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:

  • Ber ⁡ ( X Y ) = Ber ⁡ ( X ) Ber ⁡ ( Y ) {\displaystyle \operatorname {Ber} (XY)=\operatorname {Ber} (X)\operatorname {Ber} (Y)}
  • Ber ⁡ ( e X ) = e s t r ( X ) {\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

X = [ A 0 0 D ] {\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

Ber ⁡ ( X ) = det ( A ) det ( D ) − 1 {\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

X = [ A B C D ] {\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

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

or, equivalently, by

Ber ⁡ ( X ) = det ( A ) det ( D − C A − 1 B ) − 1 . {\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

D − C A − 1 B {\displaystyle D-CA^{-1}B\,}

is known as the Schur complement of A relative to [ A B C D ] . {\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

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

Then the Berezinian of X is defined as

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

Properties

  • The Berezinian of X {\displaystyle X} is always a unit in the ring R0.
  • Ber ⁡ ( X − 1 ) = Ber ⁡ ( X ) − 1 {\displaystyle \operatorname {Ber} (X^{-1})=\operatorname {Ber} (X)^{-1}}
  • Ber ⁡ ( X s t ) = Ber ⁡ ( X ) {\displaystyle \operatorname {Ber} (X^{st})=\operatorname {Ber} (X)} where X s t {\displaystyle X^{st}} denotes the supertranspose of X {\displaystyle X} .
  • Ber ⁡ ( X ⊕ Y ) = Ber ⁡ ( X ) B e r ( Y ) {\displaystyle \operatorname {Ber} (X\oplus Y)=\operatorname {Ber} (X)\mathrm {Ber} (Y)}

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

E x t A p ( R , A ) {\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.

See also

  • Berezin integration

References

  • 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