Derived Representation Schemes and Nakajima Quiver Varieties

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on $S^4$. On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.


Introduction
Nakajima quiver varieties are certain Poisson varieties constructed from linear representations of a quiver. They were firstly introduced by Nakajima ( [28], [29]) as a geometric tool to study representations of Kac-Moody algebras. They are also interesting from a purely geometric point of view, being a large class of examples of algebraic symplectic manifolds, many of which have been objects of study on their own (for example flag manifolds, framed moduli spaces of torsion free sheaves on P 2 , or a Lie algebra version of the character variety of a Riemann surfacesee [3]). More recent studies have also supported the idea that symplectic resolutions, and in particular hyperkÃd'hler reductions such as Nakajima quiver varieties, provide a bridge between enumerative geometry, representation theory and integrable systems ( [1], [33], [35], [36], [37]).
Quiver varieties are varieties of representations of a quiver: one fixes a vector space on each vertex of the quiver and then consider the linear space of representations obtained by associating to each arrow of the quiver a linear map. Kronheimer and Nakajima ([22]) have first introduced a framed version, which amounts to doubling the set of vertices and drawing a new arrow from each new vertex to its corresponding old one. One of the reasons for considering framed representations is that they appear naturally in the ADHM construction ( [2]) of solutions of self-dual or antiself-dual Yang-Mills equations on S 4 . They are also interesting from the point of view of representation theory of Lie algebras because dimension vectors of the framed vertices appear as highest weights of the representations ([30]). The framing is equivalent to a simpler operation of adding just one vertex with dimension vector 1, together with as many arrows to each vertex as the framing dimension (as pointed out in [9]), however in this paper we consider the framed version of Nakajima quiver varieties.
The framed quiver is then doubled, which means that each arrow gets doubled by an arrow that goes in the opposite direction: the linear space of representations becomes now a linear cotangent bundle M(Q, v, w) := T * L(Q fr , v, w) (where v, w are dimension vectors for, respectively, the original and framing vertices). The gauge group is a general linear group on the original vertices G = G v and there is a moment map 1.1. Outline and results. In this paper we link these varieties with some (derived) representation schemes. The idea of considering representation schemes is certainly not new, in fact it is motivated by the very first algebraic origin of these varieties (see, for example, representation schemes of preprojective algebras in [10] and [13]). However the derived version of representation schemes introduces some new invariants in a natural way. The theory of representation schemes is recalled in detail in § 2.1. To a (unital, associative) algebra A ∈ Alg k one associates Rep V (A), the scheme of finite dimensional representations into a fixed vector space V. There is a relative version in which the algebra A comes with a fixed structure ι : S → A of algebra over another algebra S with a fixed representation ρ : S → End(V) and it is natural to define Rep V (A) as the scheme of only those finite dimensional representations which are compatible with ρ.
General definitions and results on representation schemes work well over any field k of characteristic zero, but it is necessary to specialise to k = C in order to relate them to (Nakajima) quiver varieties, which are algebraic varieties over the complex numbers. The (complex) linear space of representations of a quiver Q is a representation scheme of the form Rep V (A), where A = CQ is the path algebra of the quiver. This fact is a consequence of one of the basic results in the theory of representations of quivers: There is an equivalence of categories between the category of C-linear representations of a quiver Q and the category of left CQ-modules.
The construction can be easily adapted to include the framing and the doubling of the quiver, and also the operation of taking the fiber of zero through the moment map. In other words it is possible to write the scheme µ −1 (0) as a representation scheme for the path algebra of the framed, doubled quiver, modulo the ideal I µ defined by the moment map: where C v = ⊕ a C v a is the direct sum of the vector spaces placed on the original vertices of the quiver and C w = ⊕ a C w a is the one on the framing. We denote this representation scheme also simply by Rep v,w (A). The gauge group by which we take the quotient is G = G v := a GL v a (C) ⊂ G v × G w . This group also arises naturally in the context of representation functors. It is possible to construct an invariant subfunctor by the group G and by doing so we obtain the affine Nakajima variety as the partial character variety . Now that we have such a model for this singular scheme we can try to resolve it using the machinery of model categories and in particular the theory of derived representation schemes ( [4], [5]): we consider the derived scheme where A cof ∼ A is a cofibrant replacement in the category of differential graded algebras. It is (the homotopy class of) a differential graded scheme of the form X = (X 0 , O X,• ), where X 0 ∼ = M(Q, v, w) is the vector space of linear representations of the framed, doubled quiver, and O X,• is a sheaf of dg-algebras whose zero homology gives: We exhibit an explicit (minimal) resolution A cof ∼ A for which this derived representation scheme is a well-known object when it comes to studying resolutions of a singular locus: Theorem (3.5.2 in § 3.5). There is a cofibrant resolution A cof ∼ A ∈ DGA S which gives a model for the derived representation scheme as the (spectrum of the) Koszul complex on the moment map: A somewhat natural question is whether or not there is any relationship between Nakajima resolutions (1.1) and these derived schemes, and if it is possible to obtain informations about one of the two from the other: Relationship?
A first answer is a close relationship (an equivalence) between the condition of flatness for the moment map (which assures that M χ → M 0 is indeed a resolution, for well-behaved characters χ), and the derived representation schemes to have vanishing higher homologies: Theorem We remark that in general it might not be easy to compute homologies of derived representation schemes, and even just to predict until which degree the homology is nontrivial. Nevertheless, in this special situation it is possible to give a sufficient and necessary condition for the vanishing of higher homologies based on a geometric property (flatness) of the moment map. The importance of Theorem 1.1 is that there is a combinatorial criterium on the dimension vectors v, w (proved by Crawley-Boevey, [9], based on the canonical decomposition of Kac, [20]) for the flatness of the moment map for representations of quivers.
A second answer to the question in (1.3) comes when we compare some invariants associated with the derived representation schemes with others associated with the varieties M χ . A natural choice is to consider tautological sheaves on the GIT quotient M χ constructed with the usual machinery developed by Kirwan ( § 4.2). Because of reductiveness of the gauge group G we restrict to consider only tautological sheaves of the form V λ induced from irreducible representations V λ of G. The push-forward of these sheaves in the K-theory of the affine Nakajima variety through the map (1.1) computes their (T -)equivariant Euler characteristics: where T = T w × T h is the product of the standard maximal torus in the other general linear group on the framing vertices T w ⊂ G w and a 2-dimensional torus T h rescaling the symplectic form and the cotangent direction.
On the other hand also the representation homology H • (A, v, w) (the homology of the derived representation scheme) is naturally a G-module and therefore decomposes into the direct sum of its isotypical components: The isotypical components Hom G (V λ , H • (A, v, w)) are modules over the G-invariant zeroth homology H 0 (A, v, w) G = O(µ −1 (0)) G and therefore their Euler characteristics define invariants in It is tempting to compare the invariants defined in (1.4) and (1.6), and the main results of this paper go in this direction. First of all, when we consider the trivial representation V λ = C, we prove that if the moment map is flat, then the two invariants are indeed equal: Let v, w be dimension vectors for which the moment map is flat and let χ such that M χ (Q, v, w) is a smooth variety (and therefore a resolution of M 0 (Q, v, w)). Then we have When we consider the Hilbert-PoincarÃl' series of (1.7) we obtain an integral formula for the T -character of the ring of functions on the GIT quotient M χ , that has the following form where r i = r i (x) and s j = s j (x, t) are characters for T v and T v × T , respectively, ∆(x) is the Weyl factor for G v and the integration is over the compact real form of T v (see § 4.3 for a more detailed explanation).
Integral formulas of similar flavours have already appeared under different names, both in the mathematical literature (Jeffrey-Kirwan integral/residue formula for GIT quotients - [19]) and in the physical literature (integral formula for Nekrasov partition function - [31],[32] -proven, for example, in Appendix A in [14]). We could say that this is not a coincidence, in fact recognising the right-hand side of (1.8) in the known example of the Jordan quiver (Nekrasov partition function) as the Euler characteristic of the representation homology was one of the motivations of this project.
For what concerns other tautological sheaves V λ an equality of the same flavour of (1.7) is true only for large enough λ, where the definition of largeness depends on the quiver, the dimension vectors v, w and, perhaps more importantly, also on the GIT parameter χ (see § 4.4): Let v, w be dimension vectors for which the moment map is flat, and χ a character for which M χ (Q, v, w) is smooth. For λ large enough (Definition 4.4.1) we have Once again by taking the Hilbert-PoincarÃl' series of (1.9) we obtain a second integral formula for tautological sheaves on the GIT quotient: is a product of Schur polynomials.
1.2. Layout of the paper. In § 2 we introduce the general theory of (derived) representation schemes of an algebra. First we recall the theory of representation schemes with some examples, in particular the linear space of representations of a quiver as a representation scheme for its path algebra. Then we recall the derived version introduced by [5] and [4]. We introduce a more general way to take invariant subfunctors and an equivariant version of derived representation schemes for an action of an algebraic torus which is useful for our purposes. We decompose the representation homology in isotypical components and define new invariants in the K-theory of the classical character scheme. In § 3 we recall the construction of Nakajima quiver varieties and we show how to view the affine Nakajima variety M 0 as a partial character scheme (a quotient of a representation scheme) for the algebra A := CQ fr /I µ . We construct the derived scheme associated to it and we use the invariants defined in § 2 to decompose the representation homology into classes in the K-theory of M 0 . In § 3.4 we construct an explicit cofibrant resolution A cof ∼ A that gives us a concrete model for the derived representation scheme as the (spectrum of the) Koszul complex on the moment map. Therefore we recall some classical properties of the Koszul complex and commutative complete intersections. In § 4 we explain the main results of this paper. First we observe that, using the model found in § 3.4, the derived representation scheme has vanishing higher homologies if and only if the moment map is flat, which is a combinatorical condition on the dimension vectors of the quiver ( [9]). We recall the definition of tautological sheaves on GIT quotients by the Kirwan map and prove results that compare them with the isotypical components of the representation homology ((1.7) and (1.9)). In particular we obtain some interesting integral formulas ((1.8) and (1.10)).
In § 5 we show some concrete examples, such as the quiver A 1 for which Nakajima varieties are cotangent spaces of Grassmannians, the Jordan quiver for which we obtain framed moduli space of torsion free sheaves on P 2 , and the quiver A n−1 with some special dimension vectors for which we obtain the symplectic dual (T * P n−1 )ˇ, and compute some of the integral formulas that we have proved before.
In Appendix A we construct a model structure on equivariant dg-algebras that we need in § 2.5, and in Appendix B we recall the theory of irreducible representations for a product of general linear groups as multipartitions, and set some notation that we need in § 4.4.
Notation. Throughout the paper we denote categories by the standard monospace font: Sets, Grp, Vect k , Alg k , . . . The notation used is often both standard and self-explanatory, and when this is not the case we usually recall it in the main body of the paper.
Acknowledgements. I want to express my gratitude to my advisor Giovanni Felder, who introduced me to this subject a couple of years ago and proved numerous times to be a patient, wise and resourceful guide. I also want to mention other people with whom we shared our ideas and contributed with useful comments, in particular Yuri Berest during his brief stay in Zurich, Gabriele Rembado, Matteo Felder and Xiaomeng Xu.

Derived representation schemes of an algebra
The family of schemes of finite dimensional representations {Rep n (A)} n 1 of an algebra A has been object of study for many years (see for example the early work of Procesi, [38]). With the development of noncommutative geometry, they have been seen in a new light when Kontsevich and Rosenberg ( [21]) proposed the following principle: "Any noncommutative structure of some kind on A should give an analogous commutative structure on all the representation schemes Rep n (A), n 1".
This principle seems to work well for (formally) smooth algebras, for which the representation schemes are smooth, but fails in general. The solution proposed in [5] is to find a smoothening of representation schemes by extending representation schemes to differential graded algebras, and using the general machinery of model categories to derive them. The purpose of this section is to recall in main details the construction of this derived version of representation schemes from [5] and [4], and describe some generalisations that is useful to our purposes.
2.1. Classical representation schemes. Let k be an algebraically closed field of characteristic zero (later we fix k = C). Let A ∈ Alg k be a unital, associative algebra and V ∈ Vect k a finite dimensional vector space. We consider the functor on unital commutative algebras: This functor is (co)-representable, by the commutative algebra A V := V √ A . The two functors V √ − and (−) are, respectively, the matrix reduction functor and the abelianisation functor, which are left adjoints to the followings: is the 2-sided ideal generated by the commutators. By combining the two adjunctions in (2.2) we get an adjunction for the representation functor: so that the commutative algebra A V is uniquely defined by the natural isomorphisms: Definition 2.1.1. The affine scheme associated to A V ∈ CommAlg k is the representation scheme Rep V (A) = Spec(A V ) ∈ Aff k (strictly speaking we identify it with its functor of points as we ) as the algebra of functions on the representation scheme.
We can assume that V = k n and write simply Rep n (A) = Spec(A n ) instead of Rep V (A) = Spec(A V ). Let us show some examples: (0) If A ∈ CommAlg k ⊂ Alg k is a commutative algebra then clearly from (2.4): (1) The free algebra in m generators A = F m = k x 1 , . . . , x m has no relations and therefore Rep n (F m ) is the scheme of m-tuples of n × n matrices: (2) The polynomial algebra A = k[x 1 , . . . , x m ] can be expressed as the free algebra in m generators modulo the ideal generated by all commutators [x i , x j ] and therefore its representation scheme is the closed subscheme of m-tuples of n × n matrices that pairwise commute: (3) The algebra of dual numbers A = k[x]/(x 2 ) gives the scheme of square-zero matrices: 1 , . . . , t ±1 m ] is similar to the example of commuting matrices, except that now the matrices are required to be invertible: (6) More generally writing any finitely generated algebra as a free algebra modulo some relations A = F m / r 1 , . . . , r s , r 1 , . . . , r s ∈ F m = k x 1 , . . . , x m , then its representation scheme is identified with the closed subscheme of m-tuples of n × n matrices defined by the equations r 1 , . . . , r s .
Another fundamental example is that of path algebras of (finite) quivers. These algebras come with an additional structure of algebras over the finite dimensional algebras of their empty paths, which is crucial when considering their representations, therefore we need to consider a relative version of representation schemes. Formally we fix an algebra S ∈ Alg k and we consider the under category S ↓ Alg k (also denoted by Alg S following the notation of [4] and [5]) which is the category of algebras A ∈ Alg k together with a fixed morphism S → A. We also fix a representation ρ : S → End(V).
Notation. Sometimes when we want to remark that A comes with a map from S we denote this object as S\A ∈ Alg S . However, when there is no risk of confusion, we just use A.
With these ingredients it is natural to consider only those representations A → End(V) that agree with ρ on S. In terms of functor of points this corresponds to This functor is also (co)representable, by the commutative algebra A V defined as before except for * k substituted by * S , the coproduct in Alg S . Letting A vary we obtain a relative version of the representation functor (−) V , and a similar adjunction Example 2 (Path algebra of a quiver). Let Q be a finite quiver and A = CQ ∈ Alg C its path algebra over the complex numbers. What follows works well for any field k of characteristic zero but later we are interested only in k = C. We recall that the path algebra is the free vector space on the admissible paths in the quiver, with product given by concatenation of paths. It has a set of orthogonal idempotents {e i } i∈Q 0 ⊂ A: which are the empty paths on the vertices, and their sum is the unit of the algebra: i∈Q 0 e i = 1 ∈ A. We can then consider the subalgebra generated by these idempotents with the natural inclusion ι : S → A. We now fix a dimension vector v ∈ N Q 0 and we consider the linear space of representations of the quiver Q with the complex vector space C v i placed at the vertex i ∈ Q 0 : where s, t : Q 1 → Q 0 are the source and target maps of the quiver. From the algebraic point of view we fix the following representation of S in the vector space C v := ⊕ i C v i : Proposition 2.1.1. The linear space of representations of the quiver Q with fixed dimension vector v is isomorphic to the (relative) representation scheme of its path algebra: Proof. Let us consider the complex vector space with basis given by the set of arrows of the quiver M := Span C {x γ } γ∈Q 1 . It has the structure of an S-bimodule, and its tensor algebra is the path algebra of the quiver: For a dimension vector v ∈ N Q 0 we consider the graded vector space C v = ⊕ i C v i , whose endomorphism algebra End C (C v ) is an S-bimodule via the map (2.8). By the universal property of the tensor algebra, giving a representation T S M → End C (C v ) that agrees with ρ on S, is equivalent to give a S-bimodule map M → End C (C v ): 2.2. Derived representation schemes. As already anticipated in the introduction of this Section, the noncommutative geometry principle of transferring a geometric property on an algebra A (e.g. noncommutative complete intersection, Cohen-Macaulay, etc.) on the corresponding commutative one on Rep V (A) might fail when A is not a (formally) smooth algebra. This seems to be related to the fact that the functor Rep V (−) is not exact. We discuss the following derived version of representation schemes firstly introduced in [5]. The idea is to "resolve" the singularities of the representation schemes by using the tools of homological algebra, in the sense of Quillen's derived functors on model categories.
We enlarge the category of algebras to the one of differential graded algebras DGA k (in our conventions differentials have always degree −1), and as before we consider the under category DGA S := S ↓ DGA k of dg-algebras A with a fixed morphism S → A.
We also fix a differential graded vector space V ∈ DGVect k of finite total dimension, and denote by End(V) ∈ DGA k the differential graded algebra of endomorphisms, with differential Moreover we need to fix a representation of S in V, that is a dga morphism ρ : S → End(V), which makes End(V) an object of DGA S . With these ingredients we can define a differential graded version of the representation functor for A ∈ DGA S as the functor from commutative dg-algebras: Remark 2.2.1. We use the same notation as in the non-graded case because in the particular case of S, A, V being concentrated in degree zero we recover the same functor as before (when restricted to Alg k ⊂ DGA k ).
This functor is also (co)-representable, by the object A V := V √ A constructed in the same way as before, with where * S is the free product over S, the categorical coproduct in DGA S . As before we obtain a pair of adjoint functors These categories have model structures for which this adjunction is a Quillen adjunction, and therefore produces a total right-derived functor R End(V) ⊗ k (−) , but more importantly a leftderived functor L(−) V that we use to define the derived representation scheme. We consider on DGA k and CDGA k the so-called projective model structures for which weak equivalences are quasi-isomorphisms of complexes and fibrations are degree-wise surjective maps (Theorem 4 in [4]). It is useful for later purposes to consider also the categories DGA + k and CDGA + k , which are the categories of non-negatively graded differential graded and commutative differential graded algebras, respectively, and with their projective model structures with the only difference that now fibrations are degree-wise surjective maps in all (strictly) positive degrees. All these categories are fibrant (every object is fibrant), with initial object k and final object 0.
The category DGA S is an example of an under category (category in which objects are objects of the original category coming with a fixed morphism from the object S in this case). As such it comes with a forgetful functor DGA S → DGA k and the model structure on DGA S is the one in which weak-equivalences, fibrations and cofibrations are exactly the maps which are sent to weak-equivalences, fibrations and cofibrations via the forgetful functor. Clearly also the under category DGA S is fibrant, with final object still 0 (viewed as an object of DGA S via the unique map S → 0), and initial object S (viewed as an object of DGA S via the identity map Id S : S → S).
For a model category C, we denote by Ho(C) its homotopy category and by γ : C → Ho(C) the canonical functor.
is called derived representation functor. The homology of the (homotopy class of the) commutative differential graded algebra L(A) V ∈ Ho(CDGA k ) depends only on A ∈ Alg S and V. It is called the representation homology of A with coefficients in V:

Remark 2.2.2.
By its definition, the zero-th homology recovers the classical representation scheme (see Theorem 9 in [4]): As we anticipated before, we are interested in a slightly different version of this story: if we start from a vector space V concentrated in degree 0 and S ∈ Alg k then the previous pair (2.13) restricts to a pair of functors which is still a Quillen pair, and the analogous result of Theorem 2.2.1 holds. We give a second definition of: The derived representation functor is the following functor: . The representation homology of the relative algebra A ∈ Alg S is the homology of L(A) V ∈ Ho(CDGA + k ).
where the functor ι is the obvious inclusion and the functor τ is the one that sends an unbounded differential graded algebra A ∈ DGA S to its truncation: It is straightforward to see that τ preserves fibrations and weak equivalences, and dually the map ι preserves cofibrations and weak equivalences, in particular it sends cofibrant objects to cofibrant objects. Now let A ∈ Alg S and choose a cofibrant replacement Q ∼ A ∈ DGA + S . A priori this map is only surjective in positive degrees, but because A is concentrated in degree 0, we have A = H 0 (A), and the isomorphism in homology H 0 (Q) ∼ = H 0 (A) proves that it is surjective also in degree 0, so still a fibration in DGA S . In other words the cofibrant replacement Q ∼ A is still a cofibrant replacement in DGA S and therefore it can be used to compute the derived representation functor (2. 16), showing that Definition 2.2.1 is equivalent to Definition 2.2.2.

Remark 2.2.4 (The dual language of dg-schemes).
Another reason for considering the category CDGA + k instead of CDGA k is that it is anti-equivalent to the category of differential graded schemes, as introduced by I. Ciocan-Fontanine and M. Kapranov in [8]. We recall their definition of dg-schemes (over k) as a pair X = (X 0 , O X,• ), where X 0 is an ordinary scheme over k and O X,• is a sheaf of non-negatively graded commutative dg-algebras on X 0 such that the degree zero is O X,0 = O X 0 the structure sheaf of the classical scheme X 0 and each O X,i is quasicoherent over O X,0 . A morphism of dg-schemes over k is just a morphism of dg-ringed spaces f : , and this makes DGSch k into a category. A dg-scheme X is called affine if the underlying classical scheme X 0 is affine. The full subcategory of dg-affine schemes DGAff k ⊂ DGSch k is antiequivalent to the category CDGA + k , via the the equivalence of categories: where Γ (−) is the functor taking a dg-affine X into the global sections of the sheaf O X,• (degreewise), and Spec is the dg-spectrum sending a commutative dg-algebra A to the classical scheme X 0 = Spec(A 0 ) together with the quasicoherent sheaves O X,i associated to the modules A i via the correspondence QCoh X 0 ∼ = Mod A 0 . These names are motivated by the fact that the previous equivalence restricts to the classical equivalence of categories This definition of dg-affine schemes coincides with ToÃńn-Vezzosi's definition of derived schemes d Aff op k = sCommAlg k as simplicial commutative algebras ( [41]) because over a field k of characteristic zero they are equivalent to commutative dg-algebras.
The equivalence of categories (2.22) can be trivially used to transfer the projective model structure on commutative dg-algebras to the category of dg-affine schemes. Obviously the pair (Γ (−), Spec) becomes a Quillen equivalence, i.e. an equivalence on the homotopy categories: Moreover because every object in CDGA + k is fibrant, the derived spectrum RSpec actually coincides with the underived Spec on the objects. Definition 2.2.3. The derived representation scheme of the relative algebra A ∈ Alg S in a vector space V is the object DRep V (A) ∈ Ho(DGAff k ) obtained applying to A the following composition of functors: This definition differs from the one given in [5] and [4] only from the last composition with the derived spectrum functor. The reason we have to do so is to be consistent with the notation for the classical representation scheme Rep V (A) ∈ Aff k .

Remark 2.2.5. Because every object in
to remember that we are considering the homotopy class, but we make an abuse of notation by dropping γ.

Examples 3.
In the following examples we describe explicit cofibrant resolutions for some of the algebras in the Examples 1 and give a model for their derived representation schemes with value in a vector space V concentrated in degree 0 (therefore we still use the notation (1) The free algebra in m generators A = F m is already a cofibrant object in DGA + k because it is free, therefore The commutative algebra in two variables A = k[x, y] is not cofibrant because of the relation [x, y] = 0. It turns out that it suffices to add one variable ϑ in homological degree 1 that kills this relation (dϑ = [x, y]) to obtain a cofibrant replacement: and therefore the derived representation scheme is the nothing else but the (spectrum of the) Koszul complex for the scheme of n × n commuting matrices: (3) Calabi-Yau algebras of dimension 3 (see [16, § 1.3]). Consider the free algebra F m and the commutator quotient space of cyclic words: . . , m which we can use, together with a potential Φ ∈ (F m ) cyc , to define the algebra which is the quotient of the free algebra F m by the two-sided ideal generated by the partial derivatives of the potential Φ. For example when m = 3, F 3 = k x, y, z and observe that the partial derivatives for the potential Φ = xyz − yxz give the commutators, therefore A = k[x, y, z] is the polynomial ring in 3 variables. For an algebra defined by a potential as above in (2.27) we define the following dg-algebra: Ginzburg explains in [16] how Calabi-Yau algebras of dimension 3 are all of the form (2.27) and they are exactly those for which a suitable completion of D(F, Φ) is a cofibrant resolution. This is in particular true for the example of polynomials in 3 variables (see example 6.3.2. in [4]), for which no completion is needed and: , where the variables ξ, ϑ, λ are the ones we called ϑ 1 , ϑ 2 , ϑ 3 in (2.28).

G-invariants and isotypical components.
On the (derived) representation scheme there is a natural action of the general linear group GL(V) by which one can consider the associated character scheme of invariants. Later we consider only invariants by a subgroup G ⊂ GL(V), therefore we propose the following theory of partial invariant subfunctors by G that generalises the theory introduced in [5, § 2.3.5] and in [4, § 3.4] in the absolute case S = k. However we point out that the results of this section are strongly inspired by [5] and [4], which already contain most of the material needed.
Suppose that both V and S are concentrated in degree 0, ρ : S → End(V) is a fixed representation and consider the subgroup of ρ-preserving transformations. Observe that in the absolute case S = k then G S = GL(V). Now consider any reductive subgroup G ⊂ G S , whose right action on End(V) extends to the functor: (for this we need that G consists of transformations which all preserve ρ). And consequently we obtain a left action on (−) V : DGA S → CDGA k and we can consider the invariant subfunctor As explained in [5], unlike (−) V , the functor (−) G V does not seem to have a right adjoint, so we cannot prove that it has a left derived functor from Quillen's adjunction theorem. Nevertheless we can prove that such a left derived functor exists: To prove this theorem it is convenient to recall a few notions/results. Let Ω = k[t] ⊕ k[t]dt be the algebraic de Rham complex of the affine line A 1 k (in our conventions differentials have degree −1 and therefore dt has the wrong degree −1). We define a polynomial homotopy between f, g : A → B ∈ DGA S as a morphism h : A → B ⊗ Ω ∈ DGA S , such that h(0) = f and h(1) = g, where for each a ∈ k, h(a) is the following composite map: The reason why polynomial homotopy is equivalent to the homotopy equivalence relation in DGA S is explained in Proposition B.2. in [5].
It is important to observe that, despite the misleading notation, the map h V in part (1) is not the map obtained applying the functor (−) V to the map h. The latter would in fact be a map The same thing applies for the map h G V in part (2), which is not the map obtained applying the functor (−) G V to the map h. Proof. We omit the proof because it is analogous to the proof of Lemma 2.5 in [5].
Proof of Theorem 2.3.1.
(a) By Brown's lemma (Lemma A.2 in [5]) it is sufficient to prove that the functor (−) G V maps acyclic cofibrations between cofibrant objects to weak equivalences.
and this concludes the proof.
An analogous result holds also for the functor restricted on non-negatively graded objects, and it can be actually obtained as a corollary of Theorem 2.3.1: Proof. Using Brown's lemma we just need to prove that (−) G V sends a trivial cofibrations between cofibrant objects A ∼ → B to weak equivalences. We consider the following commutative diagram: is still a trivial cofibration between cofibrant objects in DGA S , according to Remark 2.2.3. Now we can use the proof of Theorem 2.3.1 to conclude that the functor (−) G V : DGA S → CDGA k sends this map to a weak equivalence: Finally from the very construction of ι we have that this map is a weak equivalence if and only This concludes the proof of (a), while (b) follows from (a) as in Theorem 2.3.1.
Now we derive also the other isotypical components of the representation functor. Let us fix any irreducible, finite-dimensional representation U λ of the reductive group G. We consider the following functor: Then we can prove the following analogue to Theorem 2.3.1: (a) The functor (2.33) has a total left derived functor To prove it we need the following analogue of Lemma 2.3.1: ⊗ Ω ∈ DGA S be a polynomial homotopy between f, g : A → B. Then: Proof. It is essentially a corollary of Lemma 2.3.1. In fact, we can define h λ,V to be where h V is the map from part (1) of Lemma 2.3.1. The map h V was G-equivariant, and therefore also h λ, The analogous results in the non-negative case also hold: S there is a natural isomorphism: Proof. The proof follows from Theorem 2.3.2 in the same way as the proof of Corollary 2.3.1 followed from Theorem 2.3.1.

K-theoretic classes.
We use the classical G-invariant subfunctor (−) G V : Alg S → CommAlg k to define Definition 2.4.1. The partial character scheme of an algebra A ∈ Alg S in a vector space V, relative to a subgroup G ⊂ G S , is the affine quotient of the representation scheme: The name is motivated by the fact that in the absolute case S = k and G = GL(V) the full group, we would obtain the classical scheme of characters Rep The derived version is: The derived partial character scheme of A ∈ Alg S in a vector space V, relative to a subgroup G ⊂ G S , is the affine quotient of the derived representation scheme: . Let us recall that the obvious inclusion Sch k → DGSch k has for right adjoint the truncation functor π 0 : DGSch k → Sch k that associates to a dg-scheme sheaves on X 0 , and also on the closed subscheme π 0 (X) ⊂ X 0 . We can put these data together in a dg-affine scheme: which in the affine case X = Spec(A) is nothing but Spec(H • (A)).
Definition 2.4.3 (Definition 2.2.6. in [8]). A dg-scheme X is of finite type if X 0 is a scheme of finite type and each O X,i is a coherent sheaf on X 0 .
Let now come to the case of our interest, a dg-affine scheme of finite type X = Spec(B), for which the sheaves H i (O X,• ) are coherent both over X 0 and over π 0 (X) = Spec(H 0 (B)), therefore they define a class in the algebraic K-theory 1 (2.39) [H i (O X,• )] ∈ K(π 0 (X)) .
We first consider the derived scheme X = DRep V (A). Let us assume that A is an algebra such that, for each vector space V, the following two conditions are satisfied: (1) The derived representation scheme X = DRep V (A) is of finite type.
(2) The structure sheaf O X,• of the derived representation scheme is bounded, in the sense that O X,i = 0 for i 0. This is true for all algebras that we consider in this article, as we show in § 3.4 and § 3.5. The truncated scheme obtained from the derived representation scheme is the classical representation scheme, as explained in Remark 2.2.2: By condition (1) each homology defines a coherent sheaf on π 0 (X) = Rep V (A) and therefore a class . By condition (2) there is only a finite number of them nonzero, therefore in particular the following definition makes sense, because the sum in (2.40) is bounded: The virtual fundamental class -or Euler characteristic of the derived representation scheme X = DRep V (A) is the following invariant in the K-theory of the classical representation scheme: This virtual fundamental class carries an action of the group G, which is reductive, and therefore it decomposes into a direct sum of its irreducible components. To formalise this we first consider the quotient by derived partial character scheme and therefore they define coherent sheaves on Rep G V (A).

Definition 2.4.5.
The Euler characteristic of the U λ -irreducible component of the derived partial character scheme is We observe that the irreducible component corresponding to the trivial representation U 0 = k is the virtual fundamental class of the derived partial character scheme, which we denote by However, often the algebra A itself comes with an action of some algebraic torus T which helps when calculating its invariants (for example the corresponding decomposition of A might consist of finite dimensional weight spaces, allowing a graded dimensions count). In this section we explain how such an action T A induces a well-defined group scheme action T DRep V (A), in the sense that different models for the derived representation scheme are linked by T -equivariant quasi-isomorphism, and therefore their homologies (and all the other invariants, as the Euler characteristics introduced in § 2.4) carry a well-defined induced T -action. First we give a notion of a rational T -action, for an algebraic group T ∈ Grp k on any (dg,commutative) algebra. Definition 2.5.1. Let C be any of the following categories: DGVect k , DGA S , CDGA k or their nonnegatively graded versions. A rational action of an algebraic group T over k on an object A ∈ C is a morphism of groups ρ : T → Aut C (A) with the additional property that every element a ∈ A is contained in a finite dimensional T -stable vector subspace a ∈ V ⊂ A on which the induced action T → GL k (V) is a morphism of algebraic groups over k. We denote by C T the category with objects the objects in C with a rational T -action and morphisms the equivariant morphisms.
This definition is motivated by the fact that the equivalence of categories (2.22) enriches to an equivalence of categories between (CDGA + k ) T and the (opposite) category of dg-affine schemes with a group scheme action of T . Remark 2.5.1. If we denote by T the one-object groupoid associated to the group T , then a rational action on an object in C is just a functor T → C with some additional properties, and a T -equivariant morphism is a natural transformation of functors. Another way to say this is that we can view the category C T ⊂ [T, C] as a full subcategory of the category of functors. If C, D are two among the categories mentioned in 2.5.1, and F : C → D is any functor between them, then we can consider the induced functor on the functor categories F * = F • (−) : [T, C] → [T, D]. If this induced functor sends objects of C T ⊂ [T, C] into objects of D T , then it restricts to a functor that we denote by F T : C T → D T . This is true whenever F is defined purely in "algebraic terms" 2 , which is the case of all the functors we considered so far. The induced functor F T is an enrichment of the functor F in the sense that we can recover F under the natural forgetful functors: It is easy to see from the definition of the representation functor that a rational action T A induces (as explained in Remark 2.5.1), an action T A V which is still rational, and therefore a group scheme action T Rep V (A). To summarise the adjunction (2.19) enriches to an adjunction: We did not add a superscript (−) T to the enriched functors in this diagram because we want to avoid confusion with the same symbols used with a different meaning in § 2.3. From now on we restrict ourselves to the case of our interest in this paper of an algebraic torus T = (k × ) r . To do what we promised to do in the beginning of this section we need to prove that, roughly speaking, any T -equivariant algebra admits an equivariant cofibrant replacement in the model category DGA + S , and that any two such equivariant cofibrant replacements produce quasi-isomorphic representation schemes. To do it we introduce a model structure on the category (DGA + S ) T compatible with the model structures on DGA + S under the forgetful functor (in the following Theorem we explain in which sense these model structures are compatible). We recall that DGA + S is equipped with the projective model structure in which weak equivalences are quasiisomorphisms and fibrations are surjections in positive degrees. We also observe that actually the category of T -equivariant dg-algebras over S is (DGA + S ) T = S ↓ (DGA + k ) T nothing else but the under category of T -equivariant dg-algebras over k receiving a map from S if we give S the trivial action, and therefore we only need to give a model structure in the absolute case S = k. Proof. We refer the reader to Appendix A for the proof of this Theorem.
As a corollary of this result we can naturally equip the derived representation scheme of a Tequivariant algebra with a group scheme action of T . In fact, let S ∈ Alg k and (A ∈ Alg S ) T = S ↓ (Alg k ) T be a T -equivariant algebra. 2 We leave intentionally this as an intuitive, not well-defined, notion.

Corollary 2.5.1. There is a well-defined action T DRep V (A) which is compatible with the one on
Proof. First of all, we can pick up a T -equivariant cofibrant replacement Q ∼ A ∈ (DGA + S ) T using the model structure we just defined. Because of Theorem 2.5.1 (1) and (2), when we forget the T -action we still have a cofibrant replacement for A, therefore we can use this Q as a model for There is a natural T -action on this dg-scheme induced by T Q, which is compatible with the one on its truncation π 0 (Rep V (Q)) ∼ = Rep V (A). To prove that the previous definition is well posed, we show that if Q ∼ A is any another Tequivariant cofibrant replacement, then there is a T -equivariant quasi-isomorphisms of dg-schemes In fact by the general machinery of model categories we can lift the identity map 1 A : A → A to a T -equivariant (weak equivalence) between the two cofibrant replacements f : Q ∼ → Q . When we forget the T -action, this is still a weak equivalence, therefore giving an isomorphism γf in the homotopy category Ho(DGA + S ) and therefore L(γf) V is an isomorphism in Ho(CDGA + k ). But because both domain and codomain are cofibrant, L(γf) V = γf V , and therefore As a final consequence, the representation homology of a T -equivariant algebra, and all the other invariants defined in § 2.4, enrich to T -equivariant invariants. For example we can define the T -equivariant virtual fundamental class of the derived representation scheme X = DRep V (A) as the following object in the equivariant K-theory of the classical representation scheme: and also all the other U λ -irreducible components for a reductive group G by which we take the quotient (see § 2.4) as In particular for U 0 = k the trivial representation, we obtain an equivariant version of the virtual fundamental class of the derived partial character scheme X G = DRep G V (A), which we denote by:

The case of Nakajima quiver varieties
In this section we first recall the construction of Nakajima quiver varieties and secondly we construct some derived representation schemes related to them.
3.1. Nakajima quiver varieties. We already recalled in Example 2 that a finite quiver is a finite directed graph defined by its sets of vertices and edges Q = (Q 0 , Q 1 ) with two maps (source and target of an arrow) s, t : Q 1 → Q 0 .
We first frame the quiver, this means that we add a new vertex for each old one with a new arrow from the new to the old. Then we double the framed quiver, in order to obtain a cotangent (symplectic) space when we consider its representations. We denote this quiver by Q fr . To consider representations of a framed (doubled) quiver, we need to fix two dimension vectors v, w ∈ N Q 0 , and usually one assumes that (at least one of the components of) the framing vector is nonzero: w = 0. Notation. We denote the linear representations of the doubled, framed quiver by Explicitely it is the following cotangent linear space: We denote elements of this space by quadruples (X, Y, I, J) = (X γ , Y γ , I a , J a ) γ,a , where X γ , I a ∈ L(Q fr , v, w) are elements of the representation space of the framed quiver, and (Y γ , J a ) are cotangent vectors to them. The gauge group is the general linear group on the set of vertices of the original quiver Q: which acts by conjugation in a Hamiltonian fashion on M (Q, v, w). The moment map for this action is where in the above equation [X, Y] + IJ is a shortened symbol for Nakajima varieties are defined as symplectic reductions of M(Q, v, w) by this action. The affine Nakajima quiver variety is the geometric quotient: The GIT Nakajima variety is instead given by the choice of a character χ ∈ Hom Grp C (G, C × ) as the proj of the graded ring of χ-quasiinvariant functions on µ −1 (0): ) with the property f(g · p) = χ n (g)f(p) for all g ∈ G and p ∈ µ −1 (0)). The inclusion of G-invariant functions as degree zero elements of the graded ring of χ-quasiinvariant functions O(µ −1 (0)) G ⊂ O(µ −1 (0)) G,χ induces a projective morphism: which is often a symplectic resolution of singularities. Sometimes we denote these varieties simply by M χ , M 0 implicitly fixing the quiver Q, and the dimension vectors v, w.

3.2.
Derived representation schemes models. In Proposition 2.1.1 we showed how the linear space of representations of a quiver is isomorphic to the representation scheme for its path algebra. The same thing holds for the doubled, framed quiver so that To obtain the zero locus of the moment map, we consider the 2-sided ideal I µ ⊂ CQ fr generated by the |Q 0 |-elements of the path algebra described in (3.4), and consider the quotient algebra (3.10) A := CQ fr /I µ ∈ Alg S , relative to the subalgebra S ⊂ A of idempotents, with fixed representation ρ = ρ v,w : S → End C (C v ⊕ C w ) (as in (2.8)). The following result is an immediate consequence of the fact that taking the quotient by some ideal amounts simply to impose these new relations in the representation scheme (see Examples 1. (6)): Proposition 3.2.1. The zero locus of the moment map µ is the (relative) representation scheme for the path algebra of the framed, doubled quiver, modulo the Hamiltonian relation: Notation. We denote the corresponding derived representation scheme and representation homology by: The representation homology H • (A, v, w) is a graded commutative algebra, so when we view it in CDGA + C we mean that the differential is zero.
Remark 2.2.2, together with Proposition 3.2.1 tells us that the π 0 of this derived scheme X = DRep v,w (A) is the zero locus of the moment map: Proof. It follows directly from the previous observation (3.13) and the Theorem 2.3.1. More precisely:

K-theoretic classes in the affine Nakajima variety.
In § 3.4 we describe an explicit cofibrant resolution for our algebra A = CQ fr , A cof ∼ A and therefore a model for the derived representation scheme DRep v,w A = Rep v,w A cof , but we can already use Corollary 3.2.1 to define some interesting invariants in the K-theory of M 0 = Rep G v,w (A). Throughout this section we denote by X = DRep v,w (A) the derived representation scheme and by X G = DRep G v,w (A) the corresponding partial character scheme, whose π 0 (X G ) = M 0 is the affine Nakajima variety.
There is a torus, the (standard) maximal torus of the gauge group on the framing vertices T w ⊂ G w acting on the linear space of representations Rep v,w (A), and therefore as explained in § 2.4 it induces an action T w DRep v,w (A) and on its quotient by the gauge group G v : There is an additional (2-dimensional) torus T h = (C × ) 2 A acting rationally on the path algebra of the doubled framed quiver. This action can be described by assigning, respectively, the following Z 2 -weights to the arrows (x γ , y γ , i a , j a ) (see § 3.1 to recall the name of the arrows): (1, 0), (0, 1),(1, 1),(0, 0), or explicitly as As explained in § 2.5, also this torus induces actions In other words, the whole torus T := T w × T h acts on the derived representation scheme X = DRep v,w (A) and its partial character scheme . Using the definitions we gave in § 2.4 and § 2.5 we obtain the following invariants in the (equivariant) K theory of the affine Nakajima variety M 0 = Rep G v v,w (A), for example the virtual fundamental class More generally for each irreducible representation U λ of G v , the Euler characterstic of the corresponding isotypical component as

Explicit cofibrant resolution.
In this section we describe an explicit cofibrant resolution for the S-algebra A constructed in the previous Section. Let us recall that where S is the subalgebra generated by the idempotents of the path algebra of the framed quiver. The main obstruction for this object to be cofibrant is the Hamiltonian relation described by the ideal I µ . The simplest idea is then to add one more variable for each of the generating relations in I µ which kills the relation itself. This technique in general might not work due to higher homologies, but we prove that this case is one of the well-behaved cases. We construct the following quiver Q ϑ , which is obtained by adding to the framed, doubled quiver Q fr , one loop called ϑ a on each original vertex a ∈ Q 0 . In the path algebra CQ ϑ we assign homological degree 0 to the original arrows, and homological degree 1 to the new arrows ϑ a . The differential is induced by the moment map (equations as in (3.5)) dϑ a = µ a (x, y, i, j) .
It sits in the following diagram where π is the composition of the following two obvious projections: This amounts to prove that, in the diagram (3.19), the map π is an acyclic fibration, and ι is a cofibration. Proof. We need to prove that: (i) π is degreewise surjective in degrees 1 (this is obvious, because A is concentrated in degree 0).
is an isomorphism for each i 0, which becomes proving that H 0 (π) is an isomorphism, this is evident from the construction of A cof . We are left to prove that A cof has no higher homologies.
We first decompose A cof as a direct sum of the subalgebras of paths starting and finishing at a fixed couple of vertices: This decomposition preserves the differential, so we only need to prove that each P a,b has no higher homologies. Claim: If we substitute each j a , i a with the cycle c a = i a j a and prove that the resulting dg-algebras have no higher homologies, then neither P a,b have.
Proof of the claim: Let us call P a,b the dg-algebra of paths from a to b in the quiver Q ϑ , where we substitute each pair of arrows j a , i a with the cycle c a = i a j a . Then, for each fixed a, b ∈ Q 0 , we have four cases 3 : Now we consider the following filtration on the algebras P a,b : F p := Span C { paths with #x + #y 2p} .
Remember that the differential has the form "dϑ = [x, y] + c", so that the associated graded has differential of the form d gr ϑ a = c a , which involves only loops on the vertices a ∈ Q 0 . But then we can decompose the dg-algebras P a,b into their word structure 4 , and discover that the only non-trivial building blocks of which they are made of are dg-algebras of the form which have no higher homologies 5 .
Lemma 3.4.2. ι : S → A cof is a cofibration in DGA + C , or equivalently A cof is a cofibrant object in DGA + S . Proof. We need to prove that ι has the left lifting property with respect to acyclic fibrations.
Then the associated derived intersection can be defined as the derived scheme where ⊗ L R is the derived tensor product of R-modules. The algebra of functions on this derived scheme is the Koszul complex: 4 More precisely, we can decompose P a,b into the direct sum of those paths that, except for the arrows ϑ a and c ameaning that we set these to 1 -are the same. 5 An elementary argument is to observe that the derivation defined by the formula h(c) = ϑ and h(c) = 0, is a homotopy between the 0 map and the map length(−) · Id, which is an isomorphism in (homological) degrees 1. This implies that H i (L) = 0 for i 1.
A more concrete way to describe it is the following: we can view the collection of functions f = (f 1 , . . . , f m ) as a map of affine schemes f : A n C → V := A m C , and consider its dual map where R is in homological degree 0, the vector space V * is in homological degree 1, and the differential An useful classical result on the Koszul complex is Theorem 3.5.1 ([23]). The following are equivalent: (1) dim C (Spec(H 0 (K))) = n − m.
Let us turn back to the case of our interest, in which we want to recognise as the Koszul complex on the moment map. We recall that the quiver Q ϑ is constructed from the quiver Q fr by adding a new loop in homological degree 1 on each of the original vertices of the quiver Q. Therefore, a representation of the path algebra CQ ϑ is just a representation of the subalgebra CQ fr (an element of the vector space M (Q, v, w)), together with a family of endomorphisms Θ = (Θ a ) a∈Q 0 ∈ a∈Q 0 gl v a (C) = g v in homological degree 1. Putting everything together we obtain Q, v, w)) ⊗ C Λ • g v ∈ CDGA + C , which is nothing else but the Koszul complex for the zero locus defined by the moment map Its spectrum is a model for our derived representation scheme, as the derived intersection of the moment map equations: gives a model for the derived representation scheme as the (spectrum of the) Koszul complex on the moment map: In particular we can observe that this is a derived scheme of finite type (Definition 2.4.3) and that the Koszul complex is bounded, Therefore all the invariants defined in § 3.3 ((3.15),(3.16)) make sense, because the sums are bounded (by the dimension of the Lie algebra dim C g v = v 2 = v · v). Remark 3.5.1. In § 3.4 we gave a self-contained proof of why the resolution provided by the path algebra of the quiver Q ϑ obtained by adding one loop on each vertex in which the corresponding component of the moment map is considered (i.e. the original vertices) works. In § 3.5 we explained why the resulting representation scheme is the Koszul complex on the moment map. We remark that the same results can be explained in a slightly different flavour through the theory of noncommutative complete intersections (NCCI) and partial preprojective algebras ( [10], [13]).

Flat moment map and vanishing representation homology.
In this section we recall some classical results on the flatness for the moment map of Nakajima quiver varieties which are useful for our purposes. We show how flatness is equivalent to the condition of vanishing of higher representation homologies for the corresponding algebra.
Remember that for each quiver Q and for each fixed dimensions v, w ∈ N Q 0 we have the corresponding Nakajima varieties M 0 (affine) and M χ (quasiprojective), where χ ∈ Hom Grp C (G v , C × ) is a given (nontrivial) character. We also recall that the group of all characters of the gauge group In this section we use the parameter θ for the characters and denote M χ θ simply by M θ . We recall that the Cartan matrix of the quiver Q is the matrix The subset of R Q 0 of v-regular vectors is the complement of some hyperplanes. Its connected components are called chambers, and the variety M θ depends only on the chamber of θ. Let v ∈ N Q 0 be a dimension vector and θ ∈ Z Q 0 be v-regular, then any θ-semistable point in µ −1 (0) is θ-stable and M θ is a smooth, connected, variety of (complex) (with the convention that M θ = ∅ when this dimension is negative).
When we take the zero locus by µ we expect to decrease the dimension by the number of equations of µ, which is v · v and then again by v · v when taking the G v -quotient.
Let us consider for some v-regular θ the natural affinisation morphism This morphism is a Poisson morphism 6 (obviously, because ϕ * is the identity) and it is a resolution of singularities (i.e. projective and birational) ( [7]). The variety M θ depends, a priori on the chamber of θ, but actually its affinisation Spec(O(M θ )) is independent of the choice of v-regular θ. We can call this variety simply M and we obtain a diagram of the following form which is the so-called Stein factorisation ( [40]) of the proper morphism p. The pre-image of the point 0 ∈ M 0 through ψ is always 0 ∈ M. In particular the fiber p −1 (0) is equal to the central fiber ϕ −1 (0) of the affinisation morphism and therefore is a homotopy retract of the variety M θ . The combinatorial criterium for the flatness of the moment map proved in [9] is given in the setting of a non-framed quiver Γ . For any dimension vector α ∈ N Γ 0 we consider the linear space of representations of the doubled quiver L(Γ , α). The gauge group acting a priori in a non-trivial way is now G α /C × because, without the framing, the diagonal torus C × ⊂ G α acts trivially on the linear space of representations. The Lie algebra of this group can be identified with the subalgebra g α ⊂ g α = ⊕ i gl α i (C) of matrices with sum of their traces equal to zero (the notation g α is borrowed from [13]). The moment map is now Let us denote by p the following function Theorem 4.1.3 (Theorem 1.1 in [9]). The following are equivalent: (3) p(α) r t=1 p(β (t) ) for each decomposition α = β (1) + · · · + β (r) with each β (t) positive root. (4) p(α) r t=1 p(β (t) ) for each decomposition α = β (1) + · · · + β (r) with each β (t) ∈ N Γ 0 \{0}. In a remark in § 1 in [9], Crawley-Boevey explains how to adapt this setting to the situation of a framed quiver. From a quiver Q and a framing vector w we can construct a new quiver Γ := Q ∞ , which is obtained by adding only one new vertex, denoted by ∞, together with a number of w a arrows towards each vertex a ∈ Q 0 . If we fix now a dimension vector v ∈ N Q 0 and define the new vector α := (v, 1) ∈ N Γ 0 , then and we have the following criterium: Corollary 4.1.1. Consider the quiver Q fr with dimension vectors v, w ∈ N Q 0 , and the quiver Γ = Q ∞ with α = (v, 1). Then the following are equivalent: (1) µ is flat.
For the condition (2) now we can use the combinatorical test given by Theorem 4.1.3, and using this result, we can prove that the derived representation scheme has vanishing higher homologies if and only if the moment map µ is flat: The representation homology is the homology of the Koszul complex and therefore, by Theorem 3.5.1, it vanishes in degrees i 1 if and only if the dimension condition (4.5) is satisfied.
In the following examples we use Theorem 4.1.3 for some quivers and we find the combinatorical condition on the dimension vectors for the moment map to be flat. It is convenient to observe that for the quiver Γ = Q ∞ the map p is, for vectors of the form (β, 1) or (β, 0) (that is the only type of vectors that we need to decompose the dimension vector α = (v, 1)):
the following inequality holds: v(w − v) β 0 (w − β 0 ) + r − β 2 1 − · · · − β 2 r . We can observe that actually all β 1 , . . . , β r 1 and therefore the function r − β 2 1 − · · · − β 2 r reaches its maximum for β 1 = · · · = β r = 1 for which it is 0. So we just need to test that The inequality can also be rewritten as (2) The second example is a quiver with one vertex and m loops (m 1). In particular the Jordan quiver for m = 1 described in Figure 1. We show that for each choice of v 0 and w 1 the moment map is flat. The quiver Γ = Q ∞ still has 2 vertices, the first one with m loops and w arrows connecting the 2 nd to the 1 st , so that: We need to test that for each decomposition (v, 1) = (β 0 , 1) + (β 1 , 0) + · · · + (β r , 1), β 1 , . . . , β r 1 , the following inequality holds which is actually true component-wise because Therefore the moment map is always flat.
(3) The third example is the quiver Q = A n−1 with the following particular choice of vectors v = (1, . . . , 1) and w a = δ a,1 + δ a,n−1 (for which the Nakajima variety is the symplectic dual of T * P n−1 , as explained in the next Section). The resulting quiver Γ = Q ∞ is the cyclic quiver with n vertices and dimension vector α = (1, . . . , 1) constant to 1, for which it is easy to check that the moment map is flat. In fact p(α) = 1 while for any other β ∈ N n , 0 = β = α we have p(β) 0 so that condition (4) of Theorem 4.1.3 is satisfied.

Kirwan map and tautological sheaves.
Let M χ = M χ (Q, v, w) be a smooth Nakajima quiver variety (so χ = χ θ with θ being v-regular, see Theorem 4.1.1), then the locus of χ-semistable points coincides with the locus of χ-stable points, on which the action is free, and The equivariant Kirwan map (in cohomology) is the map obtained by composing the natural pullback for the inclusion due to the fact that the G-action on the χ-stable locus is free: . McGerty and Nevins have recently shown that the Kirwan map (4.6) is surjective ([24, Corollary 1.5]), and that the same holds for other generalised cohomology theories such as K-theory and elliptic cohomology. We are particularly interested in the K-theory, so the Kirwan map is Moreover the zero locus of the moment map µ −1 (0) is equivariantly contractible: where R(−) is the representation ring (over C), so the Kirwan map has the form: and it is a surjective map of R(T )-modules. K T (M χ ) is therefore generated by tautological classes, because they come from classes of topologically trivial vector bundles: if U is a G × T -module, and [U] ∈ R(G × T ) is its class, then Moreover the map (4.8) is a map of R(T )-modules, so the only non-trivial part consist in its image on vector spaces U that are only representations of G. For U = V λ irreducible representation of G, we denote by a calligraphic V λ the sheaf whose K-theoretic class is We can use these tautological classes to define invariants in the K-theory of the affine Nakajima variety by using the pushforward under the map p: It is important to recall that in general the push-forward of a proper map p in K-theory is given by the alternate sums of right-derived functors of p * . In this particular case the target variety M 0 is affine, therefore this alternate sum calculates the Euler characteristic of a sheaf F on M χ , under the natural identifications: For an irreducible representation U = V λ of G the composition (4.10) gives the Euler characteristic of the corresponding tautological sheaf V λ : The notable special case of U = V 0 the trivial 1-dimensional representation of G, has image under the Kirwan map the (K-theoretic class of the) sheaf of functions on the GIT quotient V 0 = O M χ , and its Euler characteristic:

Comparison theorem and first integral formula.
In § 3.3 we defined the virtual fundamental classes of the isotypical components of the derived character scheme (4.14) and in particular for V λ = V 0 = C:

Theorem 4.3.1.
Let v, w be dimension vectors for which the moment map is flat, and let χ = χ θ with θ v-regular, so that M χ (Q, v, w) is smooth. Then we have the following equality in the equivariant K-theory of the affine Nakajima variety : Proof. The first equality is a somewhat classical result. Firstly the (derived) pushforward in K-theory coincides with the underived pushforward because of the vanishing of higher cohomologies (Grauert-Riemenschneider theorem, ([18])). Moreover when the moment map is flat and M χ is smooth we can use Theorem (4.1.2): Finally by Theorem 1.1 the representation homology H • (A, v, w) vanishes in positive degrees, so that the Euler characteristic of its G-invariant part (4.15) is:   Q, v, w)) ⊗ Λ • g v is a resolution of O(µ −1 (0)): and the subcomplex of G-invariants is a resolution of the functions on M 0 : As a corollary of Theorem 4.3.1, we can take Hilbert-PoincarÃl' series (character for the torus) of the equality in (4.16) and obtain a equality between numerical (power) series counting the graded dimensions. Formally, if M 0 were compact, the Hilbert-PoincarÃl' would be the pushforward to the point: ch T : K T (M 0 ) → K T (pt) = R(T ), instead in general we land in the field of fractions (see, for example, §4 in [31])

Remark 4.3.2.
If we consider the only fixed point for the torus action 0 ∈ M 0 , and denote its inclusion by ι 0 : {0} → M 0 , then by functoriality we have ch T = (ι 0, * ) −1 , and this tells us that is not really necessary to invert all non-zero elements in R(T ), but only the ones of the form 1 − t β for non-zero weights β, so that we actually land in the following smaller localisation (see §2.1 and §2.3 in [34]): where α, β run over all weights of T and β = 0.
Let us denote by x ∈ T v ⊂ G the variables in the maximal torus of the gauge group (KÃd'hler variables) and by t = (a, h) ∈ T = T w × T h the equivariant variables. Then we have, by Weyl's integral formula: (W is the Weyl group of G, ∆(x) is the Weyl factor, and integrations are over the compact real forms of G, T v ) Moreover, because the Euler characteristic of the homology of a complex is equal to the Euler characteristic of the complex itself, we have (4.20) ch T v ×T (χ T (A, v, w) .
where s j are the weights of M(Q, v, w) * and r i are the weights of g: To summarise: We calculate the above expression (4.21) in some concrete examples in § 5.

Remark 4.3.3.
The right-hand side of (4.21) does not depend on the GIT parameter χ, while the left-hand side a priori does. By picking different v-regular χ, χ we obtain a combinatorical identity which we will show to be non-trivial, also in simplest quiver cases (see § 5, specifically Remark 5.1.1 in § 5.1).

Other isotypical components and second integral formula.
In this section we prove a result similar to Theorem 4.3.1 to relate other tautological sheaves with the corresponding isotypical components. Let us recall that to define M χ we fixed a character χ ∈ Hom Grp C (G, C × ). This character defines a 1-dimensional representation C χ of G, whose image under the Kirwan map is the Serre twisting sheaf For each V λ irreducible representation of G, we have a tautological sheaf V λ in the K-theory of M χ . By Serre vanishing theorem when we twist by a sufficiently large power m 0 of the twisting sheaf, higher cohomology vanish, so that Moreover, more or less by definition of the GIT quotient M χ , this is equal to the G-invariant global sections of the trivial vector bundle V λ ⊗ C χ m over the stable locus: Finally for m 0 large enough, the following natural restriction map becomes an isomorphism (see for example the proof of Lemma 3 in Appendix A of [1]): but the left-hand side is nothing else but It is worth noticing at this point that irreducible representations V λ of G are labelled by collections of partitions λ = (λ (1) , . . . , λ (n) ) and that the representation V λ ⊗ C χ m is still a irreducible representation of G, corresponding to the shifted collection of partitions: (see Appendix B for the notation). We give the following definition: Definition 4.4.1. We say that an irreducible representation V λ is large enough if λ = λ + mθ (see (4.28)) with m 0 large enough for both (4.24) and (4.26) to be true. This notion depends on the quiver Q, on the dimension vectors v, w and on the v-regular χ = χ θ .
Denoting by λ * the partition corresponding to the dual representation, we can continue equation (4.27) to recognise: the isotypical component of λ * of the (zeroth) representation homology. Finally if we observe that with flat moment map, higher homologies vanish, we obtain the following result:  Q, v, w)) .
The analogous integral formula to obtained by taking characters is where f λ (x) = ch T v (V λ ) (it is the product of Schur polynomials associated to the partitions in λ).

Examples
In this section we explain some concrete examples, mainly from the easiest quivers already considered in the previous sections. We see how such elementary quivers still produce varieties of great interest in various fields of mathematics.

Cotangent bundle of Grassmannian.
The quiver Q = A 1 with only one vertex and no arrows. The framed, doubled quiver has two vertices and two arrows connecting them in opposite directions.

Single-vertex quiver
• v w i j Therefore: Because we have only one vertex we have to choose the GIT parameter θ ∈ Z, and it is easy to check that the v-regularity condition means simply θ = 0 (independently from v). For θ = 0 we have the following identifications of the semistable locus: θ-semistable points = J injective, θ < 0 I surjective, θ > 0 and the GIT quotient is isomorphic to the cotangent bundle T * Gr(v, w) of v-planes in C w in the case θ < 0 and to T * Gr(w − v, w) in the case θ > 0. The two varieties are isomorphic to each other, but we have the following different identifications of the points in the Grassmannian: The affine quotient can be identified (using some version of the fundamental theorem of invariant theory): where A represents the composition J • I : C w → C w . The condition on the rank is due to the fact that A : W → V → W factorises through V, but sometimes it is superfluous. In fact in general  In this case in the torus T = T w × T h only the product h 1 h 2 appears and we denote it by h. We can use (4.21) for χ = χ −1 for which M χ = T * Gr(v, w) and obtain a formula for the character of the ring of functions on the cotangent bundle of Grassmannian: where in the above x = (x α ) = (x 1 , . . . , x v ) and a = (a γ ) = (a 1 , . . . , a w ).
The integral in the right-hand side can be computed by iterated residues, and by doing so we can recognise the localisation formula in equivariant K-theory as a sum over the fixed points p ∈ (T * Gr(v, w)) T of the inverse of the K-theoretic Euler class of the tangent space at that point: .
For what concerns other sheaves, let us consider the standard representation The associated tautological sheaf V on M χ −1 = T * Gr(v, w) is indeed the usual tautological sheaf of rank v. Irreducible representations are labelled by Schur functors is a integer partition of v parts, and we consider the corresponding tautological sheaves V λ . For example the (standard) tautological sheaf itself is V = V (1,0,...,0) , or powers of the Serre twisting sheaf are: A partition λ becomes large (Definition 4.4.1) in the sense that we can apply Theorem 4.3.1 when all its components are negative enough (because the character χ = χ −1 is negative), and it turns out that it suffices to have λ 1 0, that is equivalent to say that the partition is made of non-positive terms (an example is (5.3), in which for m > 0 the partition is negative and the corresponding sheaf has vanishing higher cohomologies). In this range we have (5.4) is the Schur polynomial associated to the partition λ. Again, the integral in the right-hand side can be computed by means of iterated residues, giving the localisation formula for the corresponding tautological sheaf: , where the expression s λ (a B ) means that we are evaluating the Schur polynomial s λ (x 1 , . . . , x v ) in the point x = (a β ) β∈B . . This is the case of the Jordan quiver, the quiver with one vertex and one loop, Figure 1. Therefore: For GIT paramater θ ∈ Z: When the framing is w = 1 we obtain the Hilbert-Chow morphism from the Hilbert scheme of v points on C 2 to the symmetric v-power: For general v and w the integral formula looks like: , and it is also known as the integral formula for Nekrasov partition function (proved for example in Appendix A of [14]). For other isotypical components, let us say that we fixed χ = χ 1 . Again we have a tautological sheaf of rank v, V, and other sheaves associated to irreducible representations are labelled by Schur functors V λ where λ is an integer partition of v parts. In this case the largeness condition indeed means that the partition is big enough, and it turns out that it suffices for it to be non-negative λ 1 · · · λ v 0. In this range we have: For λ 0 the Schur polynomial s λ (x) is indeed an actual polynomial (and not a Laurent polynomial), and therefore with (5.8) we recover the integral formula for Nekrasov partition function with matter fields (the matter field is represented by the sheaf V λ in this case) which was proved for example in [25]. 5.3. Symplectic dual of T * P n−1 . X = T * Gr(k, n) has a symplectic dual, Xˇ, which for the choice of parameters 2k n can be shown to be also a Nakajima quiver variety ( [39]). Specifically it is the Nakajima variety associated to the following A n−1 quiver, with dimension vectors: , k − 1, . . . , 2, 1) , We restrict to the case k = 1, for which dimension vectors are (5.9) v = (1, . . . , 1) , w = (1, 0, . . . , 0, 1) , and the corresponding Nakajima quiver variety is the symplectic dual of T * P n−1 . For n = 2 we go back to the A 1 case with dimensions v = 1 and w = 2, so we find that T * P 1 is symplectic dual to itself. Let us study the other cases n 3 which are different. As usual we denote the arrows in the quiver by x 1 , . . . , x n−2 , their dual by y 1 , . . . , y n−2 and then we have i 1 , j 1 and i n−1 , j n−1 because of the non-trivial framing at the vertices 1 and n − 1. The zero locus of the moment map is the following algebraic variety in a 2n-dimensional affine space The gauge group is a n − 1-dimensional torus G v = GL 1 (C) n−1 = (C × ) n−1 , and the affine Nakajima variety is identified with the ADE singularity of type A n−1 : where x = x 1 · · · x n−2 i 1 j n−1 , y = y 1 · · · y n−2 i n−1 j 1 , z = x 1 y 1 . We recall that the action Z n C 2 that gives the corresponding ADE singularity of type A n−1 is given by the embedding Z n ⊂ SL 2 (C) in which a n-th root of unity ξ ∈ Z n becomes the matrix diag(ξ, ξ −1 ) ∈ SL 2 (C).
We fix GIT parameter χ = χ θ + with θ + = (1, 1, . . . , 1). The corresponding smooth Nakajima quiver variety is a consecutive (n − 1 times) blowup of the singular point x = y = z = 0 in (5.10): with exceptional fiber p −1 (0) given by n − 1 copies of Riemann spheres P 1 intersecting in such a way that their underlying intersection graph is A n−1 (see [11]), as shown in Figure 4. The associated derived representation scheme is where ϑ i have homological degree 1 and differential dϑ n−1 = x n−2 y n−2 + i n−1 j n−1 , and they are invariants under the gauge group G v = GL n−1 1 , so that the associated character scheme is simply C[x, y, z 1 , . . . , z n−2 , z n−1 , z n ] xy = z 1 · · · z n [ϑ 1 , . . . , ϑ n−1 ] , where x, y are the same classes as before in (5.10), z k = x k y k for k = 1, . . . , n − 2, z n−1 = i 1 j 1 , z n = i n−1 j n−1 . We denote the variables in the equivariant torus T = T w × T h by (a,ã, h 1 , h 2 ) (where a is on the vertex 1 andã on the vertex n − 1) and we have: (5.13) Appendix A. Projective model structure on T-equivariant dg-algebras In this Appendix we give a proof of Theorem 2.5.1 that gives a projective-like model structure on the category of T -equivariant dg-algebras (DGA + k ) T , for an algebraic torus T = (k × ) r . We use the same strategy used in [6], in which the authors prove that the category of bigraded dgalgebras BiDGA k has a projective-like model structure 7 . The key observation is to recognise that BiDGA k being the category of dg-algebras with an additional non-negative (polynomial) compatible grading, is equivalent to the category of T -equivariant dg-algebras with a polynomial torus action (i.e. weight spaces are only for non-negative weights), and that the polynomial condition can be dropped, and substituted by the rational condition, in which weights can be arbitrary integers.
More precisely, weight spaces for a torus T = (k × ) r are r-tuples of integers n ∈ Z r , and we observe that the category of dg-algebras with a rational T -action (DGA + k ) T is equivalent to the category of dg-algebras A ∈ DGA + k with: (1) An additional grading of the underlying chain complex A = ⊕ n∈Z r A(n). This means that each A(n) is a complex of vector spaces preserved by the differential in A: dA(n) ⊂ A(n). (2) The grading is compatible with the multiplication in A: A(n) · A(m) ⊂ A(n + m).
In fact, on one hand if A ∈ (DGA + k ) T then for n ∈ Z r we define A(n) = {a ∈ A | t · a = t n a, ∀t ∈ T } as the corresponding weight space and the above 2 conditions are satisfied thanks to the rationality of the action (recall, Definition 2.5.1). On the other hand, obviously if we have such a decomposition we define the T -action on A accordingly by t · a := n t n a(n), where a = n a(n), and the resulting T -action is rational.
The observation that (DGA + k ) T is equivalent to the category of dg-algebras with an additional grading as described above will be also useful later, and we will use indifferently one or the other property, according to what is more convenient from time to time.
Let us also denote by k[T ] = O(T ), a Laurent polynomial ring in r variables and observe that Lemma A.0.1. The forgetful functor U : (DGA + k ) T → DGA + k is left-adjoint to the "free T -equivariant extension" functor: Proof. The adjunction is given by the natural isomorphisms: 7 Which ultimately follows the explicit proofs of the existence of the projective model structure on DGA + k by [27] or [15].
Conversely if we start from a map f : UA → B which is not necessarily T -equivariant, we can construct a T -equivariant map ϕ : A → k[T ] ⊗ B by decomposing: ϕ : and defining ϕ | A(n) : A(n) → B · t n as f | A(n) (−) · t n .
In order to prove Theorem 2.5.1 we need a few definitions and lemmas. Throughout this section of the Appendix we denote by C = DGA + k and by C T = (DGA + k ) T .

Notation.
We denote by Cof , WE, Fib the collection of cofibrations, weak equivalences, and fibrations in the projective model structure on C. So Fib are surjective maps in positive homological degrees, WE are the quasi-isomorphisms, and Cof = (WE ∩ Fib), where (−) denotes the collection of morphisms with the left lifting property with respect to another collection of morphisms. Finally recall that a fibration which is also a quasi-isomorphism is actually surjective in all homological degrees, so that WE ∩ Fib consists of surjective quasi-isomorphisms. (1) Each S * k T (V (i) ) has a differential and a compatible embedding S * k T (V (i) ) ⊂ R such that at the limit V = ∪ i V (i) : (2) Each differential has the property that d(V (i) ) ⊂ S * k T (V (i−1) ) (and for i = 0, d(V (0) ) ⊂ S).
We denote the collecion of such morphism by TE ⊂ Mor(C T ). (ii) Every Tate extension has the left lifting property with respect to morphisms that are surjective quasi-isomorphisms: TE ⊂ (U −1 (WE ∩ Fib)).
For the proof one can check that the proof of Proposition 3.1 (which relies on Proposition 2.1(ii)) of [15] can be used also in this case of T -equivariant (i.e. additionally graded) objects.
Now let x be a variable of positive homological degree as well as of some weight n ∈ Z r for the torus T , and set V x := [0 → k · x → k · dx → 0], and its tensor algebra T (V x ) ∈ C T . Extensions by objects of this form play another important role: Definition A.0.2. A morphism in C T of the form S → S * k i∈I T (V x i ), where I is any, possibly uncountably infinite, indexing set, is called a special extension. We denote the collection of special extensions by SE ⊂ Mor(C T ). To prove that a lift exists in the case (ii), thanks to Proposition A.0.1 we only need to find a lift when i is a special extension, but this is true by Lemma A.0.3(ii). (MC5) We need to prove that each morphism S → A in C T has factorisations of the form: (i) cofibration followed by an acyclic fibration, (ii) acyclic cofibration followed by a fibration. (i) follows from Lemma A.0.2(i), and (ii) follows from Lemma A.0.3(i).
(2) This follows from the fact that U is left adjoint to k[T ] ⊗ (−) (Lemma A.0.1), and the latter preserves weak equivalences and fibrations, therefore U preserves cofibrations.
Appendix B. Representation theory of G = G v In this Appendix we recall the theory of irreducible representations of (a product of) general linear groups and we fix the notation. Polynomial irreducible representations of GL v (C) are labelled by ordinary (non-negative) partitions λ = (λ 1 , . . . , λ v ). More precisely, they are obtained by applying the Schur functors S λ (−) : Vect C → Vect C to the standard representation V = C v : All irreducible rational representations of GL v (C) are of the form (B.4) for some integer-valued partition λ ∈ P v . Their characters are generalised Schur polynomials and they form a linear basis of the ring of symmetric Laurent polynomials in v variables: . . . , v n ) is a dimension vector and G v = i GL v i (C) is a product of general linear groups, then its irreducible rational representations are labelled by n-tuples of partitions λ = (λ (1) , . . . , λ (n) ) ∈ i P v i , as the external tensor product of Schur modules: (B.6) V λ := S λ (1) (C v 1 ) · · · S λ (n) (C v n ) .