# Survey of D-branes in Various Settings

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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?

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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.

answered Dec 21, 2017 by (5,140 points)

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?

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).

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?

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