A. Konechny and A. Schwarz :
“Geometry of \( N = 1 \) super Yang–Mills theory in curved superspace ,”
J. Geom. Phys.
23 : 2
(1997 ),
pp. 97–110 .
MR
1467173
Zbl
0899.53061
ArXiv
hep-th/9609081
article

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Anatoly Vladimirovich Konechny
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@article {key1467173m,
AUTHOR = {Konechny, Anatoli and Schwarz, Albert},
TITLE = {Geometry of \$N = 1\$ super {Y}ang--{M}ills
theory in curved superspace},
JOURNAL = {J. Geom. Phys.},
FJOURNAL = {Journal of Geometry and Physics},
VOLUME = {23},
NUMBER = {2},
YEAR = {1997},
PAGES = {97--110},
DOI = {10.1016/S0393-0440(96)00050-2},
NOTE = {ArXiv:hep-th/9609081. MR:1467173. Zbl:0899.53061.},
ISSN = {0393-0440},
}
A. Konechny and A. Schwarz :
“On \( (k\oplus l|g) \) -dimensional supermanifolds ,”
pp. 201–206
in
Supersymmetry and quantum field theory
(Kharkov, Ukraine, 5–7 January 1997 ).
Edited by J. Wess and V. P. Akulov .
Lecture Notes in Physics 509 .
1998 .
Proceedings of the D. Volkov Memorial Seminar.
MR
1677320
Zbl
0937.58001
incollection

Abstract
People
BibTeX
@incollection {key0937.58001z,
AUTHOR = {Konechny, A. and Schwarz, A.},
TITLE = {On \$(k\oplus l|g)\$-dimensional supermanifolds},
BOOKTITLE = {Supersymmetry and quantum field theory},
EDITOR = {Wess, Julius and Akulov, Vladimir P.},
SERIES = {Lecture Notes in Physics},
NUMBER = {509},
YEAR = {1998},
PAGES = {201--206},
NOTE = {(Kharkov, Ukraine, 5--7 January 1997).
Proceedings of the D. Volkov Memorial
Seminar. Zbl:0937.58001.},
ISSN = {0075-8450},
ISBN = {9783540646235},
}
A. Konechny and A. Schwarz :
“Supersymmetry algebra and BPS states of super Yang–Mills theories on noncommutative tori ,”
Phys. Lett., B
453 : 1–2
(1999 ),
pp. 23–29 .
MR
1690319
Zbl
1058.58500
ArXiv
hep-th/9901077
article

Abstract
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BibTeX

We consider 10-dimensional super Yang–Mills theory with topological terms compactified on a noncommutative torus. We calculate supersymmetry algebra and derive BPS energy spectra from it. The cases of \( d \) -dimensional tori with \( d = 2,\,3,\,4 \) are considered in full detail. \( SO(d,d,\mathbb{Z}) \) -invariance of the BPS spectrum and relation of new results to the previous work in this direction are discussed.

Anatoly Vladimirovich Konechny
Related

@article {key1690319m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {Supersymmetry algebra and {BPS} states
of super {Y}ang--{M}ills theories on
noncommutative tori},
JOURNAL = {Phys. Lett., B},
FJOURNAL = {Physics Letters. B},
VOLUME = {453},
NUMBER = {1--2},
YEAR = {1999},
PAGES = {23--29},
DOI = {10.1016/S0370-2693(99)00335-4},
NOTE = {ArXiv:hep-th/9901077. MR:1690319. Zbl:1058.58500.},
ISSN = {0370-2693},
}
A. Konechny and A. Schwarz :
“BPS states on non-commutative tori and duality ,”
Nucl. Phys., B
550 : 3
(1999 ),
pp. 561–584 .
MR
1693262
Zbl
0949.58004
ArXiv
hep-th/9811159
article

Abstract
People
BibTeX

We study gauge theories on non-commutative tori. It has been proved that Morita equivalence of non-commutative tori leads to a physical equivalence (\( SO(d,d|\mathbb{Z}) \) -duality) of the corresponding gauge theories [Nucl. Phys. B 534 (1998) 720]. We calculate the energy spectrum of maximally supersymmetric BPS states in these theories and show that this spectrum agrees with the . The relation of our results with those of recent calculations is discussed.

Anatoly Vladimirovich Konechny
Related

@article {key1693262m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {{BPS} states on non-commutative tori
and duality},
JOURNAL = {Nucl. Phys., B},
FJOURNAL = {Nuclear Physics. B},
VOLUME = {550},
NUMBER = {3},
YEAR = {1999},
PAGES = {561--584},
DOI = {10.1016/S0550-3213(99)00184-4},
NOTE = {ArXiv:hep-th/9811159. MR:1693262. Zbl:0949.58004.},
ISSN = {0550-3213},
}
A. Konechny and A. Schwarz :
“\( 1/4 \) -BPS states on noncommutative tori ,”
J. High Energy Phys.
1999 : 9
(1999 ).
article no. 30, 15 pages.
MR
1720696
Zbl
0957.81086
ArXiv
hep-th/9907008
article

Abstract
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Anatoly Vladimirovich Konechny
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@article {key1720696m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {\$1/4\$-{BPS} states on noncommutative
tori},
JOURNAL = {J. High Energy Phys.},
FJOURNAL = {Journal of High Energy Physics},
VOLUME = {1999},
NUMBER = {9},
YEAR = {1999},
DOI = {10.1088/1126-6708/1999/09/030},
NOTE = {article no. 30, 15 pages. ArXiv:hep-th/9907008.
MR:1720696. Zbl:0957.81086.},
ISSN = {1126-6708},
}
A. Konechny and A. Schwarz :
“Moduli spaces of maximally supersymmetric solutions on non-commutative tori and non-commutative orbifolds ,”
J. High Energy Phys.
2000 : 9
(2000 ),
pp. article no. 005, 24 pages .
MR
1789106
Zbl
0989.81623
ArXiv
hep-th/0005174
article

Abstract
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Anatoly Vladimirovich Konechny
Related

@article {key1789106m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {Moduli spaces of maximally supersymmetric
solutions on non-commutative tori and
non-commutative orbifolds},
JOURNAL = {J. High Energy Phys.},
FJOURNAL = {Journal of High Energy Physics},
VOLUME = {2000},
NUMBER = {9},
YEAR = {2000},
PAGES = {article no. 005, 24 pages},
DOI = {10.1088/1126-6708/2000/09/005},
NOTE = {ArXiv:hep-th/0005174. MR:1789106. Zbl:0989.81623.},
ISSN = {1126-6708},
}
A. Konechny and A. Schwarz :
“Compactification of M(atrix) theory on noncommutative toroidal orbifolds ,”
Nuclear Phys. B
591 : 3
(2000 ),
pp. 667–684 .
MR
1797572
Zbl
1042.81580
ArXiv
hep-th/9912185
article

Abstract
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It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal \( \mathbb{Z}_2 \) orbifolds leads to the algebra \( B_{\theta} \) that can be defined as a crossed product of noncommutative torus and the group \( \mathbb{Z}_2 \) . Our paper is devoted to the study of projective modules over \( B_{\theta} \) (\( \mathbb{Z}_2 \) -equivariant projective modules over a noncommutative torus). We analyze the Morita equivalence (duality) for \( B_{\theta} \) algebras working out the two-dimensional case in detail.

Anatoly Vladimirovich Konechny
Related

@article {key1797572m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {Compactification of {M}(atrix) theory
on noncommutative toroidal orbifolds},
JOURNAL = {Nuclear Phys. B},
FJOURNAL = {Nuclear Physics B},
VOLUME = {591},
NUMBER = {3},
YEAR = {2000},
PAGES = {667--684},
DOI = {10.1016/S0550-3213(00)00544-7},
NOTE = {ArXiv:hep-th/9912185. MR:1797572. Zbl:1042.81580.},
ISSN = {0550-3213},
}
A. Konechny and A. Schwarz :
“Theory of \( (k\oplus l|q) \) -dimensional supermanifolds ,”
Selecta Math. (N.S.)
6 : 4
(2000 ),
pp. 471–486 .
MR
1847384
Zbl
1001.58001
article

Abstract
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BibTeX
Anatoly Vladimirovich Konechny
Related

@article {key1847384m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {Theory of \$(k\oplus l|q)\$-dimensional
supermanifolds},
JOURNAL = {Selecta Math. (N.S.)},
FJOURNAL = {Selecta Mathematica. New Series},
VOLUME = {6},
NUMBER = {4},
YEAR = {2000},
PAGES = {471--486},
DOI = {10.1007/PL00001396},
NOTE = {MR:1847384. Zbl:1001.58001.},
ISSN = {1022-1824},
}
A. Konechny and A. Schwarz :
“Introduction to M(atrix) theory and noncommutative geometry ,”
Phys. Rep.
360 : 5–6
(2002 ),
pp. 353–465 .
MR
1892926
Zbl
0985.81126
ArXiv
hep-th/0012145
article

Abstract
People
BibTeX

Noncommutative geometry is based on an idea that an associative algebra can be regarded as “an algebra of functions on a noncommutative space”. The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang–Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang–Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics.

In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes’ noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and \( SO(d,d|\mathbb{Z}) \) -duality, an elementary discussion of noncommutative orbifolds, noncommutative solitons and instantons. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics.

The second part of the review devoted to solitons and instantons on noncommutative Euclidean space is almost independent of the first part.

Anatoly Vladimirovich Konechny
Related

@article {key1892926m,
AUTHOR = {Konechny, Anatoly and Schwarz, Albert},
TITLE = {Introduction to {M}(atrix) theory and
noncommutative geometry},
JOURNAL = {Phys. Rep.},
FJOURNAL = {Physics Reports. A Review Section of
Physics Letters},
VOLUME = {360},
NUMBER = {5--6},
YEAR = {2002},
PAGES = {353--465},
DOI = {10.1016/S0370-1573(01)00096-5},
NOTE = {ArXiv:hep-th/0012145. MR:1892926. Zbl:0985.81126.},
ISSN = {0370-1573},
}