Conjectural Existence of the Categorial Complex Branes for Generalized Calabi-Yau.

Geometric Langlands Duality can be formulated as follows: Let C be a Riemann surface (compact, without boundary), G be a compact reductive Lie group, G_C be its complexification, and M_flat(G, C) be the moduli space of stable flat G_C-connections on C. The Langlands dual of G is another compact reductive Lie group ^LG defined by the condition that its weight and coweight lattices are exchanged relative to G. Let Bun(^LG, C) be the moduli stack of holomorphic ^LG-bundles on C. One of the statements of Geometric Langlands Duality is that the derived category of coherent sheaves on M_flat(G, C) is equivalent to the derived category of D-modules over Bun(^LG, C).

M_flat(G, C) is mirror to another moduli space which, roughly speaking, can be described as the cotangent bundle to Bun(^LG, C). The category of A-branes on T ^∗ Bun(^LG, C) (with the canonical symplectic form) is equivalent to the category of B-branes on a noncommutative deformation of T ^∗ Bun(^LG, C). The latter is the same as the category of (analytic) D-modules on Bun(^LG, C).

So, what exactly is, the relationship between A-branes and noncommutative B-branes. This relationship arises whenever the target space X is the total space of the cotangent bundle to a complex manifold Y. It is understood that the  symplectic form ω is proportional to the canonical symplectic form on T ^∗ Y. With the B-field vanishing, and Y as a complex, we regard ω as the real part of a holomorphic symplectic form Ω. If qⁱ are holomorphic coordinates on Y, and p_i are dual coordinates on the fibers of T ^∗ Y,  Ω can be written as

Ω = 1/ħdp_i ∧ dqⁱ = dΘ

Since ω (as well as Ω) is exact, the closed A-model of X is rather trivial: there are no nontrivial instantons, and the quantum cohomology ring is isomorphic to the classical one.

We would like to understand the category of A-branes on X = T ^∗ Y. The key observation is that ∃ a natural coisotropic A-brane on X well-defined up to tensoring with a flat line bundle on X. Its curvature 2-form is exact and given by

F = Im Ω

If we denote by I the natural almost complex structure on X coming from the complex structure on Y , we have F = ωI, and therefore the endomorphism ω⁻¹F = I squares to −1. Therefore any unitary connection on a trivial line bundle over X whose curvature is F defines a coisotropic A-brane. 

Now, what about the endomorphisms of the canonical coisotropic A-brane, i.e., the algebra of BRST-closed open string vertex operators? This is easy if Y is an affine space. If one covers Y with charts each of which is an open subset of Cⁿ, and then argues that the computation can be performed locally on each chart and the results “glued together”, one gets closer to the fact that the algebra in question is the cohomology of a certain sheaf of algebras, whose local structure is the same as for Y = Cⁿ. In general, the path integral defining the correlators of vertex operators does not have any locality properties in the target space. Each term in perturbation theory depends only on the infinitesimal neighbourhood of a point. This shows that the algebra of open-string vertex operators, regarded as a formal power series in ħ, is the cohomology of a sheaf of algebras, which is locally isomorphic to a similar sheaf for X = Cⁿ × Cⁿ.

Let us apply these observations to the canonical coisotropic A-brane on X = T ^∗ Y. Locally, we can identify Y with a region in Cⁿ by means of holomorphic coordinate functions q₁, . . . , q_n. Up to BRST-exact terms, the action of the A-model on a disc Σ 􏰠takes the form

S = 1/ħ ∫_∂Σ φ ^∗ (p_idqⁱ)

where φ is a map from Σ to X. This action is identical to the action of a particle on Y with zero Hamiltonian, except that qⁱ are holomorphic coordinates on Y rather than ordinary coordinates. The BRST-invariant open-string vertex operators can be taken to be holomorphic functions of p, q. Therefore quantization is locally straightforward and gives a noncommutative deformation of the algebra of holomorphic functions on T ^∗ Y corresponding to a holomorphic Poisson bivector

P = ħ∂/∂p_i ∧ ∂/∂qⁱ

One can write an explicit formula for the deformed product:

􏰋(f ⋆ g)(p, q) = exp(􏰋ħ/2(∂²/∂p_i∂q̃ⁱ  −  ∂²/∂q^i∂p̃_i )) f(p, q) g (p̃, q̃)|_{p̃ = p, q̃ = q}

This product is known as the Moyal-Wigner product, which is a formal power series in ħ that may have zero radius of convergence. To rectify the situation, one can restrict to functions which are polynomial in the fiber coordinates p_i. Such locally-defined functions on T ^∗ Y can be thought of as symbols of differential operators; the Moyal-Wigner product in this case reduces to the product of symbols and is a polynomial in ħ. Thus locally the sheaf of open-string vertex operators is modelled on the sheaf of holomorphic differential operators on Y (provided we restrict to operators polynomial in p_i).

Locally, there is no difference between the sheaf of holomorphic differential operators D(Y ) and the sheaf of holomorphic differential operatorsD(Y, L) on a holomorphic line bundle L over Y. Thus the sheaf of open-string vertex operators could be any of the sheaves D(Y, L). Moreover, the classical problem is symmetric under p_i → −p_i combined with the orientation reversal of Σ; if we require that quantization preserve this symmetry, then the algebra of open-string vertex operators must be isomorphic to its opposite algebra. It is known that the opposite of the sheaf D(Y, L) is the sheaf D(Y, L⁻¹ ⊗ K_Y), so symmetry under p_i → −p_i requires L to be a square root of the canonical line bundle K_Y. It does not matter which square root one takes, since they all differ by flat line bundles on Y, and tensoring L by a flat line bundle does not affect the sheaf D(Y, L). The conclusion is that the sheaf of open-string vertex operators for the canonical coisotropic A-brane α on X = T ^∗ Y is isomorphic to the sheaf of noncommutative algebras D(Y, K^1/2). One can use this fact to associate Y to any A-brane β on X a twisted D-module, i.e., a sheaf of modules over D(Y, K^1/2). Consider the A-model with target X on a strip Σ = I × R, where I is a unit interval, and impose boundary conditions corresponding to branes α and β on the two boundaries of Σ. Upon quantization of this model, one gets a sheaf on vector spaces on Y which is a module over the sheaf of open-string vertex operators inserted at the α boundary. A simple example is to take β to be the zero section of T ^∗ Y with a trivial line bundle. Then the corresponding sheaf is simply the sheaf of sections of K_Y^1/2, with a tautological action of D(Y, K_Y^1/2).

One can argue that the map from A-branes to (complexes of) D-modules can be extended to an equivalence of categories of A-branes on X and the derived category of D-modules on Y. The argument relies on the conjectural existence of the category of generalized complex branes for any generalized Calabi-Yau. One can regard the Geometric Langlands Duality as a nonabelian generalization. 

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The Natural Theoretic of Electromagnetism. Thought of the Day 147.0

In Maxwell’s theory, the field strength F = 1/2F_μν dx^μ ∧ dx^ν is a real 2-form on spacetime, and thence a natural object at the same time. The homogeneous Maxwell equation dF = 0 is an equation involving forms and it has a well-known local solution F = dA’, i.e. there exists a local spacetime 1-form A’ which is a potential for the field strength F. Of course, if spacetime is contractible, as e.g. for Minkowski space, the solution is also a global one. As is well-known, in the non-commutative Yang-Mills theory case the field strength F = 1/2F^A_μν T_A ⊗ dx^μ ∧ dx^ν is no longer a spacetime form. This is a somewhat trivial remark since the transformation laws of such field strength are obtained as the transformation laws of the curvature of a principal connection with values in the Lie algebra of some (semisimple) non-Abelian Lie group G (e.g. G = SU(n), n 2 ≥ 2). However, the common belief that electromagnetism is to be intended as the particular case (for G =U(1)) of a non-commutative theory is not really physically evident. Even if we subscribe this common belief, which is motivated also by the tremendous success of the quantized theory, let us for a while discuss electromagnetism as a standalone theory.

From a mathematical viewpoint this is a (different) approach to electromagnetism and the choice between the two can be dealt with on a physical ground only. Of course the 1-form A’ is defined modulo a closed form, i.e. locally A” = A’ + dα is another solution.

How can one decide whether the potential of electromagnetism should be considered as a 1-form or rather as a principal connection on a U(1)-bundle? First of all we notice that by a standard hole argument (one can easily define compact supported closed 1-forms, e.g. by choosing the differential of compact supported functions which always exist on a paracompact manifold) the potentials A and A’ represent the same physical situation. On the other hand, from a mathematical viewpoint we would like the dynamical field, i.e. the potential A’, to be a global section of some suitable configuration bundle. This requirement is a mathematical one, motivated on the wish of a well-defined geometrical perspective based on global Variational Calculus.

The first mathematical way out is to restrict attention to contractible spacetimes, where A’ may be always chosen to be global. Then one can require the gauge transformations A” = A’ + dα to be Lagrangian symmetries. In this way, field equations select a whole equivalence class of gauge-equivalent potentials, a procedure which solves the hole argument problem. In this picture the potential A’ is really a 1-form, which can be dragged along spacetime diffeomorphism and which admits the ordinary Lie derivatives of 1-forms. Unfortunately, the restriction to contractible spacetimes is physically unmotivated and probably wrong.

Alternatively, one can restrict electromagnetic fields F, deciding that only exact 2-forms F are allowed. That actually restricts the observable physical situations, by changing the homogeneous Maxwell equations (i.e. Bianchi identities) by requiring that F is not only closed but exact. One should in principle be able to empirically reject this option.

On non-contractible spacetimes, one is necessarily forced to resort to a more “democratic” attitude. The spacetime is covered by a number of patches U_α. On each patch U_α one defines a potential A^(α). In the intersection of two patches the two potentials A^(α) and A^(β) may not agree. In each patch, in fact, the observer chooses his own conventions and he finds a different representative of the electromagnetic potential, which is related by a gauge transformation to the representatives chosen in the neighbour patch(es). Thence we have a family of gauge transformations, one in each intersection U_αβ, which obey cocycle identities. If one recognizes in them the action of U(1) then one can build a principal bundle P = (P, M, π; U(1)) and interpret the ensuing potential as a connection on P. This leads way to the gauge natural formalism.

Anyway this does not close the matter. One can investigate if and when the principal bundle P, in addition to the obvious principal structure, can be also endowed with a natural structure. If that were possible then the bundle of connections C_p (which is associated to P) would also be natural. The problem of deciding whether a given gauge natural bundle can be endowed with a natural structure is quite difficult in general and no full theory is yet completely developed in mathematical terms. That is to say, there is no complete classification of the topological and differential geometric conditions which a principal bundle P has to satisfy in order to ensure that, among the principal trivializations which determine its gauge natural structure, one can choose a sub-class of trivializations which induce a purely natural bundle structure. Nor it is clear how many inequivalent natural structures a good principal bundle may support. Though, there are important examples of bundles which support at the same time a natural and a gauge natural structure. Actually any natural bundle is associated to some frame bundle L(M), which is principal; thence each natural bundle is also gauge natural in a trivial way. Since on any paracompact manifold one can choose a global Riemannian metric g, the corresponding tangent bundle T(M) can be associated to the orthonormal frame bundle O(M, g) besides being obviously associated to L(M). Thence the natural bundle T(M) may be also endowed with a gauge natural bundle structure with structure group O(m). And if M is orientable the structure can be further reduced to a gauge natural bundle with structure group SO(m).

Roughly speaking, the task is achieved by imposing restrictions to cocycles which generate T(M) according to the prescription by imposing a privileged class of changes of local laboratories and sets of measures. Imposing the cocycle ψ_(αβ) to take its values in O(m) rather than in the larger group GL(m). Inequivalent gauge natural structures are in one-to-one correspondence with (non isometric) Riemannian metrics on M. Actually whenever there is a Lie group homomorphism ρ : GU(m) → G for some s onto some given Lie group G we can build a natural G-principal bundle on M. In fact, let (U_α, ψ_(α)) be an atlas of the given manifold M, ψ_(αβ) be its transition functions and jψ_(αβ) be the induced transition functions of L(M). Then we can define a G-valued cocycle on M by setting ρ(jψ_(αβ)) and thence a (unique up to fibered isomorphisms) G-principal bundle P(M) = (P(M), M, π; G). The bundle P(M), as well as any gauge natural bundle associated to it, is natural by construction. Now, defining a whole family of natural U(1)-bundles P_q(M) by using the bundle homomorphisms

ρ_q: GL(m) → U(1): J ↦ exp(iq ln det|J|) —– (1)

where q is any real number and In denotes the natural logarithm. In the case q = 0 the image of ρ₀ is the trivial group {I}; and, all the induced bundles are trivial, i.e. P = M x U(1).

The natural lift φ’ of a diffeomorphism φ: M → M is given by

φ'[x, e^iθ]_α = [φ(x), e^{iq ln det|J|}. e^iθ]_α —– (2)

where J is the Jacobin of the morphism φ. The bundles P_q(M) are all trivial since they allow a global section. In fact, on any manifold M, one can define a global Riemannian metric g, where the local sections glue together.

Since the bundles P_q(M) are all trivial, they are all isomorphic to M x U(1) as principal U(1)-bundles, though in a non-canonical way unless q = 0. Any two of the bundles P_q1(M) and P_q2(M) for two different values of q are isomorphic as principal bundles but the isomorphism obtained is not the lift of a spacetime diffeomorphism because of the two different values of q. Thence they are not isomorphic as natural bundles. We are thence facing a very interesting situation: a gauge natural bundle C associated to the trivial principal bundle P can be endowed with an infinite family of natural structures, one for each q ∈ R; each of these natural structures can be used to regard principal connections on P as natural objects on M and thence one can regard electromagnetism as a natural theory.

Now that the mathematical situation has been a little bit clarified, it is again a matter of physical interpretation. One can in fact restrict to electromagnetic potentials which are a priori connections on a trivial structure bundle P ≅ M x U(1) or to accept that more complicated situations may occur in Nature. But, non-trivial situations are still empirically unsupported, at least at a fundamental level.

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.

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.

Killing Fields

Let κ^a be a smooth field on our background spacetime (M, g_ab). κ^a is said to be a Killing field if its associated local flow maps Γ_s are all isometries or, equivalently, if £_κ g_ab = 0. The latter condition can also be expressed as ∇(_aκ_b) = 0.

Any number of standard symmetry conditions—local versions of them, at least can be cast as claims about the existence of Killing fields. Local, because killing fields need not be complete, and their associated flow maps need not be defined globally.

(M, g_ab) is stationary if it has a Killing field that is everywhere timelike.

(M, g_ab) is static if it has a Killing field that is everywhere timelike and locally hypersurface orthogonal.

(M, g_ab) is homogeneous if its Killing fields, at every point of M, span the tangent space.

In a stationary spacetime there is, at least locally, a “timelike flow” that preserves all spacetime distances. But the flow can exhibit rotation. Think of a whirlpool. It is the latter possibility that is ruled out when one passes to a static spacetime. For example, Gödel spacetime, is stationary but not static.

Let κ^a be a Killing field in an arbitrary spacetime (M, g_ab) (not necessarily Minkowski spacetime), and let γ : I → M be a smooth, future-directed, timelike curve, with unit tangent field ξ^a. We take its image to represent the worldline of a point particle with mass m > 0. Consider the quantity J = (P^aκ_a), where P^a = mξ^a is the four-momentum of the particle. It certainly need not be constant on γ[I]. But it will be if γ is a geodesic. For in that case, ξⁿ∇_nξ^a = 0 and hence

ξⁿ∇_nJ = m(κ_a ξⁿ∇_nξ^a + ξⁿξ^a ∇_nκ_a) = mξⁿξ^a ∇(_nκ_a) = 0

Thus, J is constant along the worldlines of free particles of positive mass. We refer to J as the conserved quantity associated with κ^a. If κ^a is timelike, we call J the energy of the particle (associated with κ^a). If it is spacelike, and if its associated flow maps resemble translations, we call J the linear momentum of the particle (associated with κ^a). Finally, if κ^a is spacelike, and if its associated flow maps resemble rotations, then we call J the angular momentum of the particle (associated with κ^a).

It is useful to keep in mind a certain picture that helps one “see” why the angular momentum of free particles (to take that example) is conserved. It involves an analogue of angular momentum in Euclidean plane geometry. Figure below shows a rotational Killing field κ^a in the Euclidean plane, the image of a geodesic (i.e., a line) L, and the tangent field ξ^a to the geodesic. Consider the quantity J = ξ^aκ_a, i.e., the inner product of ξ^a with κ^a – along L, and we can better visualize the assertion.

Figure: κ^a is a rotational Killing field. (It is everywhere orthogonal to a circle radius, and is proportional to it in length.) ξ^a is a tangent vector field of constant length on the line L. The inner product between them is constant. (Equivalently, the length of the projection of κ^a onto the line is constant.)

Let us temporarily drop indices and write κ·ξ as one would in ordinary Euclidean vector calculus (rather than ξ^aκ_a). Let p be the point on L that is closest to the center point where κ vanishes. At that point, κ is parallel to ξ. As one moves away from p along L, in either direction, the length ∥κ∥ of κ grows, but the angle ∠(κ,ξ) between the vectors increases as well. It should seem at least plausible from the picture that the length of the projection of κ onto the line is constant and, hence, that the inner product κ·ξ = cos(∠(κ , ξ )) ∥κ ∥ ∥ξ ∥ is constant.

That is how to think about the conservation of angular momentum for free particles in relativity theory. It does not matter that in the latter context we are dealing with a Lorentzian metric and allowing for curvature. The claim is still that a certain inner product of vector fields remains constant along a geodesic, and we can still think of that constancy as arising from a compensatory balance of two factors.

Let us now turn to the second type of conserved quantity, the one that is an attribute of extended bodies. Let κ^a be an arbitrary Killing field, and let T_ab be the energy-momentum field associated with some matter field. Assume it satisfies the conservation condition (∇_aT^ab = 0). Then (T^abκ_b) is divergence free:

∇_a(T^abκ_b) = κ_b∇_aT^ab + T^ab∇_aκ_b = T^ab∇(_aκ_b) = 0

(The second equality follows from the conservation condition and the symmetry of T^ab; the third follows from the fact that κ^a is a Killing field.) It is natural, then, to apply Stokes’s theorem to the vector field (T^abκ_b). Consider a bounded system with aggregate energy-momentum field T_ab in an otherwise empty universe. Then there exists a (possibly huge) timelike world tube such that T_ab vanishes outside the tube (and vanishes on its boundary).

Let S₁ and S₂ be (non-intersecting) spacelike hypersurfaces that cut the tube as in the figure below, and let N be the segment of the tube falling between them (with boundaries included).

Figure: The integrated energy (relative to a background timelike Killing field) over the intersection of the world tube with a spacelike hypersurface is independent of the choice of hypersurface.

By Stokes’s theorem,

∫_S2(T^abκ_b)dS_a – ∫_S1(T^abκ_b)dS_a = ∫_S2∩∂N(T^abκ_b)dS_a – ∫_S1∩∂N(T^abκ_b)dS_a

= ∫_∂N(T^abκ_b)dS_a = ∫_N∇_a(T^abκ_b)dV = 0

Thus, the integral ∫_S(T^abκ_b)dS_a is independent of the choice of spacelike hypersurface S intersecting the world tube, and is, in this sense, a conserved quantity (construed as an attribute of the system confined to the tube). An “early” intersection yields the same value as a “late” one. Again, the character of the background Killing field κ^a determines our description of the conserved quantity in question. If κ^a is timelike, we take ∫_S(T^abκ_b)dS_a to be the aggregate energy of the system (associated with κ^a). And so forth.

Dynamics of Point Particles: Orthogonality and Proportionality

Let γ be a smooth, future-directed, timelike curve with unit tangent field ξ^a in our background spacetime (M, g_ab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λ^a be a vector at p. Then there is a natural decomposition of λ^a into components proportional to, and orthogonal to, ξ^a:

λ^a = (λ^bξ_b)ξ^a + (λ^a −(λ^bξ_b)ξ^a) —– (1)

Here, the first part of the sum is proportional to ξ^a, whereas the second one is orthogonal to ξ^a.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λ^a relative to ξ^a (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξ^a is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

k_ab = ξ_a ξ_b —– (2)

h_ab = g_ab − ξ_a ξ_b —– (3)

then the decomposition can be expressed in the form

λ^a = k^a_b λ^b + h^a_b λ^b —– (4)

We can think of k_ab and h_ab as the relative temporal and spatial metrics determined by ξ^a. They are symmetric and satisfy

k^a_bk^b_c = k^a_c —– (5)

h^a_bh^b_c = h^a_c —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′^a. (Since ξ^a and ξ′^a are both future-directed, they are co-oriented; i.e., ξ^a ξ′^a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′^a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥h^a_b ξ′^b∥ / ∥k^a_b ξ′^b∥ —– (7)

For any vector μ^a, ∥μ^a∥ is (μ^aμ_a)^1/2 if μ is causal, and it is (−μ^aμ_a)^1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥k^a_b ξ′^b∥ = (k^a_b ξ′^b k_ac ξ′^c)^1/2 = (k_bc ξ′^bξ′^c)^1/2 = (ξ′^bξ_b)

and

∥h^a_b ξ′^b∥ = (−h^a_b ξ′^b h_ac ξ′^c)^1/2 = (−h_bc ξ′^bξ′^c)^1/2 = ((ξ′^bξ_b)² − 1)^1/2

so

v = ((ξ’^bξ_b)² − 1)^1/2 / (ξ′^bξ_b) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′^bξ_b) = 1/√(1 – v²) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector P^a that is tangent to its worldline there. The length ∥P^a∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if P^a is timelike, we can write it in the form P^a =mξ^a, where m = ∥P^a∥ > 0 and ξ^a is the four-velocity of the particle. No such decomposition is possible when P^a is null and m = ∥P^a∥ = 0.

Suppose a particle O with positive mass has four-velocity ξ^a at a point, and another particle O′ has four-momentum P^a there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose P^a in terms of ξ^a. By equations (4) and (2), we have

P^a = (P^bξ_b) ξ^a + h^a_bP^b —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = P^bξ_b. If O′ has positive mass and P^a = mξ′^a, this yields, by equation (9),

E = m (ξ′^bξ_b) = m/√(1 − v²) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc² and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term h^a_bP^b in the decomposition of P^a, i.e., the component of P^a orthogonal to ξ^a. It follows from equations (8) and (9) that it has magnitude

p = ∥h^a_b mξ′^b∥ = m((ξ′^bξ_b)² − 1)^1/2 = mv/√(1 − v²) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξ^a. Just as we understand ξ^a to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξⁿ∇_nξ^a – the directional derivative of ξ^a in the direction ξ^a – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξ^a. (This is, since ξ^a(ξⁿ∇_nξ_a) = 1/2 ξⁿ∇_n(ξ^aξ_a) = 1/2 ξⁿ∇_n (1) = 0). The magnitude ∥ξⁿ∇_nξ^a∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field F^a on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξⁿ∇_nξ^a satisfies

F^a = mξⁿ∇_nξ^a —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields F_ab. If a particle with mass m > 0, charge q, and four-velocity field ξ^a is present, the force exerted by the field on the particle at a point is given by qF^a_bξ^b. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qF^a_bξ^b = mξ^b∇_bξ^a —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξ^a. But so is the force field on the left, since ξ_a(F^a_bξ^b) = ξ^aξ^bF_ab = ξ^aξ^bF_(ab), and F_(ab) = 0 by the anti-symmetry of F_ab.

Arbitrage, or Tensors thereof…

What is an arbitrage? Basically it means ”to get something from nothing” and a free lunch after all. More strict definition states the arbitrage as an operational opportunity to make a risk-free profit with a rate of return higher than the risk-free interest rate accured on deposit.

The arbitrage appears in the theory when we consider a curvature of the connection. A rate of excess return for an elementary arbitrage operation (a difference between rate of return for the operation and the risk-free interest rate) is an element of curvature tensor calculated from the connection. It can be understood keeping in mind that a curvature tensor elements are related to a difference between two results of infinitesimal parallel transports performed in different order. In financial terms it means that the curvature tensor elements measure a difference in gains accured from two financial operations with the same initial and final points or, in other words, a gain from an arbitrage operation.

In a certain sense, the rate of excess return for an elementary arbitrage operation is an analogue of the electromagnetic field. In an absence of any uncertanty (or, in other words, in an absense of walks of prices, exchange and interest rates) the only state is realised is the state of zero arbitrage. However, if we place the uncertenty in the game, prices and the rates move and some virtual arbitrage possibilities to get more than less appear. Therefore we can say that the uncertanty play the same role in the developing theory as the quantization did for the quantum gauge theory.

What of “matter” fields then, which interact through the connection. The “matter” fields are money flows fields, which have to be gauged by the connection. Dilatations of money units (which do not change a real wealth) play a role of gauge transformation which eliminates the effect of the dilatation by a proper tune of the connection (interest rate, exchange rates, prices and so on) exactly as the Fisher formula does for the real interest rate in the case of an inflation. The symmetry of the real wealth to a local dilatation of money units (security splits and the like) is the gauge symmetry of the theory.

A theory may contain several types of the “matter” fields which may differ, for example, by a sign of the connection term as it is for positive and negative charges in the electrodynamics. In the financial stage it means different preferances of investors. Investor’s strategy is not always optimal. It is due to partially incomplete information in hands, choice procedure, partially, because of investors’ (or manager’s) internal objectives. Physics of Finance

 

 

Poincaré and Geometry of Curvature. Thought of the Day 60.0

It is not clear that Poincaré regarded Riemannian, variably curved, “geometry” as a bona fide geometry. On the one hand, his insistence on generality and the iterability of mathematical operations leads him to dismiss geometries of variable curvature as merely “analytic”. Distinctive of mathematics, he argues, is generality and the fact that induction applies to its processes. For geometry to be genuinely mathematical, its constructions must be everywhere iterable, so everywhere possible. If geometry is in some sense about rigid motion, then a manifold of variable curvature, especially where the degree of curvature depends on something contingent like the distribution of matter, would not allow a thoroughly mathematical, idealized treatment. Yet Poincaré also writes favorably about Riemannian geometries, defending them as mathematically coherent. Furthermore, he admits that geometries of constant curvature rest on a hypothesis – that of rigid body motion – that “is not a self evident truth”. In short, he seems ambivalent. Whether his conception of geometry includes or rules out variable curvature is unclear. We can surmise that he recognized Riemannian geometry as mathematical, and interesting, but as very different and more abstract than geometries of constant curvature, which are based on the further limitations discussed above (those motivated by a world satisfying certain empirical preconditions). These limitations enable key idealizations, which in turn allow constructions and synthetic proofs that we recognize as “geometric”.

Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = v_aw^a = 0. By definition, their null tangent vectors have the two-component spinor form λ^Aπ^A′, where λ^A is varying and π^A′ is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λ^Aπ^A′ μ_Aπ_A^′ = 0 = η(v, v) = λ^Aπ^A′ λ_Aπ_A^′ . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field π_A^′ is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP³. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.

Cosmology: Friedmann-Lemaître Universes

Cosmology starts by assuming that the large-scale evolution of spacetime can be determined by applying Einstein’s field equations of Gravitation everywhere: global evolution will follow from local physics. The standard models of cosmology are based on the assumption that once one has averaged over a large enough physical scale, isotropy is observed by all fundamental observers (the preferred family of observers associated with the average motion of matter in the universe). When this isotropy is exact, the universe is spatially homogeneous as well as isotropic. The matter motion is then along irrotational and shearfree geodesic curves with tangent vector ua, implying the existence of a canonical time-variable t obeying u_a = −t_,a. The Robertson-Walker (‘RW’) geometries used to describe the large-scale structure of the universe embody these symmetries exactly. Consequently they are conformally flat, that is, the Weyl tensor is zero:

C_ijkl := R_ijkl + 1/2(R_ikg_jl + R_jlg_ik − R_il g_jk − R_jkg_il) − 1/6R(g_ikg_jl − g_ilg_jk) = 0 —– (1)

this tensor represents the free gravitational field, enabling non-local effects such as tidal forces and gravitational waves which do not occur in the exact RW geometries.

Comoving coordinates can be chosen so that the metric takes the form:

ds² = −dt² + S²(t)dσ², u^a = δ^a₀ (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector u^a = dx^a/dt represent the histories of fundamental observers. The space sections {t = const} are surfaces of homogeneity and have maximal symmetry: they are 3-spaces of constant curvature K = k/S²(t) where k is the sign of K. The normalized metric dσ² characterizes a 3-space of normalized constant curvature k; coordinates (r, θ, φ) can be chosen such that

dσ² = dr² + f²(r) dθ² + sin²θdφ² —– (3)

where f (r) = {sin r, r, sinh r} if k = {+1, 0, −1} respectively. The rate of expansion at any time t is characterized by the Hubble parameter H(t) = S ̇/S.

To determine the metric’s evolution in time, one applies the Einstein Field Equations, showing the effect of matter on space-time curvature, to the metric (2,3). Because of local isotropy, the matter tensor T_ab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector u^a:

T_ab = (μ + p/c²)u_au_b + (p/c²)g_ab —– (4)

, c is the speed of light. The energy density μ(t) and pressure term p(t)/c² are the timelike and spacelike eigenvalues of T_ab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

T^ab_;b = 0 ⇔ μ ̇ + (μ + p/c²)3S ̇/S = 0 —– (5)

This becomes determinate when a suitable equation of state function w := pc²/μ relates the pressure p to the energy density μ and temperature T : p = w(μ,T)μ/c² (w may or may not be constant). Baryons have {p_bar = 0 ⇔ w = 0} and radiation has {p_radc² = μ_rad/3 ⇔ w = 1/3,μ_rad = aT⁴_rad}, which by (5) imply

μ_bar ∝ S⁻³, μ_rad ∝ S⁻⁴, T_rad ∝ S⁻¹ —– (6)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ + 3p/c²) + Λ —– (7)

, where κ is the gravitational constant and Λ is the cosmological constant. A cosmological constant can also be regarded as a fluid with pressure p related to the energy density μ by {p = −μc² ⇔ w = −1}. This shows that the active gravitational mass density of the matter and fields present is μ_grav := μ + 3p/c². For ordinary matter this will be positive:

μ + 3p/c² > 0 ⇔ w > −1/3 —– (8)

(the ‘Strong Energy Condition’), so ordinary matter will tend to cause the universe to decelerate (S ̈ < 0). It is also apparent that a positive cosmological constant on its own will cause an accelerating expansion (S ̈ > 0). When matter and a cosmological constant are both present, either result may occur depending on which effect is dominant. The first integral of equations (5, 7) when S ̇≠ 0 is the Friedmann equation

S ̇²/S² = κμ/3 + Λ/3 – k/S² —– (9)

This is just the Gauss equation relating the 3-space curvature to the 4-space curvature, showing how matter directly causes a curvature of 3-spaces. Because of the spacetime symmetries, the ten Einstein Filed Equations are equivalent to the two equations (7, 9). Models of this kind, that is with a Robertson-Walker (‘RW’) geometry with metric (2, 3) and dynamics governed by equations (5, 7, 9), are called Friedmann-Lemaître universes (‘FL’). The Friedmann equation (9) controls the expansion of the universe, and the conservation equation (5) controls the density of matter as the universe expands; when S ̇≠ 0 , equation (7) will necessarily hold if (5, 9) are both satisfied. Given a determinate matter description (specifying the equation of state w = w(μ, T) explicitly or implicitly) for each matter component, the existence and uniqueness of solutions follows both for a single matter component and for a combination of different kinds of matter, for example μ = μ_bar + μ_rad + μ_cdm + μ_ν where we include cold dark matter (cdm) and neutrinos (ν). Initial data for such solutions at an arbitrary time t₀ (eg. today) consists of,

• The Hubble constant H₀ := (S ̇/S)₀ = 100h km/sec/Mpc;

• A dimensionless density parameter Ω_i0 := κμ_i0/3H₀² for each type of matter present (labelled by i);

• If Λ ≠ 0, either Ω_Λ0 := Λ/3H²₀, or the dimensionless deceleration parameter q := −(S ̈/S) H⁻²₀.

Given the equations of state for the matter, this data then determines a unique solution {S(t), μ(t)}, i.e. a unique corresponding universe history. The total matter density is the sum of the terms Ω_i0 for each type of matter present, for example

Ω_m0 = Ω_bar0 + Ω_rad0 + Ω_cdm0 + Ω_ν0, —– (10)

and the total density parameter Ω₀ is the sum of that for matter and for the cosmological constant:

Ω₀ = Ω_m0 + Ω_Λ0 —– (11)

Evaluating the Raychaudhuri equation (7) at the present time gives an important relation between these parameters: when the pressure term p/c² can be ignored relative to the matter term μ (as is plausible at the present time, and assuming we represent ‘dark energy’ as a cosmological constant.),

q₀ = 1/2 Ω_m0 − Ω_Λ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q₀); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ω_m0, showing how matter causes a deceleration of the universe. Evaluating the Friedmann equation (9) at the time t₀, the spatial curvature is

K₀:= k/S₀² = H₀² (Ω₀ − 1) —– (13)

The value Ω₀ = 1 corresponds to spatially flat universes (K₀ = 0), separating models with positive spatial curvature (Ω₀ > 1 ⇔ K₀ > 0) from those with negative spatial curvature (Ω₀ < 1 ⇔ K₀ < 0).

The FL models are the standard models of modern cosmology, surprisingly effective in view of their extreme geometrical simplicity. One of their great strengths is their explanatory role in terms of making explicit the way the local gravitational effect of matter and radiation determines the evolution of the universe as a whole, this in turn forming the dynamic background for local physics (including the evolution of the matter and radiation).