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To define the refined Chern-Simons theory on a three-manifold M , we needed to study M-theory on Y × T N × S 1 , where Y = T ∗ M with N M5 branes on M × C × S 1 . Consider a dual description of this, by dimensionally reducing on the S 1 of the Taub-Nut space. Without M5 branes, we would obtain IIA string theory on the geometry, Y × R3 × S 1 with a D6 brane wrapping Y × S 1 and sitting at the origin of R3 . Adding the N M5 branes on M × C × S 1 , we get IIA string theory with the addition of N D4 branes, wrapping M × S 1 times a half-line R+ in R3 , ending on the D6 brane.
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