I happened to read some things about 'numerical Donaldson-Thomas invariants' and the 'integrality conjecture' (by Kontsevich and Soibelman, iirc) and I was hoping someone would shed light on what these are about, as I cannot make any connection myself.

One starts with an integer $ m \ge 1 $ and the power series algebra $ \mathbb{Q} [[ x_1,x_2 ]]$ with multiplication defined by $ (x_1^ax_2^b).(x_1^cx_2^d) = (-1)^{m(ad-bc)} x_1^{a+c} x_2^{b+d} $. For $ u,v $ natural numbers not both zero, denote by $ T_{u,v} $ the automorphism of the above algebra defined by $$ T_{u,v}(x_1) = x_1(1-x_1^ux_2^v)^{-mv} $$ $$ T_{u,v}(x_2) = x_2(1-x_1^ux_2^v)^{mu} $$ The integrality conjecture is the following statement: There are unique integers $ n_{u,v} $ such that $$ T_{1,0}.T_{0,1} = \prod_{\frac{u}{v} \uparrow} T_{u,v}^{n_{u,v}} $$ where the product is written in increasing order of the fractions from left to right. The uniquely determined integers $ n_{u,v} $ are called numerical Donaldson-Thomas invariants.

Atleast for me, it is absolutely out of the blue to see it for the first time, so perhaps somoeone could explain to me where this comes from? But for concrete questions, in what sense are these numbers 'invariants' in a geometric setting? And what is the connection with Donaldson-Thomas invariants? I think I understand them decently enough and as far as I can see, I see no counting of sheaves (or curves) on a threefold here, or any geometry at all. I'm missing something, hence this question and I'd be grateful for an answer.

I happened to read some things about 'numerical Donaldson-Thomas invariants' and the 'integrality conjecture' (by Kontsevich and Soibelman, iirc) and I was hoping someone would shed light on what these are about, as I cannot make any connection myself.

One starts with an integer $ m \ge 1 $ and the power series algebra $ \mathbb{Q} [[ x_1,x_2 ]]$ with multiplication defined by $ (x_1^ax_2^b).(x_1^cx_2^d) = (-1)^{m(ad-bc)} x_1^{a+c} x_2^{b+d} $. For $ u,v $ natural numbers not both zero, denote by $ T_{u,v} $ the automorphism of the above algebra defined by $$ T_{u,v}(x_1) = x_1(1-x_1^ux_2^v)^{-mv} $$ $$ T_{u,v}(x_2) = x_2(1-x_1^ux_2^v)^{mu} $$ The integrality conjecture is the following statement: There are unique integers $ n_{u,v} $ such that $$ T_{1,0}.T_{0,1} = \prod_{\frac{u}{v} \uparrow} T_{u,v}^{n_{u,v}} $$ where the product is written in increasing order of the fractions from left to right. The uniquely determined integers $ n_{u,v} $ are called numerical Donaldson-Thomas invariants.

Atleast for me, it is absolutely out of the blue to see it for the first time, so perhaps someone could explain to me where this comes from? But for concrete questions, in what sense are these numbers 'invariants' in a geometric setting? And what is the connection with Donaldson-Thomas invariants? I think I understand them decently enough and as far as I can see, I see no counting of sheaves (or curves) on a threefold here, or any geometry at all. I'm missing something, hence this question and I'd be grateful for an answer.