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.

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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 TCPn, and the metric is the Calabi hyperkähler metric.

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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.

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

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

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

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

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

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

(x0,…, xn) ∈ Cn+1 \ {0,…,0}

identified under the equivalence relation

(x0,…, xn) ∼ (λx0,…, λxn) for λ ∈ C

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

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

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

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

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

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

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

The Natural Theoretic of Electromagnetism. Thought of the Day 147.0

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In Maxwell’s theory, the field strength F = 1/2Fμν dxμ ∧ dxν is a real 2-form on spacetime, and thence a natural object at the same time. The homogeneous Maxwell equation dF = 0 is an equation involving forms and it has a well-known local solution F = dA’, i.e. there exists a local spacetime 1-form A’ which is a potential for the field strength F. Of course, if spacetime is contractible, as e.g. for Minkowski space, the solution is also a global one. As is well-known, in the non-commutative Yang-Mills theory case the field strength F = 1/2FAμν TA ⊗ dxμ ∧ dxν is no longer a spacetime form. This is a somewhat trivial remark since the transformation laws of such field strength are obtained as the transformation laws of the curvature of a principal connection with values in the Lie algebra of some (semisimple) non-Abelian Lie group G (e.g. G = SU(n), n 2 ≥ 2). However, the common belief that electromagnetism is to be intended as the particular case (for G =U(1)) of a non-commutative theory is not really physically evident. Even if we subscribe this common belief, which is motivated also by the tremendous success of the quantized theory, let us for a while discuss electromagnetism as a standalone theory.

From a mathematical viewpoint this is a (different) approach to electromagnetism and the choice between the two can be dealt with on a physical ground only. Of course the 1-form A’ is defined modulo a closed form, i.e. locally A” = A’ + dα is another solution.

How can one decide whether the potential of electromagnetism should be considered as a 1-form or rather as a principal connection on a U(1)-bundle? First of all we notice that by a standard hole argument (one can easily define compact supported closed 1-forms, e.g. by choosing the differential of compact supported functions which always exist on a paracompact manifold) the potentials A and A’ represent the same physical situation. On the other hand, from a mathematical viewpoint we would like the dynamical field, i.e. the potential A’, to be a global section of some suitable configuration bundle. This requirement is a mathematical one, motivated on the wish of a well-defined geometrical perspective based on global Variational Calculus.

The first mathematical way out is to restrict attention to contractible spacetimes, where A’ may be always chosen to be global. Then one can require the gauge transformations A” = A’ + dα to be Lagrangian symmetries. In this way, field equations select a whole equivalence class of gauge-equivalent potentials, a procedure which solves the hole argument problem. In this picture the potential A’ is really a 1-form, which can be dragged along spacetime diffeomorphism and which admits the ordinary Lie derivatives of 1-forms. Unfortunately, the restriction to contractible spacetimes is physically unmotivated and probably wrong.

Alternatively, one can restrict electromagnetic fields F, deciding that only exact 2-forms F are allowed. That actually restricts the observable physical situations, by changing the homogeneous Maxwell equations (i.e. Bianchi identities) by requiring that F is not only closed but exact. One should in principle be able to empirically reject this option.

On non-contractible spacetimes, one is necessarily forced to resort to a more “democratic” attitude. The spacetime is covered by a number of patches Uα. On each patch Uα one defines a potential A(α). In the intersection of two patches the two potentials A(α) and A(β) may not agree. In each patch, in fact, the observer chooses his own conventions and he finds a different representative of the electromagnetic potential, which is related by a gauge transformation to the representatives chosen in the neighbour patch(es). Thence we have a family of gauge transformations, one in each intersection Uαβ, which obey cocycle identities. If one recognizes in them the action of U(1) then one can build a principal bundle P = (P, M, π; U(1)) and interpret the ensuing potential as a connection on P. This leads way to the gauge natural formalism.

Anyway this does not close the matter. One can investigate if and when the principal bundle P, in addition to the obvious principal structure, can be also endowed with a natural structure. If that were possible then the bundle of connections Cp (which is associated to P) would also be natural. The problem of deciding whether a given gauge natural bundle can be endowed with a natural structure is quite difficult in general and no full theory is yet completely developed in mathematical terms. That is to say, there is no complete classification of the topological and differential geometric conditions which a principal bundle P has to satisfy in order to ensure that, among the principal trivializations which determine its gauge natural structure, one can choose a sub-class of trivializations which induce a purely natural bundle structure. Nor it is clear how many inequivalent natural structures a good principal bundle may support. Though, there are important examples of bundles which support at the same time a natural and a gauge natural structure. Actually any natural bundle is associated to some frame bundle L(M), which is principal; thence each natural bundle is also gauge natural in a trivial way. Since on any paracompact manifold one can choose a global Riemannian metric g, the corresponding tangent bundle T(M) can be associated to the orthonormal frame bundle O(M, g) besides being obviously associated to L(M). Thence the natural bundle T(M) may be also endowed with a gauge natural bundle structure with structure group O(m). And if M is orientable the structure can be further reduced to a gauge natural bundle with structure group SO(m).

Roughly speaking, the task is achieved by imposing restrictions to cocycles which generate T(M) according to the prescription by imposing a privileged class of changes of local laboratories and sets of measures. Imposing the cocycle ψ(αβ) to take its values in O(m) rather than in the larger group GL(m). Inequivalent gauge natural structures are in one-to-one correspondence with (non isometric) Riemannian metrics on M. Actually whenever there is a Lie group homomorphism ρ : GU(m) → G for some s onto some given Lie group G we can build a natural G-principal bundle on M. In fact, let (Uα, ψ(α)) be an atlas of the given manifold M, ψ(αβ) be its transition functions and jψ(αβ) be the induced transition functions of L(M). Then we can define a G-valued cocycle on M by setting ρ(jψ(αβ)) and thence a (unique up to fibered isomorphisms) G-principal bundle P(M) = (P(M), M, π; G). The bundle P(M), as well as any gauge natural bundle associated to it, is natural by construction. Now, defining a whole family of natural U(1)-bundles Pq(M) by using the bundle homomorphisms

ρq: GL(m) → U(1): J ↦ exp(iq ln det|J|) —– (1)

where q is any real number and In denotes the natural logarithm. In the case q = 0 the image of ρ0 is the trivial group {I}; and, all the induced bundles are trivial, i.e. P = M x U(1).

The natural lift φ’ of a diffeomorphism φ: M → M is given by

φ'[x, e]α = [φ(x), eiq ln det|J|. e]α —– (2)

where J is the Jacobin of the morphism φ. The bundles Pq(M) are all trivial since they allow a global section. In fact, on any manifold M, one can define a global Riemannian metric g, where the local sections glue together.

Since the bundles Pq(M) are all trivial, they are all isomorphic to M x U(1) as principal U(1)-bundles, though in a non-canonical way unless q = 0. Any two of the bundles Pq1(M) and Pq2(M) for two different values of q are isomorphic as principal bundles but the isomorphism obtained is not the lift of a spacetime diffeomorphism because of the two different values of q. Thence they are not isomorphic as natural bundles. We are thence facing a very interesting situation: a gauge natural bundle C associated to the trivial principal bundle P can be endowed with an infinite family of natural structures, one for each q ∈ R; each of these natural structures can be used to regard principal connections on P as natural objects on M and thence one can regard electromagnetism as a natural theory.

Now that the mathematical situation has been a little bit clarified, it is again a matter of physical interpretation. One can in fact restrict to electromagnetic potentials which are a priori connections on a trivial structure bundle P ≅ M x U(1) or to accept that more complicated situations may occur in Nature. But, non-trivial situations are still empirically unsupported, at least at a fundamental level.

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

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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.
  • limr→∞ ρi(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.

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 {Ui}i∈I of X, the transition functions between trivializations P|Ui can be lifted to bundle isomorphisms gij on double intersections Uij = Ui ∩ Uj which are projectively coherent, i.e., over each of the triple intersections Uijk = Ui ∩ Uj ∩ Uk the composition gki gjk gij is given as multiplication by a U(1)-valued function fijk : Uijk → U(1). The collection {(Uij, fijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class BP 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] ↦ BP. There is also an isomorphism

H2(X, U(1)X) → H3(X, ZX) ≅ H3(X, Z)

The image δ(P) ∈ H3(X, Z) of BP 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 BP, we will write P = PH. 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

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where we identify the cyclic group Zn 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 BP ∈ H2(X, U(1)X) comes from H2(X, (Zn)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

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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 Px = 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 = PH → 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(PH)]

The group K0(X, H) depends functorially on the pair (X, PH), 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 = PH 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 PH we can associate a bundle of compact operators EH → X given by

EH = PH ×PU K

with δ(EH) = [H]. The Banach algebra AH := C0(X, EH) of continuous sections of EH 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(AH).

We will also need a smooth version of this definition. Let AH be the smooth subalgebra of AH given by the algebra CT(X, H) = C(X, L1PH),

where L1PH = PH ×PUL1. Then the inclusion CT(X, H) → CT(X, H) induces an isomorphism KCT(X, H) → KCT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism KCT(X, H) ≅ K(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = PH 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, Mm(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 Cnm ≅ (Cm)⊕n, which is naturally an Mm(C)-module.

 

Fibrations of Elliptic Curves in F-Theory.

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F-theory compactifications are by definition compactifications of the type IIB string with non-zero, and in general non-constant string coupling – they are thus intrinsically non-perturbative. F-theory may also seen as a construction to geometrize (and thereby making manifest) certain features pertaining to the S-duality of the type IIB string.

Let us first recapitulate the most important massless bosonic fields of the type IIB string. From the NS-NS sector, we have the graviton gμν, the antisymmetric 2-form field B as well as the dilaton φ; the latter, when exponentiated, serves as the coupling constant of the theory. Moreover, from the R-R sector we have the p-form tensor fields C(p) with p = 0,2,4. It is also convenient to include the magnetic duals of these fields, B(6), C(6) and C(8) (C(4) has self-dual field strength). It is useful to combine the dilaton with the axion into one complex field:

τIIB ≡ C(0) + ie —– (1)

The S-duality then acts via projective SL(2, Z) transformations in the canonical manner:

τIIB → (aτIIB + b)/(cτIIB + d) with a, b, c, d ∈ Z and ad – bc = 1

Furthermore, it acts via simple matrix multiplication on the other fields if these are grouped into doublets (B(2)C(2)), (B(6)C(4)), while C(4) stays invariant.

The simplest F-theory compactifications are the highest dimensional ones, and simplest of all is the compactification of the type IIB string on the 2-sphere, P1. However, as the first Chern class does not vanish: C1(P1) = – 2, this by itself cannot be a good, supersymmetry preserving background. The remedy is to add extra 7-branes to the theory, which sit at arbitrary points zi on the P1, and otherwise fill the 7+1 non-compact space-time dimensions. If this is done in the right way, C1(P1) is cancelled, thereby providing a consistent background.

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Encircling the location of a 7-brane in the z-plane leads to a jump of the perceived type IIB string coupling, τIIB →τIIB  +1.

To explain how this works, consider first a single D7-brane located at an arbitrary given point z0 on the P1. A D7-brane carries by definition one unit of D7-brane charge, since it is a unit source of C(8). This means that is it magnetically charged with respect to the dual field C(0), which enters in the complexified type IIB coupling in (1). As a consequence, encircling the plane location z0 will induce a non-trivial monodromy, that is, a jump on the coupling. But this then implies that in the neighborhood of the D7-brane, we must have a non-constant string coupling of the form: τIIB(z) = 1/2πiIn[z – z0]; we thus indeed have a truly non-perturbative situation.

In view of the SL(2, Z) action on the string coupling (1), it is natural to interpret it as a modular parameter of a two-torus, T2, and this is what then gives a geometrical meaning to the S-duality group. This modular parameter τIIB = τIIB(Z) is not constant over the P1 compactification manifold, the shape of the T2 will accordingly vary along P1. The relevant geometrical object will therefore not be the direct product manifold T2 x P1, but rather a fibration of T2 over P1

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Fibration of an elliptic curve over P1, which in total makes a K3 surface.

The logarithmic behavior of τIIB(z) in the vicinity of a 7-brane means that the T2 fiber is singular at the brane location. It is known from mathematics that each of such singular fibers contributes 1/12 to the first Chern class. Therefore we need to put 24 of them in order to have a consistent type IIB background with C1 = 0. The mathematical data: “Tfibered over P1 with 24 singular fibers” is now exactly what characterizes the K3 surface; indeed it is the only complex two-dimensional manifold with vanishing first Chern class (apart from T4).

The K3 manifold that arises in this context is so far just a formal construct, introduced to encode of the behavior of the string coupling in the presence of 7-branes in an elegant and useful way. One may speculate about a possible more concrete physical significance, such as a compactification manifold of a yet unknown 12 dimensional “F-theory”. The existence of such a theory is still unclear, but all we need the K3 for is to use its intriguing geometric properties for computing physical quantities (the quartic gauge threshold couplings, ultimately).

In order to do explicit computations, we first of all need a concrete representation of the K3 surface. Since the families of K3’s in question are elliptically fibered, the natural starting point is the two-torus T2. It can be represented in the well-known “Weierstraβ” form:

WT2 = y2 + x3 + xf + g = 0 —– (2)

which in turn is invariantly characterized by the J-function:

J = 4(24f)3/(4f3 + 27g2) —– (3)

An elliptically fibered K3 surface can be made out of (2) by letting f → f8(z) and g → g12(z) become polynomials in the P1 coordinate z, of the indicated orders. The locations zi of the 7-branes, which correspond to the locations of the singular fibers where J(τIIB(zi)) → ∞, are then precisely where the discriminant

∆(z) ≡ 4f83(z) + 27g122(z)

=: ∏i=124(z –  zi) vanishes.

Interleaves

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Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t<n will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t<n is defined on the category CWn⊃∂, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S22 e3 is simply connected and has vanishing integral homology in odd dimensions. However, H3(K;Z/2) = Z/2 ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H2k(X),Q/Z) = H2k+1(X) ⊗ Q/Z ⊕ Tor(H2k(X),Q/Z) = H2k+1(X;Q/Z) = 0.

Thus H2k(X) is torsion-free, since the group Tor(H2k(X),Q/Z) is isomorphic to the torsion subgroup of H2k(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form

Vi=1kSi2ƒe4

where the homotopy class of the attaching map ƒ : S3 → Vi=1k Si2 may be viewed as a symmetric k × k matrix with integer entries, as π3(Vi=1kSi2) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H1(M;G) ≅ H1(M) ⊗ G ⊕ Tor(H0M,G) = 0, since H0(M) = Z. By Poincaré duality,

0 = H3(M) ≅ H3(M) ≅ Hom(H3M,Z) ⊕ Ext(H2M,Z),

so that H2(M) is free. This implies that Tor(H2M,G) = 0 and hence H3(M;G) ≅ H3(M) ⊗ G ⊕ Tor(H2M,G) = 0. Finally, by G-coefficient Poincaré duality,

H5(M;G) ≅ H1(M;G) ≅ Hom(H1M,G) ⊕ Ext(H0M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: Danilov’s Theorem implies that H(X;Z) is torsion-free and the map A(X) → H(X;Z) given by composing the canonical map from Chow groups to homology, Ak(X) = An−k(X) → H2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H2n−2k(X;Z) ≅ H2k(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that Hodd(X;Z) = 0. By Poincaré duality, Heven(X;Z) is free and Hodd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that Hodd(X;G) = 0 for any coefficient group G. If we only wanted to establish Hodd(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A(X) → H(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ, the corresponding proper birational morphism X(Δ) → X(Δ) induces a surjection H(X(Δ);Z) → H(X(Δ);Z) and the diagram

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commutes.

Let G be a complex, simply connected, semisimple Lie group and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π1(G) → π1(G/P) → π0(P) → π0(G) = 0,

which shows that π1(G/P) → π0(P) is a bijection. Accordingly, ∃ elements sw(P) ∈ H2l(w)(G/P;Z) (“Schubert classes,” given geometrically by Schubert cells), indexed by w ranging over a certain subset of the Weyl group of G, that form a basis for H(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular Heven(G/P;Z) is free and Hodd(G/P;Z) = 0. Thus Hodd(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x1yn+1 +···+ xny2n −xn+1y1 −···−x2nyn

on C2n × C2n are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a Borel group B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V1 ⊂···⊂ Vn−1 ⊂ Cn}

of flags of subspaces Vi with dimVi = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of k-dimensional subspaces of Cn. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations.

Is There a Philosophy of Bundles and Fields? Drunken Risibility.

The bundle formulation of field theory is not at all motivated by just seeking a full mathematical generality; on the contrary it is just an empirical consequence of physical situations that concretely happen in Nature. One among the simplest of these situations may be that of a particle constrained to move on a sphere, denoted by S2; the physical state of such a dynamical system is described by providing both the position of the particle and its momentum, which is a tangent vector to the sphere. In other words, the state of this system is described by a point of the so-called tangent bundle TS2 of the sphere, which is non-trivial, i.e. it has a global topology which differs from the (trivial) product topology of S2 x R2. When one seeks for solutions of the relevant equations of motion some local coordinates have to be chosen on the sphere, e.g. stereographic coordinates covering the whole sphere but a point (let us say the north pole). On such a coordinate neighbourhood (which is contractible to a point being a diffeomorphic copy of R2) there exists a trivialization of the corresponding portion of the tangent bundle of the sphere, so that the relevant equations of motion can be locally written in R2 x R2. At the global level, however, together with the equations, one should give some boundary conditions which will ensure regularity in the north pole. As is well known, different inequivalent choices are possible; these boundary conditions may be considered as what is left in the local theory out of the non-triviality of the configuration bundle TS2.

Moreover, much before modem gauge theories or even more complicated new field theories, the theory of General Relativity is the ultimate proof of the need of a bundle framework to describe physical situations. Among other things, in fact, General Relativity assumes that spacetime is not the “simple” Minkowski space introduced for Special Relativity, which has the topology of R4. In general it is a Lorentzian four-dimensional manifold possibly endowed with a complicated global topology. On such a manifold, the choice of a trivial bundle M x F as the configuration bundle for a field theory is mathematically unjustified as well as physically wrong in general. In fact, as long as spacetime is a contractible manifold, as Minkowski space is, all bundles on it are forced to be trivial; however, if spacetime is allowed to be topologically non-trivial, then trivial bundles on it are just a small subclass of all possible bundles among which the configuration bundle can be chosen. Again, given the base M and the fiber F, the non-unique choice of the topology of the configuration bundle corresponds to different global requirements.

A simple purely geometrical example can be considered to sustain this claim. Let us consider M = S1 and F = (-1, 1), an interval of the real line R; then ∃ (at least) countably many “inequivalent” bundles other than the trivial one Mö0 = S1 X F , i.e. the cylinder, as shown

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Furthermore the word “inequivalent” can be endowed with different meanings. The bundles shown in the figure are all inequivalent as embedded bundles (i.e. there is no diffeomorphism of the ambient space transforming one into the other) but the even ones (as well as the odd ones) are all equivalent among each other as abstract (i.e. not embedded) bundles (since they have the same transition functions).

The bundles Mön (n being any positive integer) can be obtained from the trivial bundle Mö0 by cutting it along a fiber, twisting n-times and then glueing again together. The bundle Mö1 is called the Moebius band (or strip). All bundles Mön are canonically fibered on S1, but just Mö0 is trivial. Differences among such bundles are global properties, which for example imply that the even ones Mö2k allow never-vanishing sections (i.e. field configurations) while the odd ones Mö2k+1 do not.

Geometry and Localization: An Unholy Alliance? Thought of the Day 95.0

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There are many misleading metaphors obtained from naively identifying geometry with localization. One which is very close to that of String Theory is the idea that one can embed a lower dimensional Quantum Field Theory (QFT) into a higher dimensional one. This is not possible, but what one can do is restrict a QFT on a spacetime manifold to a submanifold. However if the submanifold contains the time axis (a ”brane”), the restricted theory has too many degrees of freedom in order to merit the name ”physical”, namely it contains as many as the unrestricted; the naive idea that by using a subspace one only gets a fraction of phase space degrees of freedom is a delusion, this can only happen if the subspace does not contain a timelike line as for a null-surface (holographic projection onto a horizon).

The geometric picture of a string in terms of a multi-component conformal field theory is that of an embedding of an n-component chiral theory into its n-dimensional component space (referred to as a target space), which is certainly a string. But this is not what modular localization reveals, rather those oscillatory degrees of freedom of the multicomponent chiral current go into an infinite dimensional Hilbert space over one localization point and do not arrange themselves according according to the geometric source-target idea. A theory of this kind is of course consistent but String Theory is certainly a very misleading terminology for this state of affairs. Any attempt to imitate Feynman rules by replacing word lines by word sheets (of strings) may produce prescriptions for cooking up some mathematically interesting functions, but those results can not be brought into the only form which counts in a quantum theory, namely a perturbative approach in terms of operators and states.

String Theory is by no means the only area in particle theory where geometry and modular localization are at loggerheads. Closely related is the interpretation of the Riemann surfaces, which result from the analytic continuation of chiral theories on the lightray/circle, as the ”living space” in the sense of localization. The mathematical theory of Riemann surfaces does not specify how it should be realized; if its refers to surfaces in an ambient space, a distinguished subgroup of Fuchsian group or any other of the many possible realizations is of no concern for a mathematician. But in the context of chiral models it is important not to confuse the living space of a QFT with its analytic continuation.

Whereas geometry as a mathematical discipline does not care about how it is concretely realized the geometrical aspects of modular localization in spacetime has a very specific geometric content namely that which can be encoded in subspaces (Reeh-Schlieder spaces) generated by operator subalgebras acting onto the vacuum reference state. In other words the physically relevant spacetime geometry and the symmetry group of the vacuum is contained in the abstract positioning of certain subalgebras in a common Hilbert space and not that which comes with classical theories.

Contact Geometry and Manifolds

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Let M be a manifold of dimension 2n + 1. A contact structure on M is a distribution ξ ⊂ TM of dimension 2n, such that the defining 1-form α satisfies

α ∧ (dα)n ≠ 0 —– (1)

A 1-form α satisfying (1) is said to be a contact form on M. Let α be a contact form on M; then there exists a unique vector field Rα on M such that

α(Rα) = 1, ιRα dα = 0,

where ιRα dα denotes the contraction of dα along Rα. By definition Rα is called the Reeb vector field of the contact form α. A contact manifold is a pair (M, ξ) where M is a 2n + 1-dimensional manifold and ξ is a contact structure. Let (M, ξ) be a contact manifold and fix a defining (contact) form α. Then the 2-form κ = 1/2 dα defines a symplectic form on the contact structure ξ; therefore the pair (ξ, κ) is a symplectic vector bundle over M. A complex structure on ξ is the datum of J ∈ End(ξ) such that J2 = −Iξ.

Let α be a contact form on M, with ξ = ker α and let κ = 1/2 dα. A complex structure J on ξ is said to be κ-calibrated if gJ [x](·, ·) := κ[x](·, Jx ·) is a JxHermitian inner product on ξx for any x ∈ M.

The set of κ-calibrated complex structures on ξ will be denoted by Cα(M). If J is a complex structure on ξ = ker α, then we extend it to an endomorphism of TM by setting

J(Rα) = 0.

Note that such a J satisfies

J2 =−I + α ⊗ Rα

If J is κ-calibrated, then it induces a Riemannian metric g on M given by

g := gJ + α ⊗ α —– (2)

Furthermore the Nijenhuis tensor of J is defined by

NJ (X, Y) = [JX, JY] − J[X, JY] − J[Y, JX] + J2[X, Y] for any X, Y ∈ TM

A Sasakian structure on a 2n + 1-dimensional manifold M is a pair (α, J), where

• α is a contact form;

• J ∈ Cα(M) satisfies NJ = −dα ⊗ Rα

The triple (M, α, J) is said to be a Sasakian manifold. Let (M, ξ) be a contact manifold. A differential r-form γ on M is said to be basic if

ιRα γ = 0, LRα γ = 0,

where L denotes the Lie derivative and Rα is the Reeb vector field of an arbitrary contact form defining ξ. We will denote by ΛrB(M) the set of basic r-forms on (M, ξ). Note that

rB(M) ⊂ Λr+1B(M)

The cohomology HB(M) of this complex is called the basic cohomology of (M, ξ). If (M, α, J) is a Sasakian manifold, then

J(ΛrB(M)) = ΛrB(M), where, as usual, the action of J on r-forms is defined by

Jφ(X1,…, Xr) = φ(JX1,…, JXr)

Consequently ΛrB(M) ⊗ C splits as

ΛrB(M) ⊗ C = ⊕p+q=r Λp,qJ(ξ)

and, according with this gradation, it is possible to define the cohomology groups Hp,qB(M). The r-forms belonging to Λp,qJ(ξ) are said to be of type (p, q) with respect to J. Note that κ = 1/2 dα ∈ Λ1,1J(ξ) and it determines a non-vanishing cohomology class in H1,1B(M). The Sasakian structure (α, J) also induces a natural connection ∇ξ on ξ given by

ξX Y = (∇X Y)ξ if X ∈ ξ

= [Rα, Y] if X = Rα

where the subscript ξ denotes the projection onto ξ. One easily gets

ξX J = 0, ∇ξXgJ = 0, ∇ξX dα = 0, ∇ξX Y − ∇ξY X = [X,Y]ξ,

for any X, Y ∈ TM. Consequently we have Hol(∇ξ) ⊆ U(n).

The basic cohomology class

cB1(M) = 1/2π [ρT] ∈ H1,1B(M)

is called the first basic Chern class of (M, α, J) and, if it vanishes, then (M, α, J) is said to be null-Sasakian.

Furthermore a Sasakian manifold is called α-Einstein if there exist λ, ν ∈ C(M, R) such that

Ric = λg + να ⊗ α, where Ric is the Ricci Tensor.

A submanifold p: L ֒→ M of a 2n + 1-dimensional contact manifold (M, ξ) is said to be Legendrian if :

1) dimRL = n,

2) p(TL) ⊂ ξ

Observe that, if α is a defining form of the contact structure ξ, then condition 2) is equivalent to say that p(α) = 0. Hence Legendrian submanifolds are the analogue of Lagrangian submanifolds in contact geometry.