Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → M^i-1 →^di-1 Mⁱ →^di → Mⁱ⁺¹ → …

such that dⁱ ◦ d^i-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fⁱ : Mⁱ → Nⁱ in A, making the following diagram commute, where dⁱ_M, dⁱ_N denote the respective differentials:

We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the i^th cohomology object is the quotient

Hⁱ(M) = ker(dⁱ)/im(dⁱ⁻¹)

This operation of taking cohomology at the i^th place defines a functor

Hⁱ(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕_i Mⁱ

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d² = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hⁱ(f): Hⁱ(M) → Hⁱ(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →^f M⁰ →^d0 M¹ →^d1 M² → …

is a resolution of a module S, then by definition the morphism of complexes

is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s⁻¹: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f₁ · f₂ ·· · ·· f_r−1 · f_r where each f_i is either a morphism of C(A), or is of the form s⁻¹ for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s⁻¹ ∼ id_b and s⁻¹ · s ∼ id_a for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes

Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hⁱ(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hⁱ(−): D(A) → A

establishing that composite functor H⁰(−) ◦ J is isomorphic to the identity functor on A.

Advertisement

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

A coarse structure on a set X is defined to be a collection of subsets of X × X, called the controlled sets or entourages for the coarse structure, which satisfy some simple axioms. The most important of these states that if E and F are controlled then so is

E ◦ F := {(x, z) : ∃y, (x, y) ∈ E, (y, z) ∈ F}

Consider the metric spaces Zⁿ and Rⁿ. Their small-scale structure, their topology is entirely different, but on the large scale they resemble each other closely: any geometric configuration in Rⁿ can be approximated by one in Zⁿ, to within a uniformly bounded error. We think of such spaces as “coarsely equivalent”. The other axioms require that the diagonal should be a controlled set, and that subsets, transposes, and (finite) unions of controlled sets should be controlled. It is accurate to say that a coarse structure is the large-scale counterpart of a uniformity than of a topology.

Coarse structures and coarse spaces enjoy a philosophical advantage over coarse metric spaces, in that, all left invariant bounded geometry metrics on a countable group induce the same metric coarse structure which is therefore transparently uniquely determined by the group. On the other hand, the absence of a natural gauge complicates the notion of a coarse family, while it is natural to speak of sets of uniform size in different metric spaces it is not possible to do so in different coarse spaces without imposing additional structure.

Mikhail Leonidovich Gromov introduced the notion of coarse embedding for metric spaces. Let X and Y be metric spaces.

A map f : X → Y is said to be a coarse embedding if ∃ nondecreasing functions ρ₁ and ρ₂ from R₊ = [0, ∞) to R such that

• ρ₁(d(x,y)) ≤ d(f(x),f(y)) ≤ ρ₂(d(x,y)) ∀ x, y ∈ X.
• lim_r→∞ ρ_i(r) = +∞ (i=1, 2).

Intuitively, coarse embeddability of a metric space X into Y means that we can draw a picture of X in Y which reflects the large scale geometry of X. In early 90’s, Gromov suggested that coarse embeddability of a discrete group into Hilbert space or some Banach spaces should be relevant to solving the Novikov conjecture. The connection between large scale geometry and differential topology and differential geometry, such as the Novikov conjecture, is built by index theory. Recall that an elliptic differential operator D on a compact manifold M is Fredholm in the sense that the kernel and cokernel of D are finite dimensional. The Fredholm index of D, which is defined by

index(D) = dim(kerD) − dim(cokerD),

has the following fundamental properties:

(1) it is an obstruction to invertibility of D;

(2) it is invariant under homotopy equivalence.

The celebrated Atiyah-Singer index theorem computes the Fredholm index of elliptic differential operators on compact manifolds and has important applications. However, an elliptic differential operator on a noncompact manifold is in general not Fredholm in the usual sense, but Fredholm in a generalized sense. The generalized Fredholm index for such an operator is called the higher index. In particular, on a general noncompact complete Riemannian manifold M, John Roe (Coarse Cohomology and Index Theory on Complete Riemannian Manifolds) introduced a higher index theory for elliptic differential operators on M.

The coarse Baum-Connes conjecture is an algorithm to compute the higher index of an elliptic differential operator on noncompact complete Riemannian manifolds. By the descent principal, the coarse Baum-Connes conjecture implies the Novikov higher signature conjecture. Guoliang Yu has proved the coarse Baum-Connes conjecture for bounded geometry metric spaces which are coarsely embeddable into Hilbert space. The metric spaces which admit coarse embeddings into Hilbert space are a large class, including e.g. all amenable groups and hyperbolic groups. In general, however, there are counterexamples to the coarse Baum-Connes conjecture. A notorious one is expander graphs. On the other hand, the coarse Novikov conjecture (i.e. the injectivity part of the coarse Baum-Connes conjecture) is an algorithm of determining non-vanishing of the higher index. Kasparov-Yu have proved the coarse Novikov conjecture for spaces which admit coarse embeddings into a uniformly convex Banach space.

Game Theory and Finite Strategies: Nash Equilibrium Takes Quantum Computations to Optimality.

Finite games of strategy, within the framework of noncooperative quantum game theory, can be approached from finite chain categories, where, by finite chain category, it is understood a category C(n;N) that is generated by n objects and N morphic chains, called primitive chains, linking the objects in a specific order, such that there is a single labelling. C(n;N) is, thus, generated by N primitive chains of the form:

x₀ →^f₁ x₁ →^f₂ x₁ → … x_n-1 →^f_n x_n —– (1)

A finite chain category is interpreted as a finite game category as follows: to each morphism in a chain x_i-1 →^f_i x_i, there corresponds a strategy played by a player that occupies the position i, in this way, a chain corresponds to a sequence of strategic choices available to the players. A quantum formal theory, for a finite game category C(n;N), is defined as a formal structure such that each morphic fundament f_i of the morphic relation x_i-1 →^f_i x_i is a tuple of the form:

f_i := (H_i, P_i, P^ˆ_fi) —– (2)

where H_i is the i-th player’s Hilbert space, P_i is a complete set of projectors onto a basis that spans the Hilbert space, and P^ˆ_fi ∈ P_i. This structure is interpreted as follows: from the strategic Hilbert space H_i, given the pure strategies’ projectors P_i, the player chooses to play P^ˆ_fi .

From the morphic fundament (2), an assumption has to be made on composition in the finite category, we assume the following tensor product composition operation:

f_j ◦ f_i = f_ji —– (3)

f_ji = (H_ji = H_j ⊗ H_i, P_ji = P_j ⊗ P_i, P^ˆ_fji = P^ˆ_fj ⊗ P^ˆ_fi) —– (4)

From here, a morphism for a game choice path could be introduced as:

x₀ →^f_n…21 x_n —– (5)

f_n…21 = (H_G = ⊗_i=n¹ H_i, P_G = ⊗_i=n¹ P_i, P^ˆ _fn…21 = ⊗_i=n¹Pˆ_fi) —– (6)

in this way, the choices along the chain of players are completely encoded in the tensor product projectors P^ˆ_fn…21. There are N = ∏_i=1ⁿ dim(H_i) such morphisms, a number that coincides with the number of primitive chains in the category C(n;N).

Each projector can be addressed as a strategic marker of a game path, and leads to the matrix form of an Arrow-Debreu security, therefore, we call it game Arrow-Debreu projector. While, in traditional financial economics, the Arrow-Debreu securities pay one unit of numeraire per state of nature, in the present game setting, they pay one unit of payoff per game path at the beginning of the game, however this analogy may be taken it must be addressed with some care, since these are not securities, rather, they represent, projectively, strategic choice chains in the game, so that the price of a projector P^ˆ_fn…21 (the Arrow-Debreu price) is the price of a strategic choice and, therefore, the result of the strategic evaluation of the game by the different players.

Now, let |Ψ⟩ be a ket vector in the game’s Hilbert space H_G, such that:

|Ψ⟩ = ∑_fn…21 ψ(f_n…21)|(f_n…21⟩ —– (7)

where ψ(f_n…21) is the Arrow-Debreu price amplitude, with the condition:

∑_fn…21 |ψ(f_n…21)|² = D —– (8)

for D > 0, then, the |ψ(f_n…21)|² corresponds to the Arrow-Debreu prices for the game path f_n…21 and D is the discount factor in riskless borrowing, defining an economic scale for temporal connections between one unit of payoff now and one unit of payoff at the end of the game, such that one unit of payoff now can be capitalized to the end of the game (when the decision takes place) through a multiplication by 1/D, while one unit of payoff at the end of the game can be discounted to the beginning of the game through multiplication by D.

In this case, the unit operator, 1^ˆ = ∑_fn…21 P^ˆ_fn…21 has a similar profile as that of a bond in standard financial economics, with ⟨Ψ|1^ˆ|Ψ⟩ = D, on the other hand, the general payoff system, for each player, can be addressed from an operator expansion:

π_i^ˆ = ∑_fn…21 π_i (f_n…21) P^ˆ_fn…21 —– (9)

where each weight π_i(f_n…21) corresponds to quantities associated with each Arrow-Debreu projector that can be interpreted as similar to the quantities of each Arrow-Debreu security for a general asset. Multiplying each weight by the corresponding Arrow-Debreu price, one obtains the payoff value for each alternative such that the total payoff for the player at the end of the game is given by:

⟨Ψ|1^ˆ|Ψ⟩ = ∑_fn…21 π_i(f_n…21) |ψ(f_n…21)|²/D —– (10)

We can discount the total payoff to the beginning of the game using the discount factor D, leading to the present value payoff for the player:

PV_i = D ⟨Ψ|π_i^ˆ|Ψ⟩ = D ∑_fn…21 π_i  (f_n…21) |ψ(f_n…21)|²/D —– (11)

, where π_i  (f_n…21) represents quantities, while the ratio |ψ(f_n…21)|²/D represents the future value at the decision moment of the quantum Arrow- Debreu prices (capitalized quantum Arrow-Debreu prices). Introducing the ket

|Q⟩ ∈ H_G, such that:

|Q⟩ = 1/√D |Ψ⟩ —– (12)

then, |Q⟩ is a normalized ket for which the price amplitudes are expressed in terms of their future value. Replacing in (11), we have:

PV_i = D ⟨Q|π^ˆ_i|Q⟩ —– (13)

In the quantum game setting, the capitalized Arrow-Debreu price amplitudes ⟨f_n…21|Q⟩ become quantum strategic configurations, resulting from the strategic cognition of the players with respect to the game. Given |Q⟩, each player’s strategic valuation of each pure strategy can be obtained by introducing the projector chains:

C^ˆ_fi = ∑_{fn…i+1fi-1…1} P^ˆ_fn…i+1 ⊗ P^ˆ_fi ⊗ P^ˆ_fi-1…1 —– (14)

with ∑_fi C^ˆ_fi = 1^ˆ. For each alternative choice of the player i, the chain sums over all of the other choice paths for the rest of the players, such chains are called coarse-grained chains in the decoherent histories approach to quantum mechanics. Following this approach, one may introduce the pricing functional from the expression for the decoherence functional:

D (f_i, g_i : |Q⟩) = ⟨Q| C^ˆ†_fi C_gi|Q⟩  —– (15)

we, then have, for each player

D (f_i, g_i : |Q⟩) = 0, ∀ f_i ≠ g_i —– (16)

this is the usual quantum mechanics’ condition for an aditivity of measure (also known as decoherence condition), which means that the capitalized prices for two different alternative choices of player i are additive. Then, we can work with the pricing functional D(f_i, f_i :|Q⟩) as giving, for each player an Arrow-Debreu capitalized price associated with the pure strategy, expressed by f_i. Given that (16) is satisfied, each player’s quantum Arrow-Debreu pricing matrix, defined analogously to the decoherence matrix from the decoherent histories approach, is a diagonal matrix and can be expanded as a linear combination of the projectors for each player’s pure strategies as follows:

D_i (|Q⟩) = ∑_fi D(f_i, f_i : |Q⟩) P^ˆ_fi —– (17)

which has the mathematical expression of a mixed strategy. Thus, each player chooses from all of the possible quantum computations, the one that maximizes the present value payoff function with all the other strategies held fixed, which is in agreement with Nash.

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.

The Affinity of Mirror Symmetry to Algebraic Geometry: Going Beyond Formalism

Even though formalism of homological mirror symmetry is an established case, what of other explanations of mirror symmetry which lie closer to classical differential and algebraic geometry? One way to tackle this is the so-called Strominger, Yau and Zaslow mirror symmetry or SYZ in short.

The central physical ingredient in this proposal is T-duality. To explain this, let us consider a superconformal sigma model with target space (M, g), and denote it (defined as a geometric functor, or as a set of correlation functions), as

CFT(M, g)

In physics, a duality is an equivalence

CFT(M, g) ≅ CFT(M′, g′)

which holds despite the fact that the underlying geometries (M,g) and (M′, g′) are not classically diffeomorphic.

T-duality is a duality which relates two CFT’s with toroidal target space, M ≅ M′ ≅ T^d, but different metrics. In rough terms, the duality relates a “small” target space, with noncontractible cycles of length L < l_s, with a “large” target space in which all such cycles have length L > l_s.

This sort of relation is generic to dualities and follows from the following logic. If all length scales (lengths of cycles, curvature lengths, etc.) are greater than l_s, string theory reduces to conventional geometry. Now, in conventional geometry, we know what it means for (M, g) and (M′, g′) to be non-isomorphic. Any modification to this notion must be associated with a breakdown of conventional geometry, which requires some length scale to be “sub-stringy,” with L < l_s. To state T-duality precisely, let us first consider M = M′ = S¹. We parameterise this with a coordinate X ∈ R making the identification X ∼ X + 2π. Consider a Euclidean metric g_R given by ds² = R²dX². The real parameter R is usually called the “radius” from the obvious embedding in R². This manifold is Ricci-flat and thus the sigma model with this target space is a conformal field theory, the “c = 1 boson.” Let us furthermore set the string scale l_s = 1. With this, we attain a complete physical equivalence.

CFT(S¹, g_R) ≅ CFT(S¹, g_1/R)

Thus these two target spaces are indistinguishable from the point of view of string theory.

Just to give a physical picture for what this means, suppose for sake of discussion that superstring theory describes our universe, and thus that in some sense there must be six extra spatial dimensions. Suppose further that we had evidence that the extra dimensions factorized topologically and metrically as K₅ × S¹; then it would make sense to ask: What is the radius R of this S¹ in our universe? In principle this could be measured by producing sufficiently energetic particles (so-called “Kaluza-Klein modes”), or perhaps measuring deviations from Newton’s inverse square law of gravity at distances L ∼ R. In string theory, T-duality implies that R ≥ l_s, because any theory with R < l_s is equivalent to another theory with R > l_s. Thus we have a nontrivial relation between two (in principle) observable quantities, R and l_s, which one might imagine testing experimentally. Let us now consider the theory CFT(T^d, g), where T^d is the d-dimensional torus, with coordinates Xⁱ parameterising R^d/2πZ^d, and a constant metric tensor g_ij. Then there is a complete physical equivalence

CFT(T^d, g) ≅ CFT(T^d, g⁻¹)

In fact this is just one element of a discrete group of T-duality symmetries, generated by T-dualities along one-cycles, and large diffeomorphisms (those not continuously connected to the identity). The complete group is isomorphic to SO(d, d; Z).

While very different from conventional geometry, T-duality has a simple intuitive explanation. This starts with the observation that the possible embeddings of a string into X can be classified by the fundamental group π₁(X). Strings representing non-trivial homotopy classes are usually referred to as “winding states.” Furthermore, since strings interact by interconnecting at points, the group structure on π₁ provided by concatenation of based loops is meaningful and is respected by interactions in the string theory. Now π₁(T^d) ≅ Z^d, as an abelian group, referred to as the group of “winding numbers”.

Of course, there is another Z^d we could bring into the discussion, the Pontryagin dual of the U(1)^d of which T^d is an affinization. An element of this group is referred to physically as a “momentum,” as it is the eigenvalue of a translation operator on T^d. Again, this group structure is respected by the interactions. These two group structures, momentum and winding, can be summarized in the statement that the full closed string algebra contains the group algebra C[Z^d] ⊕ C[Z^d].

In essence, the point of T-duality is that if we quantize the string on a sufficiently small target space, the roles of momentum and winding will be interchanged. But the main point can be seen by bringing in some elementary spectral geometry. Besides the algebra structure, another invariant of a conformal field theory is the spectrum of its Hamiltonian H (technically, the Virasoro operator L₀ + L ^̄₀). This Hamiltonian can be thought of as an analog of the standard Laplacian ∆_g on functions on X, and its spectrum on T^d with metric g is

Spec ∆_g = {∑_i,j=1^d g^ijp_ip_j; p_i ∈ Z^d}

On the other hand, the energy of a winding string is (intuitively) a function of its length. On our torus, a geodesic with winding number w ∈ Z^d has length squared

L² = ∑_i,j=1^d g_ijwⁱw^j

Now, the only string theory input we need to bring in is that the total Hamiltonian contains both terms,

H = ∆_g + L² + · · ·

where the extra terms … express the energy of excited (or “oscillator”) modes of the string. Then, the inversion g → g⁻¹, combined with the interchange p ↔ w, leaves the spectrum of H invariant. This is T-duality.

There is a simple generalization of the above to the case with a non-zero B-field on the torus satisfying dB = 0. In this case, since B is a constant antisymmetric tensor, we can label CFT’s by the matrix g + B. Now, the basic T-duality relation becomes

CFT(T^d, g + B) ≅ CFT(T^d, (g + B)⁻¹)

Another generalization, which is considerably more subtle, is to do T-duality in families, or fiberwise T-duality. The same arguments can be made, and would become precise in the limit that the metric on the fibers varies on length scales far greater than l_s, and has curvature lengths far greater than l_s. This is sometimes called the “adiabatic limit” in physics. While this is a very restrictive assumption, there are more heuristic physical arguments that T-duality should hold more generally, with corrections to the relations proportional to curvatures l_s²R and derivatives l_s∂ of the fiber metric, both in perturbation theory and from world-sheet instantons.

Fréchet Spaces and Presheave Morphisms.

A topological vector space V is both a topological space and a vector space such that the vector space operations are continuous. A topological vector space is locally convex if its topology admits a basis consisting of convex sets (a set A is convex if (1 – t) + ty ∈ A ∀ x, y ∈ A and t ∈ [0, 1].

We say that a locally convex topological vector space is a Fréchet space if its topology is induced by a translation-invariant metric d and the space is complete with respect to d, that is, all the Cauchy sequences are convergent.

A seminorm on a vector space V is a real-valued function p such that ∀ x, y ∈ V and scalars a we have:

(1) p(x + y) ≤ p(x) + p(y),

(2) p(ax) = |a|p(x),

(3) p(x) ≥ 0.

The difference between the norm and the seminorm comes from the last property: we do not ask that if x ≠ 0, then p(x) > 0, as we would do for a norm.

If {p_i}_{i∈N} is a countable family of seminorms on a topological vector space V, separating points, i.e. if x ≠ 0, there is an i with p_i(x) ≠ 0, then ∃ a translation-invariant metric d inducing the topology, defined in terms of the {p_i}:

d(x, y) = ∑_i=1^∞ 1/2ⁱ p_i(x – y)/(1 + p_i(x – y))

The following characterizes Fréchet spaces, giving an effective method to construct them using seminorms.

A topological vector space V is a Fréchet space iff it satisfies the following three properties:

• it is complete as a topological vector space;
• it is a Hausdorff space;
• its topology is induced by a countable family of seminorms {p_i}_{i∈N}, i.e., U ⊂ V is open iff for every u ∈ U ∃ K ≥ 0 and ε > 0 such that {v|p_k(u – v) < ε ∀ k ≤ K} ⊂ U.

We say that a sequence (xn) in V converges to x in the Fréchet space topology defined by a family of seminorms iff it converges to x with respect to each of the given seminorms. In other words, x_n → x, iff p_i(x_n – x) → 0 for each i.

Two families of seminorms defined on the locally convex vector space V are said to be equivalent if they induce the same topology on V.

To construct a Fréchet space, one typically starts with a locally convex topological vector space V and defines a countable family of seminorms p_k on V inducing its topology and such that:

1. if x ∈ V and p_k(x) = 0 ∀ k ≥ 0, then x = 0 (separation property);
2. if (x_n) is a sequence in V which is Cauchy with respect to each seminorm, then ∃ x ∈ V such that (x_n) converges to x with respect to each seminorm (completeness property).

The topology induced by these seminorms turns V into a Fréchet space; property (1) ensures that it is Hausdorff, while the property (2) guarantees that it is complete. A translation-invariant complete metric inducing the topology on V can then be defined as above.

The most important example of Fréchet space, is the vector space C^∞(U), the space of smooth functions on the open set U ⊆ Rⁿ or more generally the vector space C^∞(M), where M is a differentiable manifold.

For each open set U ⊆ Rⁿ (or U ⊂ M), for each K ⊂ U compact and for each multi-index I , we define

||ƒ||_K,I := sup_x∈K |(∂^|I|/∂x^I (ƒ)) (x)|, ƒ ∈ C^∞(U)

Each ||.||_K,I defines a seminorm. The family of seminorms obtained by considering all of the multi-indices I and the (countable number of) compact subsets K covering U satisfies the properties (1) and (1) detailed above, hence makes C^∞(U) into a Fréchet space. The sets of the form

|ƒ ∈ C^∞(U)| ||ƒ – g||_K,I < ε

with fixed g ∈ C^∞(U), K ⊆ U compact, and multi-index I are open sets and together with their finite intersections form a basis for the topology.

All these constructions and results can be generalized to smooth manifolds. Let M be a smooth manifold and let U be an open subset of M. If K is a compact subset of U and D is a differential operator over U, then

p_K,D(ƒ) := sup_x∈K|D(ƒ)|

is a seminorm. The family of all the seminorms  p_K,D with K and D varying among all compact subsets and differential operators respectively is a separating family of seminorms endowing C_M^∞(U) with the structure of a complete locally convex vector space. Moreover there exists an equivalent countable family of seminorms, hence C_M^∞(U) is a Fréchet space. Let indeed {V_j} be a countable open cover of U by open coordinate subsets, and let, for each j, {K_j,i} be a countable family of compact subsets of V_j such that ∪_i K_j,i = V_j. We have the countable family of seminorms

p_K,I := sup_x∈K |(∂^|I|/∂x^I (ƒ)) (x)|, K ∈  {K_j,i}

inducing the topology. C_M^∞(U) is also an algebra: the product of two smooth functions being a smooth function.

A Fréchet space V is said to be a Fréchet algebra if its topology can be defined by a countable family of submultiplicative seminorms, i.e., a countable family {q_i)_i∈N of seminorms satisfying

q_i(ƒg) ≤q_i (ƒ) q_i(g) ∀ i ∈ N

Let F be a sheaf of real vector spaces over a manifold M. F is a Fréchet sheaf if:

(1)  for each open set U ⊆ M, F(U) is a Fréchet space;

(2)  for each open set U ⊆ M and for each open cover {U_i} of U, the topology of F(U) is the initial topology with respect to the restriction maps F(U) → F(U_i), that is, the coarsest topology making the restriction morphisms continuous.

As a consequence, we have the restriction map F(U) → F(V) (V ⊆ U) as continuous. A morphism of sheaves ψ: F → F’ is said to be continuous if the map F(U) → F'(U) is open for each open subset U ⊆ M.

Global Significance of Chinese Investments. My Deliberations in Mumbai (04/03/2018)

Legends:

What are fitted values in statistics?

The values for an output variable that have been predicted by a model fitted to a set of data. a statistical is generally an equation, the graph of which includes or approximates a majority of data points in a given data set. Fitted values are generated by extending the model of past known data points in order to predict unknown values. These are also called predicted values.

What are outliers in statistics?

These are observation points that are distant from other observations and may arise due to variability in the measurement  or it may indicate experimental errors. These may also arise due to heavy tailed distribution.

What is LBS (Locational Banking statistics)?

The locational banking statistics gather quarterly data on international financial claims and liabilities of bank offices in the reporting countries. Total positions are broken down by currency, by sector (bank and non-bank), by country of residence of the counterparty, and by nationality of reporting banks. Both domestically-owned and foreign-owned banking offices in the reporting countries record their positions on a gross (unconsolidated) basis, including those vis-à-vis own affiliates in other countries. This is consistent with the residency principle of national accounts, balance of payments and external debt statistics.

What is CEIC?

Census and Economic Information Centre

What are spillover effects?

These refer to the impact that seemingly unrelated events in one nation can have on the economies of other nations. since 2009, China has emerged a major source of spillover effects. This is because Chinese manufacturers have driven much of the global commodity demand growth since 2000. With China now being the second largest economy in the world, the number of countries that experience spillover effects from a Chinese slowdown is significant. China slowing down has a palpable impact on worldwide trade in metals, energy, grains and other commodities.

How does China deal with its Non-Performing Assets?

China adopted a four-point strategy to address the problems. The first was to reduce risks by strengthening banks and spearheading reforms of the state-owned enterprises (SOEs) by reducing their level of debt. The Chinese ensured that the nationalized banks were strengthened by raising disclosure standards across the board.

The second important measure was enacting laws that allowed the creation of asset management companies, equity participation and most importantly, asset-based securitization. The “securitization” approach is being taken by the Chinese to handle even their current NPA issue and is reportedly being piloted by a handful of large banks with specific emphasis on domestic investors. According to the International Monetary Fund (IMF), this is a prudent and preferred strategy since it gets assets off the balance sheets quickly and allows banks to receive cash which could be used for lending.

The third key measure that the Chinese took was to ensure that the government had the financial loss of debt “discounted” and debt equity swaps were allowed in case a growth opportunity existed. The term “debt-equity swap” (or “debt-equity conversion”) means the conversion of a heavily indebted or financially distressed company’s debt into equity or the acquisition by a company’s creditors of shares in that company paid for by the value of their loans to the company. Or, to put it more simply, debt-equity swaps transfer bank loans from the liabilities section of company balance sheets to common stock or additional paid-in capital in the shareholders’ equity section.

Let us imagine a company, as on the left-hand side of the below figure, with assets of 500, bank loans of 300, miscellaneous debt of 200, common stock of 50 and a carry-forward loss of 50. By converting 100 of its debt into equity (transferring 50 to common stock and 50 to additional paid-in capital), thereby improving the balance sheet position and depleting additional paid-in capital (or using the net income from the following year), as on the right-hand side of the figure, the company escapes insolvency. The former creditors become shareholders, suddenly acquiring 50% of the voting shares and control of the company.

The first benefit that results from this is the improvement in the company’s finances produced by the reduction in debt. The second benefit (from the change in control) is that the creditors become committed to reorganizing the company, and the scope for moral hazard by the management is limited. Another benefit is one peculiar to equity: a return (i.e., repayment) in the form of an increase in enterprise value in the future. In other words, the fact that the creditors stand to make a return on their original investment if the reorganization is successful and the value of the business rises means that, like the debtor company, they have more to gain from this than from simply writing off their loans. If the reorganization is not successful, the equity may, of course, prove worthless.

The fourth measure they took was producing incentives like tax breaks, exemption from administrative fees and transparent evaluations norms. These strategic measures ensured the Chinese were on top of the NPA issue in the early 2000s, when it was far larger than it is today. The noteworthy thing is that they were indeed successful in reducing NPAs. How is this relevant to India and how can we address the NPA issue more effectively?

For now, capital controls and the paying down of foreign currency loans imply that there are few channels through which a foreign-induced debt sell-off could trigger a collapse in asset prices. Despite concerns in 2016 over capital outflow, China’s foreign exchange reserves have stabilised.

But there is a long-term cost. China is now more vulnerable to capital outflow. Errors and omissions on its national accounts remain large, suggesting persistent unrecorded capital outflows. This loss of capital should act as a salutary reminder to those who believe that China can take the lead on globalisation or provide the investment or currency business to fuel things like a post-Brexit economy.

The Chinese government’s focus on debt management will mean tighter controls on speculative international investments. It will also provide a stern test of China’s centrally planned financial system for the foreseeable future.

Global Significance of Chinese investments

Categories of Pointwise Convergence Topology: Theory(ies) of Bundles.

Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {U_i}_i∈I of X, the transition functions between trivializations P|_Ui can be lifted to bundle isomorphisms g_ij on double intersections U_ij = U_i ∩ U_j which are projectively coherent, i.e., over each of the triple intersections U_ijk = U_i ∩ U_j ∩ U_k the composition g_ki g_jk g_ij is given as multiplication by a U(1)-valued function f_ijk : U_ijk → U(1). The collection {(U_ij, f_ijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class B_P in the sheaf cohomology group H²(X, U(1)_X). On the other hand, the sheaf cohomology H¹(X, PU(H)_X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H¹(X,PU(H)_X).

There is an isomorphism

H¹(X, PU(H)_X) →^≈ H²(X, U(1)_X) provided by the

boundary map [P] ↦ B_P. There is also an isomorphism

H²(X, U(1)_X) →^≈ H³(X, Z_X) ≅ H³(X, Z)

The image δ(P) ∈ H³(X, Z) of B_P is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H³(X, R) by a closed three-form H on X, called the H-flux of the given B-field B_P, we will write P = P_H. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H³(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Z_n with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cⁿ). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element B_P ∈ H²(X, U(1)_X) comes from H²(X, (Z_n)_X), and therefore its order divides n.

One also has δ(P₁ ⊗ P₂) = δ(P₁) + δ(P₂) and δ(P^∨) = −δ(P). This follows from the commutative diagram

and the fact that P^∨ ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H³(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that P_x = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z₂-graded Hilbert space. We define Fred⁰(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T² − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred⁰(H) ֒→ B(H) × K(H) given by T → (T, T² − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = P_H → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred⁰(P) whose fibres are Fred⁰(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K⁰(X, H) = [X, Fred⁰(P_H)]

The group K⁰(X, H) depends functorially on the pair (X, P_H), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K⁰(X, H) → K⁰(X, H′). Addition in K⁰(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K⁰(X, H₂) with [H₂] = δ(P ⊗ P(C²)) = δ(P) = [H]. Under the isomorphism H ⊗ C² ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C²) for any projective bundle P and so K⁰(X, H₂) is canonically isomorphic to K⁰(X, H). When [H] is a non-torsion element of H³(X, Z), so that P = P_H is an infinite-dimensional bundle of projective spaces, then the index map K⁰(X, H) → Z is zero, i.e., any section of Fred⁰(P) takes values in the index zero component of Fred⁰(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle P_H we can associate a bundle of compact operators E_H → X given by

E_H = P_H ×_PU K

with δ(E_H) = [H]. The Banach algebra A_H := C₀(X, E_H) of continuous sections of E_H vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K^•(X, H) of X is canonically isomorphic to the algebraic K-theory group K_•(A_H).

We will also need a smooth version of this definition. Let A^∞_H be the smooth subalgebra of A_H given by the algebra CT^∞(X, H) = C^∞(X, L¹_PH),

where L¹_PH = P_H ×_PUL¹. Then the inclusion CT^∞(X, H) → CT(X, H) induces an isomorphism K_•CT(X, H) →^≈ K_•CT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism K_•CT^∞(X, H) ≅ K^•(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = P_H over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K⁰(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, M_m(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism

δ : Br(X) →^≈ tor(H³(X, Z))

where tor(H³(X, Z)) is the torsion subgroup of H³(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K₀(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H³(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K⁰(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K⁰(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K₀(C(X)) = K⁰(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre C^nm ≅ (C^m)^⊕n, which is naturally an M_m(C)-module.

 

Embedding Branes in Minkowski Space-Time Dimensions To Decipher Them As Particles Or Otherwise

The physics treatment of Dirichlet branes in terms of boundary conditions is very analogous to that of the “bulk” quantum field theory, and the next step is again to study the renormalization group. This leads to equations of motion for the fields which arise from the open string, namely the data (M, E, ∇). In the supergravity limit, these equations are solved by taking the submanifold M to be volume minimizing in the metric on X, and the connection ∇ to satisfy the Yang-Mills equations.

Like the Einstein equations, the equations governing a submanifold of minimal volume are highly nonlinear, and their general theory is difficult. This is one motivation to look for special classes of solutions; the physical arguments favoring supersymmetry are another. Just as supersymmetric compactification manifolds correspond to a special class of Ricci-flat manifolds, those admitting a covariantly constant spinor, supersymmetry for a Dirichlet brane will correspond to embedding it into a special class of minimal volume submanifolds. Since the physical analysis is based on a covariantly constant spinor, this special class should be defined using the spinor, or else the covariantly constant forms which are bilinear in the spinor.

The standard physical arguments leading to this class are based on the kappa symmetry of the Green-Schwarz world-volume action, in which one finds that the subset of supersymmetry parameters ε which preserve supersymmetry, both of the metric and of the brane, must satisfy

φ ≡ Re ε^t Γε|_M = Vol|_M —– (1)

In words, the real part of one of the covariantly constant forms on M must equal the volume form when restricted to the brane.

Clearly dφ = 0, since it is covariantly constant. Thus,

Z(M) ≡ ∫_M φ —– (2)

depends only on the homology class of M. Thus, it is what physicists would call a “topological charge”, or a “central charge”.

If in addition the p-form φ is dominated by the volume form Vol upon restriction to any p-dimensional subspace V ⊂ T_x X, i.e.,

φ|_V ≤ Vol|_V —– (3)

then φ will be a calibration in the sense of implying the global statement

∫_M φ ≤ ∫_M Vol —– (4)

for any submanifold M . Thus, the central charge |Z (M)| is an absolute lower bound for Vol(M).

A calibrated submanifold M is now one satisfying (1), thereby attaining the lower bound and thus of minimal volume. Physically these are usually called “BPS branes,” after a prototypical argument of this type due, for magnetic monopole solutions in nonabelian gauge theory.

For a Calabi-Yau X, all of the forms ω^p can be calibrations, and the corresponding calibrated submanifolds are p-dimensional holomorphic submanifolds. Furthermore, the n-form Re e^iθΩ for any choice of real parameter θ is a calibration, and the corresponding calibrated submanifolds are called special Lagrangian.

This generalizes to the presence of a general connection on M, and leads to the following two types of BPS branes for a Calabi-Yau X. Let n = dim_R M, and let F be the (End(E)-valued) curvature two-form of ∇.

The first kind of BPS D-brane, based on the ω^p calibrations, is (for historical reasons) called a “B-type brane”. Here the BPS constraint is equivalent to the following three requirements:

1. M is a p-dimensional complex submanifold of X.
2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|^p−1_M ∧ F = c · ω|^p_Mfor some real constant c.
4. F satisfies Im e^iφ(ω|_M + il_s²F)^p = 0 for some real constant φ, where l_s is the correction.

The second kind of BPS D-brane, based on the Re e^iθΩ calibration, is called an “A-type” brane. The simplest examples of A-branes are the so-called special Lagrangian submanifolds (SLAGs), satisfying

(1) M is a Lagrangian submanifold of X with respect to ω.

(2) F = 0, i.e., the vector bundle E is flat.

(3) Im e^iα Ω|_M = 0 for some real constant α.

More generally, one also has the “coisotropic branes”. In the case when E is a line bundle, such A-branes satisfy the following four requirements:

(1)  M is a coisotropic submanifold of X with respect to ω, i.e., for any x ∈ M the skew-orthogonal complement of T_xM ⊂ T_xX is contained in T_xM. Equivalently, one requires ker ω_M to be an integrable distribution on M.

(2)  The 2-form F annihilates ker ω_M.

(3)  Let F M be the vector bundle T M/ ker ω_M. It follows from the first two conditions that ω_M and F descend to a pair of skew-symmetric forms on FM, denoted by σ and f. Clearly, σ is nondegenerate. One requires the endomorphism σ⁻¹f : FM → FM to be a complex structure on FM.

(4)  Let r be the complex dimension of FM. r is even and that r + n = dim_R M. Let Ω be the holomorphic trivialization of K_X. One requires that Im e^iαΩ|_M ∧ F^r/2 = 0 for some real constant α.

Coisotropic A-branes carrying vector bundles of higher rank are still not fully understood. Physically, one must also specify the embedding of the Dirichlet brane in the remaining (Minkowski) dimensions of space-time. The simplest possibility is to take this to be a time-like geodesic, so that the brane appears as a particle in the visible four dimensions. This is possible only for a subset of the branes, which depends on which string theory one is considering. Somewhat confusingly, in the type IIA theory, the B-branes are BPS particles, while in IIB theory, the A-branes are BPS particles.

Intuitive Algebra (Groupoid/Categorical Structure) of Open Strings As Morphisms

A geometric Dirichlet brane is a triple (L, E, ∇_E) – a submanifold L ⊂ M, carrying a vector bundle E, with connection ∇_E.

The real dimension of L is also often brought into the nomenclature, so that one speaks of a Dirichlet p-brane if p = dim_RL.

An open string which stretches from a Dirichlet brane (L, E, ∇_E) to a Dirichlet brane (K, F, ∇_F), is a map X from an interval I ≅ [0,1] to M, such that X(0) ∈ L and X(1) ∈ K. An “open string history” is a map from R into open strings, or equivalently a map from a two-dimensional surface with boundary, say Σ ≡ I × R, to M , such that the two boundaries embed into L and K.

The quantum theory of these open strings is defined by a functional integral over these histories, with a weight which depends on the connections ∇_E and ∇_F. It describes the time evolution of an open string state which is a wave function in a Hilbert space H_B,B′ labelled by the two choices of brane B = (L, E, ∇_E) and B′ = (K, F, ∇_F).

Distinct Dirichlet branes can embed into the same submanifold L. One way to represent this would be to specify the configurations of Dirichlet branes as a set of submanifolds with multiplicity. However, we can also represent this choice by using the choice of bundle E. Thus, a set of N identical branes will be represented by tensoring the bundle E with C^N. The connection is also obtained by tensor product. An N-fold copy of the Dirichlet brane (L, E, ∇_E) is thus a triple (L, E ⊗C^N, ∇_E ⊗ id_N).

In physics, one visualizes this choice by labelling each open string boundary with a basis vector of C^N, which specifies a choice among the N identical branes. These labels are called Chan-Paton factors. One then uses them to constrain the interactions between open strings. If we picture such an interaction as the joining of two open strings to one, the end of the first to the beginning of the second, we require not only the positions of the two ends to agree, but also the Chan-Paton factors. This operation is the intuitive algebra of open strings.

Mathematically, an algebra of open strings can always be tensored with a matrix algebra, in general producing a noncommutative algebra. More generally, if there is more than one possible boundary condition, then, rather than an algebra, it is better to think of this as a groupoid or categorical structure on the boundary conditions and the corresponding open strings. In the language of groupoids, particular open strings are elements of the groupoid, and the composition law is defined only for pairs of open strings with a common boundary. In the categorical language, boundary conditions are objects, and open strings are morphisms. The simplest intuitive argument that a non-trivial choice can be made here is to call upon the general principle that any local deformation of the world-sheet action should be a physically valid choice. In particular, particles in physics can be charged under a gauge field, for example the Maxwell field for an electron, the color Yang-Mills field for a quark, and so on. The wave function for a charged particle is then not complex-valued, but takes values in a bundle E.

Now, the effect of a general connection ∇_E is to modify the functional integral by modifying the weight associated to a given history of the particle. Suppose the trajectory of a particle is defined by a map φ : R → M; then a natural functional on trajectories associated with a connection ∇ on M is simply its holonomy along the trajectory, a linear map from E|_φ(t₁) to E|_φ(t₂). The functional integral is now defined physically as a sum over trajectories with this holonomy included in the weight.

The simplest way to generalize this to a string is to consider the l_s → 0 limit. Now the constraint of finiteness of energy is satisfied only by a string of vanishingly small length, effectively a particle. In this limit, both ends of the string map to the same point, which must therefore lie on L ∩ K.

The upshot is that, in this limit, the wave function of an open string between Dirichlet branes (L, E, ∇) and (K, F, ∇_F) transforms as a section of E^∨ ⊠ F over L ∩ K, with the natural connection on the direct product. In the special case of (L, E, ∇_E) ≅ (K, F, ∇_F), this reduces to the statement that an open string state is a section of EndE. Open string states are sections of a graded vector bundle End E ⊗ Λ•T∗L, the degree-1 part of which corresponds to infinitesimal deformations of ∇_E. In fact, these open string states are the infinitesimal deformations of ∇_E, in the standard sense of quantum field theory, i.e., a single open string is a localized excitation of the field obtained by quantizing the connection ∇_E. Similarly, other open string states are sections of the normal bundle of L within X, and are related in the same way to infinitesimal deformations of the submanifold. These relations, and their generalizations to open strings stretched between Dirichlet branes, define the physical sense in which the particular set of Dirichlet branes associated to a specified background X can be deduced from string theory.