Berezinian

In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold.

Definition

The Berezinian is uniquely determined by two defining properties:

• [math]\displaystyle{ \operatorname{Ber}(XY) = \operatorname{Ber}(X)\operatorname{Ber}(Y) }[/math]
• [math]\displaystyle{ \operatorname{Ber}(e^X) = e^{\operatorname{str(X)}}\, }[/math]

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

[math]\displaystyle{ X = \begin{bmatrix}A & 0 \\ 0 & D\end{bmatrix} }[/math]

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

[math]\displaystyle{ \operatorname{Ber}(X) = \det(A)\det(D)^{-1} }[/math]

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

[math]\displaystyle{ X = \begin{bmatrix}A & B \\ C & D\end{bmatrix} }[/math]

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

[math]\displaystyle{ \operatorname{Ber}(X) = \det(A-BD^{-1}C)\det(D)^{-1} }[/math]

or, equivalently, by

[math]\displaystyle{ \operatorname{Ber}(X) = \det(A)\det(D-CA^{-1}B)^{-1}. }[/math]

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

[math]\displaystyle{ D-CA^{-1}B \, }[/math]

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

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

[math]\displaystyle{ J = \begin{bmatrix}0 & I \\ -I & 0\end{bmatrix}. }[/math]

Then the Berezinian of X is defined as

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

Properties

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

Berezinian module

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

[math]\displaystyle{ Ext_{A}^p (R,A) }[/math]

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

See also

• Berezin integration

References

• Berezin, Feliks Aleksandrovich (1966), The method of second quantization, Pure and Applied Physics, 24, Boston, MA: Academic Press, ISBN 978-0-12-089450-5, https://books.google.com/books?id=fAlRAAAAMAAJ 
• Deligne, Pierre; Morgan, John W. (1999), "Notes on supersymmetry (following Joseph Bernstein)", in Deligne, Pierre; Etingof, Pavel; Freed, Daniel S. et al., 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, https://books.google.com/books?id=TQIsyvw1KnsC&pg=PA41 
• Manin, Yuri Ivanovich (1997), Gauge Field Theory and Complex Geometry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-61378-7