Research project (May 2003)

Prof. Francesco Toppan

leader of the CNPq-approved group

"Algebraic Structures in Field Theory"

The project involves both the group researchers and the external collaborators. Its aim is the algebraic classification and characterization

of dynamical systems, with investigations concerning supergravity, M-theory, integrable systems, as well as the different applications of the

representation theory of algebras.

The following research topics are currently under investigation:.

1) The role of the exceptional structures (such as the exceptional Lie and Jordan

algebras, etc.) in the fundamental physics

2) A classification of the generalized supersymmetries.

3) An algebraic characterization of the integrable systems.

4) The braid group representations in application to cryptography.

1) This item, currently under intense investigation (several suggestions have been

made pointing out the role of the exceptional algebras in the framework of the

unification of the interactions, see e.g. the works listed at the point i), is

studied in terms of the division algebra of the octonions (the mathematical

responsibles for the existence of these structures).

In a joint work with the group collaborators, new results have ben found,

see e.g. ii e iii.

i) E. Witten, Quest for Unification, hep-ph/0207124.

P. Ramond, Exceptional Groups and Physics,hep-th/0301050.

L. Smolin, The Exceptional Jordan algebra and the Matrix String, hep-th/0104050.

ii) F. Toppan, JHEP PRHEP unesp/2002/39.

iii) H.L. Carrion, M. Rojas and F. Toppan, hep-th/0302113.

2) This item is based on the division-algebra classification of supersymmetries. The

non-associative division algebra of the octonions play a special role,

see the works in i and ii, as well as tghe recently discovered, iii, octonionic M-

theory. This line is under intense investigation, see e.g. the works iv and v.

i) A. Pashnev and F. Toppan, JMP 42 (2001) 5257-5271.

ii) F. Toppan, NPB (Pr. Suppl.) 102&103 (2001) 270-277.

H.L. Carrion, M. Rojas and F. Toppan, MPLA 18 (2003) 787-798.

iii) J. Lukierski and F. Toppan, PLB 539 (2002) 266-276; hep-th/0212201.

iv) L. Boya, hep-th/0301037.

v) S.J. Gates Jr., W.D. Linch III and J. Phillips, hep-th/0211034.

3) I have started thsi line with the works in i. My most recent papers are listed in ii.

In iii we constructed potentially integrable systems associated with non-associative

algebras. It is under investigation their possible role in the description of

dynamical systems constrained to move on the S7 sphere, regarded as a

compactification manifold for the 11-dimensional supergravity and the M-theory.

i) F. Toppan, PLB 327 (1994) 249-256.

F. Toppan, IJMPA 10 (1995) 895-922.

ii) E. Ivanov, S. Krivonos and F. Toppan, MPLA (1999) 2673-2688.

A. Das, A. Constandache and F. Toppan, LMP 60 (2002) 197-209.

Z. Popowicz and F. Toppan, nlin.SI/0302061.

iii) H.L. Carrion, M. Rojas and F. Toppan, JPA (2003) 3809-3820.

4) This line is based on the works in i, where the linearization of the Artin

representation of the braid group was constructed. These works have been referred

by mathematicians, see e.g. (ii) and even by the US Navy (iii).

Ten years ago the fundamental motivation concerned the possible applications to the

2+1-dimensional anyonic theories and the Quantum Hall Effect.

At present another, even more concrete, application is under investigation. The braid

group is in fact used in cryptography (iv e v). In vi the weakness of some codes

(cracked by the simplest representation, the so-called Burau representation) has been

shown. Since our linear Artin representation is(unlike the Burau rep.) faithful, it

could be potentially used to study more efficiently and in full generality this

weakness problem.

i) F. Constantinescu and F. Toppan, JKTR 2 (1993) 399-412.

M. Luedde and F. Toppan, PLB 288 (1992) 321-330.

ii) R. Haering-Oldenburg, JKTR 9 (2000) 1005-1009.

iii) web.usna.navy.mil/~wdj/rep

iv) K. Ko, S. Lee, J. Cheon, J. Han, J. Kang and C. Park, New public-key

cryptosystems using braid groups, Techn. Rep. KAISTT, Korea (2000).

v) www.tcs.hut.fi/~helger/crypto

vi) J. Hughes, A Linear Algebraic Attack on the AAFG1 Braid Group

Cryptosystem, ACISP (2002).