[SOLVED] Pi-stability and Bridgeland conjecture

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I am reading Aspinwall\'s "D-branes on Calabi-Yau Manifolds",\nhep-th/0403166. Aspinwall discusses Pi-stability and the Bridgeland\nconjecture on pp. 79. Unfortunately I cannot access Bridgeland\'s\noriginal papers right now, so here are a couple of questions:\n\nIf I understand correctly, given any state of D-branes E (i.e. an\nobject E in the derived category D(X) describing these branes), there\nis expected to be kind of a Bridgeland-functor B which sends this to\nthe state\n\nB(E) = \\osum_i A_i(E)\n\nrepresenting a collection of stable D-branes A_i(E) into which E\ndecays.\n\nI understand that the existence of B is conjectural but considered\nvery plausibel. Assuming B does exist\n\n- is it expected to be single valued up to marginal stability? I.e.\nis B(E) unique up to decays of marginally stable states?\n\n- is B expected to be compatible with direct sums in the sense that\n\nB( E1 \\oplus B( E2 \\oplus E3) )\n= B( B(E1 \\oplus E2) \\oplus E3 )\n= B( E1 \\oplus E2 \\oplus E3 )\n?\n\nIn categorical terms what I am trying to get at is this:\n\nThe derived category D(X) knows direct sums. The cone construction\ncorresponds to subtraction. Given the direct sum of two D-branes, the\nchoice of cone, if any, between them which might make them react, is\ngiven by Pi-stability.\n\nSo we can subtract in D(X) by performing a direct sum followed by the\nBridgeland operator. For this to be well defined the answer to the\nabove two questions must be "Yes". Is it?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>I am reading Aspinwall's "D-branes on Calabi-Yau Manifolds",
http://www.arxiv.org/abs/hep-th/0403166. Aspinwall discusses $\Pi-stability$ and the Bridgeland
conjecture on pp. 79. Unfortunately I cannot access Bridgeland's
original papers right now, so here are a couple of questions:

If I understand correctly, given any state of D-branes E (i.e. an
object E in the derived category D(X) describing these branes), there
is expected to be kind of a Bridgeland-functor B which sends this to
the state

B(E) $= \osum_i A_i(E)$

representing a collection of stable D-branes $A_i(E)$ into which E
decays.

I understand that the existence of B is conjectural but considered
very plausibel. Assuming B does exist

- is it expected to be single valued up to marginal stability? I.e.
is B(E) unique up to decays of marginally stable states?

- is B expected to be compatible with direct sums in the sense that

B( $E1 \oplus B( E2 \oplus E3) )= B( B(E1 \oplus E2) \oplus E3 )= B( E1 \oplus E2 \oplus E3 )$
?

In categorical terms what I am trying to get at is this:

The derived category D(X) knows direct sums. The cone construction
corresponds to subtraction. Given the direct sum of two D-branes, the
choice of cone, if any, between them which might make them react, is
given by $\Pi-stability$.

So we can subtract in D(X) by performing a direct sum followed by the
Bridgeland operator. For this to be well defined the answer to the
above two questions must be "Yes". Is it?

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