# Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

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

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

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that SG is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is Hp. To say that Hp depends only on the G-bundle pEG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p0EG → p1EG is the same as the set of homotopy classes of paths from p0 to p1. When S = S1 the connected components of the space of maps correspond to the conjugacy classes in G: each bundle Pg corresponds to a specific point pg in the mapping space, and a group element h defines a specific path from pg to phgh−1 .

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

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

A G-equivariant TFT gives us for each element g ∈ G a vector space Cg, associated to the circle equipped with the bundle pg whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to pg1 and pg2 on the two incoming circles, and to pg1g2 on the outgoing circle, induces a product

Cg1 ⊗ Cg2 → Cg1g2 —– (1)

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

Cg ⊗ Cg−1 → C

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

αh : Cg → Chgh−1 —– (2)

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

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

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ1 ∈ Cg1 and φ2 ∈ Cg2 then

αg212 = φ2φ1 —– (3)

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

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

TrCh 􏰌Lψαg􏰍 = TrCg−1 􏰌Rψαh􏰍 —– (4)

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

h = ∑ηi ⊗ ηi ∈ Ch ⊗ Ch−1. Then (4) is easily seen to be equivalent to

∑αhii = 􏰟 ∑ηiαgi) —– (5)

in which both sides are elements of Chgh−1g−1.

# Closed String Algebra as a Graded-Commutative Algebra C: Cochain Complex Differentials: Part 2, Note Quote.

The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom HY1⊔Y2 ≅ HY1 ⊗ HY2 only makes sense if there is an involutory canonical isomorphism HY1 ⊗ HY2 ≅ HY2 ⊗ HY1 .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V0 ⊕ V1, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)deg v deg ww ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spinc theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spinc structure”.

A spinc structure on a surface with a conformal structure is a pair of holomorphic line bundles L1, L2 with an isomorphism L1 ⊗ L2 ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L1 = L2. On a 1-manifold S a spinc structure means a spinc structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space HS;L1,L2 to each 1-manifold S with spinc structure, and an operator

US0;L1,L2: HS0;L1,L2 → HS1;L1,L2

to each spinc-cobordism from S0 to S1. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin1-manifold (S;L1,L2). Let G = Aut(L1) denote the group of bundle isomorphisms L1 → L1 which cover diffeomorphisms of S. (We can identify this group with Aut(L2).) It has a homomorphism onto the group Diff+(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L1, which can be identified with the group of smooth maps from S to C×. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff+(S) by the commutative Lie algebra Ω0(S) of smooth real-valued functions on S. Let Λ0S;L1,L2 denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ1S;L1,L2 = Γ(L1) ⊕ Γ(L2). The bracket Λ1 ⊗ Λ1 → Λ0 is completely determined by the property that elements of Γ(L1) and of Γ(L2) anticommute among themselves, while the composite

Γ(L1) ⊗ Γ(L2) → Λ0 → VectC(S)

takes (λ12) to λ1λ2 ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L1,L2) to act on the vector space HS;L1,L2, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L1 and L2. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spinc structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spinc-structure.

# 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.

# 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:

rV, U : C(V) → C(U), ƒ ↦ ƒ|U

which is such that

i) rU, U = id

ii) rW, U = rV, U ○ rW, V

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

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 CM and CN, 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*: CN(V) →  CM-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,…., xn]/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 ↦ OX(U)

where OX(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 ↦ OX(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, OX) is another example of a (locally) ringed space.

# Canonical Actions on Bundles – Philosophizing Identity Over Gauge Transformations.

In physical applications, fiber bundles often come with a preferred group of transformations (usually the symmetry group of the system). The modem attitude of physicists is to regard this group as a fundamental structure which should be implemented from the very beginning enriching bundles with a further structure and defining a new category.

A similar feature appears on manifolds as well: for example, on ℜ2 one can restrict to Cartesian coordinates when we regard it just as a vector space endowed with a differentiable structure, but one can allow also translations if the “bigger” affine structure is considered. Moreover, coordinates can be chosen in much bigger sets: for instance one can fix the symplectic form w = dx ∧ dy on ℜ2 so that ℜ2 is covered by an atlas of canonical coordinates (which include all Cartesian ones). But ℜ2 also happens to be identifiable with the cotangent bundle T*ℜ so that we can restrict the previous symplectic atlas to allow only natural fibered coordinates. Finally, ℜ2 can be considered as a bare manifold so that general curvilinear coordinates should be allowed accordingly; only if the full (i.e., unrestricted) manifold structure is considered one can use a full maximal atlas. Other choices define instead maximal atlases in suitably restricted sub-classes of allowed charts. As any manifold structure is associated with a maximal atlas, geometric bundles are associated to “maximal trivializations”. However, it may happen that one can restrict (or enlarge) the allowed local trivializations, so that the same geometrical bundle can be trivialized just using the appropriate smaller class of local trivializations. In geometrical terms this corresponds, of course, to impose a further structure on the bare bundle. Of course, this newly structured bundle is defined by the same basic ingredients, i.e. the same base manifold M, the same total space B, the same projection π and the same standard fiber F, but it is characterized by a new maximal trivialization where, however, maximal refers now to a smaller set of local trivializations.

Examples are: vector bundles are characterized by linear local trivializations, affine bundles are characterized by affine local trivializations, principal bundles are characterized by left translations on the fiber group. Further examples come from Physics: gauge transformations are used as transition functions for the configuration bundles of any gauge theory. For these reasons we give the following definition of a fiber bundle with structure group.

A fiber bundle with structure group G is given by a sextuple B = (E, M, π; F ;>.., G) such that:

• (E, M, π; F) is a fiber bundle. The structure group G is a Lie group (possibly a discrete one) and λ : G —–> Diff(F) defines a left action of G on the standard fiber F .
• There is a family of preferred trivializations {(Uα, t(α)}α∈I of B such that the following holds: let the transition functions be gˆ(αβ) : Uαβ —–> Diff(F) and let eG be the neutral element of G. ∃ a family of maps g(αβ) : Uαβ —–> G such

that, for each x ∈ Uαβγ = Uα ∩ Uβ ∩ Uγ

g(αα)(x) = eG

g(αβ)(x) = [g(βα)(x)]-1

g(αβ)(x) . g(βγ)(x) . g(γα)(x) = eG

and

(αβ)(x) = λ(g(αβ)(x)) ∈ Diff(F)

The maps g(αβ) : Uαβ —–> G, which depend on the trivialization, are said to form a cocycle with values in G. They are called the transition functions with values in G (or also shortly the transition functions). The preferred trivializations will be said to be compatible with the structure. Whenever dealing with fiber bundles with structure group the choice of a compatible trivialization will be implicitly assumed.

Fiber bundles with structure group provide the suitable framework to deal with bundles with a preferred group of transformations. To see this, let us begin by introducing the notion of structure bundle of a fiber bundle with structure group B = (B, M, π; F; x, G).

Let B = (B, M, π; F; x, G) be a bundle with a structure group; let us fix a trivialization {(Uα, t(α)}α∈I and denote by g(αβ) : Uαβ —–> G its transition functions. By using the canonical left action L : G —–> Diff(G) of G onto itself, let us define gˆ(αβ) : Uαβ —–> Diff(G) given by gˆ(αβ)(x) = L (g(αβ)(x)); they obviously satisfy the cocycle properties. Now by constructing a (unique modulo isomorphisms) principal bundle PB = P(B) having G as structure group and g(αβ) as transition functions acting on G by left translation Lg : G —> G.

The principal bundle P(B) = (P, M, p; G) constructed above is called the structure bundle of B = (B, M, π; F; λ, G).

Notice that there is no similar canonical way of associating a structure bundle to a geometric bundle B = (B, M, π; F), since in that case the structure group G is at least partially undetermined.

Each automorphism of P(B) naturally acts over B.

Let, in fact, {σ(α)}α∈I be a trivialization of PB together with its transition functions g(αβ) : Uαβ —–> G defined by σ(β) = σ(α) . g(αβ). Then any principal morphism Φ = (Φ, φ) over PB is locally represented by local maps ψ(α) : Uα —> G such that

Φ : [x, h]α ↦ [φ(α)(x), ψ(α)(x).h](α)

Since Φ is a global automorphism of PB for the above local expression, the following property holds true in Uαβ.

φ(α)(x) = φ(β)(x) ≡ x’

ψ(α)(x) = g(αβ)(x’) . ψ(β)(x) . g(βα)(x)

By using the family of maps {(φ(α), ψ(α))} one can thence define a family of global automorphisms of B. In fact, using the trivialization {(Uα, t(α)}α∈I, one can define local automorphisms of B given by

Φ(α)B : (x, y) ↦ (φ(α)(x), [λ(ψ(α)(x))](y))

These local maps glue together to give a global automorphism ΦB of the bundle B, due to the fact that g(αβ) are also transition functions of B with respect to its trivialization {(Uα, t(α)}α∈I.

In this way B is endowed with a preferred group of transformations, namely the group Aut(PB) of automorphisms of the structure bundle PB, represented on B by means of the canonical action. These transformations are called (generalized) gauge transformations. Vertical gauge transformations, i.e. gauge transformations projecting over the identity, are also called pure gauge transformations.

# Hypersurfaces

Let (S, CS) and (M, CM) be manifolds of dimension k and n, respectively, with 1 ≤ k ≤ n. A smooth map : S → M is said to be an imbedding if it satisfies the following three conditions.

(I1) Ψ is injective.

(I2) At all points p in S, the associated (push-forward) linear map (Ψp) : Sp → MΨ(p) is injective.

(I3) ∀ open sets O1 in S, Ψ[O1] = [S] ∩ O2 for some open set O2 in M. (Equivalently, the inverse map Ψ−1 : Ψ[S] → S is continuous with respect to the relative topology on [S].)

Several comments about the definition are in order. First, given any point p in S, (I2) implies that (Ψp)[Sp] is a k-dimensional subspace of MΨ(p). So the condition cannot be satisfied unless k ≤ n. Second, the three conditions are independent of one another. For example, the smooth map Ψ : R → R2 defined by (s) = (cos(s), sin(s)) satisfies (I2) and (I3) but is not injective. It wraps R round and round in a circle. On the other hand, the smooth map : R → R defined by (s) = s3 satisfies (I1) and (I3) but is not an imbedding because (Ψ0) : R0 → R0 is not injective. (Here R0 is the tangent space to the manifold R at the point 0). Finally, a smooth map : S → M can satisfy (I1) and (I2) but still have an image that “bunches up on itself.” It is precisely this possibility that is ruled out by condition (I3). Consider, for example, a map : R → R2 whose image consists of part of the image of the curve y = sin(1/x) smoothly joined to the segment {(0, y) : y < 1}, as in the figure below. It satisfies conditions (I1) and (I2) but is not an imbedding because we can find an open interval O1 in R such that given any open set O2 in R2, Ψ[O1] ≠ O2 ∩ Ψ[R].

Suppose(S, CS) and (M, CM) are manifolds with S ⊆ M. We say that (S, CS) is an imbedded submanifold of (M, CM) if the identity map id: S → M is an imbedding. If, in addition, k = n − 1 (where k and n are the dimensions of the two manifolds), we say that (S, CS) is a hypersurface in (M, CM). Let (S, CS) be a k-dimensional imbedded submanifold of the n-dimensional manifold (M, CM), and let p be a point in S. We need to distinguish two senses in which one can speak of “tensors at p.” There are tensors over the vector space Sp (call them S-tensors at p) and ones over the vector space Mp (call them M-tensors at p). So, for example, an S-vector ξ ̃a at p makes assignments to maps of the form f ̃: O ̃ → R where O ̃ is a subset of S that is open in the topology induced by CS, and f ̃ is smooth relative to CS. In contrast, an M-vector ξa at p makes assignments to maps of the form f : O → R where O is a subset of M that is open in the topology induced by CM, and f is smooth relative to CM. Our first task is to consider the relation between S-tensors at p and M-tensors there.

Let us say that ξa ∈ (Mp)a is tangent to S if ξa ∈ (idp)[(Sp)a]. (This makes sense. We know that (idp)[(Sp)a] is a k-dimensional subspace of (Mp)a; ξa either belongs to that subspace or it does not.) Let us further say that ηa in (Mp)a is normal to S if ηaξa =0 ∀ ξa ∈ (Mp)a that are tangent to S. Each of these classes of vectors has a natural vector space structure. The space of vectors ξa ∈ (Mp)a tangent to S has dimension k. The space of co-vectors ηa ∈ (Mp)a normal to S has dimension (n − k).

# Abelian Categories, or Injective Resolutions are Diagrammatic. Note Quote.

Jean-Pierre Serre gave a more thoroughly cohomological turn to the conjectures than Weil had. Grothendieck says

Anyway Serre explained the Weil conjectures to me in cohomological terms around 1955 – and it was only in these terms that they could possibly ‘hook’ me …I am not sure anyone but Serre and I, not even Weil if that is possible, was deeply convinced such [a cohomology] must exist.

Specifically Serre approached the problem through sheaves, a new method in topology that he and others were exploring. Grothendieck would later describe each sheaf on a space T as a “meter stick” measuring T. The cohomology of a given sheaf gives a very coarse summary of the information in it – and in the best case it highlights just the information you want. Certain sheaves on T produced the Betti numbers. If you could put such “meter sticks” on Weil’s arithmetic spaces, and prove standard topological theorems in this form, the conjectures would follow.

By the nuts and bolts definition, a sheaf F on a topological space T is an assignment of Abelian groups to open subsets of T, plus group homomorphisms among them, all meeting a certain covering condition. Precisely these nuts and bolts were unavailable for the Weil conjectures because the arithmetic spaces had no useful topology in the then-existing sense.

At the École Normale Supérieure, Henri Cartan’s seminar spent 1948-49 and 1950-51 focussing on sheaf cohomology. As one motive, there was already de Rham cohomology on differentiable manifolds, which not only described their topology but also described differential analysis on manifolds. And during the time of the seminar Cartan saw how to modify sheaf cohomology as a tool in complex analysis. Given a complex analytic variety V Cartan could define sheaves that reflected not only the topology of V but also complex analysis on V.

These were promising for the Weil conjectures since Weil cohomology would need sheaves reflecting algebra on those spaces. But understand, this differential analysis and complex analysis used sheaves and cohomology in the usual topological sense. Their innovation was to find particular new sheaves which capture analytic or algebraic information that a pure topologist might not focus on.

The greater challenge to the Séminaire Cartan was, that along with the cohomology of topological spaces, the seminar looked at the cohomology of groups. Here sheaves are replaced by G-modules. This was formally quite different from topology yet it had grown from topology and was tightly tied to it. Indeed Eilenberg and Mac Lane created category theory in large part to explain both kinds of cohomology by clarifying the links between them. The seminar aimed to find what was common to the two kinds of cohomology and they found it in a pattern of functors.

The cohomology of a topological space X assigns to each sheaf F on X a series of Abelian groups HnF and to each sheaf map f : F → F′ a series of group homomorphisms Hnf : HnF → HnF′. The definition requires that each Hn is a functor, from sheaves on X to Abelian groups. A crucial property of these functors is:

HnF = 0 for n > 0

for any fine sheaf F where a sheaf is fine if it meets a certain condition borrowed from differential geometry by way of Cartan’s complex analytic geometry.

The cohomology of a group G assigns to each G-module M a series of Abelian groups HnM and to each homomorphism f : M →M′ a series of homomorphisms HnF : HnM → HnM′. Each Hn is a functor, from G-modules to Abelian groups. These functors have the same properties as topological cohomology except that:

HnM = 0 for n > 0

for any injective module M. A G-module I is injective if: For every G-module inclusion N M and homomorphism f : N → I there is at least one g : M → I making this commute

Cartan could treat the cohomology of several different algebraic structures: groups, Lie groups, associative algebras. These all rest on injective resolutions. But, he could not include topological spaces, the source of the whole, and still one of the main motives for pursuing the other cohomologies. Topological cohomology rested on the completely different apparatus of fine resolutions. As to the search for a Weil cohomology, this left two questions: What would Weil cohomology use in place of topological sheaves or G-modules? And what resolutions would give their cohomology? Specifically, Cartan & Eilenberg defines group cohomology (like several other constructions) as a derived functor, which in turn is defined using injective resolutions. So the cohomology of a topological space was not a derived functor in their technical sense. But a looser sense was apparently current.

I have realized that by formulating the theory of derived functors for categories more general than modules, one gets the cohomology of spaces at the same time at small cost. The existence follows from a general criterion, and fine sheaves will play the role of injective modules. One gets the fundamental spectral sequences as special cases of delectable and useful general spectral sequences. But I am not yet sure if it all works as well for non-separated spaces and I recall your doubts on the existence of an exact sequence in cohomology for dimensions ≥ 2. Besides this is probably all more or less explicit in Cartan-Eilenberg’s book which I have not yet had the pleasure to see.

Here he lays out the whole paper, commonly cited as Tôhoku for the journal that published it. There are several issues. For one thing, fine resolutions do not work for all topological spaces but only for the paracompact – that is, Hausdorff spaces where every open cover has a locally finite refinement. The Séminaire Cartan called these separated spaces. The limitation was no problem for differential geometry. All differential manifolds are paracompact. Nor was it a problem for most of analysis. But it was discouraging from the viewpoint of the Weil conjectures since non-trivial algebraic varieties are never Hausdorff.

The fact that sheaf cohomology is a special case of derived func- tors (at least for the paracompact case) is not in Cartan-Sammy. Cartan was aware of it and told [David] Buchsbaum to work on it, but he seems not to have done it. The interest of it would be to show just which properties of fine sheaves we need to use; and so one might be able to figure out whether or not there are enough fine sheaves in the non-separated case (I think the answer is no but I am not at all sure!).

So Grothendieck began rewriting Cartan-Eilenberg before he had seen it. Among other things he preempted the question of resolutions for Weil cohomology. Before anyone knew what “sheaves” it would use, Grothendieck knew it would use injective resolutions. He did this by asking not what sheaves “are” but how they relate to one another. As he later put it, he set out to:

consider the set13 of all sheaves on a given topological space or, if you like, the prodigious arsenal of all the “meter sticks” that measure it. We consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so to speak “right in front of your nose”; that is what we call the structure of a “category”…From here on, this kind of “measuring superstructure” called the “category of sheaves” will be taken as “incarnating” what is most essential to that space.

The Séminaire Cartan had shown this structure in front of your nose suffices for much of cohomology. Definitions and proofs can be given in terms of commutative diagrams and exact sequences without asking, most of the time, what these are diagrams of.  Grothendieck went farther than any other, insisting that the “formal analogy” between sheaf cohomology and group cohomology should become “a common framework including these theories and others”. To start with, injectives have a nice categorical sense: An object I in any category is injective if, for every monic N → M and arrow f : N → I there is at least one g : M → I such that

Fine sheaves are not so diagrammatic.

Grothendieck saw that Reinhold Baer’s original proof that modules have injective resolutions was largely diagrammatic itself. So Grothendieck gave diagrammatic axioms for the basic properties used in cohomology, and called any category that satisfies them an Abelian category. He gave further diagrammatic axioms tailored to Baer’s proof: Every category satisfying these axioms has injective resolutions. Such a category is called an AB5 category, and sometimes around the 1960s a Grothendieck category though that term has been used in several senses.

So sheaves on any topological space have injective resolutions and thus have derived functor cohomology in the strict sense. For paracompact spaces this agrees with cohomology from fine, flabby, or soft resolutions. So you can still use those, if you want them, and you will. But Grothendieck treats paracompactness as a “restrictive condition”, well removed from the basic theory, and he specifically mentions the Weil conjectures.

Beyond that, Grothendieck’s approach works for topology the same way it does for all cohomology. And, much further, the axioms apply to many categories other than categories of sheaves on topological spaces or categories of modules. They go far beyond topological and group cohomology, in principle, though in fact there were few if any known examples outside that framework when they were given.

# Quantum Geometrodynamics and Emergence of Time in Quantum Gravity

It is clear that, like quantum geometrodynamics, the functional integral approach makes fundamental use of a manifold. This means not just that it uses mathematical continua, such as the real numbers (to represent the values of coordinates, or physical quantities); it also postulates a 4-dimensional manifold M as an ‘arena for physical events’. However, its treatment of this manifold is very different from the treatment of spacetime in general relativity in so far as it has a Euclidean, not Lorentzian metric (which, apart from anything else, makes the use of the word ‘event’ distinctly problematic). Also, we may wish to make a summation over different such manifolds, it is in general necessary to consider complex metrics in the functional integral (so that the ‘distance squared’ between two spacetime points can be a complex number), whereas classical general relativity uses only real metrics.

Thus one might think that the manifold (or manifolds!) does not (do not) deserve the name ‘spacetime’. But what is in a name?! Let us in any case now ask how spacetime as understood in present-day physics could emerge from the above use of Riemannian manifolds M, perhaps taken together with other theoretical structures.

In particular: if we choose to specify the boundary conditions using the no-boundary proposal, this means that we take only those saddle-points of the action as contributors (to the semi-classical approximation of the wave function) that correspond to solutions of the Einstein field equations on a compact manifold M with a single boundary Σ and that induce the given values h and φ0 on Σ.

In this way, the question of whether the wave function defined by the functional integral is well approximated by this semi-classical approximation (and thus whether it predicts classical spacetime) turns out to be a question of choosing a contour of integration C in the space of complex spacetime metrics. For the approximation to be valid, we must be able to distort the contour C into a steepest-descents contour that passes through one or more of these stationary points and elsewhere follows a contour along which |e−I| decreases as rapidly as possible away from these stationary points. The wave function is then given by:

Ψ[h, φ0, Σ] ≈ ∑p e−Ip/ ̄h

where Ip are the stationary points of the action through which the contour passes, corresponding to classical solutions of the field equations satisfying the given boundary conditions. Although in general the integral defining the wave function will have many saddle-points, typically there is only a small number of saddle-points making the dominant contribution to the path integral.

For generic boundary conditions, no real Euclidean solutions to the classical Einstein field equations exist. Instead we have complex classical solutions, with a complex action. This accords with the account of the emergence of time via the semiclassical limit in quantum geometrodynamics.

On the Emergence of Time in Quantum Gravity

# Diffeomorphism Invariance: General Relativity Spacetime Points Cannot Possess Haecceity.

Eliminative or radical ontic structural realism (ROSR) offers a radical cure—appropriate given its name—to what it perceives to be the ailing of traditional, object-based realist interpretations of fundamental theories in physics: rid their ontologies entirely of objects. The world does not, according to this view, consist of fundamental objects, which may or may not be individuals with a well-defined intrinsic identity, but instead of physical structures that are purely relational in the sense of networks of ‘free-standing’ physical relations without relata.

Advocates of ROSR have taken at least three distinct issues in fundamental physics to support their case. The quantum statistical features of an ensemble of elementary quantum particles of the same kind as well as the features of entangled elementary quantum (field) systems as illustrated in the violation of Bell-type inequalities challenge the standard understanding of the identity and individuality of fundamental physical objects: considered on their own, an elementary quantum particle part of the above mentioned ensemble or an entangled elementary quantum system (that is, an elementary quantum system standing in a quantum entanglement relation) cannot be said to satisfy genuine and empirically meaningful identity conditions. Thirdly, it has been argued that one of the consequences of the diffeomorphism invariance and background independence found in general relativity (GTR) is that spacetime points should not be considered as traditional objects possessing some haecceity, i.e. some identity on their own.

The trouble with ROSR is that its main assertion appears squarely incoherent: insofar as relations can be exemplified, they can only be exemplified by some relata. Given this conceptual dependence of relations upon relata, any contention that relations can exist floating freely from some objects that stand in those relations seems incoherent. If we accept an ontological commitment e.g. to universals, we may well be able to affirm that relations exist independently of relata – as abstracta in a Platonic heaven. The trouble is that ROSR is supposed to be a form of scientific realism, and as such committed to asserting that at least certain elements of the relevant theories of fundamental physics faithfully capture elements of physical reality. Thus, a defender of ROSR must claim that, fundamentally, relations-sans-relata are exemplified in the physical world, and that contravenes both the intuitive and the usual technical conceptualization of relations.

The usual extensional understanding of n-ary relations just equates them with subsets of the n-fold Cartesian product of the set of elementary objects assumed to figure in the relevant ontology over which the relation is defined. This extensional, ultimately set-theoretic, conceptualization of relations pervades philosophy and operates in the background of fundamental physical theories as they are usually formulated, as well as their philosophical appraisal in the structuralist literature. The charge then is that the fundamental physical structures that are represented in the fundamental physical theories are just not of the ‘object-free’ type suggested by ROSR.

While ROSR should not be held to the conceptual standards dictated by the metaphysical prejudices it denies, giving up the set-theoretical framework and the ineliminable reference to objects and relata attending its characterizations of relations and structure requires an alternative conceptualization of these notions so central to the position. This alternative conceptualization remains necessary even in the light of ‘metaphysics first’ complaints, which insist that ROSR’s problem must be confronted, first and foremost, at the metaphysical level, and that the question of how to represent structure in our language and in our theories only arises in the wake of a coherent metaphysical solution. But the radical may do as much metaphysics as she likes, articulate her theory and her realist commitments she must, and in order to do that, a coherent conceptualization of what it is to have free-floating relations exemplified in the physical world is necessary.

ROSR thus confronts a dilemma: either soften to a more moderate structural realist position or else develop the requisite alternative conceptualizations of relations and of structures and apply them to fundamental physical theories. A number of structural realists have grabbed the first leg and proposed less radical and non-eliminative versions of ontic structural realism (OSR). These moderate cousins of ROSR aim to take seriously the difficulties of the traditional metaphysics of objects for understanding fundamental physics while avoiding these major objections against ROSR by keeping some thin notion of object. The picture typically offered is that of a balance between relations and their relata, coupled to an insistence that these relata do not possess their identity intrinsically, but only by virtue of occupying a relational position in a structural complex. Because it strikes this ontological balance, we term this moderate version of OSR ‘balanced ontic structural realism’ (BOSR).

But holding their ground may reward the ROSRer with certain advantages over its moderate competitors. First, were the complete elimination of relata to succeed, then structural realism would not confront any of the known headaches concerning the identity of these objects or, relatedly, the status of the Principle of the Identity of Indiscernibles. To be sure, this embarrassment can arguably be avoided by other moves; but eliminating objects altogether simply obliterates any concerns whether two objects are one and the same. Secondly, and speculatively, alternative formulations of our fundamental physical theories may shed light on a path toward a quantum theory of gravity.

For these presumed advantages to come to bear, however, the possibility of a precise formulation of the notion of ‘free-standing’ (or ‘object-free’) structure, in the sense of a network of relations without relata (without objects) must thus be achieved.  Jonathan Bain has argued that category theory provides the appropriate mathematical framework for ROSR, allowing for an ‘object-free’ notion of relation, and hence of structure. This argument can only succeed, however, if the category-theoretical formulation of (some of the) fundamental physical theories has some physical salience that the set-theoretical formulation lacks, or proves to be preferable qua formulation of a physical theory in some other way.

F. A. Muller has argued that neither set theory nor category theory provide the tools necessary to clarify the “Central Claim” of structural realism that the world, or parts of the world, have or are some structure. The main reason for this arises from the failure of reference in the contexts of both set theory and category theory, at least if some minimal realist constraints are imposed on how reference can function. Consequently, Muller argues that an appropriately realist stucturalist is better served by fixing the concept of structure by axiomatization rather than by (set-theoretical or category-theoretical) definition.