Schematic Grothendieck Representation

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

 

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

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

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

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

0 → ρ(M) → M → ρ⁻¹(M) → 0

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

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

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

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

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

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

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

Introducing a binding process

A + B ↔ C —– (1)

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

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

Thus, there are binding processes

B + B̄ ↔ Z₁ + Z₂ + · · · .

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

B + B̄ ↔ 0

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

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

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

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

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

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

I(L₁, L₂) = #([L₁] ∩ [L₂]) —– (2)

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

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

∫C^{(p + 1)} —–(3)

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

∫_{R × X} C^{(p + 1)} ∧ ω_L —– (4)

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

∫_LC = ∫_XC ∧ ω_L —– (5)

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

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

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

(L − Λ)α = 0, α ∈ ⊕_p H^p(X, Q)

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

Interleaves

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 CW_n⊃∂, 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 = S² ∪₂ 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(H_2k(X),Q/Z) = H_2k+1(X) ⊗ Q/Z ⊕ Tor(H_2k(X),Q/Z) = H_2k+1(X;Q/Z) = 0.

Thus H_2k(X) is torsion-free, since the group Tor(H_2k(X),Q/Z) is isomorphic to the torsion subgroup of H_2k(X).

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

V_i=1^kS_i²∪_ƒe⁴

where the homotopy class of the attaching map ƒ : S³ → V_i=1^k S_i² may be viewed as a symmetric k × k matrix with integer entries, as π₃(V_i=1^kS_i²) ≅ 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, A^k(X) = A_n−k(X) → H_2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H_2n−2k(X;Z) ≅ H_2k(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 H_odd(X;Z) = 0. By Poincaré duality, H_even(X;Z) is free and H_odd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that H_odd(X;G) = 0 for any coefficient group G. If we only wanted to establish H_odd(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

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 s_w(P) ∈ H_2l(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 H_even(G/P;Z) is free and H_odd(G/P;Z) = 0. Thus H_odd(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

x₁y_n+1 +···+ x_ny_{2n −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 ⊂···⊂ V_n−1 ⊂ C_n}

of flags of subspaces V_i with dimV_i = 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.

Weyl, “To understand nature, start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations . . .”

Gauge transformations appear of primarily descriptive nature only if we consider them in their function as changes of local (in the mathematical sense) changes of trivializations. In this function they are comparable to the transformations of the coordinates in a differentiable manifold, which also seem to have a purely “descriptive” function. But the coordinate changes stand in close relation to (local) diffeomorphisms. Therefore the postulate of coordinate independence of natural laws, or of the Lagrangian density, can and is being restated in terms of diffeomorphism invariance in general relativity. Similarly, the local changes of trivializations may be read as local descriptions.

The question as to whether or not the automorphisms express crucial physical properties  has nothing to do with the specific gauge nature of the groups, but hinges on the more overarching question of physical adequateness and physical content of the theory. The question of whether or why gauge symmetries can express physical content is not much different from the Kretschmann question of whether or why coordinate invariance of the laws, respectively coordinate covariance description of a physical theory, can have physical content. In the latter case the answer to the question has been dealt with in the philosophy of physics literature in great detail. Weyl’s answer is contained in his thoughts on the distinction of physical and mathematical automorphisms.

Let us shed a side-glance at gravitational gauge theories not taken into account by Weyl. In Einstein-Cartan gravity, which later turned out to be equivalent to Kibble-Sciama gravity, the localized rotational degrees of freedom lead to a conserved spin current and a non-symmetric energy tensor. This is a structurally pleasing effect, fitting roughly into the Noether charge paradigm, although with a peculiar “crossover” of the two Noether currents and the currents feeding the dynamical equations, inherited from Einstein gravity and Cartan’s identification of translational curvature with torsion. The rotational current, spin, feeds the dynamical equation of translational curvature; the translational current, energy-momentum, feeds the rotational curvature in the (generalized) Einstein equation. It may acquire physical relevance only if energy densities surpass the order of magnitude 10³⁸ times the density of neutron stars. By this reason the current cannot yet be considered a physically striking effect. It may turn into one, if gravitational fields corresponding to extremely high energy densities acquire empirical relevance. For the time being, the rotational current can safely be neglected, Einstein-Cartan gravity reduces effectively to Einstein gravity, and Weyl’s argument for the symmetry of the energy-momentum tensor remains the most “striking consequence” in the sense of  rotational degrees of freedom.

On the other hand, the translational degrees of freedom give a more direct expression for the Noether currents of energy-momentum than the diffeomorphisms. The physical consequences for the diffeomorphism degrees of freedom reduce to the invariance constraint for the Lagrangian density for Einstein gravity considered as a special case of the Einstein-Cartan theory (with effectively vanishing spin). Besides these minor shifts, it may be more interesting to realize that the approach of Kibble and Sciama agreed nicely with Weyl’s methodological remark that for understanding nature we better “start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations . . . ”. This describes quite well what Sciama and Kibble did. They started to explore the consequences of localizing (in the physical sense) the translational and rotational degrees of freedom of special relativity. Their theory was built around the generalized automorphism group arising from localizing the Poincaré group.