[PENDING] 2 Moduli spaces of quiver representations

We also describe constructions of submanifolds (known as branes [35]) in these hyperk¨ahler module spaces with particularly rich holomorphic and complex geometries. In Section 4, we introduce hyperk¨ahler module spaces and survey some constructions of interesting submanifolds known as branes.

Quiver representations over a field

In Section 5, we give the proof of Crawley-Boevey and Van den Bergh relating the counting of absolutely indecomposable quiver representations to the Betti numbers of connected hyperk¨ahler module spaces, and then sketch how Schiffmann extends this to bundles . One can also compute Ext groups of quiver representations by obtaining projective resolutions of the associated modulo k(Q).

GIT construction of moduli spaces

The GLd-invariant sections of positive powers of Lθ are also used to determine a GIT notion of stability with respect to χθ (see [37, Definition 2.1] for k = k, where we note that the notion of stability is modified to take height for the presence of the global stabilizer Using the Hilbert-Mumford criterion to relate GIT semistability of geometric points with stability for 1-parameter subgroupsλ:G→GLd, King proves that the GIT concept of (semi)stability can translates into a notion of (semi)stability for d-dimensional representations of Q.

Symplectic construction of complex quiver varieties

In particular, there is a moment map µR: Repd(Q) → u∗d for the action of Ud on Repd(Q) given by. Consequently, we can consider the derivative dχθ|Ud: ud → u(1) ∼= 2πiR as a converging fixed point of u∗d (often we also denote it byθ or χθ); this joint fixed point can be used to shift the moment map.

Construction of moduli spaces of vector bundles

The unitary gauge group G acts on A of unitary connections, and we can relate this to the action of the complex gauge group GC on C using the following isomorphism. The sign appears here because of our sign conventions for the infinitesimal lift property of the moment map. The symplectic reduction of a G-action to A at the central value iµ(E)IdE ∈ LieG is a space of modules.

If the GIT semistability and stability of the Gd action on µ−1(η) with respect to χ agree, then the variety µ−1(η)//χGLd is a smooth algebraic symplectic variety with algebraic symplectic form induced by the Liouville form onT∗Repd (Q) by an algebraic version of the Marsden-Weinstein theorem (see [17]).

Hyperk¨ ahler quiver varieties

More generally, we can perform a hyperkäahler reduction of the cotangent bundle of a complex vector space M = Cn. Using the identification C×C∼=H given by (m, α)7→ x−jα in each coordinate we obtain an identificationT∗M ∼=M×M ∼=Hnwhich we can use to equip T∗M with a hyperk¨ Ahler structure. More precisely, we obtain complex structures I, J and K corresponding to the appropriate multiplication by i, j and k on Hn, and the hyperkähler metric is the real part of the quaternionic inner product.

By the Kempf-Ness theorem, this hyperk¨ahler reduction is homeomorphic to the GIT quotient of G acting on µ−1(η) with respect to the character of G obtained from χ by exponentiation and complexization; So.

Moduli spaces of Higgs bundles

We recall that the theoretical construction of the moduli space M = MssX(n, d) of semistable vector bundles is the space of S-equivalence classes of complex gauge orbits in the space of semistable holomorphic structures Css. In fact, T∗C is by nature an infinite-dimensional planar hyperkähler manifold, as via the Atiyah–. The zero-level set of the hyperkähler moment map is the set of solutions of Hitchin's self-duality equations [24].

We recall that the box moduli space Mθ-ssd (Q) has a hyperk¨ahler analogue given by the moduli space Mθ-ssd Q,R0.

Branes

For both box representations and vector bundles, there is a surprising relation between the counts of absolutely indecomposable objects over finite fields and the Betti numbers of the (complex) hyperk¨ahler moduli spaces described above. The zero level set of the moment map defines relations R0 on the doubled box Q such that µ−1(0) = Repd Q,R0. Let us write the hyperk¨ahler moment map for the action of the maximally compact subgroup Ud

For any converging fixed point θ ∈u∗d we obtain a local continuous trivialization of the hyperk¨ahler moment map.

Purity of the special fibre X 0 via the scaling action

A key feature of semi-projective Gm-actions is that they give rise to a Bia lynicki-Birula decomposition [7] of Z, which gives a description of the cohomology (and other invariants such as Chow groups and motif) of Z in terms of this fixed site of it. Since the fixed locus is smooth and projective, we will conclude in Section 5.3 that Z is (cohomologically) pure. Since p is projective and Gm-equivariant, the fixed locus X0Gm =p−1(x0) is projective and the flow under the Gm-action ast→0 exists for all points in X0.

In particular, the cohomology of X0 can be described in terms of the cohomology of the smooth projective variety p−1(x0).

Purity and point counting over finite fields

The fixed points of the relative Frobenius on Z are exactly the set of Fq points in Z and similarly the fixed points of FrnZ are Z(Fqn). The last part of the Weil conjectures was Deligne's proof of the Riemann hypothesis: for a smooth and projective Fq manifold Zal eigenvalues of FrZ onHci(Z,Ql) have absolute valueqi/2 (for any choice of embedding Ql ,→C) . We can now provide a standard proof of the purity of a smooth quasi-projective manifold with a semi-projective Gm action using the Bia lynicki-Birula decomposition [7].

In particular, this will provide a proof of the purity of the Fq-manifold X0 mentioned at the end of Section 5.2.

Point count for the general fibre and absolutely indecomposable representations

Let Z be a smooth variety defined over Fq which is pure and has count of polynomial points over Fqr; which is|Z(Fqr)|=P(qr)for a polynomialP(t)∈Z[t]. In fact, we will want to compare the Betti cohomology of a complex variety with the number of points of a reduction of this variety in a finite field using a theorem of Katz, which appears as an appendix in [22]. A final technical tool needed to relate the number of points of X to absolutely indecomposable representations of finite Qover fields with large characteristic is Burnside's formula for the number of orbits under a finite group action.

It remains for us to describe the endomorphism ring of an absolutely indecomposable representation.

Kac’s theorem on absolutely indecomposable quiver representations

For the polynomial behavior of AQ,d(q), it suffices to prove that the number IQ,d(q) of isomorphism classes of insoluble-dimensional Fq representations of Q is given by a polynomial inq using standard reductions that involving Galois descent. Kac computes MQ,d(q) using Burnside's theorem, where one must sum over all conjugacy classes of GLd for all by enumerating all possible Jordan normal forms using polynomials (giving the partition field) and partitions (giving the size of Jordan -blocks). Kac proves the independence of the orientation of Qusing reflection functions and the fact that insoluble representations correspond to trajectories in Repd(Q) with a unipotent stabilizer group.

Compute the Kac polynomial AQ,d(q) for each of the following quivers and dimension vectors: . a) The ground plane with dimension vector n∈N. b).

Specialisation and relating the cohomology of the special fibre and general fibre

Then there is a nonempty open subscheme U ⊂S over which the formation of the GIT semistable set and GIT quotient commute with base change; that is, for all points s: Speck→U, we have. Using the (topological) triviality of the family X → A1 over C and the comparison theorem together with Deligne's change of basis result for direct images, we derive that for p0 and `6=p there are isomorphisms. Let Fq be a finite field with sufficiently large characteristic p such that the construction of the GIT quotient X commutes with base change and the family X → A1 is even.

Now consider the family X→A1 over SpecZN for sufficiently large N indivisible byp, then by changing the basis we can get the Fp-variety X0 × FqFp and the complex manifold X0,C and these basis changes commute to form the GIT quotient.

A brief survey of Schiffmann’s results for bundles

Putting all of the above together, we get the Crawley-Boevey and Van den Bergh proof. For bundles on curves, we would like to understand the behavior of this count as X varies in the moduli space of curves over Fq, and give a representation-theoretic interpretation of the associated polynomial. A description of the behavior of this counter as X varies in the space of moduli of curves of genus g was achieved by Schiffmann in [58]: he proves that there exists a polynomial (depending on (n, d) and the genus of the curve) in the Weil numbers of the curve over a finite field which gives the count An,d(X) for any curve X over a finite field evaluating at the Weil numbers X;.

Therefore, a slightly perturbed model of the Higgs group modulus space is needed to compare with the indecomposable vector bundles.

Representation theoretic interpretations of the Kac polynomials

The moduli space of stable Higgs bundles over a finite field is already known to be cohomologically pure (for example, see [23, Section 1.3]), so (5.5) also gives an explicit formula for the `-adic Poincar´e polynomial of the moduli space of stable Higgs bundles on X. C(n, d) is independent of the choice of the smooth projective curve XC of genus g (since they are all diffeomorphic to the sign variety genusg for GLn), this allows Schiffmann to relate Ag,n to the Poincar polynomial HssX. In the case of curves, the representation-theoretic interpretation of Ag,nin involves a spherical Hall algebra constructed analogously to the above, but replacing the characteristic functions of simple vibrational representations with constant functions on a stack of modules of rank n of degree d of coherent bundles with n ≤ 1.

There is also a corresponding 2d-cohomological Hall algebra in the cases of curves associated with the stack of Higgs sheaves, which can be seen as the cotangent stack of the modulo stack of coherent sheaves [56].