M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich :
“The geometry of the master equation and topological quantum field theory Int. J. Mod. Phys. A
12 : 7
(1997 ),
pp. 1405–1429 .
MR
1432574 Zbl
1073.81655 ArXiv
hep-th/9502010 article
Abstract
People
BibTeX
In Batalin–Vilkovisky formalism, a classical mechanical system is specified by means of a solution to the classical master equation . Geometrically, such a solution can be considered as a \( QP \) -manifold, i.e. a supermanifold equipped with an odd vector field \( Q \) obeying \( \{Q,Q\} = 0 \) and with \( Q \) -invariant odd symplectic structure. We study geometry of \( QP \) -manifolds. In particular, we describe some construction of \( QP \) -manifolds and prove a classification theorem (under certain conditions).
We apply these geometric constructions to obtain in a natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern–Simons theory in BV-formalism arises as a sigma-model with target space \( \Pi\mathcal{G} \) . (Here \( \mathcal{G} \) stands for a Lie algebra and \( \Pi \) denotes parity inversion.)
@article {key1432574m,
AUTHOR = {Alexandrov, M. and Schwarz, A. and Zaboronsky,
O. and Kontsevich, M.},
TITLE = {The geometry of the master equation
and topological quantum field theory},
JOURNAL = {Int. J. Mod. Phys. A},
FJOURNAL = {International Journal of Modern Physics
A},
VOLUME = {12},
NUMBER = {7},
YEAR = {1997},
PAGES = {1405--1429},
DOI = {10.1142/S0217751X97001031},
NOTE = {ArXiv:hep-th/9502010. MR:1432574. Zbl:1073.81655.},
ISSN = {0217-751X},
}
Y. Chen, M. Kontsevich, and A. Schwarz :
“Symmetries of WDVV equations Nuclear Phys. B
730 : 3
(2005 ),
pp. 352–363 .
MR
2180009 Zbl
1276.81096 ArXiv
hep-th/0508221 article
Abstract
People
BibTeX
We say that a function \( F(\tau) \) obeys WDVV equations, if for a given invertible symmetric matrix \( \eta^{\alpha \beta} \) and all \( \tau \in \mathcal{T} \subset \mathbb{R}^n \) , the expressions
\[ c_{\beta\gamma}^{\alpha}(\tau) = \eta^{\alpha\lambda}\partial_{\lambda}\partial_{\beta}\partial_{\gamma}F \]
can be considered as structure constants of commutative associative algebra; the matrix \( \eta_{\alpha \beta} \) inverse to \( \eta^{\alpha \beta} \) determines an invariant scalar product on this algebra. A function \( x^{\alpha}(z,\tau) \) obeying
\[ \partial_{\alpha}\partial_{\beta} \,x^{\gamma}(z,\tau) = z^{-1}c_{\alpha\beta}^{\epsilon}\partial_{\epsilon}\,x^{\gamma}(z,\tau) \]
is called a calibration of a solution of WDVV equations. We show that there exists an infinite-dimensional group acting on the space of calibrated solutions of WDVV equations (in different form such a group was constructed in [Givental 2004]. We describe the action of Lie algebra of this group.
@article {key2180009m,
AUTHOR = {Chen, Yujun and Kontsevich, Maxim and
Schwarz, Albert},
TITLE = {Symmetries of {WDVV} equations},
JOURNAL = {Nuclear Phys. B},
FJOURNAL = {Nuclear Physics B},
VOLUME = {730},
NUMBER = {3},
YEAR = {2005},
PAGES = {352--363},
DOI = {10.1016/j.nuclphysb.2005.09.025},
NOTE = {ArXiv:hep-th/0508221. MR:2180009. Zbl:1276.81096.},
ISSN = {0550-3213},
}
M. Kontsevich, N. Nekrasov, A. Schwarz, M. Shubin, and D. Sternheimer :
“Foreword 1
in
Special volume dedicated to the memory of F. A. Berezin ,
published as Lett. Math. Phys.
74 : 1 .
Issue edited by N. A. Nekrasov, A. Schwarz, M. Shubin, D. Sternheimer, and M. Kontsevich .
Springer Netherlands (Dordrecht ),
2005 .
MR
2193544 Zbl
1146.00304 incollection
People
BibTeX
@article {key2193544m,
AUTHOR = {Kontsevich, Maxim and Nekrasov, Nikita
and Schwarz, Albert and Shubin, Mikhail
and Sternheimer, Daniel},
TITLE = {Foreword},
JOURNAL = {Lett. Math. Phys.},
FJOURNAL = {Letters in Mathematical Physics},
VOLUME = {74},
NUMBER = {1},
YEAR = {2005},
PAGES = {1},
DOI = {10.1007/s11005-005-0013-y},
NOTE = {\textit{Special volume dedicated to
the memory of {F}.~{A}. {B}erezin}.
Issue edited by N. A. Nekrasov,
A. Schwarz, M. Shubin, D. Sternheimer,
and M. Kontsevich. MR:2193544.
Zbl:1146.00304.},
ISSN = {0377-9017},
}
Special volume dedicated to the memory of F. A. Berezin Lett. Math. Phys.
74 : 1 .
Issue edited by N. A. Nekrasov, A. Schwarz, M. Shubin, D. Sternheimer, and M. Kontsevich .
Springer Netherlands (Dordrecht ),
October 2005 .
book
People
BibTeX
@book {key68150180,
TITLE = {Special volume dedicated to the memory
of {F}.~{A}. {B}erezin},
EDITOR = {Nekrasov, N. A. and Schwarz, A. and
Shubin, M. and Sternheimer, Daniel and
Kontsevich, Maxim},
PUBLISHER = {Springer Netherlands},
ADDRESS = {Dordrecht},
MONTH = {October},
YEAR = {2005},
PAGES = {109},
URL = {https://link.springer.com/journal/11005/volumes-and-issues/74-1},
NOTE = {Published as \textit{Lett. Math. Phys.}
\textbf{74}:1.},
ISSN = {0377-9017},
}
M. Kontsevich, A. Schwarz, and V. Vologodsky :
“Integrality of instanton numbers and \( p \) -adic B-model Phys. Lett. B
637 : 1–2
(2006 ),
pp. 97–101 .
MR
2230876 Zbl
1247.14058 ArXiv
hep-th/0603106 article
Abstract
People
BibTeX
Vadim Aleksandrovich Vologodsky
Related
Maksim Lvovich Kontsevich
Related
@article {key2230876m,
AUTHOR = {Kontsevich, Maxim and Schwarz, Albert
and Vologodsky, Vadim},
TITLE = {Integrality of instanton numbers and
\$p\$-adic {B}-model},
JOURNAL = {Phys. Lett. B},
FJOURNAL = {Physics Letters. B. Particle Physics,
Nuclear Physics and Cosmology},
VOLUME = {637},
NUMBER = {1--2},
YEAR = {2006},
PAGES = {97--101},
DOI = {10.1016/j.physletb.2006.04.012},
NOTE = {ArXiv:hep-th/0603106. MR:2230876. Zbl:1247.14058.},
ISSN = {0370-2693},
}
A. I. Aptekarev, V. M. Buchstaber, V. A. Vassiliev, M. L. Gromov, Yu. S. Ilyashenko, B. S. Kashin, V. M. Keselman, V. V. Kozlov, M. L. Kontsevich, I. M. Krichever, N. G. Kruzhilin, S. K. Lando, Yu. I. Manin, G. A. Margulis, S. Yu. Nemirovski, S. P. Novikov, Yu. G. Reshetnyak, Ya. G. Sinai, S. P. Suetin, D. V. Treschev, D. B. Fuchs, A. G. Khovanskii, E. M. Chirka, A. S. Schwarz, and A. N. Shiryaev :
“Vladimir Antonovich Zorich (on his 80th birthday) Usp. Mat. Nauk
73 : 5(443)
(2018 ),
pp. 193–196 .
An English translation was published in Russ. Math. Surv. 73 :5 (2018)
MR
3859406 article
People
BibTeX
@article {key3859406m,
AUTHOR = {Aptekarev, A. I. and Buchstaber, V.
M. and Vassiliev, V. A. and Gromov,
M. L. and Ilyashenko, Yu. S. and Kashin,
B. S. and Keselman, V. M. and Kozlov,
V. V. and Kontsevich, M. L. and Krichever,
I. M. and Kruzhilin, N. G. and Lando,
S. K. and Manin, Yu. I. and Margulis,
G. A. and Nemirovski, S. Yu. and Novikov,
S. P. and Reshetnyak, Yu. G. and Sinai,
Ya. G. and Suetin, S. P. and Treschev,
D. V. and Fuchs, D. B. and Khovanskii,
A. G. and Chirka, E. M. and Schwarz,
A. S. and Shiryaev, A. N.},
TITLE = {Vladimir {A}ntonovich {Z}orich (on his
80th birthday)},
JOURNAL = {Usp. Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk},
VOLUME = {73},
NUMBER = {5(443)},
YEAR = {2018},
PAGES = {193--196},
DOI = {10.4213/rm9828},
NOTE = {An English translation was published
in \textit{Russ. Math. Surv.} \textbf{73}:5
(2018). MR:3859406.},
ISSN = {0042-1316},
}
A. I. Aptekarev, V. M. Buchstaber, V. A. Vassiliev, M. L. Gromov, Yu. S. Ilyashenko, B. S. Kashin, V. M. Keselman, V. V. Kozlov, M. L. Kontsevich, I. M. Krichever, N. G. Kruzhilin, S. K. Lando, Yu. I. Manin, G. A. Margulis, S. Yu. Nemirovski, S. P. Novikov, Yu. G. Reshetnyak, Ya. G. Sinai, S. P. Suetin, D. V. Treschev, D. B. Fuchs, A. G. Khovanskii, E. M. Chirka, A. S. Schwarz, and A. N. Shiryaev :
“Vladimir Antonovich Zorich (on his 80th birthday) Russ. Math. Surv.
73 : 5
(2018 ),
pp. 935–939 .
English translation of Russian original published in Usp. Mat. Nauk 73 :5(443) (2018)
Zbl
1417.01018 article
People
BibTeX
@article {key1417.01018z,
AUTHOR = {Aptekarev, A. I. and Buchstaber, V.
M. and Vassiliev, V. A. and Gromov,
M. L. and Ilyashenko, Yu. S. and Kashin,
B. S. and Keselman, V. M. and Kozlov,
V. V. and Kontsevich, M. L. and Krichever,
I. M. and Kruzhilin, N. G. and Lando,
S. K. and Manin, Yu. I. and Margulis,
G. A. and Nemirovski, S. Yu. and Novikov,
S. P. and Reshetnyak, Yu. G. and Sinai,
Ya. G. and Suetin, S. P. and Treschev,
D. V. and Fuchs, D. B. and Khovanskii,
A. G. and Chirka, E. M. and Schwarz,
A. S. and Shiryaev, A. N.},
TITLE = {Vladimir {A}ntonovich {Z}orich (on his
80th birthday)},
JOURNAL = {Russ. Math. Surv.},
FJOURNAL = {Russian Mathematical Surveys},
VOLUME = {73},
NUMBER = {5},
YEAR = {2018},
PAGES = {935--939},
DOI = {10.1070/RM9828},
NOTE = {English translation of Russian original
published in \textit{Usp. Mat. Nauk}
\textbf{73}:5(443) (2018). Zbl:1417.01018.},
ISSN = {0036-0279},
}
A. Schwarz :
Scattering matrix and inclusive scattering matrix in algebraic quantum field theory Preprint ,
August 2019 .
To Maxim Kontsevich on the occasion of his 55th birthday with love and admiration.
ArXiv
1908.09388 techreport
Abstract
People
BibTeX
We study the scattering of particles and quasiparticles in the framework of algebraic quantum field theory. The main novelty is the construction of inclusive scattering matrix related to inclusive cross-sections. The inclusive scattering matrix can be expressed in terms of generalized Green functions by a formula similar to the LSZ formula for the conventional scattering matrix.
The consideration of inclusive scattering matrix is necessary in quantum field theory if a unitary scattering matrix does not exist (if the theory does not have particle interpretation). It is always necessary if we want to consider collisions of quasiparticles.
@techreport {key1908.09388a,
AUTHOR = {Schwarz, Albert},
TITLE = {Scattering matrix and inclusive scattering
matrix in algebraic quantum field theory},
TYPE = {Preprint},
MONTH = {August},
YEAR = {2019},
PAGES = {14},
URL = {https://arxiv.org/pdf/1908.09388.pdf},
NOTE = {To Maxim Kontsevich on the occasion
of his 55th birthday with love and admiration.
ArXiv:1908.09388.},
}
