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Quantum Invariants

RRP $372.99

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern-Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants.


Sheaves On Graphs, Their Homological Invariants, And A Proof Of The Hanna Neumann Conjecture

RRP $245.99

In this paper the author establishes some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. He then uses these ideas to prove the Hanna Neumann Conjecture of the 1950s; in fact, he proves a strengthened form of the conjecture.


A Theory Of Generalized Donaldson-thomas Invariants

RRP $281.99

This book studies generalized Donaldson-Thomas invariants $bar{DT}{}^alpha(tau)$. They are rational numbers which 'count' both $tau$-stable and $tau$-semistable coherent sheaves with Chern character $alpha$ on $X$; strictly $tau$-semistable sheaves must be counted with complicated rational weights. The $bar{DT}{}^alpha(tau)$ are defined for all classes $alpha$, and are equal to $DT^alpha(tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $tau$. To prove all this, the authors study the local structure of the moduli stack $mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $mathfrak M$ may be written locally as $mathrm{Crit}(f)$ for $f:Uto{mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $nu_mathfrak M$. They compute the invariants $bar{DT}{}^alpha(tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $mathrm{mod}$-$mathbb{C}Qbackslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.



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