perhaps difficult, but very intriguing. Boris Pioline has also

explained me various relations of these mathematical objects to the p-adic

numbers and adels (which are some ordered composite objects made of many

p-adic numbers). Well, this is where the next big conceptual discoveries

about M-theory may hide. I am certainly among those who believe that a

proper (generalized) geometric understanding of these crazy exceptional

duality groups of M-theory may hide a key to reveal something very deep

about string/M-theory, perhaps something that would allow us, at least in

principle, study also the realistic backgrounds in a non-perturbative and

complete fashion.

Replies including disagreement welcome.

Best regards

Lubos

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