By M. Aizenman (Chief Editor)

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It is much more accurately described as a relabeling process. Since stability is a physical quality, we are forced to relabel B since it has decayed. The obvious candidate in the above case is to call it C. Thus monodromy would transform B into C. Life can be more complicated than this however. If we have an open string f : A → C, then, since ϕ(B) = ϕ(C) when B decays to C + A[1], C will immediately decay further to D = Cone(f : A → C) plus another A[1]. Suppose A is “spherical” in the sense of [20] which means Hom(A, A[m]) = C for m = 0 or 3, and Hom(A, A[m]) = 0 otherwise.

Bg . One of these constants can be put equal to zero. We set C0 = 0. This problem also has a unique solution. 1) in terms ˜ of the modified Green function G(z, ζ ). The definition of the latter is similar to that of the G(z, ζ ) but differs in two respects: ˜ ˜ (G1) G(z, ζ ) is required to have zero periods over the boundaries b1 , . . ) vanishes on the boundary. ˜ Under the condition that G(z, ζ ) = 0 on b0 such a function is unique. 7) just meets these requirements. We conclude that the modified Green function is expressed through the dual tau-function F˜ .

4 In [17] it was suggested that the monodromy action on the derived category should be translated by 2 to undo this action on A. Since monodromy is a relabeling process, one is free to do this, but it looks unnatural from the perspective of associating monodromy with -stability. 2. General monodromies. , if more than just a single D-brane A becomes massless. In order to answer this we need to set up a general description of how one might analyze monodromy in a multi-dimensional moduli space. There are two paradigms for monodromy — both of which are useful: 1.

### Communications in Mathematical Physics - Volume 259 by M. Aizenman (Chief Editor)

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