Schematic Grothendieck Representation

A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R^(Ring) we have a commutative diagram in R^:



If Rep is schematic, then, P : Top(A) → R^ is a presheaf with values in R^ over the lattice Top(A)o, for every A in R.

The modality is to restrict attention to Tors(Rep(A)); that is, a lattice in the usual sense; and hence this should be viewed as the commutative shadow of a suitable noncommutative theory.

For obtaining the complete lattice Q(A), a duality is expressed by an order-reversing bijection: (−)−1 : Q(A) → Q((Rep(A))o). (Rep(A))o is not a Grothendieck category. It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. If a comparison functor is constructed via Hom(P, −) as a functor to the category of sets, it works well for the category of set-valued sheaves over a Grothendieck topology.

Now (−)−1 is defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))o, implying we write (Top(A))−1 for the image of Top(A) in Q((Rep( A))o). This is encoded in the exact sequence in Rep(A):

0 → ρ(M) → M → ρ−1(M) → 0

(reversed in (Rep(A))o). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)−1 is not in Tors((Rep(A))op). If notationally, TT(G) is the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (TT(G))−1 ≅ TT(Gop). Hence we may view Tors(G)−1 as a complete sublattice of TT(Gop).

Conjectural Existence of the Categorial Complex Branes for Generalized Calabi-Yau.

Geometric Langlands Duality can be formulated as follows: Let C be a Riemann surface (compact, without boundary), G be a compact reductive Lie group, GC be its complexification, and Mflat(G, C) be the moduli space of stable flat GC-connections on C. The Langlands dual of G is another compact reductive Lie group LG defined by the condition that its weight and coweight lattices are exchanged relative to G. Let Bun(LG, C) be the moduli stack of holomorphic LG-bundles on C. One of the statements of Geometric Langlands Duality is that the derived category of coherent sheaves on Mflat(G, C) is equivalent to the derived category of D-modules over Bun(LG, C).

Mflat(G, C) is mirror to another moduli space which, roughly speaking, can be described as the cotangent bundle to Bun(LG, C). The category of A-branes on T Bun(LG, C) (with the canonical symplectic form) is equivalent to the category of B-branes on a noncommutative deformation of T Bun(LG, C). The latter is the same as the category of (analytic) D-modules on Bun(LG, C).

So, what exactly is, the relationship between A-branes and noncommutative B-branes. This relationship arises whenever the target space X is the total space of the cotangent bundle to a complex manifold Y. It is understood that the  symplectic form ω is proportional to the canonical symplectic form on T Y. With the B-field vanishing, and Y as a complex, we regard ω as the real part of a holomorphic symplectic form Ω. If qi are holomorphic coordinates on Y, and pi are dual coordinates on the fibers of T Y,  Ω can be written as

Ω = 1/ħdpi ∧ dqi = dΘ

Since ω (as well as Ω) is exact, the closed A-model of X is rather trivial: there are no nontrivial instantons, and the quantum cohomology ring is isomorphic to the classical one.

We would like to understand the category of A-branes on X = T Y. The key observation is that ∃ a natural coisotropic A-brane on X well-defined up to tensoring with a flat line bundle on X. Its curvature 2-form is exact and given by

F = Im Ω

If we denote by I the natural almost complex structure on X coming from the complex structure on Y , we have F = ωI, and therefore the endomorphism ω−1F = I squares to −1. Therefore any unitary connection on a trivial line bundle over X whose curvature is F defines a coisotropic A-brane. 

Now, what about the endomorphisms of the canonical coisotropic A-brane, i.e., the algebra of BRST-closed open string vertex operators? This is easy if Y is an affine space. If one covers Y with charts each of which is an open subset of Cn, and then argues that the computation can be performed locally on each chart and the results “glued together”, one gets closer to the fact that the algebra in question is the cohomology of a certain sheaf of algebras, whose local structure is the same as for Y = Cn. In general, the path integral defining the correlators of vertex operators does not have any locality properties in the target space. Each term in perturbation theory depends only on the infinitesimal neighbourhood of a point. This shows that the algebra of open-string vertex operators, regarded as a formal power series in ħ, is the cohomology of a sheaf of algebras, which is locally isomorphic to a similar sheaf for X = Cn × Cn.

Let us apply these observations to the canonical coisotropic A-brane on X = T Y. Locally, we can identify Y with a region in Cn by means of holomorphic coordinate functions q1, . . . , qn. Up to BRST-exact terms, the action of the A-model on a disc Σ 􏰠takes the form

S = 1/ħ ∫∂Σ φ (pidqi)

where φ is a map from Σ to X. This action is identical to the action of a particle on Y with zero Hamiltonian, except that qi are holomorphic coordinates on Y rather than ordinary coordinates. The BRST-invariant open-string vertex operators can be taken to be holomorphic functions of p, q. Therefore quantization is locally straightforward and gives a noncommutative deformation of the algebra of holomorphic functions on T Y corresponding to a holomorphic Poisson bivector

P = ħ∂/∂pi ∧ ∂/∂qi

One can write an explicit formula for the deformed product:

􏰋(f ⋆ g)(p, q) = exp(􏰋ħ/2(∂2/∂pi∂q̃i  −  ∂2/∂qi∂p̃i )) f(p, q) g (p̃, q̃)|p̃ = p, q̃ = q

This product is known as the Moyal-Wigner product, which is a formal power series in ħ that may have zero radius of convergence. To rectify the situation, one can restrict to functions which are polynomial in the fiber coordinates pi. Such locally-defined functions on T Y can be thought of as symbols of differential operators; the Moyal-Wigner product in this case reduces to the product of symbols and is a polynomial in ħ. Thus locally the sheaf of open-string vertex operators is modelled on the sheaf of holomorphic differential operators on Y (provided we restrict to operators polynomial in pi).

Locally, there is no difference between the sheaf of holomorphic differential operators D(Y ) and the sheaf of holomorphic differential operatorsD(Y, L) on a holomorphic line bundle L over Y. Thus the sheaf of open-string vertex operators could be any of the sheaves D(Y, L). Moreover, the classical problem is symmetric under pi → −pi combined with the orientation reversal of Σ; if we require that quantization preserve this symmetry, then the algebra of open-string vertex operators must be isomorphic to its opposite algebra. It is known that the opposite of the sheaf D(Y, L) is the sheaf D(Y, L−1 ⊗ KY), so symmetry under pi → −pi requires L to be a square root of the canonical line bundle KY. It does not matter which square root one takes, since they all differ by flat line bundles on Y, and tensoring L by a flat line bundle does not affect the sheaf D(Y, L). The conclusion is that the sheaf of open-string vertex operators for the canonical coisotropic A-brane α on X = T Y is isomorphic to the sheaf of noncommutative algebras D(Y, K1/2). One can use this fact to associate Y to any A-brane β on X a twisted D-module, i.e., a sheaf of modules over D(Y, K1/2). Consider the A-model with target X on a strip Σ = I × R, where I is a unit interval, and impose boundary conditions corresponding to branes α and β on the two boundaries of Σ. Upon quantization of this model, one gets a sheaf on vector spaces on Y which is a module over the sheaf of open-string vertex operators inserted at the α boundary. A simple example is to take β to be the zero section of T Y with a trivial line bundle. Then the corresponding sheaf is simply the sheaf of sections of KY1/2, with a tautological action of D(Y, KY1/2).

One can argue that the map from A-branes to (complexes of) D-modules can be extended to an equivalence of categories of A-branes on X and the derived category of D-modules on Y. The argument relies on the conjectural existence of the category of generalized complex branes for any generalized Calabi-Yau. One can regard the Geometric Langlands Duality as a nonabelian generalization. 

A Sheaf of Modules is a Geometric Generalization of a Module over a Ring – A Case Derivative of Abelian Closure


A coherent sheaf is a generalization of, on the one hand, a module over a ring, and on the other hand, a vector bundle over a manifold. Indeed, the category of coherent sheaves is the “abelian closure” of the category of vector bundles on a variety.

Given a field which we always take to be the field of complex numbers C, an affine algebraic variety X is the vanishing locus

X = 􏰐(x1,…, xn) : fi(x1,…, xn) = 0􏰑 ⊂ An

of a set of polynomials fi(x1,…, xn) in affine space An with coordinates x1,…, xn. Associated to an affine variety is the ring A = C[X] of its regular functions, which is simply the ring C[x1,…, xn] modulo the ideal ⟨fi⟩ of the defining polynomials. Closed subvarieties Z of X are defined by the vanishing of further polynomials and open subvarieties U = X \ Z are the complements of closed ones; this defines the Zariski topology on X. The Zariski topology is not to be confused with the complex topology, which comes from the classical (Euclidean) topology of Cn defined using complex balls; every Zariski open set is also open in the complex topology, but the converse is very far from being true. For example, the complex topology of A1 is simply that of C, whereas in the Zariski topology, the only closed sets are A1 itself and finite point sets.

Projective varieties X ⊂ Pn are defined similarly. Projective space Pn is the set of lines in An+1 through the origin; an explicit coordinatization is by (n + 1)-tuples

(x0,…, xn) ∈ Cn+1 \ {0,…,0}

identified under the equivalence relation

(x0,…, xn) ∼ (λx0,…, λxn) for λ ∈ C

Projective space can be decomposed into a union of (n + 1) affine pieces (An)i = 􏰐[x0,…, xn] : xi ≠ 0􏰑 with n affine coordinates yj = xj/xi. A projective variety X is the locus of common zeros of a set {fi(x1,…, xn)} of homogeneous polynomials. The Zariski topology is again defined by choosing for closed sets the loci of vanishing of further homogeneous polynomials in the coordinates {xi}. The variety X is covered by the standard open sets Xi = X ∩ (An)i ⊂ X, which are themselves affine varieties. A variety􏰭 X is understood as a topological space with a finite open covering X = ∪i Ui, where every open piece Ui ⊂ An is an affine variety with ring of global functions Ai = C[Ui]; further, the pieces Ui are glued together by regular functions defined on open subsets. The topology on X is still referred to as the Zariski topology. X also carries the complex topology, which again has many more open sets.

Given affine varieties X ⊂ An, Y ⊂ Am, a morphism f : X → Y is given by an m-tuple of polynomials {f1(x1, . . . , xn), . . . , fm(x1, . . . , xn)} satisfying the defining relations of Y. Morphisms on projective varieties are defined similarly, using homogeneous polynomials of the same degree. Morphisms on general varieties are defined as morphisms on their affine pieces, which glue together in a compatible way.

If X is a variety, points P ∈ X are either singular or nonsingular. This is a local notion, and hence, it suffices to define a nonsingular point on an affine piece Ui ⊂ An. A point P ∈ Ui is nonsingular if, locally in the complex topology, a neighbourhood of P ∈ Ui is a complex submanifold of Cn.

The motivating example of a coherent sheaf of modules on an algebraic variety X is the structure sheaf or sheaf of regular functions OX. This is a gadget with the following properties:

  1. On every open set U ⊂ X, we are given an abelian group (or even a commutative ring) denoted OX(U), also written Γ(U, OX), the ring of regular functions on U.
  2. Restriction: if V ⊂ U is an open subset, a restriction map resUV : OX(U) → OX(V) is defined, which simply associates to every regular function f defined over U, the restriction of this function to V. If W ⊂ V ⊂ U are open sets, then the restriction maps clearly satisfy resUW = resVW ◦ resUV.
  3. Sheaf Property: suppose that an open subset U ⊂ X is covered by a collection of open subsets {Ui}, and suppose that a set of regular functions fi ∈ OX(Ui) is given such that whenever Ui and Uj intersect, then the restrictions of fi and fj to Ui ∩ Uj agree. Then there is a unique function f ∈ OX(U) whose restriction to Ui is fi.

In other words, the sheaf of regular functions consists of the collection of regular functions on open sets, together with the obvious restriction maps for open subsets; moreover, this data satisfies the Sheaf Property, which says that local functions, agreeing on overlaps, glue in a unique way to a global function on U.

A sheaf F on the algebraic variety X is a gadget satisfying the same formal properties; namely, it is defined by a collection {F(U)} of abelian groups on open sets, called sections of F over U, together with a compatible system of restriction maps on sections resUV : F(U) → F(V) for V ⊂ U, so that the Sheaf Property is satisfied: sections are locally defined just as regular functions are. But, what of sheaves of OX-modules? The extra requirement is that the sections F(U) over an open set U form a module over the ring of regular functions OX(U), and all restriction maps are compatible with the module structures. In other words, we multiply local sections by local functions, so that multiplication respects restriction. A sheaf of OX-modules is defined by the data of an A-module for every open subset U ⊂ X with ring of functions A = OX(U), so that these modules are glued together compatibly with the way the open sets glue. Hence, a sheaf of modules is indeed a geometric generalization of a module over a ring.

Hochschild Cohomology Tethers to Closed String Algebra by way of Cyclicity.


When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξa = ia(ξ) ∈ Oaa of each object a of B, and η ◦ ξa = ξb ◦ η for each morphism η ∈ Oba from a to b. The family {ξa} thus constitutes a natural transformation from the identity functor 1B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1B. It is automatically commutative, for if {ξa}, {ηa} ∈ E then ξa ◦ ηa = ηa ◦ ξa by the definition of naturality. (A natural transformation from 1B to 1B is a collection of elements {ξa ∈ Oaa} such that ξa ◦ f = f ◦ ξb for each morphism f ∈ Oab from b to a. But we can take a = b and f = ηa.) If B is a Frobenius category then there is a map πab : Obb → Oaa for each pair of objects a, b, and we can define jb : Obb → E by jb(η)a = πab(η) for η ∈ Obb. In other words, jb is defined so that the Cardy condition ιa ◦ jb = πab holds. But the question arises whether we can define a trace θ : E → C to make E into a Frobenius algebra, and with the property that

θaa(ξ)η) = θ(ξja(η)) —– (1)

∀ ξ ∈ E and η ∈ Oaa. This is certainly true if B is a semisimple Frobenius category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(εa) = θa(1a)2, where a is an irreducible object, and εa ∈ E is the characteristic function of the point a in the spectrum of E. Nevertheless, a Frobenius category need not be semisimple, and we cannot, unfortunately, take E as the closed string algebra in the general case. If, for example, B has just one object a, and Oaa is a commutative local ring of dimension greater than 1, then E = Oaa, and so ιa : E → Oaa is an isomorphism, and its adjoint map ja ought to be an isomorphism too. But that contradicts the Cardy condition, as πaa is multiplication by ∑ψiψi, which must be nilpotent.

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH0(B) of B in degree 0. The groups HHp(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH0(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is πba(1a) ∈ Obb. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH0(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y0φ1 Y1φ2 ··· →φk Yk —– (2)

assigns F(φ1,…,φk) ∈ Hom(Y0,Yk). The differential in the complex is defined by

(dF)(φ1,…,φk+1) = F(φ2,…,φk+1) ◦ φ1 + ∑i=1k(−1)i F(φ1,…,φi+1 ◦ φi,…,φk+1) + (−1)k+1φk+1 ◦ F(φ1,…,φk) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism FY to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ1, φ2) of the composition law (φ1, φ2) ↦ φ2 ◦ φ1 of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O. But if O is a Frobenius algebra it is isomorphic as a bimodule to O, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Yk, Y0) is the dual space of Hom(Y0, Yk) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ0 ∈ Hom(Yk, Y0). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→φ0 Y0φ1 Y1 →φ2···→φk Ykφ0··· —– (4)

assigns a complex number F(φ01,…,φk).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ01,…,φk) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.

Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → Mi-1 →di-1 Mi →di → Mi+1 → …

such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:


We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the ith cohomology object is the quotient

Hi(M) = ker(di)/im(di−1)

This operation of taking cohomology at the ith place defines a functor

Hi(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕i Mi

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →f M0 →d0 M1 →d1 M2 → …

is a resolution of a module S, then by definition the morphism of complexes


is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes


Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hi(−): D(A) → A

establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.