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Title:
New Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 4-folds
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Abstract:
Donaldson and Thomas (1998) proposed that it should be possible to define interesting invariants "counting" moduli spaces of "Spin(7) instantons" on a Calabi-Yau 4-fold Y (or a compact Spin(7)-manifold), which can be viewed as a complexified version of Donaldson invariants of real 4-manifolds, just as DT invariants of Calabi-Yau 3-folds are a complexified version of Casson invariants of real 3-manifolds.
Carrying out this programme using gauge theory is very difficult and currently unsolved, because one needs to somehow compactify the moduli spaces of Spin(7) instantons by including singular solutions, and have a nice enough geometric structure on this compactification to define a virtual cycle.
We outline an alternative approach using (derived) algebraic geometry in which these difficulties do not occur. If a complex vector bundle E --> Y with c_1(E) . [\omega]^3=0 admits a holomorphic vector bundle structure, then just as topological spaces, the moduli spaces of Spin(7) instantons and of Hermitian-Yang-Mills connections coincide, though as non-reduced schemes (in a C-infinity sense) the HYM moduli space is a proper subscheme, as the HYM conditions impose more equations. In fact, the HYM equations are overdetermined elliptic in 4 complex dimensions, so one does not expect to be able to define a virtual cycle for the HYM moduli space.
In the Derived Algebraic Geometry of Toen-Vezzosi, suppose we are given a derived moduli scheme M of coherent sheaves (or complexes of coherent sheaves) on Y. Then by work of Pantev-Toen-Vaquie-Vessozi, M has a "-2-shifted symplectic structure". Thus the shifted symplectic Darboux Theorem of Bussi-Brav-Joyce gives Zariski local models for M.
We will explain how to use these local models to give the underlying complex analytic topological space M* of M the structure of a "derived smooth manifold" M*, whose real virtual dimension is half the expected real virtual dimension of M (!). Heuristically speaking, this is because there is a "Lagrangian fibration" M --> M*, and the dimension of the base of a Lagrangian fibration is half the dimension of the symplectic manifold.
Now compact, oriented derived manifolds admit virtual cycles, in homology or bordism. This means that if we are given a proper derived moduli scheme M of coherent sheaves on Y of complex virtual dimension d, plus an "orientation" (similar to orientations for instanton moduli spaces in Donaldson theory), we can define a virtual cycle [M*]_{virt} in the homology H_d(M*;Z). This is what we need to define Donaldson-Thomas style invariants "counting" (semi)stable coherent sheaves on a Calabi-Yau 4-fold Y, which will be unchanged under deformations of the complex structure of Y.
Here is how this deals with the issue of compactification of moduli spaces of gauge-theoretic Spin(7) instantons. The moduli spaces of (say) stable algebraic vector bundles, and of HYM connections, agree by Hitchin-Kobayashi; and the moduli spaces of HYM connections and Spin(7) instantons agree as above. However, algebraic geometry often gives us compactifications of moduli spaces of stable algebraic vector bundles, as moduli spaces of stable torsion-free coherent sheaves, essentially for free. Our algebraic geometry approach does not care about the difference between moduli of vector bundles and moduli of coherent sheaves, but in gauge theory the difference is crucial.
This is joint work with Dennis Borisov, and relies on earlier joint work with Bussi and Brav. It is also related to papers by Cao and Leung.
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