It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang–Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the Chern–Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and D=11 supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in an Ashtekar formalism.

1.
M. Green, V. Schwarz, and E. Witten, Superstrings Theory (Cambridge University Press, Cambridge, 1987), Vols. I and II;
M. Kaku, Introduction to Superstrings (Spring-Verlag, Berlin 1990).
2.
H.
Nishino
and
E.
Sezgin
,
Phys. Lett. B
388
,
569
(
1996
),
H.
Nishino
and
E.
Sezgin
, hep-th/9607185.
3.
P. K.
Townsend
, “Four lectures on M theory,” Proceedings of the ICTP on the Summer School on High Energy Physics and Cosmology, June, 1996,
P. K.
Townsend
, hep-th/9612121.
4.
M. J.
Duff
,
Int. J. Mod. Phys. A
11
,
5623
(
1996
).
5.
P.
Horava
and
E.
Witten
,
Nucl. Phys. B
460
,
506
(
1996
).
6.
M. J.
Duff
,
R. R.
Khuri
, and
J. X.
Lu
,
Phys. Rep.
259
,
213
(
1995
).
7.
E.
Witten
,
Nucl. Phys. B
463
,
383
(
1996
).
8.
J. H.
Schwarz
,
Phys. Lett. B
360
,
13
(
1995
).
9.
J. H.
Schwarz
,
Nucl. Phys. B (Proc. Suppl.)
55
,
1
(
1997
).
10.
E.
Bergshoeff
,
E.
Sezgin
, and
P. K.
Townsend
,
Phys. Lett. B
189
,
75
(
1987
).
11.
E.
Witten
,
Nucl. Phys. B
463
,
383
(
1996
).
12.
M. J.
Duff
,
B. E. W.
Nilsson
, and
C. N.
Pope
,
Phys. Rep.
130
,
1
(
1986
).
13.
D.
Kutasov
and
E.
Martinec
,
Nucl. Phys. B
477
,
652
(
1996
);
D.
Kutasov
,
E.
Martinec
, and
M.
O’Loughlin
,
Nucl. Phys. B
477
,
675
(
1996
).
14.
T.
Banks
,
W.
Fischler
,
S. H.
Shenker
, and
L.
Susskind
,
Phys. Rev. D
55
,
5112
(
1997
).
15.
E.
Martinec
, “
Matrix theory and N=(2,1) strings
,” hep-th/9706194.
16.
C. M.
Hull
,
J. High Energy Phys.
11
,
17
(
1998
).
17.
J.
Khoury
and
H.
Verlinde
, “
On open/closed string duality
,” hep-th/0001056.
18.
E.
Cremmer
and
J.
Sherk
,
Nucl. Phys. B
50
,
222
(
1972
).
19.
J. M.
Maldacena
,
Adv. Theor. Math. Phys.
2
,
231
(
1998
);
J. M.
Maldacena
,
Int. J. Theor. Phys.
38
,
1113
(
1999
).
20.
H.
Whitney
,
Am. J. Math.
57
,
509
(
1935
).
21.
J. A.
Nieto
,
Rev. Mex. Fis.
44
,
358
(
1998
).
22.
F.
Englert
,
Phys. Lett. B
119
,
339
(
1982
).
23.
W. T.
Tutte
,
Trans. Am. Math. Soc.
88
,
144
(
1958
);
W. T.
Tutte
,
Trans. Am. Math. Soc.
88
,
161
(
1958
).
24.
I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers; An Elementary Introduction to Algebras (Spring-Verlag New York, 1989).
25.
M. P.
Blecowe
and
M. J.
Duff
,
Nucl. Phys. B
310
,
387
(
1988
).
26.
M.
Thistlethwaite
,
Topology
26
,
297
(
1987
).
27.
E.
Witten
,
Commun. Math. Phys.
121
,
352
(
1989
).
28.
G.
Birkhoff
,
Am. J. Math.
57
,
800
(
1935
).
29.
N. White, in Combinatorial Geometries (Cambridge University Press, Cambridge, 1987).
30.
S.
MacLane
,
Am. J. Math.
58
,
236
(
1936
).
31.
D. J. A. Welsh, Martroid Theory (Academic, London, 1976).
32.
E. L. Lawler, Combinatory Optimization: Networks and Matroids (Holt, Rinehart, and Winston, New York, 1976).
33.
W. T. Tutte, Introduction to the Theory of Matroids (Elsevier, New York, 1971).
34.
R. J. Wilson, Introduction to Graph Theory, 3rd. ed. (Wiley, New York, 1985).
35.
J. P. S. Kung, A Source Book in Matroid Theory (Birkhauser, Boston, 1986).
36.
K. Ribnikov, Análisis Combinatorio (Editorial Mir, Moscú, 1988).
37.
F.
Gursey
and
C.
Tze
,
Phys. Lett. B
127
,
191
(
1983
).
38.
J. F.
Adams
,
Ann. Math.
72
,
20
(
1960
).
39.
N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, Princeton, NJ, 1970).
40.
M. F. Atiyah. K-Theory (Benjamin, New York, 1979).
41.
E. Guanagnini, The Link Invariants of the Chern–Simons Field Theory (Walter de Gruyter, Berlin, 1993).
42.
T.
Zaslavsky
,
Discrete Appl. Math.
4
,
47
(
1982
).
43.
L. H.
Kauffman
, “New invariants in the theory of knots,” Lectures given in Rome, June 1986;
L. H.
Kauffman
,
Asterisque
163–164
,
137
(
1988
).
44.
L.
Smolin
, “
M-theory as a matrix extension of Chern–Simons theory
,” hep-th/0002009.
45.
A.
Connes
, “
A short survey of noncommutative geometry
,” hep-th/0003006.
46.
E.
Witten
,
Selecta Mat.
1
,
383
(
1995
),
E.
Witten
, hep-th/9505186.
47.
R. Gambini and J. Pullin, Loops, Knots, Gauge Theory and Quantum Gravity (Cambrige University Press, Cambrige, 1996).
This content is only available via PDF.
You do not currently have access to this content.