Abstract
Recall the following facts from Chaps. 1 and 2: The Schrödinger algebra \(\mathfrak{s}\mathfrak{c}\mathfrak{h}\) is defined as the algebra of projective Lie symmetries of the free Schrödinger equation in (1 + 1)-dimensions, \((-2\mathrm{i}\mathcal{M}{\partial }_{t} - {\partial }_{r}^{2})\psi = 0\).
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Notes
- 1.
This apparently abstract extension becomes important for the explicit calculation of the two-time correlation function in phase-ordering kinetics [59].
- 2.
For example, \(\mathrm{ad}N({Y }_{\frac{1} {2} }) = [N,{Y }_{\frac{1} {2} }] = -{Y }_{\frac{1} {2} }\) or \(\mathrm{ad}N({Y }_{-\frac{1} {2} }) = [N,{Y }_{-\frac{1} {2} }] = 0\).
- 3.
This fixing of the scaling function through the additional N-covariance prompted us to consider \(\widetilde{\mathfrak{s}\mathfrak{v}}\)-primary fields instead of \(\mathfrak{s}\mathfrak{v}\)-primary fields in our vertex construction (see Sect. 6.1).
- 4.
The Schrödinger-invariance of a free non-relativistic particle of spin S is proven in [49].
- 5.
In non-commutative space-time, extended supersymmetries still persist, but scale- and Galilei-invariance are broken [93].
- 6.
An isomorphic Lie superalgebra was first constructed by Gauntlett et al. [37].
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© 2012 Springer-Verlag Berlin Heidelberg
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Unterberger, J., Roger, C. (2012). Supersymmetric Extensions of the Schrödinger–Virasoro Algebra. In: The Schrödinger-Virasoro Algebra. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22717-2_11
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DOI: https://doi.org/10.1007/978-3-642-22717-2_11
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