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D-brane wrapping in the geometric transition for type-IIB on a Calabi-Yau manifold

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In Stefan Metzger's thesis (https://arxiv.org/abs/hep-th/0512285) on page 9, the following statement appears in connection with the compactifications type-IIB string theory on two Calabi-Yau manifolds that are related by a geometric transition. [The geometric transition is described on the previous page in the thesis.]

It is now interesting to see what happens if we compactify Type IIB string theory on two Calabi-Yau manifolds that are related by such a transition. Since one is interested in $\mathcal{N} = 1$ effective theories it is suitable to add either fluxes or branes in order to further break supersymmetry. It is then very natural to introduced D5-branes wrapping the two-spheres in the case of the small resolution of the singularity (Edit: small resolution refers to blowing up the singularity using an $S^2$). The manifold with a deformed singularity (Edit: deformed singularity refers to blowing up the singularity using an $S^3$) has no suitable cycles around which D-branes might wrap, so we are forced to switch on flux in order to break supersymmety.

So first of all, when can we wrap a D$p$-brane around a $k$-cycle (i.e. for what $p$ and $k$)? I thought $p \geq k$ is all we needed?

More specifically, what are the internal cycles of an $S^2$ and an $S^3$ and why can't these D-branes be wrapped around them? Is the reasoning obvious?


This post imported from StackExchange Physics at 2016-11-13 14:36 (UTC), posted by SE-user leastaction

asked Nov 8, 2016 in Theoretical Physics by leastaction (425 points) [ revision history ]
edited Nov 13, 2016 by Dilaton

1 Answer

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In the context of the question, the 10-dimensional spacetime is $\mathbb{R}^{1,3} \times X$ where $X$ is a Calabi-Yau 3-fold. Considering type IIB string theory on this geometry, without branes and with no fluxes turned on, gives rise to an effective theory on $\mathbb{R}^{1,3}$ with $\mathcal{N}=2$ supersymmetry. If one wants to go from $\mathcal{N}=2$ to $\mathcal{N}=1$ supersymmetry on $\mathbb{R}^{1,3}$, one can try to add D-branes. If one wants to preserve Poincaré invariance of $\mathbb{R}^{1,3}$, the D-brane has to fill in entirely $\mathbb{R}^{1,3}$ (it is interesting to consider cases where one does not preserve Poincaré invariance of $\mathbb{R}^{1,3}$ but I don't think that it is the context of the question). So the only possibilities, given that $Dp$-branes in IIB string theory have $p$ odd, are $D3$-branes wrapping 0-cycles in $X$, $D5$-branes wrapping 2-cycles in $X$ and $D7$-branes wrapping 4-cycles in $X$. In particular, there is nothing to wrap around a non-trivial 3-cycle.

Starting with $D5$-branes wrapping some non-trivial 2-cycle $S^2$ in $X$ and realizing the geometric transition consisting in shrinking $S^2$ and growing up some $S^3$, one can ask what happens to the theory: the $D5$ branes disappear, so how is it possible to still have $\mathcal{N}=1$ (and not $\mathcal{N}=2$)? The answer is that now there is a non-zero flux through $S^3$ (it is a field strength flux for the 2-form gauge field present in the Ramond-Ramond sector of IIB. Remark that the $D5$-branes are magnetic sources for this 2-form gauge field  and the idea is that during the geometric transition, the $D5$-branes disappear but the corresponding magnetic field remains).

answered Nov 13, 2016 by 40227 (4,660 points) [ revision history ]

In your first sentence, the dimensions don't sum to 10.

@ArnoldNeumaier $\mathbb{R}^{1,3}$ has 4 dimensions $\times X$ which has 6 real dimensions is 10 dimensions, no?

So X is the real 6-fold from a complex 3-fold. OK. (wikipedia talks here about a Calabi-Yau 6-fold.)

@40227, thanks for the explanation! Why do you have 0-cycles, 2-cycles and 4-cycles in X? Also, what is the justification for the magnetic field remaining when it's source (the D5-brane) has disappeared?

First, to be precise, when I write "cycle", I mean topologically non-trivial cycle. Topologically non-trivial $k$-cycles in $X$ up to topologically trivial cycles are classified by homology groups $H_k(X,\mathbb{Z})$. So the question, why do we have 0-cycles, 2-cycles, 4-cycles in $X$ is equivalent to why do we have $H_0(X,\mathbb{Z})$, $H_2(X,\mathbb{Z})$ and $H_4(X,\mathbb{Z})$ non-zero. It is easy for $k=0$: we always have points in $X$ as topologically non-trivial 0-cycles. For $k=2$, it is less obvious: there exists manifolds with $H_2=0$ but for a compact Calabi-Yau manifold (in the usual sense, in particular Kähler), it is a mathematical fact that there are always non-trivial 2-cycles (for example an compact submanifold of real dimension 2 which is holomorphic) and similarly a compact Calabi-Yau manifold has always non-trivial 4-cycles (for example a compact submanifold of real dimension 4 which is holomorphic).

About the magnetic field: a gauge field configuration can be produced by a source or can exist by itself if it has a non-zero flux through a topologically non-trivial cycle. The geometric transition exchanges these two possibilities: in the first geometry, there is no appropriate cycle to support a flux but there is a D-brane sourcing the field, whereas in the second geometry, there is no longer D-branes but there is now a cycle able to support a flux. This possibility for a dynamical object like a D-brane to transform in a topologically non-trivial configuration with fluxes is one of the characteristic feature of string theory (AdS/CFT is another example of this: you can start with $N$ D3 branes in IIB on flat $\mathbb{R}^{1,9}$ and in some appropriate (near horizon) limit this becomes IIB, without branes, on curved and topologically non-trivial $AdS_5 \times S^5$ with $N$ units of flux through $S^5$).

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