| Summary: | Abstract We explore the path integration — upon the contour of hermitian (non-auxliary) field configurations — of topologically twisted N = 2 $$ \mathcal{N}=2 $$ Chern-Simons-matter theory (TTCSM) on S 2 $$ {\mathbb{S}}_2 $$ times a segment. In this way, we obtain the formula for the 3D topologically twisted index, first as a convolution of TTCSM on S 2 $$ {\mathbb{S}}_2 $$ times halves of S 1 $$ {\mathbb{S}}_1 $$ , second as TTCSM on S 2 $$ {\mathbb{S}}_2 $$ times S 1 $$ {\mathbb{S}}_1 $$ — with a puncture, — and third as TTCSM on S 2 × S 1 $$ {\mathbb{S}}_2\times {\mathbb{S}}_1 $$ . In contradistinction to the first two cases, in the third case, the vector multiplet auxiliary field D is constrained to be anti-hermitian.
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