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?