Celebratio Mathematica

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

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.)

Oleg V. Zaboronski	Related
Mikhail Dmitrievich Alexandrov	Related
Maksim Lvovich Kontsevich	Related

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

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.

Yujun Chen	Related
Maksim Lvovich Kontsevich	Related

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

Vadim Aleksandrovich Vologodsky	Related
Maksim Lvovich Kontsevich	Related

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

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.

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Albert S. Schwarz

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