Supersymmetry nonrenormalization theorems

In differential equations, the Laplace invariant of any of certain differential operators is a certain function of the coefficients and their derivatives. Consider a bivariate hyperbolic differential operator of the second order

${\partial }_x{\partial }_y+a{\partial }_x+b{\partial }_y+c,$

whose coefficients

$a=a(x,y),b=c(x,y),c=c(x,y),$

are smooth functions of two variables. Its Laplace invariants have the form

$\hat{a}=c-ab-a_x\text{and}\hat{b}=c-ab-b_y.$

Their importance is due to the classical theorem:

Theorem: Two operators of the form are equivalent under gauge transformations if and only if their Laplace invariants coincide pairwise.

Here the operators

$A\text{and}\tilde{A}$

are called equivalent if there is a gauge transformation that takes one to the other:

$\tilde{A}g=e^{-\phi }A(e^{\phi }g)\equiv A_{\phi }g.$

Laplace invariants can be regarded as factorization "remainders" for the initial operator A:

${\partial }_x{\partial }_y+a{\partial }_x+b{\partial }_y+c=\{\begin{array}{c}({\partial }_x+b)({\partial }_y+a)-ab-a_x+c,\\ ({\partial }_y+a)({\partial }_x+b)-ab-b_y+c.\end{array}$

If at least one of Laplace invariants is not equal to zero, i.e.

$c-ab-a_x\ne 0\text{and/or}c-ab-b_y\ne 0,$

then this representation is a first step of the Laplace–Darboux transformations used for solving non-factorizable bivariate linear partial differential equations (LPDEs).

If both Laplace invariants are equal to zero, i.e.

$c-ab-a_x=0\text{and}c-ab-b_y=0,$

then the differential operator A is factorizable and corresponding linear partial differential equation of second order is solvable.

Laplace invariants have been introduced for a bivariate linear partial differential operator (LPDO) of order 2 and of hyperbolic type. They are a particular case of generalized invariants which can be constructed for a bivariate LPDO of arbitrary order and arbitrary type; see Invariant factorization of LPDOs.

See also

• Partial derivative
• Invariant (mathematics)
• Invariant theory

References

• G. Darboux, "Leçons sur la théorie général des surfaces", Gauthier-Villars (1912) (Edition: Second)
• G. Tzitzeica G., "Sur un theoreme de M. Darboux". Comptes Rendu de l'Academie des Aciences 150 (1910), pp. 955–956; 971–974
• L. Bianchi, "Lezioni di geometria differenziale", Zanichelli, Bologna, (1924)
• A. B. Shabat, "On the theory of Laplace–Darboux transformations". J. Theor. Math. Phys. Vol. 103, N.1,pp. 170–175 (1995) [1]
• A.N. Leznov, M.P. Saveliev. "Group-theoretical methods for integration on non-linear dynamical systems" (Russian), Moscow, Nauka (1985). English translation: Progress in Physics, 15. Birkhauser Verlag, Basel (1992)