Summary Conjecture

I make the conjecture precise here with no reference to justification. Two linked branes of dimensions m≥0 and n≥0 are linked in a flat space of k=m+n+1 dimensions. The linking is topologically characterized by an integer winding number w. The double integral over the two branes of dp∧dq∧r/|r|^k is w times the content of S^m+n. The content of the unit k-ball is C_k = π^k/2/(k/2)! and the content of its boundary (which is S^k−1) is kπ^k/2/(k/2)!.

Here is a table of C_k:

k 0 1 2 3 4 5 6
C_k 1 2 π (4/3)π π²/2 (8/15)π² π³/6
The integral for a simply linked m-brane and n-brane is thus (m+n+1)C_m+n+1.

For m=n=1 we have k=3 and 3C₃ = 4π as previously developed.

For m=0 and n=1 we get 2π. I presume here that S⁰ is a pair of points, here embedded in R². We corroborate this by letting the first brane be {(0, 0), (2, 0)} and the 2nd be the unit circle:{(x, y) | x² + y² = 1}.

For m=n=0 we get 2. There are two places on S⁰ to view the branes. Let brane P = {0, 2} and Q = {1, 3}. The (degenerate) integral involves all combinations of a ray from a point of P thru a point of Q. Starting from 0 (on P) we have two rays going to the right thru the respective points of Q. Their contributions cancel for the two points of Q have opposite orientations. Starting from point 2 of P, the two rays each encounter Q with the same orientation. The sum is thus 2. This case has excessive hand-waving! Some improved definitions of branes and linking are required to make this case good math.