Survey of D-branes in Various Settings

Question

This is a somewhat silly question, but I think it would be great to have a nice answer in one place.   There's various things I'm confused about regarding D-branes in topological string theories vs. physical string theories.  In particular, conventions about dimensions as well as relations of A-model and B-model topological strings to Type IIA, Type IIB physical strings.  

I think I understand the following points:

1. In general, a Dp-brane has $p$ spatial dimensions and a (p+1)-dimensional worldvolume in ten dimensional space $\mathbb{R}^{1,3} \times X$.

2. In Type IIA, the Dp-branes only exist for $p$ odd and are called 'A-branes.'

3. In Type IIB, the Dp-branes only exist for $p$ even and are called 'B-branes.'

4. In topological string theory, the convention is that a Dp-brane has $p$ spatial dimensions in the compact space and any number in the non-compact space.

5. Topological A-branes are D3-branes wrapping Lagrangian submanifolds of a Calabi-Yau $Y$ while topological B-branes are D0, D2, D4, D6-branes wrapping holomorphic cycles in the mirror Calabi-Yau $X$.  

So...for example, does this mean that a D2-brane in B-model topological string theory may correspond to a D2, D3, D4, D5, or D6-brane in some physical string theory?  If so, this leads to my next confusion about which physical string theory...

I'm confused about how topological A and B-branes relate to Type IIA vs. Type IIB theories.  I've heard that Witten (somehow) coincidentally named things such that A-branes arise from twisting Type IIA compactified on $Y$ while B-branes arise from twisting Type IIB on the mirror $X$.  But I also recall clearly hearing the exact opposite, i.e. that A-branes are related to TypeIIB visa versa.  Can someone maybe set me straight here?  

Answer 1

First, some terminology (sometimes different from the one used in the question): I will say "A-brane" for "topological A-brane", "B-brane" for "topological B-brane" and Dp-brane for a Dp-brane in physical string theory (so in IIA for p even and IIB for p odd). For the topological A-model on a Calabi-Yau 3-fold $X$, A-branes are Lagrangians cycles in $X$ and for the topological B-model on a Calabi-Yau 3-fold $X$, B-branes are holomorphic cycles in $X$.

The relation between branes in topological and physical string theories is indeed subtle: depending on the number of wrapped non-compact dimensions, a topological brane can lead to various physical branes, in either IIA or IIB theory. There is no simple recipe like "IIA corresponds to the A-model" or "IIA corresponds to the B-model".

The correct slogans are:

"The vector multiplet geometry of IIA on X is related to the topological A-model on X".

"The hypermultiplet geometry of IIA on X is related to the topological B-model on X".

"The vector multiplet geometry of IIB on X is related to the topological B-model on X".

"The hypermultiplet geometry of IIB on X is related to the topological A-model on X".

Under mirror symmetry, vector and hyper are preserved, IIA and IIB are exchanged, topological A- and B-models are exchanged, and so everything is compatible.

Comments

Thanks a lot, I think you perfectly answered my question!  That all makes sense.  Maybe I can comment that at the top of page 31 in (https://arxiv.org/pdf/math/0412328.pdf) the authors write " A-type D-branes occur in IIB string theory and B-type D-branes in IIA string theory."  Is this mistaken, or are they speaking of some special scenario algebraic geometers care about perhaps?  

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In the paper you are referring to, the authors have in mind BPS particles in the $\mathcal{N}=2$ 4d theory. D-branes construction of these BPS states involves one non-compact direction (the worldline of the BPS particle in 4d), and so A-branes in IIB (D3 branes around Lagrangian cycles) and B-branes in IIA (D0-D2-D4-D6 branes around holomorphic cycles).

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Awesome.  Finally, the meaning of 'A-brane' and 'B-brane'  in this paper are necessarily topological branes, as in your answer above?  In other words, these BPS particles in the $4d$ theory sit in the topological sector?   

Answer 2

Let me add a few points to the answer by 40227.

First, there are not only Lagrangian p-branes in 2p-dimensional Calabi-Yau in the A-model -- more general p+2k-dimensional coisotropic branes with non-zero curvature of the gauge connection living on them, are also allowed (see the original paper by Orlov and Kapustin Remarks on A-branes, Mirror Symmetry, and the Fukaya category for a survey).

Second, the relation between the branes in topological string theories and the physical theories (as well as with certain terms in the effective supersymmetric theories arising from Calabi-Yau compactification of string theoкies) is indeed an extensive subject, details of which could be found, for example, in Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes and Superstrings and Topological Strings at Large N.