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I am looking for as gentle of possible of an introduction to Kontsevich-Soibelman's theory of motivic DT-invariants. Specifically I am interested in the algebraic aspects of the theory and the relation with cluster categories. Obviously there is Kontsevich and Soibelman's 150 page paper on the subject, but words like etale and stack tend to make me panic. I am much more comfortable with 3-Calabi-Yau categories than 3 dimensional Calabi-Yau varieties.

flag	asked Jun 25 at 0:54 Steve 1,016●1●5●16

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Props for inventing a new term "Algebro-Geophobic". – David Corwin Jun 25 at 1:59

I wish such material existed! (although I am quite algebrophobic to be honest, so it's the gentleness I'd be interested in) For non-refined DT invariants I find Bridgeland's Introduction to Motivic Hall Algebras great. In the end etale just means local diffeomorphism (or relative Kahler differentials zero if that's clearer to you o_O) and stacks are just a fact of life and one ends up thinking about them just as any other space (I guess). At least the idea of a motivic Hall algebra is quite simple. – Yosemite Sam Jun 25 at 3:02

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Bernhard Keller has notes from an algebraic perspective here: arXiv:1102.4148. – Chris Brav Jun 25 at 8:12

I am afraid there is no easy path to the definition of motivic DT-invariants. You need moduli stacks of stable objects, the Grothendieck group of (classes of) motives, the construction of the motivic Milnor fibre. Having said that, I think there are two silver linings: 1. None of these concepts are quite as scary as they are sometimes made out to be. (For example, the Grothendieck group of classes of motives is a simpler concept than any category of motives.) 2. The wall-crossing behaviour of DT-invariants is the interesting and purely algebraic part of the story. See e.g. Keller's notes. – ABayer Jun 25 at 11:42

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There is one in-road to this subject which is more algebraic than geometric and does not require so much machinery. I'm talking about categories of representations of a quiver with super-potential. These are CY3 categories and admit motivic DT invariants, wall-crossing, etcetera. But everything is fairly concrete, stability comes from ordinary GIT stability, and since there is a global super-potential, you don't need all the $A_{\infty}$ business to generate local super-potentials. The definition of the motivic invariants is fairly easy to understand. – Jim Bryan Jun 25 at 23:57

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Motivic DT-Invariants for the Algebro-Geophobic

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