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It is known that the topological A-model allows the existence of $\frac{1}{2} \left[ D + \mathrm{rank} \left( B \right) \right]$-dimensional branes, where $D$ is a dimensionality of spacetime, and $B$ is a B-field.

Witten showed that the A-model with the target space being the cotangent bundle $T^*M$ to some 3-fold $M$ is equivalent to the Chern-Simons theory defined on this space which is interpreted as an effective theory living on the stack of 3-branes wrapping the base $M$. More general 3-branes configurations are possible if these branes wrap a Lagrangian submanifold of the embedding space. Generically, in accordance to what stated above, 5-branes are also allowed in a CY 3-fold if one has a non-zero $B$-field.

Question: Could anybody recommend any literature on these higher-dimensional topological branes and their world volume theories?

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  • $\begingroup$ I'm trying to grasp the basics of string theory and am struggling to conceptualise it physically. Is $M$ here, spacetime? Then a stack of 3-branes is ambiently spread throughout spacetime - this means that each brane is as large as spacetime, like a field. But a stack implies a separation, how can they then be separated in space? $\endgroup$
    – Mozibur Ullah
    Jan 18, 2018 at 12:16
  • $\begingroup$ Have you come across Nlab? - they are a mine of information. $\endgroup$
    – Mozibur Ullah
    Jan 18, 2018 at 12:37

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Some useful information on the subject could be found in the paper by Manfred Herbst "On Higher Rank Coisotropic A-branes", but it is not exhaustive, so the question is still relevant.

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The spectrum of higher dimensional branes in topological string theory is very rich. Maybe it would be better if you asked a specific question about a particular object.

General comments can be made.

  1. It's a basic starting point to read about A/B model S-duality over the same Calabi Yau threefold in S-duality and Topological Strings because the very existence of A model topological NS5-branes and their electric B model NS2 counterparts were predicted as a beautiful consequence of the duality. Those classes of defects are extraordinarily important as sources of non-perturbative terms.

  2. General lagrangian topological A branes are introduced in D-branes as Defects in the Calabi-Yau Crystal.

  3. A very important technicality in topological string theory concerns to how to develop techniques to define appropriate indices in order to extend the Donaldson-Thomas D6/D2/D0 BPS-counting (Quantum Foam and Topological Strings) to more general configurations involving D4 branes on toric Calabi-Yau threefolds. The answer (as far as I understand) is given in the beautiful Jafferis PhD Thesis(see also https://mathoverflow.net/questions/269554/incorporating-divisors-d4-branes-into-donaldson-thomas-theory) int the case of the vertex geometry and Crystal Melting and Toric Calabi-Yau Manifolds for arbitrary threefolds.

4.- Witten has used topological coisotropic branes extensively in several important developments (examples are this this and this). To understand the origin of those seemely "esoteric" objects I recommend to read this and this.

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