Ringel-Hall algebras of Calabi-Yau 3-folds and Donaldson-Thomas theory

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Calabi-Yau 3-folds are a special kind of 6-dimensional curved space, with a lot of geometrical structure. They are of great interest to mathematicians working in Algebraic and Differential Geometry, and to physicists working in String Theory. The greatest problem in fundamental physics is to find a single theory which successfully combines Einstein's General Relativity -- the physics of very large things, such as galaxies -- and Quantum Theory -- the physics of very small things, such as atoms. String Theory is the leading candidate for doing this. It predicts that the dimension of space-time is not 4 (3 space plus one time), but 10. The extra 6 dimensions are rolled up in a Calabi-Yau 3-fold, with very small radius. So according to String Theory, Calabi-Yau 3-folds describe the vacuum of space itself. Using physical reasoning, String Theorists made extraordinary mathematical predictions about Calabi-Yau 3-folds, known as Mirror Symmetry , which have been verified in many cases, and cause much excitement among mathematicians. Mirror Symmetry says that two quite different Calabi-Yau 3-folds X, X* can have identical Quantum Theories (so far, this is not a well-defined idea), and when this happens, we can set up a correspondence between aspects of the geometry of X and X* which affect their Quantum Theories. Often these correspondences relate objects which seem quite different -- a non-mathematical analogue would be to conjecture a one-to-one correspondence between giraffes in Kenya and bananas in Zambia. One chapter of the Mirror Symmetry story which is still work in progress relates two kinds of invariants on X and X*: the Donaldson-Thomas invariants of X, which are numbers counting algebraic objects called coherent sheaves on X, should be equal to other invariants counting special Lagrangian 3-folds on X*. Special Lagrangian 3-folds are non-algebraic objects, superficially as different from coherent sheaves as giraffes are from bananas. When mathematicians talk about invariants they mean a number, such as 42, computed by counting some kind of geometric object, which has the important property that you can make big changes to the underlying geometry, but for mysterious reasons, the number remains the same. This invariance property makes mathematicians very excited (perhaps we should get out more?) as it suggests there is some underlying mathematical reality which is independent of these big changes, which we don't yet understand, and we want to know what it is. Donaldson-Thomas invariants have this kind of invariance property. Funded by another EPSRC grant, the Principal Investigator has recently proved that if we deform a different part of the geometry of the Calabi-Yau 3-fold, Donaldson-Thomas invariants are not fixed, but change by a rigid wall-crossing formula . That is, when we cross a wall in the space of Calabi-Yau 3-folds, the Donaldson-Thomas invariants on one side of the wall can be written as sums of products of Donaldson-Thomas invariants on the other side. The goal of this project is to prove some conjectures which will first help to explain this wall-crossing formula, and secondly allow us to generalize Donaldson-Thomas invariants to a larger class of new invariants containing much more information, which will also satisfy a wall-crossing formula of a similar shape. It turns out that a very nice way of understanding multiplicative properties of invariants, such as Donaldson-Thomas invariants, is to encode them in an algebra morphism from a very large universal algebra , which is far too big to understand or compute, to a much smaller, explicit algebra, where the invariants take their values. Previous work by the Principal Investigator constructed a Lie algebra morphism from a subspace of the universal algebra. We want to extend this to an algebra morphism on the full universal algebra, and generalize it to morphisms to some larger explicit algebras.