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| In the theory of [[superalgebra]]s, if ''A'' is a [[commutative superalgebra]], ''V'' is a free right ''A''-[[supermodule]] and ''T'' is an [[endomorphism]] from ''V'' to itself, then the '''supertrace''' of ''T'', str(''T'') is defined by the following [[trace diagram]]:
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| :[[Image:Trace.png]]
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| More concretely, if we write out ''T'' in [[block matrix]] form after the decomposition into even and odd subspaces as follows,
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| :<math>T=\begin{pmatrix}T_{00}&T_{01}\\T_{10}&T_{11}\end{pmatrix}</math>
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| then the supertrace
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| :str(''T'') = the ordinary [[trace (matrix)|trace]] of ''T''<sub>0 0</sub> − the ordinary trace of ''T''<sub>11</sub>.
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| Let us show that the supertrace does not depend on a basis.
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| Suppose '''e'''<sub>1</sub>, ..., '''e'''<sub>p</sub> are the even basis vectors and '''e'''<sub>''p''+1</sub>, ..., '''e'''<sub>''p''+''q''</sub> are the odd basis vectors. Then, the components of ''T'', which are elements of ''A'', are defined as
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| :<math>T(\mathbf{e}_j)=\mathbf{e}_i T^i_j.\,</math>
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| The grading of ''T''<sup>''i''</sup><sub>''j''</sub> is the sum of the gradings of ''T'', '''e'''<sub>''i''</sub>, '''e'''<sub>''j''</sub> mod 2.
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| A change of basis to '''e'''<sub>1'</sub>, ..., '''e'''<sub>p'</sub>, '''e'''<sub>(''p''+1)'</sub>, ..., '''e'''<sub>(''p''+''q'')'</sub> is given by the [[supermatrix]]
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| :<math>\mathbf{e}_{i'}=\mathbf{e}_i A^i_{i'}</math>
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| and the inverse supermatrix | |
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| :<math>\mathbf{e}_i=\mathbf{e}_{i'} (A^{-1})^{i'}_i,\,</math>
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| where of course, ''AA''<sup>−1</sup> = ''A''<sup>−1</sup>''A'' = '''1''' (the identity).
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| We can now check explicitly that the supertrace is [[basis independent]]. In the case where ''T'' is even, we have
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| :<math>\operatorname{str}(A^{-1} T A)=(-1)^{|i'|} (A^{-1})^{i'}_j T^j_k A^k_{i'}=(-1)^{|i'|}(-1)^{(|i'|+|j|)(|i'|+|j|)}T^j_k A^k_{i'} (A^{-1})^{i'}_j=(-1)^{|j|} T^j_j
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| =\operatorname{str}(T).</math>
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| In the case where ''T'' is odd, we have
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| :<math>\operatorname{str}(A^{-1} T A)=(-1)^{|i'|} (A^{-1})^{i'}_j T^j_k A^k_{i'}=(-1)^{|i'|}(-1)^{(1+|j|+|k|)(|i'|+|j|)}T^j_k (A^{-1})^{i'}_j A^k_{i'} =(-1)^{|j|} T^j_j
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| =\operatorname{str}(T).</math>
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| The ordinary trace is not basis independent, so the appropriate trace to use in the '''Z'''<sub>2</sub>-graded setting is the supertrace.
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| The supertrace satisfies the property
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| :<math>\operatorname{str}(T_1 T_2) = (-1)^{|T_1||T_2|} \operatorname{str}(T_2 T_1)</math> | |
| for all ''T''<sub>1</sub>, ''T''<sub>2</sub> in End(''V''). In particular, the supertrace of a supercommutator is zero.
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| In fact, one can define a supertrace more generally for any associative superalgebra ''E'' over a commutative superalgebra ''A'' as a linear map tr: ''E'' -> ''A'' which vanishes on supercommutators.<ref name="berline_getzler_vergne">N. Berline, E. Getzler, M. Vergne, ''Heat Kernels and Dirac Operators'', Springer-Verlag, 1992, ISBN 0-387-53340-0, p. 39.</ref> Such a supertrace is not uniquely defined; it can always at least be modified by multiplication by an element of ''A''.
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| ==Physics Applications==
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| In supersymmetric quantum field theories, in which the action integral is invariant under a set of symmetry transformations (known as supersymmetry transformations) whose algebras are superalgebras, the supertrace has a variety of applications. In such a context, the supertrace of the mass matrix for the theory can be written as a sum over spins of the traces of the mass matrices for particles of different spin:<ref name="martin">S. Martin, ''A Supesymmetry Primer'', in ''Perspectives on supersymmetry'', G. L. Kane, ed., p. 1-98 [arXiv:hep-ph/9709356].</ref>
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| :<math>\operatorname{str}[M^2]=\sum_s(-1)^{2s} (2s+1)\operatorname{tr}[m_s^2].</math>
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| In anomaly-free theories where only renormalizable terms appear in the superpotential, the above supertrace can be shown to vanish, even when supersymmetry is spontaneously broken.
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| The contribution to the effective potential arising at one loop (sometimes referred to as the Coleman-Weinberg potential<ref name="CW">S. Coleman and E. Weinberg, ''Radiative Corrections as the Origin of Spontaneous Symmetry Breaking'', Phys. Rev. D7, p. 1888-1910, 1973.</ref>) can also be written in terms of a supertrace. If <math>M</math> is the mass matrix for a given theory, the one-loop potential can be written as
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| :<math>V_{eff}^{1-loop}=\dfrac{1}{64\pi^2}\operatorname{str}\bigg[M^4\ln\Big(\dfrac{M^2}{\Lambda^2}\Big)\bigg] =
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| \dfrac{1}{64\pi^2}\operatorname{tr}\bigg[m_{B}^4\ln\Big(\dfrac{m_{B}^2}{\Lambda^2}\Big)-
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| m_{F}^4\ln\Big(\dfrac{m_{F}^2}{\Lambda^2}\Big)\bigg]</math>
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| where <math>m_B</math> and <math>m_F</math> are the respective tree-level mass matrices for the separate bosonic and fermionic degrees of freedom in the theory and <math>\Lambda</math> is a cutoff scale.
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| ==See also==
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| * [[Berezinian]].
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| ==References==
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| {{reflist}}
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| [[Category:Super linear algebra]]
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