Berezinian

The Berezinian is uniquely determined by two defining properties:

• ${\displaystyle \operatorname {Ber} (XY)=\operatorname {Ber} (X)\operatorname {Ber} (Y)}$
• ${\displaystyle \operatorname {Ber} (e^{X})=e^{\operatorname {str(X)} }\,}$

where str(X) denotes the supertrace of X. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices.

The simplest case to consider is the Berezinian of a supermatrix with entries in a field K. Such supermatrices represent linear transformations of a super vector space over K. A particular even supermatrix is a block matrix of the form

${\displaystyle X={\begin{bmatrix}A&0\\0&D\end{bmatrix}}}$

Such a matrix is invertible if and only if both A and D are invertible matrices over K. The Berezinian of X is given by

${\displaystyle \operatorname {Ber} (X)=\det(A)\det(D)^{-1}}$

For a motivation of the negative exponent see the substitution formula in the odd case.

More generally, consider matrices with entries in a supercommutative algebra R. An even supermatrix is then of the form

${\displaystyle X={\begin{bmatrix}A&B\\C&D\end{bmatrix}}}$

where A and D have even entries and B and C have odd entries. Such a matrix is invertible if and only if both A and D are invertible in the commutative ring R₀ (the even subalgebra of R). In this case the Berezinian is given by

${\displaystyle \operatorname {Ber} (X)=\det(A-BD^{-1}C)\det(D)^{-1}}$

or, equivalently, by

${\displaystyle \operatorname {Ber} (X)=\det(A)\det(D-CA^{-1}B)^{-1}.}$

These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring R₀. The matrix

${\displaystyle D-CA^{-1}B\,}$

is known as the Schur complement of A relative to ${\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}.}$

An odd matrix X can only be invertible if the number of even dimensions equals the number of odd dimensions. In this case, invertibility of X is equivalent to the invertibility of JX, where

${\displaystyle J={\begin{bmatrix}0&I\\-I&0\end{bmatrix}}.}$

Then the Berezinian of X is defined as

${\displaystyle \operatorname {Ber} (X)=\operatorname {Ber} (JX)=\det(C-DB^{-1}A)\det(-B)^{-1}.}$

• The Berezinian of ${\displaystyle X}$ is always a unit in the ring R₀.
• ${\displaystyle \operatorname {Ber} (X^{-1})=\operatorname {Ber} (X)^{-1}}$
• ${\displaystyle \operatorname {Ber} (X^{st})=\operatorname {Ber} (X)}$ where ${\displaystyle X^{st}}$ denotes the supertranspose of ${\displaystyle X}$.
• ${\displaystyle \operatorname {Ber} (X\oplus Y)=\operatorname {Ber} (X)\mathrm {Ber} (Y)}$

The determinant of an endomorphism of a free module M can be defined as the induced action on the 1-dimensional highest exterior power of M. In the supersymmetric case there is no highest exterior power, but there is a still a similar definition of the Berezinian as follows.

Suppose that M is a free module of dimension (p,q) over R. Let A be the (super)symmetric algebra S*(M*) of the dual M* of M. Then an automorphism of M acts on the ext module

${\displaystyle Ext_{A}^{p}(R,A)}$

(which has dimension (1,0) if q is even and dimension (0,1) if q is odd)) as multiplication by the Berezinian.

• Berezin integration

• Berezin, Feliks Aleksandrovich (1966) [1965], The method of second quantization, Pure and Applied Physics, vol. 24, Boston, MA: Academic Press, ISBN 978-0-12-089450-5, MR 0208930
• Deligne, Pierre; Morgan, John W. (1999), "Notes on supersymmetry (following Joseph Bernstein)", in Deligne, Pierre; Etingof, Pavel; Freed, Daniel S.; Jeffrey, Lisa C.; Kazhdan, David; Morgan, John W.; Morrison, David R.; Witten., Edward (eds.), Quantum fields and strings: a course for mathematicians, Vol. 1, Providence, R.I.: American Mathematical Society, pp. 41–97, ISBN 978-0-8218-1198-6, MR 1701597
• Manin, Yuri Ivanovich (1997), Gauge Field Theory and Complex Geometry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-61378-7