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To obtain the topologically twisted version, we then do an additional step. The theory has a global $R$-symmetry, meaning that all fields also transform in representations of this group, and hence may be viewed as sections of vector bundles. We then activate a connection (called an $R$ gauge field) for this group and covariantize derivatives with respect to this connection as well.

If $M$ is flat, then the spin connection on $M$ is trivial so the second step does nothing, and the twisted theory is the same as the ordinary untwisted one.

The theory has a global $R$-symmetry, meaning that all fields also transform in representations of some group Lie Group $G_{R}$, and hence may be viewed as sections of vector bundles. $G_{R}$ also has the important property that enters the supersymmetry algebra non-trivially, with the supercharges also in representations of $G_{R}$.

To obtain the topologically twisted version, we now do an additional step utilizing $G_{R}$. We activate a connection (called an $R$ gauge field) for $G_{R}$ and covariantize derivatives with respect to this connection as well.

If $M$ is flat, then the spin connection on $M$ is trivial so the second step does nothing, and the twisted theory on $M$ is the same as the ordinary untwisted theory on $M$.

Warning: Your questions require answers using quantum field theory. Thus, in some cases in the following I refer to the functional integral. In the situation of interest for your question, this concept may be made precise though I do not do this here.

Now we let $L$ tend to infinity. The operator $H$ is positive definite and bounded below by zero. Hence, as $L$ tends to infinity, $e^{-H L}$ is a projection operator onto the subsector of the Hilbert space with $H=0$. This is exactly the subspace of vacuum states.

Warning: Your questions require answers using quantum field theory. Thus, in some cases in the following I refer to the functional integral and some of its properties. In the situation of interest for your question, these concepts may be made precise, though I do not do this here.

Now we let $L$ tend to infinity. The operator $H$ is positive definite and bounded below by zero. Hence, as $L$ tends to infinity, $e^{-H L}$ is a projection operator onto the subspace of the Hilbert space with $H=0$. This is exactly the subspace of vacuum states.

To obtain the topologically twisted version, we then do an additional step. The theory has a global $R$-symmetry, meaning that all fields also transform in representations of this group, and hence may be viewed as sections of (usually trivial) vector bundles. We then activate a connection (called an $R$ gauge field) for this group and covariantize derivatives with respect to this connection as well.

To obtain the topologically twisted version, we then do an additional step. The theory has a global $R$-symmetry, meaning that all fields also transform in representations of this group, and hence may be viewed as sections of vector bundles. We then activate a connection (called an $R$ gauge field) for this group and covariantize derivatives with respect to this connection as well.