Catastrophe, Gestalt and Thom’s Natural Philosophy of 3-D Space as Underlying All Abstract Forms – Thought of the Day 157.0

The main result of mathematical catastrophe theory consists in the classification of unfoldings (= evolutions around the center (the germ) of a dynamic system after its destabilization). The classification depends on two sorts of variables:

(a) The set of internal variables (= variables already contained in the germ of the dynamic system). The cardinal of this set is called corank,

(b) the set of external variables (= variables governing the evolution of the system). Its cardinal is called codimension.

The table below shows the elementary catastrophes for Thom:

The A-unfoldings are called cuspoids, the D-unfoldings umbilics. Applications of the E-unfoldings have only been considered in A geometric model of anorexia and its treatment. By loosening the condition for topological equivalence of unfoldings, we can enlarge the list, taking in the family of double cusps (X₉) which has codimension 8. The unfoldings A₃(the cusp) and A₅ (the butterfly) have a positive and a negative variant A₊₃, A_-3, A₊₅, A_-5.

We obtain Thorn’s original list of seven “catastrophes” if we consider only unfoldings up to codimension 4 and without the negative variants of A₃ and A₅.

Thom argues that “gestalts” are locally con­stituted by maximally four disjoint constituents which have a common point of equilibrium, a common origin. This restriction is ultimately founded in Gibb’s law of phases, which states that in three-dimensional space maximally four independent systems can be in equilibrium. In Thom’s natural philosophy, three-dimensional space is underlying all abstract forms. He, therefore, presumes that the restriction to four constituents in a “gestalt” is a kind of cognitive universal. In spite of the plausibility of Thom’s arguments there is a weaker assumption that the number of constituents in a gestalt should be finite and small. All unfoldings with codimension (i.e. number of external variables) smaller than or equal to 5 have simple germs. The unfoldings with corank (i.e. number of internal variables) greater than two have moduli. As a matter of fact the most prominent semantic archetypes will come from those unfoldings considered by René Thom in his sketch of catastrophe theoretic semantics.

Ricci-Flat Metric & Diffeomorphism – Does there Exist a Possibility of a Complete Construction of a Metric if the Surface isn’t a Smooth Manifold? Note Quote.

Using twistors, the Gibbons-Hawking ansatz is generalized to investigate 4n-dimensional hyperkähler metrics admitting an action of the n-torus Tⁿ. The hyperkähler case could further admit a tri-holomorphic free action of the n-torus Tⁿ. It turns out that the metric may be written in coordinates adapted to the torus action, in a form similar to the Gibbons-Hawking ansatz in dimension 4, and such that the non-linear Einstein equations reduce to a set of linear equations (essentially saying that certain functions on Euclidean 3-space are harmonic). In the case of 8-manifolds (n = 2), the solutions can be described geometrically, in terms of arrangements of 3-dimensional linear subspaces in Euclidean 6-space.

There are in fact many explicit examples known of metrics on non-compact manifolds with SU(n) or Sp(2n) holonomy. The other holonomy groups automatically yielding Ricci-flat metrics are the special holonomy groups G₂ in dimension 7 and Spin(7) in dimension 8. Until fairly recently only three explicit examples of complete metrics (in dimension 7) with G₂-holonomy and one explicit example (in dimension 8) with Spin(7)-holonomy were known. The G₂-holonomy examples are asymptotically conical and live on the bundle of self-dual two-forms over S⁴, the bundle of self-dual two-forms over CP², and the spin bundle of S³ (topologically R⁴ × S³), respectively. The metrics are of cohomogeneity one with respect to the Lie groups SO(5), SU(3) and SU(2) × SU(2) × SU(2) respectively. A cohomogeneity-one metric has a Lie group acting via isometries, with general (principal) orbits of real codimension one. In particular, if the metric is complete, then X is the holomorphic cotangent bundle of projective n-space T^∗CPⁿ, and the metric is the Calabi hyperkähler metric.

The G₂-holonomy examples are all examples in which a Lie group G acts with low codimension orbits. This is a general feature of explicit examples of Einstein metrics. The simplest case of such a situation would be when there is a single orbit of a group action, in which case the metric manifold is homogeneous. For metrics on homogeneous manifolds, the Einstein condition may be expressed purely algebraically. Moreover, all homogeneous Ricci-flat manifolds are flat, and so no interesting metrics occur. Then what about cohomogeneity one with respect to G, i.e., the orbits of G are codimension one in general? Here, the Einstein condition reduces to a system of non-linear ordinary differential equations in one variable, namely the parameter on the orbit space. In the Ricci-flat case, Cheeger-Gromoll theorem implies that the manifold has at most one end. In the non-compact case, the orbit space is R₊ and there is just one singular orbit. Geometrically, if the principal orbit is of the form G/K, the singular orbit (the bolt) is G/H for some subgroup H ⊃ K; if G is compact, a necessary and sufficient condition for the space to be a smooth manifold is that H/K is diffeomorphic to a sphere. In many cases, this is impossible because of the form of the group G, and so any metric constructed will not be complete.

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.

Network Theoretic of the Fermionic Quantum State – Epistemological Rumination. Thought of the Day 150.0

In quantum physics, fundamental particles are believed to be of two types: fermions or bosons, depending on the value of their spin (an intrinsic ‘angular moment’ of the particle). Fermions have half-integer spin and cannot occupy a quantum state (a configuration with specified microscopic degrees of freedom, or quantum numbers) that is already occupied. In other words, at most one fermion at a time can occupy one quantum state. The resulting probability that a quantum state is occupied is known as the Fermi-Dirac statistics.

Now, if we want to convert this into a model with maximum entropy, where the real movement is defined topologically, then we require a reproduction of heterogeneity that is observed. The starting recourse is network theory with an ensemble of networks where each vertex i has the same degree k_i as in the real network. This choice is justified by the fact that, being an entirely local topological property, the degree is expected to be directly affected by some intrinsic (non-topological) property of vertices. The caveat is that the real shouldn’t be compared with the randomized, which could otherwise lead to interpreting the observed as ‘unavoidable’ topological constraints, in the sense that the violation of the observed values would lead to an ‘impossible’, or at least very unrealistic values.

The resulting model is known as the Configuration Model, and is defined as a maximum-entropy ensemble of graphs with given degree sequence. The degree sequence, which is the constraint defining the model, is nothing but the ordered vector k of degrees of all vertices (where the i^th component k_i is the degree of vertex i). The ordering preserves the ‘identity’ of vertices: in the resulting network ensemble, the expected degree ⟨k_i⟩ of each vertex i is the same as the empirical value k_i for that vertex. In the Configuration Model, the graph probability is given by

P(A) = ∏_i<jq_ij(a_ij) =  ∏_i<jp_ij^a_ij (1 – p_ij)^1-a_ij —– (1)

where q_ij(a) = p_ij^{a (1 – p_ij)1-a} is the probability that particular entry of the adjacency matrix A takes the value a_ij = a, which is a Bernoulli process with different pairs of vertices characterized by different connection probabilities p_ij. A Bernoulli trial (or Bernoulli process) is the simplest random event, i.e. one characterized by only two possible outcomes. One of the two outcomes is referred to as the ‘success’ and is assigned a probability p. The other outcome is referred to as the ‘failure’, and is assigned the complementary probability 1 − p. These probabilities read

⟨a_ij⟩ = p_ij = (x_ix_j)/(1 + x_ix_j) —– (2)

where x_i is the Lagrange multiplier obtained by ensuring that the expected degree of the corresponding vertex i equals its observed value: ⟨k_i⟩ = k_i ∀ i. As always happens in maximum-entropy ensembles, the probabilistic nature of configurations implies that the constraints are valid only on average (the angular brackets indicate an average over the ensemble of realizable networks). Also note that p_ij is a monotonically increasing function of x_i and x_j. This implies that ⟨k_i⟩ is a monotonically increasing function of x_i. An important consequence is that two variables i and j with the same degree k_i = kj must have the same value x_i = x_j.

(2) provides an interesting connection with quantum physics, and in particular the statistical mechanics of fermions. The ‘selection rules’ of fermions dictate that only one particle at a time can occupy a single-particle state, exactly as each pair of vertices in binary networks can be either connected or disconnected. In this analogy, every pair i, j of vertices is a ‘quantum state’ identified by the ‘quantum numbers’ i and j. So each link of a binary network is like a fermion that can be in one of the available states, provided that no two objects are in the same state. (2) indicates the expected number of particles/links in the state specified by i and j. With no surprise, it has the same form of the so-called Fermi-Dirac statistics describing the expected number of fermions in a given quantum state. The probabilistic nature of links allows also for the presence of empty states, whose occurrence is now regulated by the probability coefficients (1 − p_ij). The Configuration Model allows the whole degree sequence of the observed network to be preserved (on average), while randomizing other (unconstrained) network properties. now, when one compares the higher-order (unconstrained) observed topological properties with their expected values calculated over the maximum-entropy ensemble, it should be indicative of the fact that the degree of sequence is informative in explaining the rest of the topology, which is a consequent via probabilities in (2). Colliding these into a scatter plot, the agreement between model and observations can be simply assessed as follows: the less scattered the cloud of points around the identity function, the better the agreement between model and reality. In principle, a broadly scattered cloud around the identity function would indicate the little effectiveness of the chosen constraints in reproducing the unconstrained properties, signaling the presence of genuine higher-order patterns of self-organization, not simply explainable in terms of the degree sequence alone. Thus, the ‘fermionic’ character of the binary model is the mere result of the restriction that no two binary links can be placed between any two vertices, leading to a mathematical result which is formally equivalent to the one of quantum statistics.

The Canonical of a priori and a posteriori Variational Calculus as Phenomenologically Driven. Note Quote.

The expression variational calculus usually identifies two different but related branches in Mathematics. The first aimed to produce theorems on the existence of solutions of (partial or ordinary) differential equations generated by a variational principle and it is a branch of local analysis (usually in Rⁿ); the second uses techniques of differential geometry to deal with the so-called variational calculus on manifolds.

The local-analytic paradigm is often aimed to deal with particular situations, when it is necessary to pay attention to the exact definition of the functional space which needs to be considered. That functional space is very sensitive to boundary conditions. Moreover, minimal requirements on data are investigated in order to allow the existence of (weak) solutions of the equations.

On the contrary, the global-geometric paradigm investigates the minimal structures which allow to pose the variational problems on manifolds, extending what is done in Rⁿ but usually being quite generous about regularity hypotheses (e.g. hardly ever one considers less than C^∞-objects). Since, even on manifolds, the search for solutions starts with a local problem (for which one can use local analysis) the global-geometric paradigm hardly ever deals with exact solutions, unless the global geometric structure of the manifold strongly constrains the existence of solutions.

A further a priori different approach is the one of Physics. In Physics one usually has field equations which are locally given on a portion of an unknown manifold. One thence starts to solve field equations locally in order to find a local solution and only afterwards one tries to find the maximal analytical extension (if any) of that local solution. The maximal extension can be regarded as a global solution on a suitable manifold M, in the sense that the extension defines M as well. In fact, one first proceeds to solve field equations in a coordinate neighbourhood; afterwards, one changes coordinates and tries to extend the found solution out of the patches as long as it is possible. The coordinate changes are the cocycle of transition functions with respect to the atlas and they define the base manifold M. This approach is essential to physical applications when the base manifold is a priori unknown, as in General Relativity, and it has to be determined by physical inputs.

Luckily enough, that approach does not disagree with the standard variational calculus approach in which the base manifold M is instead fixed from the very beginning. One can regard the variational problem as the search for a solution on that particular base manifold. Global solutions on other manifolds may be found using other variational principles on different base manifolds. Even for this reason, the variational principle should be universal, i.e. one defines a family of variational principles: one for each base manifold, or at least one for any base manifold in a “reasonably” wide class of manifolds. The strong requirement, which is physically motivated by the belief that Physics should work more or less in the same way regardless of the particular spacetime which is actually realized in Nature. Of course, a scenario would be conceivable in which everything works because of the particular (topological, differentiable, etc.) structure of the spacetime. This position, however, is not desirable from a physical viewpoint since, in this case, one has to explain why that particular spacetime is realized (a priori or a posteriori).

In spite of the aforementioned strong regularity requirements, the spectrum of situations one can encounter is unexpectedly wide, covering the whole of fundamental physics. Moreover, it is surprising how the geometric formalism is effectual for what concerns identifications of basic structures of field theories. In fact, just requiring the theory to be globally well-defined and to depend on physical data only, it often constrains very strongly the choice of the local theories to be globalized. These constraints are one of the strongest motivations in choosing a variational approach in physical applications. Another motivation is a well formulated framework for conserved quantities. A global- geometric framework is a priori necessary to deal with conserved quantities being non-local.

In the modem perspective of Quantum Field Theory (QFT) the basic object encoding the properties of any quantum system is the action functional. From a quantum viewpoint the action functional is more fundamental than field equations which are obtained in the classical limit. The geometric framework provides drastic simplifications of some key issues, such as the definition of the variation operator. The variation is deeply geometric though, in practice, it coincides with the definition given in the local-analytic paradigm. In the latter case, the functional derivative is usually the directional derivative of the action functional which is a function on the infinite-dimensional space of fields defined on a region D together with some boundary conditions on the boundary ∂D. To be able to define it one should first define the functional space, then define some notion of deformation which preserves the boundary conditions (or equivalently topologize the functional space), define a variation operator on the chosen space, and, finally, prove the most commonly used properties of derivatives. Once one has done it, one finds in principle the same results that would be found when using the geometric definition of variation (for which no infinite dimensional space is needed). In fact, in any case of interest for fundamental physics, the functional derivative is simply defined by means of the derivative of a real function of one real variable. The Lagrangian formalism is a shortcut which translates the variation of (infinite dimensional) action functionals into the variation of the (finite dimensional) Lagrangian structure.

Another feature of the geometric framework is the possibility of dealing with non-local properties of field theories. There are, in fact, phenomena, such as monopoles or instantons, which are described by means of non-trivial bundles. Their properties are tightly related to the non-triviality of the configuration bundle; and they are relatively obscure when regarded by any local paradigm. In some sense, a local paradigm hides global properties in the boundary conditions and in the symmetries of the field equations, which are in turn reflected in the functional space we choose and about which, it being infinite dimensional, we do not know almost anything a priori. We could say that the existence of these phenomena is a further hint that field theories have to be stated on bundles rather than on Cartesian products. This statement, if anything, is phenomenologically driven.

When a non-trivial bundle is involved in a field theory, from a physical viewpoint it has to be regarded as an unknown object. As for the base manifold, it has then to be constructed out of physical inputs. One can do that in (at least) two ways which are both actually used in applications. First of all, one can assume the bundle to be a natural bundle which is thence canonically constructed out of its base manifold. Since the base manifold is identified by the (maximal) extension of the local solutions, then the bundle itself is identified too. This approach is the one used in General Relativity. In these applications, bundles are gauge natural and they are therefore constructed out of a structure bundle P, which, usually, contains extra information which is not directly encoded into the spacetime manifolds. In physical applications the structure bundle P has also to be constructed out of physical observables. This can be achieved by using gauge invariance of field equations. In fact, two local solutions differing by a (pure) gauge transformation describe the same physical system. Then while extending from one patch to another we feel free both to change coordinates on M and to perform a (pure) gauge transformation before glueing two local solutions. Then coordinate changes define the base manifold M, while the (pure) gauge transformations form a cocycle (valued in the gauge group) which defines, in fact, the structure bundle P. Once again solutions with different structure bundles can be found in different variational principles. Accordingly, the variational principle should be universal with respect to the structure bundle.

Local results are by no means less important. They are often the foundations on which the geometric framework is based on. More explicitly, Variational Calculus is perhaps the branch of mathematics that possibilizes the strongest interaction between Analysis and Geometry.

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

… → M^i-1 →^di-1 Mⁱ →^di → Mⁱ⁺¹ → …

such that dⁱ ◦ d^i-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 dⁱ_M, dⁱ_N 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 i^th cohomology object is the quotient

Hⁱ(M) = ker(dⁱ)/im(dⁱ⁻¹)

This operation of taking cohomology at the i^th 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 ·· · ·· f_r−1 · f_r where each f_i 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 ∼ id_b and s⁻¹ · s ∼ id_a 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.

Coarse Philosophies of Coarse Embeddabilities: Metric Space Conjectures Act Algorithmically On Manifolds – Thought of the Day 145.0

A coarse structure on a set X is defined to be a collection of subsets of X × X, called the controlled sets or entourages for the coarse structure, which satisfy some simple axioms. The most important of these states that if E and F are controlled then so is

E ◦ F := {(x, z) : ∃y, (x, y) ∈ E, (y, z) ∈ F}

Consider the metric spaces Zⁿ and Rⁿ. Their small-scale structure, their topology is entirely different, but on the large scale they resemble each other closely: any geometric configuration in Rⁿ can be approximated by one in Zⁿ, to within a uniformly bounded error. We think of such spaces as “coarsely equivalent”. The other axioms require that the diagonal should be a controlled set, and that subsets, transposes, and (finite) unions of controlled sets should be controlled. It is accurate to say that a coarse structure is the large-scale counterpart of a uniformity than of a topology.

Coarse structures and coarse spaces enjoy a philosophical advantage over coarse metric spaces, in that, all left invariant bounded geometry metrics on a countable group induce the same metric coarse structure which is therefore transparently uniquely determined by the group. On the other hand, the absence of a natural gauge complicates the notion of a coarse family, while it is natural to speak of sets of uniform size in different metric spaces it is not possible to do so in different coarse spaces without imposing additional structure.

Mikhail Leonidovich Gromov introduced the notion of coarse embedding for metric spaces. Let X and Y be metric spaces.

A map f : X → Y is said to be a coarse embedding if ∃ nondecreasing functions ρ₁ and ρ₂ from R₊ = [0, ∞) to R such that

• ρ₁(d(x,y)) ≤ d(f(x),f(y)) ≤ ρ₂(d(x,y)) ∀ x, y ∈ X.
• lim_r→∞ ρ_i(r) = +∞ (i=1, 2).

Intuitively, coarse embeddability of a metric space X into Y means that we can draw a picture of X in Y which reflects the large scale geometry of X. In early 90’s, Gromov suggested that coarse embeddability of a discrete group into Hilbert space or some Banach spaces should be relevant to solving the Novikov conjecture. The connection between large scale geometry and differential topology and differential geometry, such as the Novikov conjecture, is built by index theory. Recall that an elliptic differential operator D on a compact manifold M is Fredholm in the sense that the kernel and cokernel of D are finite dimensional. The Fredholm index of D, which is defined by

index(D) = dim(kerD) − dim(cokerD),

has the following fundamental properties:

(1) it is an obstruction to invertibility of D;

(2) it is invariant under homotopy equivalence.

The celebrated Atiyah-Singer index theorem computes the Fredholm index of elliptic differential operators on compact manifolds and has important applications. However, an elliptic differential operator on a noncompact manifold is in general not Fredholm in the usual sense, but Fredholm in a generalized sense. The generalized Fredholm index for such an operator is called the higher index. In particular, on a general noncompact complete Riemannian manifold M, John Roe (Coarse Cohomology and Index Theory on Complete Riemannian Manifolds) introduced a higher index theory for elliptic differential operators on M.

The coarse Baum-Connes conjecture is an algorithm to compute the higher index of an elliptic differential operator on noncompact complete Riemannian manifolds. By the descent principal, the coarse Baum-Connes conjecture implies the Novikov higher signature conjecture. Guoliang Yu has proved the coarse Baum-Connes conjecture for bounded geometry metric spaces which are coarsely embeddable into Hilbert space. The metric spaces which admit coarse embeddings into Hilbert space are a large class, including e.g. all amenable groups and hyperbolic groups. In general, however, there are counterexamples to the coarse Baum-Connes conjecture. A notorious one is expander graphs. On the other hand, the coarse Novikov conjecture (i.e. the injectivity part of the coarse Baum-Connes conjecture) is an algorithm of determining non-vanishing of the higher index. Kasparov-Yu have proved the coarse Novikov conjecture for spaces which admit coarse embeddings into a uniformly convex Banach space.

The Affinity of Mirror Symmetry to Algebraic Geometry: Going Beyond Formalism

Even though formalism of homological mirror symmetry is an established case, what of other explanations of mirror symmetry which lie closer to classical differential and algebraic geometry? One way to tackle this is the so-called Strominger, Yau and Zaslow mirror symmetry or SYZ in short.

The central physical ingredient in this proposal is T-duality. To explain this, let us consider a superconformal sigma model with target space (M, g), and denote it (defined as a geometric functor, or as a set of correlation functions), as

CFT(M, g)

In physics, a duality is an equivalence

CFT(M, g) ≅ CFT(M′, g′)

which holds despite the fact that the underlying geometries (M,g) and (M′, g′) are not classically diffeomorphic.

T-duality is a duality which relates two CFT’s with toroidal target space, M ≅ M′ ≅ T^d, but different metrics. In rough terms, the duality relates a “small” target space, with noncontractible cycles of length L < l_s, with a “large” target space in which all such cycles have length L > l_s.

This sort of relation is generic to dualities and follows from the following logic. If all length scales (lengths of cycles, curvature lengths, etc.) are greater than l_s, string theory reduces to conventional geometry. Now, in conventional geometry, we know what it means for (M, g) and (M′, g′) to be non-isomorphic. Any modification to this notion must be associated with a breakdown of conventional geometry, which requires some length scale to be “sub-stringy,” with L < l_s. To state T-duality precisely, let us first consider M = M′ = S¹. We parameterise this with a coordinate X ∈ R making the identification X ∼ X + 2π. Consider a Euclidean metric g_R given by ds² = R²dX². The real parameter R is usually called the “radius” from the obvious embedding in R². This manifold is Ricci-flat and thus the sigma model with this target space is a conformal field theory, the “c = 1 boson.” Let us furthermore set the string scale l_s = 1. With this, we attain a complete physical equivalence.

CFT(S¹, g_R) ≅ CFT(S¹, g_1/R)

Thus these two target spaces are indistinguishable from the point of view of string theory.

Just to give a physical picture for what this means, suppose for sake of discussion that superstring theory describes our universe, and thus that in some sense there must be six extra spatial dimensions. Suppose further that we had evidence that the extra dimensions factorized topologically and metrically as K₅ × S¹; then it would make sense to ask: What is the radius R of this S¹ in our universe? In principle this could be measured by producing sufficiently energetic particles (so-called “Kaluza-Klein modes”), or perhaps measuring deviations from Newton’s inverse square law of gravity at distances L ∼ R. In string theory, T-duality implies that R ≥ l_s, because any theory with R < l_s is equivalent to another theory with R > l_s. Thus we have a nontrivial relation between two (in principle) observable quantities, R and l_s, which one might imagine testing experimentally. Let us now consider the theory CFT(T^d, g), where T^d is the d-dimensional torus, with coordinates Xⁱ parameterising R^d/2πZ^d, and a constant metric tensor g_ij. Then there is a complete physical equivalence

CFT(T^d, g) ≅ CFT(T^d, g⁻¹)

In fact this is just one element of a discrete group of T-duality symmetries, generated by T-dualities along one-cycles, and large diffeomorphisms (those not continuously connected to the identity). The complete group is isomorphic to SO(d, d; Z).

While very different from conventional geometry, T-duality has a simple intuitive explanation. This starts with the observation that the possible embeddings of a string into X can be classified by the fundamental group π₁(X). Strings representing non-trivial homotopy classes are usually referred to as “winding states.” Furthermore, since strings interact by interconnecting at points, the group structure on π₁ provided by concatenation of based loops is meaningful and is respected by interactions in the string theory. Now π₁(T^d) ≅ Z^d, as an abelian group, referred to as the group of “winding numbers”.

Of course, there is another Z^d we could bring into the discussion, the Pontryagin dual of the U(1)^d of which T^d is an affinization. An element of this group is referred to physically as a “momentum,” as it is the eigenvalue of a translation operator on T^d. Again, this group structure is respected by the interactions. These two group structures, momentum and winding, can be summarized in the statement that the full closed string algebra contains the group algebra C[Z^d] ⊕ C[Z^d].

In essence, the point of T-duality is that if we quantize the string on a sufficiently small target space, the roles of momentum and winding will be interchanged. But the main point can be seen by bringing in some elementary spectral geometry. Besides the algebra structure, another invariant of a conformal field theory is the spectrum of its Hamiltonian H (technically, the Virasoro operator L₀ + L ^̄₀). This Hamiltonian can be thought of as an analog of the standard Laplacian ∆_g on functions on X, and its spectrum on T^d with metric g is

Spec ∆_g = {∑_i,j=1^d g^ijp_ip_j; p_i ∈ Z^d}

On the other hand, the energy of a winding string is (intuitively) a function of its length. On our torus, a geodesic with winding number w ∈ Z^d has length squared

L² = ∑_i,j=1^d g_ijwⁱw^j

Now, the only string theory input we need to bring in is that the total Hamiltonian contains both terms,

H = ∆_g + L² + · · ·

where the extra terms … express the energy of excited (or “oscillator”) modes of the string. Then, the inversion g → g⁻¹, combined with the interchange p ↔ w, leaves the spectrum of H invariant. This is T-duality.

There is a simple generalization of the above to the case with a non-zero B-field on the torus satisfying dB = 0. In this case, since B is a constant antisymmetric tensor, we can label CFT’s by the matrix g + B. Now, the basic T-duality relation becomes

CFT(T^d, g + B) ≅ CFT(T^d, (g + B)⁻¹)

Another generalization, which is considerably more subtle, is to do T-duality in families, or fiberwise T-duality. The same arguments can be made, and would become precise in the limit that the metric on the fibers varies on length scales far greater than l_s, and has curvature lengths far greater than l_s. This is sometimes called the “adiabatic limit” in physics. While this is a very restrictive assumption, there are more heuristic physical arguments that T-duality should hold more generally, with corrections to the relations proportional to curvatures l_s²R and derivatives l_s∂ of the fiber metric, both in perturbation theory and from world-sheet instantons.

Categories of Pointwise Convergence Topology: Theory(ies) of Bundles.

Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {U_i}_i∈I of X, the transition functions between trivializations P|_Ui can be lifted to bundle isomorphisms g_ij on double intersections U_ij = U_i ∩ U_j which are projectively coherent, i.e., over each of the triple intersections U_ijk = U_i ∩ U_j ∩ U_k the composition g_ki g_jk g_ij is given as multiplication by a U(1)-valued function f_ijk : U_ijk → U(1). The collection {(U_ij, f_ijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class B_P in the sheaf cohomology group H²(X, U(1)_X). On the other hand, the sheaf cohomology H¹(X, PU(H)_X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H¹(X,PU(H)_X).

There is an isomorphism

H¹(X, PU(H)_X) →^≈ H²(X, U(1)_X) provided by the

boundary map [P] ↦ B_P. There is also an isomorphism

H²(X, U(1)_X) →^≈ H³(X, Z_X) ≅ H³(X, Z)

The image δ(P) ∈ H³(X, Z) of B_P is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H³(X, R) by a closed three-form H on X, called the H-flux of the given B-field B_P, we will write P = P_H. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H³(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Z_n with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cⁿ). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element B_P ∈ H²(X, U(1)_X) comes from H²(X, (Z_n)_X), and therefore its order divides n.

One also has δ(P₁ ⊗ P₂) = δ(P₁) + δ(P₂) and δ(P^∨) = −δ(P). This follows from the commutative diagram

and the fact that P^∨ ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H³(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that P_x = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z₂-graded Hilbert space. We define Fred⁰(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T² − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred⁰(H) ֒→ B(H) × K(H) given by T → (T, T² − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = P_H → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred⁰(P) whose fibres are Fred⁰(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K⁰(X, H) = [X, Fred⁰(P_H)]

The group K⁰(X, H) depends functorially on the pair (X, P_H), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K⁰(X, H) → K⁰(X, H′). Addition in K⁰(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K⁰(X, H₂) with [H₂] = δ(P ⊗ P(C²)) = δ(P) = [H]. Under the isomorphism H ⊗ C² ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C²) for any projective bundle P and so K⁰(X, H₂) is canonically isomorphic to K⁰(X, H). When [H] is a non-torsion element of H³(X, Z), so that P = P_H is an infinite-dimensional bundle of projective spaces, then the index map K⁰(X, H) → Z is zero, i.e., any section of Fred⁰(P) takes values in the index zero component of Fred⁰(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle P_H we can associate a bundle of compact operators E_H → X given by

E_H = P_H ×_PU K

with δ(E_H) = [H]. The Banach algebra A_H := C₀(X, E_H) of continuous sections of E_H vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K^•(X, H) of X is canonically isomorphic to the algebraic K-theory group K_•(A_H).

We will also need a smooth version of this definition. Let A^∞_H be the smooth subalgebra of A_H given by the algebra CT^∞(X, H) = C^∞(X, L¹_PH),

where L¹_PH = P_H ×_PUL¹. Then the inclusion CT^∞(X, H) → CT(X, H) induces an isomorphism K_•CT(X, H) →^≈ K_•CT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism K_•CT^∞(X, H) ≅ K^•(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = P_H over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K⁰(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, M_m(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism

δ : Br(X) →^≈ tor(H³(X, Z))

where tor(H³(X, Z)) is the torsion subgroup of H³(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K₀(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H³(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K⁰(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K⁰(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K₀(C(X)) = K⁰(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre C^nm ≅ (C^m)^⊕n, which is naturally an M_m(C)-module.

 

Ringed Spaces (1)

A ringed space is a broad concept in which we can fit most of the interesting geometrical objects. It consists of a topological space together with a sheaf of functions on it.

Let M be a differentiable manifold, whose topological space is Hausdorff and second countable. For each open set U ⊂ M , let C^∞(U) be the R-algebra of smooth functions on U .

The assignment

U ↦ C^∞(U)

satisfies the following two properties:

(1) If U ⊂ V are two open sets in M, we can define the restriction map, which is an algebra morphism:

r_{V, U} : C^∞(V) → C^∞(U), ƒ ↦ ƒ|_U

which is such that

i) r_{U, U} = id

ii) r_{W, U} = r_{V, U} ○ r_{W, V}

(2) Let {U_i}_i∈I be an open covering of U and let {ƒ_i}_i∈I, ƒ_i ∈ C^∞(U_i) be a family such that ƒ_i|_{Ui ∩ Uj} = ƒ_j| _{Ui ∩ Uj} ∀ i, j ∈ I. In other words the elements of the family {ƒ_i}_i∈I agree on the intersection of any two open sets U_i ∩ U_j. Then there exists a unique ƒ ∈ C^∞(U) such that ƒ|_Ui = ƒ_i.

Such an assignment is called a sheaf. The pair (M, C^∞), consisting of the topological space M, underlying the differentiable manifold, and the sheaf of the C^∞ functions on M is an example of locally ringed space (the word “locally” refers to a local property of the sheaf of C^∞ functions.

Given two manifolds M and N, and the respective sheaves of smooth functions C_M^∞ and C_N^∞, a morphism ƒ from M to N, viewed as ringed spaces, is a morphism |ƒ|: M → N of the underlying topological spaces together with a morphism of algebras,

ƒ^*: C_N^∞(V) →  C_M^∞(ƒ^-1(V)), ƒ^*(φ)(x) = φ(|ƒ|(x))

compatible with the restriction morphisms.

Notice that, as soon as we give the continuous map |ƒ| between the topological spaces, the morphism ƒ^* is automatically assigned. This is a peculiarity of the sheaf of smooth functions on a manifold. Such a property is no longer true for a generic ringed space and, in particular, it is not true for supermanifolds.

A morphism of differentiable manifolds gives rise to a unique (locally) ringed space morphism and vice versa.

Moreover, given two manifolds, they are isomorphic as manifolds iff they are isomorphic as (locally) ringed spaces. In the language of categories, we say we have a fully faithful functor from the category of manifolds to the category of locally ringed spaces.

The generalization of algebraic geometry to the super-setting comes somehow more naturally than the similar generalization of differentiable geometry. This is because the machinery of algebraic geometry was developed to take already into account the presence of (even) nilpotents and consequently, the language is more suitable to supergeometry.

Let X be an affine algebraic variety in the affine space Aⁿ over an algebraically closed field k and let O(X) = k[x₁,…., x_n]/I be its coordinate ring, where the ideal I is prime. This corresponds topologically to the irreducibility of the variety X. We can think of the points of X as the zeros of the polynomials in the ideal I in Aⁿ. X is a topological space with respect to the Zariski topology, whose closed sets are the zeros of the polynomials in the ideals of O(X). For each open U in X, consider the assignment

U ↦ O_X(U)

where O_X(U) is the k-algebra of regular functions on U. By definition, these are the functions ƒ X → k that can be expressed as a quotient of two polynomials at each point of U ⊂ X. The assignment U ↦ O_X(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, O_X) is another example of a (locally) ringed space.