Lie brackets on the homology of moduli spaces, and wall-crossing formulae

Let K be a field, and M be the “projective linear" moduli stack of objects in a suitable K-linear abelian category A (such as the coherent sheaves coh(X) on a smooth projective K-scheme X) or triangulated category T (such as the derived category D^bcoh(X)). I will explain how to define a Lie bracket [ , ] on the homology H_*(M) (with a nonstandard grading), making H_*(M) into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra. There is also a differential-geometric version of this: if X is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, G2 manifold, Spin(7) manifold) then we can define Lie brackets both on the homology of the moduli spaces of all U(n) or SU(n) connections on X for all n, and on the homology of the moduli spaces of instanton U(n) or SU(n) connections on X for all n. All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition. Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with b^2_+=1), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for CY 3-folds, Fano 3-folds, and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear" moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs. I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra (H_*(M), [ , ]).

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