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(G^op). Hence we may view Tors(G)⁻¹ as a complete sublattice of TT(G^op).

Advertisement

Homological Algebra – Does A∞ Algebra Compensate for any Loss of Information in the Study of Chain Complexes? 1.0

In an abelian category, homological algebra is the homotopy theory of chain complexes up to quasi-isomorphism of chain complexes.  When considering nonnegatively graded chain complexes, homological algebra may be viewed as a linearized version of the homotopy theory of homotopy types or infinite groupoids. When considering unbounded chain complexes, it may be viewed as a linearized and stabilized version. Conversely, we may view homotopical algebra as a nonabelian generalization of homological algebra.

Suppose we have a topological space X and a “multiplication map” m₂ : X × X → X. This map may or may not be associative; imposing associativity is an extra condition. An A_∞ space imposes a weaker structure, which requires m₂ to be associative up to homotopy, along with “higher order” versions of this. Indeed, there are very standard situations where one has natural multiplication maps which are not associative, but obey certain weaker conditions.

The standard example is when X is the loop space of another space M, i.e., if m₀ ∈ M is a chosen base point,

X = {x : [0,1] → M |x continuous, x(0) = x(1) = m₀}.

Composition of loops is then defined, with

x₂x₁(t) = x₂(2t), when 0 ≤ t ≤ 1/2

= x₁(2t−1), when  1/2 ≤ t ≤ 1

However, this composition is not associative, but x₃(x₂x₁) and (x₁x₂)x₃ are homotopic loops.

On the left, we first traverse x₃ from time 0 to time 1/2, then traverse x₂ from time 1/2 to time 3/4, and then x₁ from time 3/4 to time 1. On the right, we first traverse x₃ from time 0 to time 1/4, x₂ from time 1/4 to time 1/2, and then x₁ from time 1/2 to time 1. By continuously deforming these times, we can homotop one of the loops to the other. This homotopy can be represented by a map

m₃ : [0, 1] × X × X × X → X such that

{0} × X × X × X → X is given by (x₃, x₂, x₁) 􏰀→ m₂(x₃, m₂(x₂, x₁)) and

{1} × X × X × X → X is given by (x₃, x₂, x₁) 􏰀→ m₂(m₂(x₃, x₂), x₁)

What, if we have four elements x₁, . . . , x₄ of X? Then there are a number of different ways of putting brackets in their product, and these are related by the homotopies defined by m₃. Indeed, we can relate

((x₄x₃)x₂)x₁ and x₄(x₃(x₂x₁))

in two different ways:

((x₄x₃)x₂)x₁ ∼ (x₄x₃)(x₂x₁) ∼ x₄(x₃(x₂x₁))

and

((x₄x₃)x₂)x₁ ∼ (x₄(x₃x₂))x₁ ∼ x₄((x₃x₂)x₁) ∼ x₄(x₃(x₂x₁)).

Here each ∼ represents a homotopy given by m₃.

Schematically, this is represented by a polygon, S₄, with each vertex labelled by one of the ways of associating x₄x₃x₂x₁, and the edges represent homotopies between them

The homotopies m₃ yield a map ∂S₄ × X₄ → X which is defined using appropriate combinations of m₂ and m₃ on each edge of the boundary of S₄. For example, restricting to the edge with vertices ((x₄x₃)x₂)x₁ and (x₄(x₃x₂))x₁, this map is given by (s, x₄, . . . , x₁) 􏰀→ m₂(m₃(s, x₄, x₃, x₂), x₁).

Thus the conditionality on the structure becomes: this map extend across S₄, giving a map

m₄ : S₄ × X₄ → X.

As homological algebra seeks to study complexes by taking quotient modules to obtain the homology, the question arises as to whether any information is lost in this process. This is equivalent to asking whether it is possible to reconstruct the original complex (up to quasi-isomorphism) given its homology or whether some additional structure is needed in order to be able to do this. The additional structure that is needed is an A_∞-structure constructed on the homology of the complex…

 

Define Operators Corresponding to Cobordisms Only Iff Each Connected Component of the Cobordism has Non-empty Outgoing Boundary. Drunken Risibility.

 

Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is O_YZ = Ext^∗_H^∗(X)(H^∗(Y), H^∗(Z)) – the cohomology, as usual, has complex coefficients, and H^∗(Y) and H^∗(Z) are regarded as H^∗(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that O_YZ is dual to O_ZY. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then O_YZ is a symmetric algebra, and is not self-dual as a vector space.)

We can do better by defining a cochain complex O’_YZ of morphisms by

O’_YZ = B_Ω(X)(Ω(Y), Ω(Z)) —– (1)

where Ω(X) denotes the usual de Rham complex of a manifold X, and B_A(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution

Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · ·  —– (2)

in which the differential is given by

dƒ(a₁ ⊗ · · · ⊗ a_k ⊗ b) = 􏰝a₁ ƒ(a₂ ⊗ · · · ⊗ a_k ⊗ b) + ∑(-1)ⁱ ƒ(a₁ ⊗ · · · ⊗ a_ia_i+1 ⊗ a_k ⊗ b) + (-1)^k ƒ(a₁ ⊗ · · · ⊗ a_k-1 ⊗ a_kb) —– (3)

whose cohomology is Ext_A(B,C). This is different from O_YZ = Ext^∗_H^∗(X)(H^∗(Y), H^∗(Z)), but related to it by a spectral sequence whose E₂-term is O_YZ and which converges to H^∗(O’_YZ) = Ext_Ω(X)(Ω(Y), Ω(Z)). But more important is that H^∗(O’_YZ) is the homology of the space P_YZ of paths in X which begin in Y and end in Z. To be precise, H_p(O’_YZ) ≅ H_p+dZ(P_YZ), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A_∞ category B’. The corresponding composition of homology groups

H_i(P_YZ) × H_j(P_ZW) → H_i+j−dZ(P_YW) —— (4)

is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product P_YZ × P_ZW with the concatenation map M → P_YW.

Now let’s attempt to fit the closed string cochain algebra C to this A_∞ category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension d_X of X, so that the cohomology H_i(C) is potentially non-zero for −d_X ≤ i < ∞. There is a map H_i(X) → H_−i(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L₀X, based at any chosen point in X.

The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H^∗(O’_YZ). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary. 

Complicated Singularities – Why Should the Discriminant Locus Change Under Dualizing?

Consider the surface S ⊆ (C^∗)² defined by the equation z₁ + z₂ + 1 = 0. Define the map log : (C^∗)² → R² by log(z₁, z₂) = (log|z₁|, log|z₂|). Then log(S) can be seen as follows. Consider the image of S under the absolute value map.

The line segment r₁ + r₂ = 1 with r₁, r₂ ≥ 0 is the image of {(−a, a−1)|0 < a < 1} ⊆ S; the ray r₂ = r₁ + 1 with r₁ ≥ 0 is the image of {(−a, a−1)|a < 0} ⊆ S; and the ray r₁ = r₂ + 1 is the image of {(−a, a−1)|a > 1} ⊆ S. The map S → |S| is one-to-one on the boundary of |S| and two-to-one in the interior, with (z₁, z₂) and (z̄₁, z̄₂) mapping to the same point in |S|. Taking the logarithm of this picture, we obtain the amoeba of S, log(S) as depicted below.

Now consider S = S × {0} ⊆ Y = (C^∗)² × R = T² × R³. We can now obtain a six-dimensional space X, with a map π : X → Y, an S¹-bundle over Y\S degenerating over S, so that π⁻¹(S) →^≅ S. We then have a T³-fibration on X, f : X → R³, by composing π with the map (log, id) : (C^∗)² × R → R³ = B. Clearly the discriminant locus of f is log(S) × {0}. If b is in the interior of log(S) × {0}, then f⁻¹(b) is obtained topologically by contracting two circles {p₁} × S¹ and {p₂} × S¹ on T³ = T² × S¹ to points. These are the familiar conical singularities seen in the special Lagrangian situation.

If b ∈ ∂(log(S) × {0}), then f⁻¹(b) has a slightly more complicated singularity, but only one. Let us examine how the “generic” singular fiber fits in here. In particular, for b in the interior of log(S) × {0}, locally this discriminant locus splits B into two regions, and these regions represent two different possible smoothings of f⁻¹(b).

Assume now that f : X → B is a special Lagrangian fibration with topology and discriminant locus ∆ being an amoeba. Let b ∈ Int(∆), and set M = f⁻¹(b). Set M^o = M\{x₁, x₂}, where x₁, x₂ are the two conical singularities of M. Suppose that the tangent cones to these two conical singularities, C₁ and C₂, are both cones of the form M⁰. Then the links of these cones, Σ₁ and Σ₂, are T²’s, and one expects that topologically these can be described as follows. Note that M^o ≅ (T²\{y₁, y₂}) × S¹ where y₁, y₂ are two points in T². We assume that the link Σ_i takes the form γ_i × S¹, where γ_i is a simple loop around y_i. If these assumptions hold, then to see how M can be smoothed, we consider the restriction maps in cohomology

H¹(M^o, R) → H¹(Σ₁, R) ⊕ H¹(Σ₂, R)

The image of this map is two-dimensional. Indeed, if we write a basis eⁱ₁, eⁱ₂ of H¹(Σ_i, R) where eⁱ₁ is Poincaré dual to [γ_i] × pt and eⁱ₂ is Poincaré dual to pt × S¹, it is not difficult to see the image of the restriction map is spanned by {(e¹₁, e²₁)} and {(e¹₂, −e²₂)}. Now this model of a topological fibration is not special Lagrangian, so in particular we don’t know exactly how the tangent cones to M at x₁ and x₂ are sitting inside C³, and thus can’t be compared directly with an asymptotically conical smoothing. So to make a plausibility argument, choose new bases fⁱ₁, fⁱ₂ of H¹(Σ_i, R) so that if M(a,0,0), M(0,a,0) and M(0,0,a) are the three possible smoothings of the two singular tangent cones at the singular points x₁, x₂ of M. Then Y(M_i^(a,0,0)) = πafⁱ₁, Y(M_i^(0,a,0)) = πafⁱ₂, and Y(M_i(0,0,a)) = −πa(fⁱ₁ + fⁱ₂).

Suppose that in this new basis, the image of the restriction map is spanned by the pairs (f¹₁, rf²₂) and (rf¹₂, f²₁) for r > 0, r ≠ 1. Then, there are two possible ways of smoothing M, either by gluing in M₁^(a,0,0) and M₂^(0,ra,0) at the singular points x₁ and x₂ respectively, or by gluing in M₁^(0,ra,0) and M₂^(a,0,0) at x₁ and x₂ respectively. This could correspond to deforming M to a fiber over a point on one side of the discriminant locus of f or the other side. This at least gives a plausibility argument for the existence of a special Lagrangian fibration of the topological type given by f. To date, no such fibrations have been constructed, however.

On giving a special Lagrangian fibration with codimension one discriminant and singular fibers with cone over T² singularities, one is just forced to confront a codimension one discriminant locus in special Lagrangian fibrations. This leads inevitably to the conclusion that a “strong form” of the Strominger-Yau-Zaslow conjecture cannot hold. In particular, one is forced to conclude that if f : X → B and f’ : X’ → B are dual special Lagrangian fibrations, then their discriminant loci cannot coincide. Thus one cannot hope for a fiberwise definition of the dualizing process, and one needs to refine the concept of dualizing fibrations. Let us see why the discriminant locus must change under dualizing. The key lies in the behaviour of the positive and negative vertices, where in the positive case the critical locus of the local model of the fibration is a union of three holomorphic curves, while in the negative case the critical locus is a pair of pants. In a “generic” special Lagrangian fibration, we expect the critical locus to remain roughly the same, but its image in the base B will be fattened out. In the negative case, this image will be an amoeba. In the case of the positive vertex, the critical locus, at least locally, consists of a union of three holomorphic curves, so that we expect the discriminant locus to be the union of three different amoebas. The figure below shows the new discriminant locus for these two cases.

Now, under dualizing, positive and negative vertices are interchanged. Thus the discriminant locus must change. This is all quite speculative, of course, and underlying this is the assumption that the discriminant loci are just fattenings of the graphs. However, it is clear that a new notion of dualizing is necessary to cover this eventuality.

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, G_C be its complexification, and M_flat(G, C) be the moduli space of stable flat G_C-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 M_flat(G, C) is equivalent to the derived category of D-modules over Bun(^LG, C).

M_flat(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 p_i are dual coordinates on the fibers of T ^∗ Y,  Ω can be written as

Ω = 1/ħdp_i ∧ 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, . . . , q_n. Up to BRST-exact terms, the action of the A-model on a disc Σ 􏰠takes the form

S = 1/ħ ∫_∂Σ φ ^∗ (p_idqⁱ)

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∂/∂p_i ∧ ∂/∂qⁱ

One can write an explicit formula for the deformed product:

􏰋(f ⋆ g)(p, q) = exp(􏰋ħ/2(∂²/∂p_i∂q̃ⁱ  −  ∂²/∂q^i∂p̃_i )) 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 p_i. 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 p_i).

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 p_i → −p_i 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 ⊗ K_Y), so symmetry under p_i → −p_i requires L to be a square root of the canonical line bundle K_Y. 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, K^1/2). 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, K^1/2). 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 K_Y^1/2, with a tautological action of D(Y, K_Y^1/2).

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. 

Grothendieckian Construction of K-Theory with a Bundle that is Topologically Trivial and Class that is Torsion.

All relativistic quantum theories contain “antiparticles,” and allow the process of particle-antiparticle annihilation. This inspires a physical version of the Grothendieck construction of K-theory. Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Algebraic K-theory started from K_i defined for i, with relations to classical constructions in algebra and number theory, followed by Quillen’s homotopy-theoretic definition ∀ i. The connections to algebra and number theory often persist for larger values of i, but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

One could also use the conserved charges of a configuration which can be measured at asymptotic infinity. By definition, these are left invariant by any physical process. Furthermore, they satisfy quantization conditions, of which the prototype is the Dirac condition on allowed electric and magnetic charges in Maxwell theory.

There is an elementary construction which, given a physical theory T, produces an abelian group of conserved charges K(T). Rather than considering the microscopic dynamics of the theory, all that is needed to be known is a set S of “particles” described by T, and a set of “bound state formation/decay processes” by which the particles combine or split to form other particles. These are called “binding processes.” Two sets of particles are “physically equivalent” if some sequence of binding processes convert the one to the other. We then define the group K(T) as the abelian group Z^S of formal linear combinations of particles, quotiented by this equivalence relation.

Suppose T contains the particles S = {A,B,C}.

If these are completely stable, we could clearly define three integral conserved charges, their individual numbers, so K(T) ≅ Z³.

Introducing a binding process

A + B ↔ C —– (1)

with the bidirectional arrow to remind us that the process can go in either direction. Clearly K(T) ≅ Z² in this case.

One might criticize this proposal on the grounds that we have assumed that configurations with a negative number of particles can exist. However, in all physical theories which satisfy the constraints of special relativity, charged particles in physical theories come with “antiparticles,” with the same mass but opposite charge. A particle and antiparticle can annihilate (combine) into a set of zero charge particles. While first discovered as a prediction of the Dirac equation, this follows from general axioms of quantum field theory, which also hold in string theory.

Thus, there are binding processes

B + B̄ ↔ Z₁ + Z₂ + · · · .

where B̄ is the antiparticle to a particle B, and Z_i are zero charge particles, which must appear by energy conservation. To define the K-theory, we identify any such set of zero charge particles with the identity, so that

B + B̄ ↔ 0

Thus the antiparticles provide the negative elements of K(T).

Granting the existence of antiparticles, this construction of K-theory can be more simply rephrased as the Grothendieck construction. We can define K(T) as the group of pairs (E, F) ∈ (Z^S, Z^S), subject to the relations (E, F) ≅ (E+B, F +B) ≅ (E+L, F +R) ≅ (E+R, F +L), where (L, R) are the left and right hand side of a binding process (1).

Thinking of these as particles, each brane B must have an antibrane, which we denote by B̄. If B wraps a submanifold L, one expects that B̄ is a brane which wraps a submanifold L of opposite orientation. A potential problem is that it is not a priori obvious that the orientation of L actually matters physically, especially in degenerate cases such as L a point.

Now, let us take X as a Calabi-Yau threefold for definiteness. A physical A-brane, which are branes of the A-model topological string and thereby a TQFT shadow of the D-branes of the superstring, is specified by a pair (L, E) of a special Lagrangian submanifold L with a flat bundle E. The obvious question could be: When are (L₁, E₁) and (L₂, E₂) related by a binding process? A simple heuristic answer to this question is given by the Feynman path integral. Two configurations are connected, if they are connected by a continuous path through the configuration space; any such path (or a small deformation of it) will appear in the functional integral with some non-zero weight. Thus, the question is essentially topological. Ignoring the flat bundles for a moment, this tells us that the K-theory group for A-branes is H₃(Y, Z), and the class of a brane is simply (rank E)·[L] ∈ H₃(Y, Z). This is also clear if the moduli space of flat connections on L is connected.

But suppose it is not, say π₁(L) is torsion. In this case, we need deeper physical arguments to decide whether the K-theory of these D-branes is H₃(Y, Z), or some larger group. But a natural conjecture is that it will be K₁(Y), which classifies bundles on odd-dimensional submanifolds. Two branes which differ only in the choice of flat connection are in fact connected in string theory, consistent with the K-group being H₃(Y, Z). For Y a simply connected Calabi-Yau threefold, K₁(Y) ≅ H₃(Y, Z), so the general conjecture is borne out in this case

There is a natural bilinear form on H₃(Y, Z) given by the oriented intersection number

I(L₁, L₂) = #([L₁] ∩ [L₂]) —– (2)

It has symmetry (−1)ⁿ. In particular, it is symplectic for n = 3. Furthermore, by Poincaré duality, it is unimodular, at least in our topological definition of K-theory.

D-branes, which are extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. The product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. Saying that a D-brane has RR-charge means that it is a source for an “RR potential,” a generalized (p + 1)-form gauge potential in ten-dimensional space-time, which can be verified from its world-volume action that contains a minimal coupling term,

∫C^{(p + 1)} —–(3)

where C^{(p + 1)} denotes the gauge potential, and the integral is taken over the (p+1)-dimensional world-volume of the brane. For p = 0, C⁽¹⁾ is a one-form or “vector” potential (as in Maxwell theory), and thus the D0-brane is an electrically charged particle with respect to this 10d Maxwell theory. Upon further compactification, by which, the ten dimensions are R⁴ × X, and a Dp-brane which wraps a p-dimensional cycle L; in other words its world-volume is R × L where R is a time-like world-line in R⁴. Using the Poincaré dual class ω_L ∈ H^2n−p(X, R) to L in X, to rewrite (3) as an integral

∫_{R × X} C^{(p + 1)} ∧ ω_L —– (4)

We can then do the integral over X to turn this into the integral of a one-form over a world-line in R⁴, which is the right form for the minimal electric coupling of a particle in four dimensions. Thus, such a wrapped brane carries a particular electric charge which can be detected at asymptotic infinity. Summarizing the RR-charge more formally,

∫_LC = ∫_XC ∧ ω_L —– (5)

where C ∈ H∗(X, R). In other words, it is a class in H_p(X, R).

In particular, an A-brane (for n = 3) carries a conserved charge in H₃(X, R). Of course, this is weaker than [L] ∈ H₃(X, Z). To see this physically, we would need to see that some of these “electric” charges are actually “magnetic” charges, and study the Dirac-Schwinger-Zwanziger quantization condition between these charges. This amounts to showing that the angular momentum J of the electromagnetic field satisfies the quantization condition J = ħn/2 for n ∈ Z. Using an expression from electromagnetism, J⃗ = E⃗ × B⃗ , this is precisely the condition that (2) must take an integer value. Thus the physical and mathematical consistency conditions agree. Similar considerations apply for coisotropic A-branes. If X is a genuine Calabi-Yau 3-fold (i.e., with strict SU(3) holonomy), then a coisotropic A-brane which is not a special Lagrangian must be five-dimensional, and the corresponding submanifold L is rationally homologically trivial, since H⁵(X, Q) = 0. Thus, if the bundle E is topologically trivial, the homology class of L and thus its K-theory class is torsion.

If X is a torus, or a K3 surface, the situation is more complicated. In that case, even rationally the charge of a coisotropic A-brane need not lie in the middle-dimensional cohomology of X. Instead, it takes its value in a certain subspace of ⊕_p H^p(X, Q), where the summation is over even or odd p depending on whether the complex dimension of X is even or odd. At the semiclassical level, the subspace is determined by the condition

(L − Λ)α = 0, α ∈ ⊕_p H^p(X, Q)

where L and Λ are generators of the Lefschetz SL(2, C) action, i.e., L is the cup product with the cohomology class of the Kähler form, and Λ is its dual.

Philosophical Identity of Derived Correspondences Between Smooth Varieties.

Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γ_f ⊂ X × Y of f, which (as a set) is defined as

Γ_f = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections p_X, p_Y from X × Y to the factors X, Y. Restricted to the subvariety Γ_f, p_X is an isomorphism (since f is a morphism). The fibres of p_Y restricted to Γ_f are just the fibres of f; so for example f is proper iff p_Y | Γ_f is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f_∗ : H(X) → H(Y)

This map can be expressed in terms of the graph Γ_f and the projection maps as

f_∗(α) = p_Y∗ (p_X^∗(α) ∪ [Γ_f]) —– (2)

where [Γ_f] ∈ H (X × Y) is the fundamental class of the subvariety [Γ_f]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that p_X be an isomorphism with some weaker assumption, such as p_X |Γ_f, p_Y | Γ_f finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ_∗ : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf O_Γ on X × Y. Associated to the projection maps p_X, p_Y, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ D^b(X × Y) with support which is proper over both factors. A derived correspondence defines a functor Φ_F by

Φ_F : D^b(X) → D^b(Y)
(−) ↦ Rp_Y∗(Lp_X^∗(−) ⊗^L F)

where (−) could refer to both objects and morphisms in D^b(X). F is sometimes called the kernel of the functor Φ_F.

The functor Φ_F is exact, as it is defined as a composite of exact functors. Since the projection p_X is flat, the derived pullback Lp_X^∗ is the same as ordinary pullback p_X^∗. Given derived correspondences E ∈ D^b(X × Y), F ∈ D^b(Y × Z), we obtain functors Φ_E : D^b(X) → D^b(Y), Φ_F : D^b(Y) → D^b(Z), which can then be composed to get a functor

Φ_F ◦ Φ_E : D^b(X) → D^b(Z)

which is a two-sided identity with respect to composition of kernels.

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,…, x_n) : f_i(x₁,…, x_n) = 0􏰑 ⊂ Aⁿ

of a set of polynomials f_i(x₁,…, x_n) in affine space Aⁿ with coordinates x₁,…, x_n. Associated to an affine variety is the ring A = C[X] of its regular functions, which is simply the ring C[x₁,…, x_n] modulo the ideal ⟨f_i⟩ 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,…, x_n) ∈ Cⁿ⁺¹ \ {0,…,0}

identified under the equivalence relation

(x₀,…, x_n) ∼ (λx₀,…, λx_n) for λ ∈ C^∗

Projective space can be decomposed into a union of (n + 1) affine pieces (Aⁿ)_i = 􏰐[x₀,…, x_n] : x_i ≠ 0􏰑 with n affine coordinates y_j = x_j/x_i. A projective variety X is the locus of common zeros of a set {f_i(x₁,…, x_n)} 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 {x_i}. The variety X is covered by the standard open sets X_i = 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 U_i, where every open piece U_i ⊂ Aⁿ is an affine variety with ring of global functions A_i = C[U_i]; further, the pieces U_i 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 ⊂ A^m, a morphism f : X → Y is given by an m-tuple of polynomials {f₁(x₁, . . . , x_n), . . . , f_m(x₁, . . . , x_n)} 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 U_i ⊂ Aⁿ. A point P ∈ U_i is nonsingular if, locally in the complex topology, a neighbourhood of P ∈ U_i 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 O_X. 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 O_X(U), also written Γ(U, O_X), the ring of regular functions on U.
2. Restriction: if V ⊂ U is an open subset, a restriction map res_UV : O_X(U) → O_X(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 res_UW = res_VW ◦ res_UV.
3. Sheaf Property: suppose that an open subset U ⊂ X is covered by a collection of open subsets {U_i}, and suppose that a set of regular functions f_i ∈ O_X(U_i) is given such that whenever U_i and U_j intersect, then the restrictions of f_i and f_j to U_i ∩ U_j agree. Then there is a unique function f ∈ O_X(U) whose restriction to U_i is f_i.

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 res_UV : 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 O_X-modules? The extra requirement is that the sections F(U) over an open set U form a module over the ring of regular functions O_X(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 O_X-modules is defined by the data of an A-module for every open subset U ⊂ X with ring of functions A = O_X(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.

Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category S_G whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of S_G by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In S_G the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S¹ we have the bundle P_g → S¹ obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and P_g is isomorphic to P_g^‘ iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space C_g = H(S,P_g) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category S_G a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that S_G is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is H_p. To say that H_p depends only on the G-bundle p^∗EG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p^∗₀EG → p^∗₁EG is the same as the set of homotopy classes of paths from p₀ to p₁. When S = S¹ the connected components of the space of maps correspond to the conjugacy classes in G: each bundle P_g corresponds to a specific point p_g in the mapping space, and a group element h defines a specific path from p_g to p_hgh⁻¹ .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using Vladimir Turaev‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.

Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S¹. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ₁ ◦ γ₂) = h(γ₁) · h(γ₂).

A G-equivariant TFT gives us for each element g ∈ G a vector space C_g, associated to the circle equipped with the bundle p_g whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to p_g1 and p_g2 on the two incoming circles, and to p_g1g2 on the outgoing circle, induces a product

C_g1 ⊗ C_g2 → C_g1g2 —– (1)

making C := ⊕_g∈GC_g into a G-graded algebra. Also there is a trace θ: C1  → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

C_g ⊗ C_g⁻¹ → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

α_h : C_g → C_hgh⁻¹ —– (2)

Diagramatically, α_h is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ C_h then α_h(φ) = φ. The reason for this is diagrammatically in the below figure.

If the holonomy along path P₂ is h then the holonomy along path P₁ is 1. However, a Dehn twist around the inner circle maps P₁ into P₂. Therefore, α_h(φ) = α₁(φ) = φ, if φ ∈ C_h.

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ₁ ∈ C_g1 and φ₂ ∈ C_g2 then

α_g2 (φ₁)φ₂ = φ₂φ₁ —– (3)

The necessity of this condition is illustrated in the figure below.

The trace of the identity map of C_g is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of α_g on C₁. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of α_g on C_h is the trace of α_h on C_g⁻¹. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh⁻¹g⁻¹. We can cut this torus along either of its generating circles to get a cobordism operator from C_c ⊗ C_h to C_h or from C_g⁻¹ ⊗ C_c to C_g⁻¹. If ψ ∈ C_{hgh⁻¹g⁻¹}. Let us introduce two linear transformations L_ψ, R_ψ associated to left- and right-multiplication by ψ. On the one hand, L_ψα_g : φ􏰀 ↦ ψ_αg(φ) is a map C_h → C_h. On the other hand R_ψα_h : φ ↦ α_h(φ)ψ is a map C_g⁻¹ → C_g⁻¹. The last sewing condition states that these two endomorphisms must have equal traces:

TrC_h 􏰌L_ψα_g􏰍 = TrC_g⁻¹ 􏰌R_ψα_h􏰍 —– (4)

(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆_g ∈ C_g ⊗ C_g⁻¹ be the “duality” element corresponding to the identity cobordism of (S¹,P_g) with both ends regarded as outgoing. We have ∆_g = ∑ξ_i ⊗ ξⁱ, where ξ_i and ξⁱ ru􏰟n through dual bases of C_g and C_g⁻¹. Let us also write

∆_h = ∑η_i ⊗ ηⁱ ∈ C_h ⊗ C_h⁻¹. Then (4) is easily seen to be equivalent to

∑α_h(ξ_i)ξⁱ = 􏰟 ∑η_iα_g(ηⁱ) —– (5)

in which both sides are elements of C_{hgh⁻¹g⁻¹}.

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 ξ_a = i_a(ξ) ∈ O_aa of each object a of B, and η ◦ ξ_a = ξ_b ◦ η for each morphism η ∈ O_ba from a to b. The family {ξ_a} thus constitutes a natural transformation from the identity functor 1_B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1_B. It is automatically commutative, for if {ξ_a}, {η_a} ∈ E then ξ_a ◦ η_a = η_a ◦ ξ_a by the definition of naturality. (A natural transformation from 1_B to 1_B is a collection of elements {ξ_a ∈ O_aa} such that ξ_a ◦ f = f ◦ ξ_b for each morphism f ∈ O_ab from b to a. But we can take a = b and f = η_a.) If B is a Frobenius category then there is a map π_a^b : O_bb → O_aa for each pair of objects a, b, and we can define j^b : O_bb → E by j^b(η)_a = π_a^b(η) for η ∈ O_bb. In other words, j^b is defined so that the Cardy condition ι_a ◦ j^b = π_a^b 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

θ_a(ι_a(ξ)η) = θ(ξj^a(η)) —– (1)

∀ ξ ∈ E and η ∈ O_aa. 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(ε_a) = θ_a(1_a)², where a is an irreducible object, and ε_a ∈ 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 O_aa is a commutative local ring of dimension greater than 1, then E = O_aa, and so ι_a : E → O_aa is an isomorphism, and its adjoint map j^a ought to be an isomorphism too. But that contradicts the Cardy condition, as π_a^a is multiplication by ∑ψ_iψⁱ, 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 HH^p(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 π_b^a(1_a) ∈ O_bb. 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

Y₀ →^φ₁ Y₁ →^φ₂ ··· →^φ_k Y_k —– (2)

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

(dF)(φ₁,…,φ_k+1) = F(φ₂,…,φ_k+1) ◦ φ₁ + ∑_i=1^k(−1)ⁱ F(φ₁,…,φ_i+1 ◦ φ_i,…,φ_k+1) + (−1)^k+1φ_k+1 ◦ F(φ₁,…,φ_k) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism F_Y 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(Y_k, Y₀) is the dual space of Hom(Y₀, Y_k) 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(Y_k, Y₀). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→^φ₀ Y₀ →^φ₁ Y1 →^φ₂···→^φ_k Y_k →^φ₀··· —– (4)

assigns a complex number F(φ₀,φ₁,…,φ_k).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ₀,φ₁,…,φ_k) 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.