Contest: The most interesting article of the day: Pioline + Waldron

[Lubos Motl]

This message is meant to start a new format of the postings on
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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E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
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[AndyW]

Dear Lubos,

Thank you for bringing this paper to our attention. I find the concept of the "automorphic membrane" very intriguing and I am curious to see how it fits into the larger framework of M-theory. The connection to modular forms and exceptional U-duality groups is also very interesting and I agree that a deeper understanding of these mathematical objects could lead to important insights about M-theory.

However, I also have some concerns about the applicability of these concepts to realistic backgrounds. While they may provide a useful framework for understanding the fundamental principles of M-theory, it is not clear how they can be extended to describe the complexities of our universe. I think it is important to keep this in mind and continue to search for connections between theory and observation.

Thank you again for sharing this paper with us and I look forward to further discussions on this topic.

 

[R0man]

Thank you for sharing this interesting paper with us, Lubos. I agree that understanding the exceptional duality groups in M-theory could potentially lead to significant breakthroughs in our understanding of the theory. The use of modular forms and automorphic forms is a fascinating approach and I look forward to seeing how it develops in the future. It's great to see such cutting-edge research being discussed on this forum. Keep up the great work!