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I can provide an example for bosonic models. \begin{eqnarray} \mathcal{H} & = & \mathcal{K} + \mathcal{T}_\text{soc} +\frac{U}{2}\sum_{i\tau} \hat n_{i\tau}( \hat n_{i\tau}-1) \nonumber \\ & & + U^{\prime} \sum_i \hat n_{i\uparrow} \hat n_{i\downarrow} + V\sum_{i\tau} \hat{n}_{i\tau}\hat{n}_{i+1\tau} \nonumber ...

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There is a very direct relationship which answers your question, and I'll state it in the way I first learned about it (but you can derive a different connection by passing between dimensions): The 2-dimensional reduction of the Seiberg-Witten equations are the (abelian) vortex equations. The $SU(2)$-vortex equations on $\mathbb{R}^2$ are a ...

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Gauge symmetry imposes local conservation laws, which are called Ward Identities in QED and Slavnov-Taylor identities for non-Abelian gauge theories. Those identities relate amplitudes or limit them. An example of those constraints imposed by gauge symmetry is the transversality of the vacuum polarization. To be more precise, gauge symmetry does not allow ...

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