Probability Space Intertwines Random Walks – Thought of the Day 144.0

Many deliberations of stochasticity start with “let (Ω, F, P) be a probability space”. One can actually follow such discussions without having the slightest idea what Ω is and who lives inside. So, what is “Ω, F, P” and why do we need it? Indeed, for many users of probability and statistics, a random variable X is synonymous with its probability distribution μ_X and all computations such as sums, expectations, etc., done on random variables amount to analytical operations such as integrations, Fourier transforms, convolutions, etc., done on their distributions. For defining such operations, you do not need a probability space. Isn’t this all there is to it?

One can in fact compute quite a lot of things without using probability spaces in an essential way. However the notions of probability space and random variable are central in modern probability theory so it is important to understand why and when these concepts are relevant.

From a modelling perspective, the starting point is a set of observations taking values in some set E (think for instance of numerical measurement, E = R) for which we would like to build a stochastic model. We would like to represent such observations x₁, . . . , x_n as samples drawn from a random variable X defined on some probability space (Ω, F, P). It is important to see that the only natural ingredient here is the set E where the random variables will take their values: the set of events Ω is not given a priori and there are many different ways to construct a probability space (Ω, F, P) for modelling the same set of observations.

Sometimes it is natural to identify Ω with E, i.e., to identify the randomness ω with its observed effect. For example if we consider the outcome of a dice rolling experiment as an integer-valued random variable X, we can define the set of events to be precisely the set of possible outcomes: Ω = {1, 2, 3, 4, 5, 6}. In this case, X(ω) = ω: the outcome of the randomness is identified with the randomness itself. This choice of Ω is called the canonical space for the random variable X. In this case the random variable X is simply the identity map X(ω) = ω and the probability measure P is formally the same as the distribution of X. Note that here X is a one-to-one map: given the outcome of X one knows which scenario has happened so any other random variable Y is completely determined by the observation of X. Therefore using the canonical construction for the random variable X, we cannot define, on the same probability space, another random variable which is independent of X: X will be the sole source of randomness for all other variables in the model. This also show that, although the canonical construction is the simplest way to construct a probability space for representing a given random variable, it forces us to identify this particular random variable with the “source of randomness” in the model. Therefore when we want to deal with models with a sufficiently rich structure, we need to distinguish Ω – the set of scenarios of randomness – from E, the set of values of our random variables.

Let us give an example where it is natural to distinguish the source of randomness from the random variable itself. For instance, if one is modelling the market value of a stock at some date T in the future as a random variable S₁, one may consider that the stock value is affected by many factors such as external news, market supply and demand, economic indicators, etc., summed up in some abstract variable ω, which may not even have a numerical representation: it corresponds to a scenario for the future evolution of the market. S₁(ω) is then the stock value if the market scenario which occurs is given by ω. If the only interesting quantity in the model is the stock price then one can always label the scenario ω by the value of the stock price S₁(ω), which amounts to identifying all scenarios where the stock S₁ takes the same value and using the canonical construction. However if one considers a richer model where there are now other stocks S₂, S₃, . . . involved, it is more natural to distinguish the scenario ω from the random variables S₁(ω), S₂(ω),… whose values are observed in these scenarios but may not completely pin them down: knowing S₁(ω), S₂(ω),… one does not necessarily know which scenario has happened. In this way one reserves the possibility of adding more random variables later on without changing the probability space.

These have the following important consequence: the probabilistic description of a random variable X can be reduced to the knowledge of its distribution μ_X only in the case where the random variable X is the only source of randomness. In this case, a stochastic model can be built using a canonical construction for X. In all other cases – as soon as we are concerned with a second random variable which is not a deterministic function of X – the underlying probability measure P contains more information on X than just its distribution. In particular, it contains all the information about the dependence of the random variable X with respect to all other random variables in the model: specifying P means specifying the joint distributions of all random variables constructed on Ω. For instance, knowing the distributions μ_X, μ_Y of two variables X, Y does not allow to compute their covariance or joint moments. Only in the case where all random variables involved are mutually independent can one reduce all computations to operations on their distributions. This is the case covered in most introductory texts on probability, which explains why one can go quite far, for example in the study of random walks, without formalizing the notion of probability space.

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Fibrations of Elliptic Curves in F-Theory.

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

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

τ_IIB ≡ C⁽⁰⁾ + ie^-φ —– (1)

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

τ_IIB → (aτ_IIB + b)/(cτ_IIB + d) with a, b, c, d ∈ Z and ad – bc = 1

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

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

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

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

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

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

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

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

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

W_T² = y² + x³ + xf + g = 0 —– (2)

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

J = 4(24f)³/(4f³ + 27g²) —– (3)

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

∆(z) ≡ 4f₈³(z) + 27g₁₂²(z)

=: ∏_i=1²⁴(z –  z_i) vanishes.

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

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

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

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

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

The bundles Mö_n (n being any positive integer) can be obtained from the trivial bundle Mö₀ by cutting it along a fiber, twisting n-times and then glueing again together. The bundle Mö₁ is called the Moebius band (or strip). All bundles Mö_n are canonically fibered on S¹, but just Mö₀ is trivial. Differences among such bundles are global properties, which for example imply that the even ones Mö_2k allow never-vanishing sections (i.e. field configurations) while the odd ones Mö_2k+1 do not.

Categorial Functorial Monads

Algebraic constructs (A,U), such as Vec, Grp, Mon, and Lat, can be fully described by the following data, called the monad associated with (A,U):

1. the functor T : Set → Set, where T = U ◦ F and F : Set → A is the associated free functor,

2. the natural transformation η : id_Set → T formed by universal arrows, and

3. the natural transformation μ : T ◦ T → T given by the unique homomorphism μ_X : T(TX) → TX that extends id_TX.

In these cases, there is a canonical concrete isomorphism K between (A,U) and the full concrete subcategory of Alg(T) consisting of those T-algebras TX →^x X that satisfy the equations x ◦ η_X = id_X and x ◦ Tx = x ◦ μ_X. The latter subcategory is called the Eilenberg-Moore category of the monad (T, η, μ). The above observation makes it possible, in the following four steps, to express the “degree of algebraic character” of arbitrary concrete categories that have free objects:

Step 1: With every concrete category (A,U) over X that has free objects (or, more generally, with every adjoint functor A →^U X) one can associate, in an essentially unique way, an adjoint situation (η, ε) : F -|U : A → X.

Step 2: With every adjoint situation (η, ε) : F -|U : A → X one can associate a monad T = (T, η, μ) on X, where T = U ◦ F : X → X.

Step 3: With every monad T = (T, η, μ) on X one can associate a concrete subcategory of Alg(T) denoted by (X^T, U^T) and called the category of T-algebras.

Step 4:  With every concrete category (A,U) over X that has free objects one can associate a distinguished concrete functor (A,U) →^K (X^T , U^T) into the associated category of T-algebras called the comparison functor for (A, U).

Concrete categories that are concretely isomorphic to a category of T-algebras for some monad T have a distinct “algebraic flavor”. Such categories (A,U) and their forgetful functors U are called monadic. It turns out that a concrete category (A, U ) is monadic iff it has free objects and its associated comparison functor (A,U) →^K (X^T , U^T) is an isomorphism. Thus, for concrete categories (A,U) that have free objects, the associated comparison functor can be considered as a means of measuring the “algebraic character” of (A,U); and the associated category of T-algebras can be considered to be the “algebraic part” of (A,U). In particular,

(a) every finitary variety is monadic,

(b) the category TopGrp, considered as a concrete category

1. over Top, is monadic,
2. over Set, is not monadic; the associated comparison functor is the forgetful functor TopGrp → Grp, so that the construct Grp may be considered as the “algebraic part” of the construct TopGrp,

(c) the construct Top is not monadic; the associated comparison functor is the forgetful functor Top → Set itself, so that the construct Set may be considered as the “algebraic part” of the construct Top; hence the construct Top may be considered as having a trivial “algebraic part”.

Among constructs, monadicity captures the idea of “algebraicness” rather well. Unfortunately, however, the behavior of monadic categories in general is far from satisfactory. Monadic functors can fail badly to reflect properties of the base category (e.g., the existence of colimits or of suitable factorization structures), and they are not closed under composition.

Right-(Left-)derived Functors

Fix an abelian category A, let J be a Δ-subcategory of K(A), let D_J be the corresponding derived category, and let

Q = Q_J : J → D_J

be the canonical Δ-functor. For any Δ-functors F and G from J to another Δ-category E, or from D_J to E, Hom(F, G) will denote the abelian group of Δ-functor morphisms from F to G.

A Δ-functor F : J → E is right-derivable if there exists a Δ-functor

RF : D_J → E

and a morphism of Δ-functors

ζ : F → RF ◦ Q

such that for every Δ-functor G : D_J → E the composed map

Hom(RF, G) →^natural Hom(RF ◦ Q, G ◦ Q) →^{via ζ} Hom(F, G ◦ Q)

is an isomorphism, (the map “via ζ” is an isomorphism). The Δ-functor F is left-derivable if there exists a Δ-functor

LF : D_J → E

and a morphism of Δ-functors

ζ : LF ◦ Q → F

such that for every Δ-functor G : D_J → E the composed map

Hom(G, LF) →^natural Hom(G ◦ Q, LF ◦ Q) →^{via ζ} Hom(G ◦ Q, F)

is an isomorphism (the map “via ζ” is an isomorphism).

Such a pair (RF, ζ) and (LF, ζ) are called the right-derived and left-derived functors of F respectively. Composition with Q gives an embedding of Δ-functor categories

Hom_∆(D_J, E)  ֒→ Hom_∆(J, E),

with image the full subcategory whose objects are the Δ-functors which transform quasi-isomorphisms into isomorphisms. Consequently we can regard a right-(left-)derived functor of F as an initial (terminal) object in the category of Δ-functor morphisms F → G′ (G′ → F ) where G′ ranges over all Δ-functors from J to E which transform quasi-isomorphisms into isomorphisms. As such, the pair (RF, ζ) (or (Lf, ζ)) – if it exists – is unique up to canonical isomorphism.

Let A′ be another abelian category. Any additive functor F : A → A′ extends to a Δ-functor F ̄ : K(A) → K(A′ ). Q′ : K(A′) → D(A′) being the canonical map, we will refer to derived functors of Q′F ̄, or of the restriction of Q′F ̄ to some specified Δ-subcategory J of K(A), as being “derived functors of F” and denote them by RF or LF.

Let A′ be an abelian category, and suppose that E is a Δ-subcategory of K(A′) or of D(A′). If RF exists we can set

RⁱF(A):= Hⁱ(RF(A)) (A ∈ J, i ∈ Z)

Since RF is a Δ-functor, any triangle A → B → C → A[1] in J is transformed by RF into a triangle in E, and hence we have an exact homology sequence:

…. → Rⁱ⁻¹F(C) → RⁱF(A) → RⁱF(B) → RⁱF(C) → Rⁱ⁺¹F(A) → ….

implying an exact sequence of A-complexes

0 → A → B → C → 0  (A, B, C ∈ J)

Local Lifts into Period Domains: Holonomies: Philosophies of Conjugacy. Part 2.

Let F = G^C/P be a flag manifold. Then there is a unique inner symmetric space G-space N associated to F together with a finite number of homogeneous fibrations F → N.

Let us emphasise that this construction depends on nothing but the conjugacy class of p ⊂ g^C and the choice of compact real form g. Equivalently, it depends solely on the choice of invariant complex structure on F.

Every flag manifold fibres over an inner symmetric space. Conversely, every inner symmetric space is the target of the canonical fibrations of at least one flag manifold. Let us now see how this story relates to the geometry of J(N).

So let p : F → N be a canonical fibration. By construction, the fibres of p are complex submanifolds of F and this allows us to define a fibre map i_p : F → J(N) as follows: at f ∈ F we have an orthogonal splitting of T_fF into horizontal and vertical subspaces both of which are invariant under the complex structure of F. Then dp restricts to give an isomorphism of the horizontal part with T_p(f)N and therefore induces an almost Hermitian structure on T_p(f)N : this is i_p(f) ∈ J_p(f)N. Such a construction is possible whenever we have a Riemannian submersion of a Hermitian manifold with complex submanifolds as fibres.

i_p : F → J(N) is a G-equivariant holomorphic embedding. This implies that i_p (F) is an almost complex submanifold of J(N) on which J is integrable.

If j ∈ Z ⊂ J(N) then G · j is a flag manifold canonically fibred over N. In fact, G · j = i_p(F ) for some canonical fibration p : F → N of a flag manifold F .

For this, the main observation is the following: at π(j), we have the symmetric decomposition g = k ⊕ q

with q ≅ T_π(j)N. If q⁻ is the (0,1)-space for j then [q⁻, q⁻] ⊕ q⁻

is the nilradical of a parabolic subalgebra p, where G · j is equivariantly biholomorphic to the corresponding flag manifold G^C/P. Each canonical fibration of a flag manifold gives rise to a G-orbit in Z for some inner symmetric G-space N and that all such orbits arise in this way. But, for fixed G, there are only a finite number of biholomorphism types of flag manifold (they are in bijective correspondence with the conjugacy classes of parabolic subalgebras of g^C) and each flag manifold admits but a finite number of canonical fibrations. Thus Z is composed of a finite number of G-orbits all of which are closed. In this way, we obtain a geometric interpretation of the purely algebraic construction of the canonical fibrations: they are just the restrictions of the projection π : J(N) → N to the various realisations of F as an orbit in Z.

For each non-compact real form G_R of a complex semisimple group Lie group G^C, there is a unique Riemannian symmetric space G_R/K of non-compact type. The corresponding involution is called the Cartan involution of G_R. Consider now the orbits of such a G_R on the various flag manifolds F = G^C/P. Those orbits which are open subsets of F are called flag domains: an orbit is a flag domain precisely when the stabilisers contain a compact Cartan subgroup of G_R. It turns out that the presence of this compact Cartan subgroup is precisely what we need to define a canonical element of g_R and thus an involution of g_R just as in the compact case. However the involution is not necessarily a Cartan involution (i.e. the associated symmetric space need not be Riemmanian). In case that the involution is a Cartan involution, the flag domain is a canonical flag domain which is then exponentiated such that the involution gets to a Riemannian symmetric space of non-compact type and a canonical fibration of canonical flag domain over it.

Let X be a flag manifold or canonical flag domain and π : X → N be a homogeneous fibration onto a Riemannian symmetric space. If φ : M → X is a holomorphic map of a Kahler manifold with image tangent to the super-horizontal distribution then π ◦ φ : M → N is harmonic.

Maps φ : M → X satisfying such hypotheses of being super-horizontal are holomorphic maps.

For instance, let X = SU(n + 1)/S(U(1) × ··· × U(1)) = F(1,…,1; C_n+1). There are n+1 homogeneous fibrations π_i : X → CPⁿ, i = 1,…,n, with π₀ holomorphic and π_n anti-holomorphic. Now a super-horizontal holomorphic map φ : S² → X is essentially just the Frenet frame of the holomorphic map π₀ ◦ φ : S² → CPⁿ while each π ◦ φ is a harmonic map S² → CPⁿ. It is the content of the classification theorem for harmonic 2-spheres in CPⁿ that all such harmonic maps arise in this way. On the non-compact side of the fence, the super-horizontal distribution is precisely that which defines the infinitesimal period relation. Thus the (local lifts of) period maps are precisely the super-horizontal holomorphic maps into period domains.

Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Id_α : Sets^Cop → Shvs(C^op,J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves Sets^Top. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define Y^X as the set of functions X → Y. Then in the category of presheaves,

Sets^CopY^X(U) ≅ Hom(h_U,Y^X) ≅ Hom(h_U × X,Y)

, where h_U is the representable sheaf h_U(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf Y^X is actually a sheaf. Finally, for the existence of the subobject-classifier Ω_Sets^Cop, it can be defined as

Ω_Sets^Cop(U) ≅ Hom(h_U,Ω) ≅ {sub-presheaves of h_U} ≅ {sieves on U}, or alternatively, for the category of proper sheaves S_hvs(C,J), as

Ω_Shvs(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

f^∗_rs(S) ∈ J(r) ⇒ f_rs ∈ S

, where f_rs : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1^∗_s(S) = f^∗_ss(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1_s ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφ_p(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φ_p(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

Quantum Groupoid

A (finite) quantum groupoid over k is a finite-dimensional k-vector space H with the structures of an associative algebra (H, m, 1) with multiplication m : H ⊗_k H → H and unit 1 ∈ H and a coassociative coalgebra (H, ∆, ε) with comultiplication ∆ : H → H ⊗_k H and counit ε : H → k such that:

1. The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that

(∆ ⊗ id)∆(1) = (∆(1) ⊗ 1) (1 ⊗ ∆(1)) = (1 ⊗ ∆(1)) (∆(1) ⊗ 1) —– (1)

2.  The counit is a k-linear map satisfying the identity:

ε(fgh) = ε(fg₍₁₎)ε(g₍₂₎h) = ε(f_g(2))ε(_g(1)h), (2) ∀ f, g, h ∈ H —– (2)

3.   There is an algebra and coalgebra anti-homomorphism S : H → H, called an antipode, such that, ∀ h ∈ H ,

m(id ⊗ S) ∆(h) = (ε ⊗ id) ∆(1)(h ⊗ 1) —– (3)

m(S ⊗ id) ∆(h) = (id ⊗ ε)(1 ⊗ h) ∆(1) —– (4)

A quantum groupoid is a Hopf algebra iff one of the following equivalent conditions holds: (i) the comultiplication is unit preserving or (ii) the counit is a homomorphism of algebras.

A morphism of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.

The set of axioms is self-dual. This allows to define a natural quantum groupoid  structure on the dual vector space H’ = Hom_k (H, k) by “reversing the arrows”:

⟨h,φ ψ⟩ = ∆(h), φ ⊗ ψ —– (5)

⟨g ⊗ h, ∆'(φ)⟩ = ⟨gh, φ⟩ —– (6)

⟨h, S'(φ)⟩ = ⟨S(h), φ⟩ —– (7)

∀ φ, ψ ∈ H’, g, h ∈ H. The unit 1ˆ ∈ H’ is ε and counit ε’ is φ → ⟨φ,1⟩. The linear endomorphisms of H defined by

h → m(id ⊗ S) ∆(h), h → m(S ⊗ id) ∆(h) —– (8)

are called the target and source counital maps and denoted ε_t and ε_s, respectively.

From axioms (3) and (4),

ε_t(h) = (ε ⊗ id) ∆(1)(h ⊗ 1), ε_s(h) = (id ⊗ ε) (1 ⊗ h)∆(1) . (9)

In the Hopf algebra case ε_t(h) = ε_s(h) = ε(h)1.

We have S ◦ ε_s = ε_t ◦ S and ε_s ◦ S = S ◦ ε_t. The images of these maps ε_t and ε_s

H_t = ε_t (H) = {h ∈ H | ∆(h) =∆(1)(h ⊗ 1)} —– (10)

H_s = ε_s (H) = {h ∈ H | ∆(h) = (1⊗h) ∆(1)} —– (11)

are subalgebras of H, called the target (respectively source) counital subalgebras. They play the role of ground algebras for H. They commute with each other and

H_t = {(φ ⊗ id) ∆(1)|φ ∈ H’,

H_s = (id ⊗ φ) ∆(1)| φ ∈ H’,

i.e., H_t (respectively H_s) is generated by the right (respectively left) tensorands of ∆(1). The restriction of S defines an algebra anti-isomorphism between H_t and H_s. Any morphism H → K of quantum groupoids preserves counital subalgebras, i.e., H_t ≅ K_t and H_s ≅ K_s.

In what follows we will use the Sweedler arrows, writing ∀ h ∈ H , φ ∈ H’:

h ⇀ φ = φ₍₁₎⟨h, φ₍₂₎⟩,

φ ↼ h = ⟨h, φ₍₁₎⟩φ₍₂₎ —– (12)

∀ h ∈ H, φ ∈ H’. Then the map z → (z ⇀ ε) is an algebra isomorphism between H_t and H. Similarly, the map y → (ε ↼ y) is an algebra isomorphism between H and H’_t. Thus, the counital subalgebras of H’ are canonically anti-isomorphic to those of H. A quantum groupoid H is called connected if H_s ∩ Z(H) = k, or, equivalently, H_t ∩ Z(H ) = k, where Z(H) denotes the center of H. A k-algebra A is separable if the multiplication epimorphism m : A ⊗_k A → A has a right inverse as an A − A bimodule homomorphism. When the characteristic of k is 0, this is equivalent to the existence of a separability element e ∈ A ⊗_k A such that m(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a), (1 ⊗ a)e = e(a ⊗ 1) ∀ a ∈ A. The counital subalgebras H_t and H_s are separable, with separability elements e_t = (S ⊗ id)∆(1) and e_s = (id ⊗S)∆(1), respectively. Observe that the adjoint actions of 1 ∈ H give rise to non-trivial maps

H → H : h → 1₍₁₎h_S(1₍₂₎) = Ad^l₁(h), h → S(1₍₁₎)h1₍₂₎ = Ad^r1(h), h ∈ H —– (13) …….

 

Odd symplectic + Odd Poisson Geometry as a Generalization of Symplectic (Poisson) Geometry to the Supercase

A symplectic structure on a manifold M is defined by a non-degenerate closed two-form ω. In a vicinity of an arbitrary point one can consider coordinates (x¹, . . . , x²ⁿ) such that ω = ∑ⁿ_i=1 dxⁱdxⁱ⁺ⁿ. Such coordinates are called Darboux coordinates. To a symplectic structure corresponds a non-degenerate Poisson structure { , }. In Darboux coordinates {xⁱ,x^j} = 0 if |i−j| ≠ n and {xⁱ,xⁱ⁺ⁿ} = −{xⁱ⁺ⁿ,xⁱ} = 1. The condition of closedness of the two-form ω corresponds to the Jacobi identity {f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0

for the Poisson bracket. If a symplectic or Poisson structure is given, then every function f defines a vector field (the Hamiltonian vector field) D_f such that D_fg = {f,g} = −ω(D_f,D_g).

A Poisson structure can be defined independently of a symplectic structure. In general it can be degenerate, i.e., there exist non-constant functions f such that D_f = 0. In the case when a Poisson structure is non-degenerate (corresponds to a symplectic structure), the map from T∗M to T M defined by the relation f → D_f is an isomorphism.

One can straightforwardly generalize these constructions to the supercase and consider symplectic and Poisson structures (even or odd) on supermanifolds. An even (odd) symplectic structure on a supermanifold is defined by an even (odd) non-degenerate closed two-form. In the same way as the existence of a symplectic structure on an ordinary manifold implies that the manifold is even-dimensional (by the non-degeneracy condition for the form ω), the existence of an even or odd symplectic structure on a supermanifold implies that the dimension of the supermanifold is equal either to (2p.q) for an even structure or to (m.m) for an odd structure. Darboux coordinates exist in both cases. For an even structure, the two-form in Darboux coordinates

z^A = (x¹,…, x^2p ;θ₁,…, θ_q) has the form ∑_i=1^p dxⁱ dx^p+i + ∑_a=1^q ε_adθ_adθ_a,

where ε_a = ±1. For an odd structure, the two-form in Darboux coordinates z^A = (x¹,…,x^m;θ₁,…,θ_m) has the form ∑_i=1^m dxⁱdθ_i.

The non-degenerate odd Poisson bracket corresponding to an odd symplectic structure has the following appearance in Darboux coordinates: {xⁱ, x^j} = 0, {θ_i,θ_j} = 0 for all i,j and {xⁱ,θ_j} = −{θ_j,xⁱ} = δ_jⁱ. Thus for arbitrary two functions f, g

where we denote by p(f) the parity of a function f (p(xⁱ) = 0, p(θ_j) = 1). Similarly one can write down the formulae for the non-degenerate even Poisson structure corresponding to an even symplectic structure.

A Poisson structure (odd or even) can be defined on a supermanifold independently of a symplectic structure as a bilinear operation on functions (bracket) satisfying the following relations taken as axioms:

where ε is the parity of the bracket (ε = 0 for an even Poisson structure and ε = 1 for an odd one). The correspondence between functions and Hamiltonian vector fields is defined in the same way as on ordinary manifolds: D_fg = {f, g}. Notice a possible parity shift: p(D_f) = p(f) + ε. Every Hamiltonian vector field D_f defines an infinitesimal transformation preserving the Poisson structure (and the corresponding symplectic structure in the case of a non-degenerate Poisson bracket).

Notice that even or odd Poisson structures on an arbitrary supermanifold can be obtained as “derived” brackets from the canonical symplectic structure on the cotangent bundle, in the following way.

Let M be a supermanifold and T∗M be its cotangent bundle. By changing parity of coordinates in the fibres of T∗M we arrive at the supermanifold ΠT ∗M. If z^A are arbitrary coordinates on the supermanifold M, then we denote by (zA,pB) the corresponding coordinates on the supermanifold T∗M and by (z^A,z_∗B) the corresponding coordinates on ΠT∗M: p(z^A) = p(p_A) = p(z_∗A) + 1. If (z^A′) are another coordinates on M, zA = zA(z′), then the coordinates z∗A transform in the same way as the coordinates p_A (and as the partial derivatives ∂/∂z^A):

p^A′ = ∂z^B(z′)/∂z^A′ p_B and z_∗A^′ = ∂z^B(z′)/∂z^A′ z_∗B

One can consider the canonical non-degenerate even Poisson structure { , }₀ (the canonical even symplectic structure) on T∗M defined by the relations {z^A,z^B}₀ = {p_C,p_D}₀ = 0, {z^A,p_B}0 = δ_B^A, and, respectively, the canonical non-degenerate odd Poisson structure { , }₁ (the canonical odd symplectic structure) on ΠT∗M defined by the relations {z^A,z^B}₀ = {z_∗C,z_∗D}₀ = 0, {z^A,z_∗B}₀ = δ^A_B.

Now consider Hamiltonians on T∗M or on ΠT∗M that are quadratic in coordinates of the fibres. An arbitrary odd quadratic Hamiltonian on T∗M (an arbitrary even quadratic Hamiltonian on ΠT∗M):

S(z,p) = S^ABp_Ap_B (p(S) = 1) or S(z,z∗) = S^ABz_∗Az_∗B (p(S) = 0) —– (1)

satisfying the condition that the canonical Poisson bracket of this Hamiltonian with itself vanishes:

{S,S}₀ = 0 or {S,S}₁ = 0 —– (2)

defines an odd Poisson structure (an even Poisson structure) on M by the formula

{f,g}^S_ε+1 = {f,{S,g}_ε}_ε —–(3)

The Hamiltonian S which generates an odd (even) Poisson structure on M via the canonical even (odd) Poisson structure on T∗M (ΠT∗M) can be called the master Hamiltonian. The bracket is a “derived bracket”. The Jacobi identity for it is equivalent to the vanishing of the canonical Poisson bracket for the master Hamiltonian. One can see that an arbitrary Poisson structure on a supermanifold can be obtained as a derived bracket.

What happens if we change the parity of the master Hamiltonian in (3)? The answer is the following. If S is an even quadratic Hamiltonian on T∗M (an odd quadratic Hamiltonian on ΠT∗M), then the condition of vanishing of the canonical even Poisson bracket { , }₀ (the canonical odd Poisson bracket { , }₁) becomes empty (it is obeyed automatically) and the relation (3) defines an even Riemannian metric (an odd Riemannian metric) on M.

Formally, odd symplectic (and odd Poisson) geometry is a generalization of symplectic (Poisson) geometry to the supercase. However, there are unexpected analogies between the constructions in odd symplectic geometry and in Riemannian geometry. The construction of derived brackets could explain close relations between odd Poisson structures in supermathematics and the Riemannian geometry.

Symplectic Manifolds

The canonical example of the n-symplectic manifold is that of the frame bundle, so the question is whether this formalism can be generalized to other principal bundles, and distinguished from the quantization arising from symplectic geometry on the prototype manifold, the bundle of linear frames, a good place to motivate the formalism.

Let us start with an n-dimensional manifold M, and let π : LM → M be the space of linear frames over a base manifold M, the set of pairs (m,e_k), where m ∈ M and {e_k},k = 1,···,n is a linear frame at m. This gives LM dimension n(n + 1), with GL(n,R) as the structure group acting freely on the right. We define local coordinates on LM in terms of those on the manifold M – for a chart on M with coordinates {xⁱ}, let

qⁱ(m,e_k) = xⁱ ◦ π(m,e_k) = xⁱ(m)

π_jⁱ(m,e_k) = e^j ∂/∂xj

where {e^j} denotes the coframe dual to {e_j}. These coordinates are analogous to those on the cotangent bundle, except, instead of a single momentum coordinate, we now have a momentum frame. We want to place some kind of structure on LM, which is the prototype of n-symplectic geometry that is similar to symplectic geometry of the cotangent bundle T∗M. The structure equation for symplectic geometry

df= _| X dθ

gives Hamilton’s equations for the phase space of a particle, where θ is the canonical symplectic 2-form. There is a naturally defined Rⁿ-valued 1-form on LM, the soldering form, given by

θ(X) ≡ u⁻¹[π∗(X)] ∀X ∈ T_uLM

where the point u = (m,e_k) ∈ LM gives the isomorphism u : Rⁿ → T_π(u)M by ξⁱr_i → ξⁱe_i, where {r_i} is the standard basis of Rⁿ. The Rⁿ-valued 2-form dθ can be shown to be non-degenerate, that is,

X _| dθ = 0 ⇔ X = 0

where we mean that each component of X dθ is identically zero. Finally, since there is also a structure group on LM, there are also group transformation properties. Let ρ be the standard representation of GL(n, R) on Rn. Then it can be shown that the pullback of dθ under right translation by g ∈ GL (n,R) is R_g^∗ dθ = ρ(g−1) · dθ.

Thus, we have an Rⁿ-valued generalization of symplectic geometry, which motivates the following definition.

Let P be a principal fiber bundle with structure group G over an m-dimensional manifold M . Let ρ : G → GL(n, R) be a linear representation of G. An n-symplectic structure on P is a Rⁿ-valued 2-form ω on P that is (i) closed and non-degenerate, in the sense that

X _| ω = 0 ⇔ X = 0

for a vector field X on P, and (ii) ω is equivariant, such that under the right action of G, R_g^∗ ω = ρ(g−1) · ω. The pair (P, ω) is called an n-symplectic manifold.

Here, we have modeled n-symplectic geometry after the frame bundle by defining the general n-symplectic manifold as a principal bundle. There is no reason, however, to limit ourselves to this, since we can let P be any manifold with a group action defined on it. One example of this would be to look at the action of the conformal group on R⁴. Since this group is locally isomorphic to O(2, 4), which is not a subgroup of GL(4, R), then forming a O(2,4) bundle over R⁴ cannot be thought of as simply a reduction of the frame bundle.