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

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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′ ≅ Td, but different metrics. In rough terms, the duality relates a “small” target space, with noncontractible cycles of length L < ls, with a “large” target space in which all such cycles have length L > ls.

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 ls, 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 < ls. 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 gR 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 ls = 1. With this, we attain a complete physical equivalence.

CFT(S1, gR) ≅ CFT(S1, g1/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 ≥ ls, because any theory with R < ls is equivalent to another theory with R > ls. Thus we have a nontrivial relation between two (in principle) observable quantities, R and ls, which one might imagine testing experimentally. Let us now consider the theory CFT(Td, g), where Td is the d-dimensional torus, with coordinates Xi parameterising Rd/2πZd, and a constant metric tensor gij. Then there is a complete physical equivalence

CFT(Td, g) ≅ CFT(Td, 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(Td) ≅ Zd, as an abelian group, referred to as the group of “winding numbers”.

Of course, there is another Zd we could bring into the discussion, the Pontryagin dual of the U(1)d of which Td 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 Td. 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[Zd] ⊕ C[Zd].

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 Td with metric g is

Spec ∆= {∑i,j=1d gijpipj; pi ∈ Zd}

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 ∈ Zd has length squared

L2 = ∑i,j=1d gijwiwj

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(Td, g + B) ≅ CFT(Td, (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 ls, and has curvature lengths far greater than ls. 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 ls2R and derivatives ls∂ of the fiber metric, both in perturbation theory and from world-sheet instantons.

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 = dimRL.

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.

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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 HB,B′ labelled by the two choices of brane B = (L, E, ∇E) and B′ = (K, F, ∇F).

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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 CN. 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 ⊗CN, ∇E ⊗ idN).

In physics, one visualizes this choice by labelling each open string boundary with a basis vector of CN, 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 ls → 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.

Matter Fields

In classical relativity theory, one generally takes for granted that all there is, and all that happens, can be described in terms of various “matter fields,” each of which is represented by one or more smooth tensor (or spinor) fields on the spacetime manifold M. The latter are assumed to satisfy particular “field equations” involving the spacetime metric gab.

Associated with each matter field F is a symmetric smooth tensor field Tab characterized by the property that, for all points p in M, and all future-directed, unit timelike vectors ξa at p, Tabξb is the four-momentum density of F at p as determined relative to ξa.

Tab is called the energy-momentum field associated with F. The four- momentum density vector Tabξb at a point can be further decomposed into its temporal and spatial components relative to ξa,

Tabξb = (Tmbξmξba + Tmbhmaξb

where the first term on the RHS is the energy density, while the second term is the three-momentum density. A number of assumptions about matter fields can be captured as constraints on the energy-momentum tensor fields with which they are associated.

Weak Energy Condition (WEC): Given any timelike vector ξa at any point in M, Tabξaξb ≥ 0.

Dominant Energy Condition (DEC): Given any timelike vector ξa at any point in M, Tabξaξb ≥ 0 and Tabξb is timelike or null.

Strengthened Dominant Energy Condition (SDEC): Given any timelike vector ξa at any point in M, Tabξaξb ≥ 0 and, if Tab ≠ 0 there, then Tabξb is timelike.

Conservation Condition (CC): ∇aTab = 0 at all points in M.

The WEC asserts that the energy density of F, as determined by any observer at any point, is non-negative. The DEC adds the requirement that the four-momentum density of F, as determined by any observer at any point, is a future-directed causal (i.e., timelike or null) vector. We can understand this second clause to assert that the energy of F does not propagate with superluminal velocity. The strengthened version of the DEC just changes “causal” to “timelike” in the second clause. It avoids reference to “point particles.” Each of the listed energy conditions is strictly stronger than the ones that precede it.

The CC, finally, asserts that the energy-momentum carried by F is locally conserved. If two or more matter fields are present in the same region of space-time, it need not be the case that each one individually satisfies the condition. Interaction may occur. But it is a fundamental assumption that the composite energy-momentum field formed by taking the sum of the individual ones satisfies it. Energy-momentum can be transferred from one matter field to another, but it cannot be created or destroyed. The stated conditions have a number of consequences that support the interpretations.

A subset S of M is said to be achronal if there do not exist points p and q in S such that p ≪ q. Let γ : I → M be a smooth curve. We say that a point p in M is a future-endpoint of γ if, for all open sets O containing p, there exists an s0 in I such that, ∀ s ∈ I, if s ≥ s0, then γ(s) ∈ O; i.e., γ eventually enters and remains in O. Now let S be an achronal subset of M. The domain of dependence D(S) of S is the set of all points p in M with this property: given any smooth causal curve without (past- or future-) endpoint, if its image contains p, then it intersects S. So, in particular, S ⊆ D(S).

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Let S be an achronal subset of M. Further, let Tab be a smooth, symmetric field on M that satisfies both the dominant energy and conservation conditions. Finally, assume Tab = 0 on S. Then Tab = 0 on all of D(S).

The intended interpretation of the proposition is clear. If energy-momentum cannot propagate (locally) outside the null-cone, and if it is conserved, and if it vanishes on S, then it must vanish throughout D(S). After all, how could it “get to” any point in D(S)? According to interpretive principle free massive point particles traverse (images of) timelike geodesics. It turns out that if the energy-momentum content of each body in the sequence satisfies appropriate conditions, then the convergence point will necessarily traverse (the image of) a timelike geodesic.

Let γ: I → M be smooth curve. Suppose that, given any open subset O of M containing γ[I], ∃ a smooth symmetric field Tab on M such that the following conditions hold.

(1) Tab satisfies the SDEC.
(2) Tab satisfies the CC.
(3) Tab = 0 outside of O.
(4) Tab ≠ 0 at some point in O.

Then γ is timelike and can be reparametrized so as to be a geodesic. This might be paraphrased another way. Suppose that for some smooth curve γ , arbitrarily small bodies with energy-momentum satisfying conditions (1) and (2) can contain the image of γ in their worldtubes. Then γ must be a timelike geodesic (up to reparametrization). Bodies here are understood to be “free” if their internal energy-momentum is conserved (by itself). If a body is acted on by a field, it is only the composite energy-momentum of the body and field together that is conserved.

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But, this formulation for granted that we can keep the background spacetime metric gab fixed while altering the fields Tab that live on M. This is justifiable only to the extent that we are dealing with test bodies whose effect on the background spacetime structure is negligible.

We have here a precise proposition in the language of matter fields that, at least to some degree, captures the interpretive principle. Similarly, it is possible to capture the behavior of light, wherein the behavior of solutions to Maxwell’s equations in a limiting regime (“the optical limit”) where wavelengths are small. It asserts, in effect, that when one passes to this limit, packets of electromagnetic waves are constrained to move along (images of ) null geodesics.

Hypersurfaces

Let (S, CS) and (M, CM) be manifolds of dimension k and n, respectively, with 1 ≤ k ≤ n. A smooth map : S → M is said to be an imbedding if it satisfies the following three conditions.

(I1) Ψ is injective.

(I2) At all points p in S, the associated (push-forward) linear map (Ψp) : Sp → MΨ(p) is injective.

(I3) ∀ open sets O1 in S, Ψ[O1] = [S] ∩ O2 for some open set O2 in M. (Equivalently, the inverse map Ψ−1 : Ψ[S] → S is continuous with respect to the relative topology on [S].)

Several comments about the definition are in order. First, given any point p in S, (I2) implies that (Ψp)[Sp] is a k-dimensional subspace of MΨ(p). So the condition cannot be satisfied unless k ≤ n. Second, the three conditions are independent of one another. For example, the smooth map Ψ : R → R2 defined by (s) = (cos(s), sin(s)) satisfies (I2) and (I3) but is not injective. It wraps R round and round in a circle. On the other hand, the smooth map : R → R defined by (s) = s3 satisfies (I1) and (I3) but is not an imbedding because (Ψ0) : R0 → R0 is not injective. (Here R0 is the tangent space to the manifold R at the point 0). Finally, a smooth map : S → M can satisfy (I1) and (I2) but still have an image that “bunches up on itself.” It is precisely this possibility that is ruled out by condition (I3). Consider, for example, a map : R → R2 whose image consists of part of the image of the curve y = sin(1/x) smoothly joined to the segment {(0, y) : y < 1}, as in the figure below. It satisfies conditions (I1) and (I2) but is not an imbedding because we can find an open interval O1 in R such that given any open set O2 in R2, Ψ[O1] ≠ O2 ∩ Ψ[R].

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Suppose(S, CS) and (M, CM) are manifolds with S ⊆ M. We say that (S, CS) is an imbedded submanifold of (M, CM) if the identity map id: S → M is an imbedding. If, in addition, k = n − 1 (where k and n are the dimensions of the two manifolds), we say that (S, CS) is a hypersurface in (M, CM). Let (S, CS) be a k-dimensional imbedded submanifold of the n-dimensional manifold (M, CM), and let p be a point in S. We need to distinguish two senses in which one can speak of “tensors at p.” There are tensors over the vector space Sp (call them S-tensors at p) and ones over the vector space Mp (call them M-tensors at p). So, for example, an S-vector ξ ̃a at p makes assignments to maps of the form f ̃: O ̃ → R where O ̃ is a subset of S that is open in the topology induced by CS, and f ̃ is smooth relative to CS. In contrast, an M-vector ξa at p makes assignments to maps of the form f : O → R where O is a subset of M that is open in the topology induced by CM, and f is smooth relative to CM. Our first task is to consider the relation between S-tensors at p and M-tensors there.

Let us say that ξa ∈ (Mp)a is tangent to S if ξa ∈ (idp)[(Sp)a]. (This makes sense. We know that (idp)[(Sp)a] is a k-dimensional subspace of (Mp)a; ξa either belongs to that subspace or it does not.) Let us further say that ηa in (Mp)a is normal to S if ηaξa =0 ∀ ξa ∈ (Mp)a that are tangent to S. Each of these classes of vectors has a natural vector space structure. The space of vectors ξa ∈ (Mp)a tangent to S has dimension k. The space of co-vectors ηa ∈ (Mp)a normal to S has dimension (n − k).

Metric. Part 1.

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A (semi-Riemannian) metric on a manifold M is a smooth field gab on M that is symmetric and invertible; i.e., there exists an (inverse) field gbc on M such that gabgbc = δac.

The inverse field gbc of a metric gab is symmetric and unique. It is symmetric since

gcb = gnb δnc = gnb(gnm gmc) = (gmn gnb)gmc = δmb gmc = gbc

(Here we use the symmetry of gnm for the third equality.) It is unique because if g′bc is also an inverse field, then

g′bc = g′nc δnb = g′nc(gnm gmb) = (gmn g′nc) gmb = δmc gmb = gcb = gbc

(Here again we use the symmetry of gnm for the third equality; and we use the symmetry of gcb for the final equality.) The inverse field gbc of a metric gab is smooth. This follows, essentially, because given any invertible square matrix A (over R), the components of the inverse matrix A−1 depend smoothly on the components of A.

The requirement that a metric be invertible can be given a second formulation. Indeed, given any field gab on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field gbc on M such that gabgbc = δac.

(2) ∀ p in M, and all vectors ξa at p, if gabξa = 0, then ξa =0.

(When the conditions obtain, we say that gab is non-degenerate.) To see this, assume first that (1) holds. Then given any vector ξa at any point p, if gab ξa = 0, it follows that ξc = δac ξa = gbc gab ξa = 0. Conversely, suppose that (2) holds. Then at any point p, the map from (Mp)a to (Mp)b defined by ξa → gab ξa is an injective linear map. Since (Mp)a and (Mp)b have the same dimension, it must be surjective as well. So the map must have an inverse gbc defined by gbc(gab ξa) = ξc or gbc gab = δac.

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In the presence of a metric gab, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξa at some point, we write gab ξa as ξb; and given a covariant vector ηb, we write gbc ηb as ηc. The notation is evidently consistent in the sense that first lowering and then raising the index of a vector (or vice versa) leaves the vector intact.

One would like to extend this notational convention to tensors with more complex index structure. But now one confronts a problem. Given a tensor αcab at a point, for example, how should we write gmc αcab? As αmab? Or as αamb? Or as αabm? In general, these three tensors will not be equal. To get around the problem, we introduce a new convention. In any context where we may want to lower or raise indices, we shall write indices, whether contravariant or covariant, in a particular sequence. So, for example, we shall write αabc or αacb or αcab. (These tensors may be equal – they belong to the same vector space – but they need not be.) Clearly this convention solves our problem. We write gmc αabc as αabm; gmc αacb as αamb; and so forth. No ambiguity arises. (And it is still the case that if we first lower an index on a tensor and then raise it (or vice versa), the result is to leave the tensor intact.)

We claimed in the preceding paragraph that the tensors αabc and αacb (at some point) need not be equal. Here is an example. Suppose ξ1a, ξ2a, … , ξna is a basis for the tangent space at a point p. Further suppose αabc = ξia ξjb ξkc at the point. Then αacb = ξia ξjc ξkb. Hence, lowering indices, we have αabc =ξia ξjb ξkc but αacb =ξia ξjc ξib at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δba and the Riemann curvature field Rabcd (associated with some derivative operator). Our convention will be to write these as δab and Rabcd – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δab = δba. (It does matter with the second.) To verify the equality, it suffices to observe that the two fields have the same action on an arbitrary field αb:

δbaαb = (gbngamδnmb = gbnganαb = gbngnaαb = δabαb

Now suppose gab is a metric on the n-dimensional manifold M and p is a point in M. Then there exists an m, with 0 ≤ m ≤ n, and a basis ξ1a, ξ2a,…, ξna for the tangent space at p such that

gabξia ξib = +1 if 1≤i≤m

gabξiaξib = −1 if m<i≤n

gabξiaξjb = 0 if i ≠ j

Such a basis is called orthonormal. Orthonormal bases at p are not unique, but all have the same associated number m. We call the pair (m, n − m) the signature of gab at p. (The existence of orthonormal bases and the invariance of the associated number m are basic facts of linear algebraic life.) A simple continuity argument shows that any connected manifold must have the same signature at each point. We shall henceforth restrict attention to connected manifolds and refer simply to the “signature of gab

A metric with signature (n, 0) is said to be positive definite. With signature (0, n), it is said to be negative definite. With any other signature it is said to be indefinite. A Lorentzian metric is a metric with signature (1, n − 1). The mathematics of relativity theory is, to some degree, just a chapter in the theory of four-dimensional manifolds with Lorentzian metrics.

Suppose gab has signature (m, n − m), and ξ1a, ξ2a, . . . , ξna is an orthonormal basis at a point. Further, suppose μa and νa are vectors there. If

μa = ∑ni=1 μi ξia and νa = ∑ni=1 νi ξia, then it follows from the linearity of gab that

gabμa νb = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

In the special case where the metric is positive definite, this comes to

gabμaνb = μ1ν1 +…+ μnνn

And where it is Lorentzian,

gab μaνb = μ1ν1 − μ2ν2 −…− μnνn

Metrics and derivative operators are not just independent objects, but, in a quite natural sense, a metric determines a unique derivative operator.

Suppose gab and ∇ are both defined on the manifold M. Further suppose

γ : I → M is a smooth curve on M with tangent field ξa and λa is a smooth field on γ. Both ∇ and gab determine a criterion of “constancy” for λa. λa is constant with respect to ∇ if ξnnλa = 0 and is constant with respect to gab if gab λa λb is constant along γ – i.e., if ξnn (gab λa λb = 0. It seems natural to consider pairs gab and ∇ for which the first condition of constancy implies the second. Let us say that ∇ is compatible with gab if, for all γ and λa as above, λa is constant w.r.t. gab whenever it is constant with respect to ∇.

Arbitrage, or Tensors thereof…

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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