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.

Hochschild Cohomology Tethers to Closed String Algebra by way of Cyclicity.

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When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξa = ia(ξ) ∈ Oaa of each object a of B, and η ◦ ξa = ξb ◦ η for each morphism η ∈ Oba from a to b. The family {ξa} thus constitutes a natural transformation from the identity functor 1B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1B. It is automatically commutative, for if {ξa}, {ηa} ∈ E then ξa ◦ ηa = ηa ◦ ξa by the definition of naturality. (A natural transformation from 1B to 1B is a collection of elements {ξa ∈ Oaa} such that ξa ◦ f = f ◦ ξb for each morphism f ∈ Oab from b to a. But we can take a = b and f = ηa.) If B is a Frobenius category then there is a map πab : Obb → Oaa for each pair of objects a, b, and we can define jb : Obb → E by jb(η)a = πab(η) for η ∈ Obb. In other words, jb is defined so that the Cardy condition ιa ◦ jb = πab holds. But the question arises whether we can define a trace θ : E → C to make E into a Frobenius algebra, and with the property that

θaa(ξ)η) = θ(ξja(η)) —– (1)

∀ ξ ∈ E and η ∈ Oaa. This is certainly true if B is a semisimple Frobenius category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(εa) = θa(1a)2, where a is an irreducible object, and εa ∈ E is the characteristic function of the point a in the spectrum of E. Nevertheless, a Frobenius category need not be semisimple, and we cannot, unfortunately, take E as the closed string algebra in the general case. If, for example, B has just one object a, and Oaa is a commutative local ring of dimension greater than 1, then E = Oaa, and so ιa : E → Oaa is an isomorphism, and its adjoint map ja ought to be an isomorphism too. But that contradicts the Cardy condition, as πaa is multiplication by ∑ψiψi, which must be nilpotent.

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH0(B) of B in degree 0. The groups HHp(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH0(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is πba(1a) ∈ Obb. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH0(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y0φ1 Y1φ2 ··· →φk Yk —– (2)

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

(dF)(φ1,…,φk+1) = F(φ2,…,φk+1) ◦ φ1 + ∑i=1k(−1)i F(φ1,…,φi+1 ◦ φi,…,φk+1) + (−1)k+1φk+1 ◦ F(φ1,…,φk) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism FY to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ1, φ2) of the composition law (φ1, φ2) ↦ φ2 ◦ φ1 of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O. But if O is a Frobenius algebra it is isomorphic as a bimodule to O, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Yk, Y0) is the dual space of Hom(Y0, Yk) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ0 ∈ Hom(Yk, Y0). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→φ0 Y0φ1 Y1 →φ2···→φk Ykφ0··· —– (4)

assigns a complex number F(φ01,…,φk).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ01,…,φk) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.

Quantum Geometrodynamics and Emergence of Time in Quantum Gravity

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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

von Neumann & Dis/belief in Hilbert Spaces

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.

— John von Neumann, letter to Garrett Birkhoff, 1935.

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The mathematics: Let us consider the raison d’ˆetre for the Hilbert space formalism. So why would one need all this ‘Hilbert space stuff, i.e. the continuum structure, the field structure of complex numbers, a vector space over it, inner-product structure, etc. Why? According to von Neumann, he simply used it because it happened to be ‘available’. The use of linear algebra and complex numbers in so many different scientific areas, as well as results in model theory, clearly show that quite a bit of modeling can be done using Hilbert spaces. On the other hand, we can also model any movie by means of the data stream that runs through your cables when watching it. But does this mean that these data streams make up the stuff that makes a movie? Clearly not, we should rather turn our attention to the stuff that is being taught at drama schools and directing schools. Similarly, von Neumann turned his attention to the actual physical concepts behind quantum theory, more specifically, the notion of a physical property and the structure imposed on these by the peculiar nature of quantum observation. His quantum logic gave the resulting ‘algebra of physical properties’ a privileged role. All of this leads us to … the physics of it. Birkhoff and von Neumann crafted quantum logic in order to emphasize the notion of quantum superposition. In terms of states of a physical system and properties of that system, superposition means that the strongest property which is true for two distinct states is also true for states other than the two given ones. In order-theoretic terms this means, representing states by the atoms of a lattice of properties, that the join p ∨ q of two atoms p and q is also above other atoms. From this it easily follows that the distributive law breaks down: given atom r ≠ p, q with r < p ∨ q we have r ∧ (p ∨ q) = r while (r ∧ p) ∨ (r ∧ q) = 0 ∨ 0 = 0. Birkhoff and von Neumann as well as many others believed that understanding the deep structure of superposition is the key to obtaining a better understanding of quantum theory as a whole.

For Schrödinger, this is the behavior of compound quantum systems, described by the tensor product. While the quantum information endeavor is to a great extend the result of exploiting this important insight, the language of the field is still very much that of strings of complex numbers, which is akin to the strings of 0’s and 1’s in the early days of computer programming. If the manner in which we describe compound quantum systems captures so much of the essence of quantum theory, then it should be at the forefront of the presentation of the theory, and not preceded by continuum structure, field of complex numbers, vector space over the latter, etc, to only then pop up as some secondary construct. How much quantum phenomena can be derived from ‘compoundness + epsilon’. It turned out that epsilon can be taken to be ‘very little’, surely not involving anything like continuum, fields, vector spaces, but merely a ‘2D space’ of temporal composition and compoundness, together with some very natural purely operational assertion, including one which in a constructive manner asserts entanglement; among many other things, trace structure then follows.

1 + 2 + 3 + … = -1/12. ✓✓✓

The Bernoulli numbers B_n are a sequence of signed rational numbers that can be defined by the exponential generating function

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These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.

The Bernoulli number  can be defined by the contour integral

numberedequation2

where the contour encloses the origin, has radius less than 2pi (to avoid the poles at +/-2pii), and is traversed in a counterclockwise direction.

bernoullinumberdigits_1000

The numbers of digits in the numerator of B_n for the n=2, 4, … are 1, 1, 1, 1, 1, 3, 1, 4, 5, 6, 6, 9, 7, 11, … , while the numbers of digits in the corresponding denominators are 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 4, 1, 3, 5, 3, …. Both of these are plotted above.

The denominator of B_(2n) is given by

numberedequation4

where the product is taken over the primes p, a result which is related to the von Staudt-Clausen theorem.

In 1859 Riemann published a paper giving an explicit formula for the number of primes up to any preassigned limit—a decided improvement over the approximate value given by the prime number theorem. However, Riemann’s formula depended on knowing the values at which a generalized version of the zeta function equals zero. (The Riemann zeta function is defined for all complex numbers—numbers of the form x + iy, where i = (−1), except for the line x = 1.) Riemann knew that the function equals zero for all negative even integers −2, −4, −6, … (so-called trivial zeros), and that it has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1, and he also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1/2. Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis. In 1900 the German mathematician David Hilbert called the Riemann hypothesis one of the most important questions in all of mathematics, as indicated by its inclusion in his influential list of 23 unsolved problems with which he challenged 20th-century mathematicians. In 1915 the English mathematician Godfrey Hardy proved that an infinite number of zeros occur on the critical line, and by 1986 the first 1,500,000,001 nontrivial zeros were all shown to be on the critical line. Although the hypothesis may yet turn out to be false, investigations of this difficult problem have enriched the understanding of complex numbers.

Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory. What properties might you want such a distribution to have? Well, if you’re doing number theory then you want to think of the prime numbers as acting “independently”: knowing that a number is divisible by p should give you no information about whether it’s divisible by q

That quickly leads you to the following realization: you should choose the exponent of each prime in the prime factorization independently. So how should you choose these? It turns out that the probability distribution on the non-negative integers with maximum entropy and a given mean is a geometric distribution. So let’s take the probability that the exponent of p is k to be equal to (1−rp)rpk  for some constant rp

This gives the probability that a positive integer n = p1e1…pkek occurs as

C ∏ki=1 rpei

, where

C = ∏p(1-rp)

So we need to choose rp such that this product converges. Now, we’d like the probability that n occurs to be monotonically decreasing as a function of n. It turns out that this is true iff r= p−s for some s > 1 (since C has to converge), which gives the probability that n occurs as

1/ns/ζ(s), 

ζ(s) is the zeta function.

The Riemann-Zeta function is a complex function that tells us many things about the theory of numbers. Its mystery is increased by the fact it has no closed form – i.e. it can’t be expressed a single formula that contains other standard (elementary) functions.

riemannzetareimabs

riemannzetaridges_700

The plot above shows the “ridges” of  inline4 for  0 < x < 1 and 0 < y < 100. The fact that the ridges appear to decrease monotonically for 0 ≤ x ≤ 1/2 is not a coincidence since it turns out that monotonic decrease implies the Riemann hypothesis. 

On the real line with x>1, the Riemann-Zeta function can be defined by the integral

numberedequation1-1

where Gamma(x) is the gamma function. If x is an integer n, then we have the identity,

inline14

=  inline17

=  inline20

so,

numberedequation2-1

The Riemann zeta function can also be defined in the complex plane by the contour integral

numberedequation16

inline78 , where the contour is illustrated below

riemannzetafunctiongamma_1000

Zeros of inline79 come in (at least) two different types. So-called “trivial zeros” occur at all negative even integers , , , …, and “nontrivial zeros” at certain

numberedequation17

for s in the “critical strip0<sigma<1. The Riemann hypothesis asserts that the nontrivial Riemann zeta function zeros of zeta(s) all have real part sigma=R[s]=1/2, a line called the “critical line.” This is now known to be true for the first 250×10^9 roots.

riemannzetacriticalstrip_700

The plot above shows the real and imaginary parts of zeta(1/2+iy) (i.e., values of zeta(z) along the critical line) as y is varied from 0 to 35.

Now consider this John Cook’s take…

S_p(n) = \sum_{k=1}^n k^p

where p is a positive integer. Here looking at what happens when p becomes a negative integer and we let n go to infinity.

If p < -1, then the limit as n goes to infinity of Sp(n) is ζ(-p). That is, for s > 1, the Riemann-Zeta function ζ(s) is defined by

\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}

We don’t have to limit ourselves to real numbers s > 1; the definition holds for complex numbers s with real part greater than 1. That’ll be important below.

When s is a positive even number, there’s a formula for ζ(s) in terms of the Bernoulli numbers:

\zeta(2n) = (-1)^{n-1} 2^{2n-1} \pi^{2n} \frac{B_{2n}}{(2n)!}

The best-known special case of this formula is that

1 + 1/4 + 1/9 + 1/16 + … = π2 / 6.

It’s a famous open problem to find a closed-form expression for ζ(3) or any other odd argument.

The formula relating the zeta function and Bernoulli tells us a couple things about the Bernoulli numbers. First, for n ≥ 1 the Bernoulli numbers with index 2n alternate sign. Second, by looking at the sum defining ζ(2n) we can see that it is approximately 1 for large n. This tells us that for large n, |B2n| is approximately (2n)! / 22n-1 π2n.

We said above that the sum defining the Riemann zeta function is valid for complex numbers s with real part greater than 1. There is a unique analytic extension of the zeta function to the rest of the complex plane, except at s = 1. The zeta function is defined, for example, at negative integers, but the sum defining zeta in the half-plane Re(s) > 1 is not valid.

One must have seen the equation

1 + 2 + 3 + … = -1/12.

This is an abuse of notation. The sum on the left clearly diverges to infinity. But if the sum defining ζ(s) for Re(s) > 1 were valid for s = -1 (which it is not) then the left side would equal ζ(-1). The analytic continuation of ζ is valid at -1, and in fact ζ(-1) = -1/12. So the equation above is true if you interpret the left side, not as an ordinary sum, but as a way of writing ζ(-1). The same approach could be used to make sense of similar equations such as

12 + 22 + 32 + … = 0

and

13 + 23 + 33 + … = 1/120.