On page 132 of "Introduction to Supergravity" by Horiatiu Nastase, the author says:

On $M = CY_3$ (Calabi-Yau space) there are $b_3$ topologically nontrivial 3-surfaces, for which we can define a basis $(A_I, B^J)$, where $I, J = 1, \ldots, b_3/2$, such that $A_I \cap A_J = B^I \cap B^J = 0$ and $A_I \cap B^J = -B^J \cap A_I = \delta_I^J$.

What does the intersection symbol denote here? I understand that $A_I$ and $B^J$ are $(b_3/2, 0)$ and $(0, b_3/2)$ forms.

What does the minus sign in $A_I \cap B^J = -B^J \cap A_I = \delta_I^J$ mean?


I guess you have to read this in the context of intersection theory. On a manifold of dimension $2n$ we have the $n$-th homology group, which can informally be thought of as being generated by equivalence classes of $n$-dimensional submanifolds. Intersecting two of these should generally give a discrete set of points. Intuitively the intersection product can be thought of as the number of points in the intersection, but it is not exactly the same as the set-theoretic intersection.

To make this concrete, think of a genus $g$ Riemann surface, which has dimension 2. Its first cohomology group has rank $2g$, and as our generators $A$ we can take curves around the holes, and generators $B$ are curves cutting open a hole.

genus 2 surface, downloaded from dlmf.nist.gov

$A_i$ and $B_j$ intersect in a single point exactly when $i = j$, otherwise they don't. The actual value is a sum of the intersection multiplicities, which are defined in a quite technical way, over all points in the intersection. Note that since we are talking about homology classes, not even the number of points is well-defined. By counting some points with negative multiplicity, we can still obtain a uniquely defined number. As an illustrative example, consider this triple intersection on a torus:

triple intersection

This should have the same value as the intersection of the big horizontal circle that crosses the other only once, and which is homologous to this more wiggly representative. You see that in all but one of the points the intersection multiplicities cancel each other.

  • $\begingroup$ Thank you for your answer doetoe. Is there a way to see the anticommutative structure in the picture you drew? $\endgroup$ – leastaction Jun 7 '15 at 14:02
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    $\begingroup$ @leastaction The actual number associated to each point in the intersection is called the intersection multiplicity. When the intersection is transversal, this is $\pm 1$ and the sign depends on the orientation of the basis of tangent vectors at the point ordered by first taking the tangent vectors to the first operand, then those of the second. In the $n=1$ (embedded) case we could take the orientation such that positive orientation corresponds with the cross product pointing outward. $\endgroup$ – doetoe Jun 7 '15 at 14:17
  • $\begingroup$ What, then, is the implication of this structure for Calami-Yau/string compactification? I notice that other authors who introduce this basis do not explicitly indicate this anti-commutative intersection theory idea. I want to understand why it is significant for what we're trying to do here. $\endgroup$ – leastaction Jun 7 '15 at 14:19
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    $\begingroup$ @leastaction I'll edit the answer to include a picture $\endgroup$ – doetoe Jun 7 '15 at 14:23
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    $\begingroup$ @leastaction Did you notice btw that your question was copied to physicsoverflow physicsoverflow.org/31683/…? It got an answer there as well $\endgroup$ – doetoe Jun 7 '15 at 15:07

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