Secondary Calculus
and
Cohomological Physics
An Introduction

Secondary calculus acts on the space of solutions of a system of partial differential equations (usually non linear equations). When the number of independent variables is zero, i.e. the equations are algebraic ones, secondary calculus reduces to classical differential calculus.

All objects in secondary calculus are coomology classes of differential complexes growing on diffieties. The last ones are, in the framework of secondary calculus, the analog of differential varieties.

All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differential varieties. The Euler operator, which associates to each variational problem the corresponding Euler-Lagrange equation, is the analog of classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like of infinite order. More generally, the analog of differential forms in secondary calculus are the elements of the first term of so-called C-Spectral sequence. And so on.

The simplest diffieties are infinite prolongations of partial differential equations, which are sub varieties of infinite jet spaces. The last ones are infinite dimensional varieties that can not be studied by means of standard functional analysis. On the contrary, the most natural language to study these objects is differential calculus over commutative algebras. Therefore, the last one must be regarded as a fundamental tool of secondary calculus. On the other hand, secondary calculus gives the possibility to develop algebraic geometry as if it were differential geometry. A large number of modern mathematical theories are very naturally and harmoniously embedded in the framework of secondary calculus, for instance: Commutative Algebra and Algebraic Geometry, Homological Algebra and Differential Topology, Lie Group and Lie Algebra Theory, Differential Geometry, etc.

In this way Mathematics, which has been studied "in vitro" for most part of the last century, is again alive. Applications of secondary calculus which already exist and the ones which can be seen at the horizon, go from arithmetic algebraic geometry to quantum field theory and its recent generalizations.

Coomological Physics was born with Gauss Theorem, describing the electric charge contained inside a given surface in terms of the flux of the electric field through the surface itself. Flux is the integral of a differential form and, consequently, a de Rham coomology class. It is not by chance the formulas of this kind, such as the well known Stokes formula, though being a natural part of classical differential calculus, have entered in modern Mathematics from Physics.

Most recent developments of particle Physics, based on quantum field theories and its generalizations, have lead to understand the deep coomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, it was understood that observables in field theory are classes in horizontal de Rham coomology which are invariant under the corresponding gauge group and so on. This current in modern theoretical physics is actually growing and it is called Coomological Physics. It is relevant that secondary calculus and coomological physics, which developed for twenty years independently from each other, arrived to the same results. Their confluency took place to the international conference Secondary Calculus and Coomological Physics (Moscow, August 24-30, 1997). For more information see this page at the Diffiety Institute Site, or check the items below in the Diffiety Institute Preprint Server, DIPS.

Essential Bibliography

1. I. S. Krasil'shchik, Calculus over Commutative Algebras: a concise user's guide, Acta Appl. Math. 49 (1997), 235--248; DIPS
2. I. S. Krasil'shchik, A. M. Verbovetsky, Homological Methods in Equations of Mathematical Physics, Open Ed. and Sciences, Opava (Czech Rep.), 1998; DIPS
3. I. S. Krasil'shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach, New York, 1986.
4. I. S. Krasil'shchik, A. M. Vinogradov (eds.), Symmetries and conservation laws for differential equations of mathematical physics, Translations of Math. Monographs, 182 Amer. Math. Soc., 1999.
5. A. M. Vinogradov, On algebro-geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977), 1200--1204.
6. A. M. Vinogradov, A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978), 144--148.
7. A. M. Vinogradov, The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. II. The nonlinear theory J. Math. Anal. Appl. 100 (1984), 1--129.
8. A. M. Vinogradov, From symmetries of partial differential equations towards secondary (`quantized') calculus, J. Geom. Phys. 14 (1994), 146--194; DIPS
9. A. M. Vinogradov, Introduction to Secondary Calculus, Proc. Conf. Secondary Calculus and Cohomology Physics (M. Henneaux, I. S. Krasil'shchik, and A. M. Vinogradov, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 1998; DIPS

Questions and suggestions should go to diffiety @ tiros.dmi.unisa.it