^{2}should be expressible in terms of symplectic areas of triangles assignable to a set of n-points and satisfy the duality rules of conformal field theories guaranteing associativity. The crucial prediction is that symplectic n-point functions vanish whenever two arguments co-incide. This provides a mechanism guaranteing the finiteness of quantum TGD implied by very general arguments relying on non-locality of the theory at the level of 3-D surfaces.

The classical picture suggests that the generators of the fusion algebra formed by fields at different point of S^{2} have this point as a continuous index. Finite quantum measurement resolution and category theoretic thinking in turn suggest that only the points of S^{2} corresponding the strands of number theoretic braids are involved. It turns out that the category theoretic option works and leads to explicit hierarchy of fusion algebras forming a good candidate for a representation of so called little disk operad whereas the first option has difficulties.

** 1. Fusion rules**

Symplectic fusion rules are non-local and express the product of fields at two points s_{k} an s_{l} of S^{2} as an integral over fields at point s_{r}, where integral can be taken over entire S^{2} or possibly also over a 1-D curve which is symplectic invariant in some sense. Also discretized version of fusion rules makes sense and is expected serve as a correlate for finite measurement resolution.

By using the fusion rules one can reduce n-point functions to convolutions of 3-point functions involving a sequence of triangles such that two subsequent triangles have one vertex in common. For instance, 4-point function reduces to an expression in which one integrates over the positions of the common vertex of two triangles whose other vertices have fixed. For n-point functions one has n-3 freely varying intermediate points in the representation in terms of 3-point functions.

The application of fusion rules assigns to a line segment connecting the two points s_{k} and s_{l} a triangle spanned by s_{k}, s_{l} and s_{r}. This triangle should be symplectic invariant in some sense and its symplectic area A_{klm} would define the basic variable in terms of which the fusion rule could be expressed as C_{klm} = f(A_{klm}), where f is fixed by some constraints. Note that A_{klm} has also interpretations as solid angle and magnetic flux.

**2. What conditions could fix the symplectic triangles?**

The basic question is how to identify the symplectic triangles. The basic criterion is certainly the symplectic invariance: if one has found N-D symplectic algebra, symplectic transformations of S^{2} must provide a new one. This is guaranteed if the areas of the symplectic triangles remain invariant under symplectic transformations. The questions are how to realize this condition and whether it might be replaced with a weaker one. There are two approaches to the problem.

** 2.1 Physics inspired approach**

In the first approach inspired by classical physics symplectic invariance for the edges is interpreted in the sense that they correspond to the orbits of a charged particle in a magnetic field defined by the Kähler form. Symplectic transformation induces only a U(1) gauge transformation and leaves the orbit of the charged particle invariant if the vertices are not affected since symplectic transformations are not allowed to act on the orbit directly in this approach. The general functional form of the structure constants C_{klm} as a function f(A_{klm}) of the symplectic area should guarantee fusion rules.

If the action of the symplectic transformations does not affect the areas of the symplectic triangles, the construction is invariant under general symplectic transformations. In the case of uncharged particle this is not the case since the edges are pieces of geodesics: in this case however fusion algebra however trivializes so that one cannot conclude anything. In the case of charged particle one might hope that the area remains invariant under general symplectic transformations whose action is induced from the action on vertices. The equations of motion for a charged particle involve a Kähler metric determined by the symplectic structure and one might hope that this is enough to achieve this miracle. If this is not the case - as it might well be - one might hope that although the areas of the triangles are not preserved, the triangles are mapped to each other in such a manner that the fusion algebra rules remain intact with a proper choice of the function f(A_{klm}). One could also consider the possibility that the function f(A_{klm} is dictated from the condition that the it remains invariance under symplectic transformations.

**2.2 Category theoretical approach**

The second realization is guided by the basic idea of category theoretic thinking: the properties of an object are determined its relationships to other objects. Rather than postulating that the symplectic triangle is something which depends solely on the three points involved via some geometric notion like that of geodesic line of orbit of charged particle in magnetic field, one assumes that the symplectic triangle reflects the properties of the fusion algebra, that is the relations of the symplectic triangle to other symplectic triangles. Thus one must assign to each triplet (s_{1},s_{2},s_{3}) of points of S^{2} a triangle just from the requirement that braided associativity holds true for the fusion algebra.

Symplectic triangles would not be unique in this approach. All symplectic transformations leaving the N points fixed and thus generated by Hamiltonians vanishing at these points would give new gauge equivalent realizations of the fusion algebra and deform the edges of the symplectic triangles without affecting their area. One could even say that symplectic triangulation defines a new kind geometric structure in S^{2}.

The elegant feature of this approach is that one can in principle construct the fusion algebra without any reference to its geometric realization just from the braided associativity and nilpotency conditions and after that search for the geometric realizations. Fusion algebra has also a hierarchy of discrete variant in which the integral over intermediate points in fusion is replaced by a sum over a fixed discrete set of points and this variant is what finite measurement resolution implies. In this case it is relatively easy to see if the geometric realization of a given abstract fusion algebra is possible.

The two approaches do not exclude each other if the motion of charged particle in S^{2} selects one representative amongst all possible candidates for the edge of the symplectic triangle. Kind of gauge choice would be in question. This aspect encourages to consider seriously also the first option. It however turns out that the physics based approach does not look plausible.

** 3. Associativity conditions and braiding**

The generalized fusion rules follow from the associativity condition for n-point functions modulo phase factor if one requires that the factor assignable to n-point function has interpretation as n-point function. Without this condition associativity would be trivially satisfied by using a product of various bracketing structures for the n fields appearing in the n-point function. In conformal field theories the phase factor defining the associator is expressible in terms of the phase factor associated with permutations represented as braidings and the same is expected to be true also now.

- Already in the case of 4-point function there are three different choices corresponding to the 4 possibilities to connect the fixed points s
_{k}and the varying point s_{r}by lines. The options are (1-2, 3-4), (1-3,2-4), and (1-4,2-3) and graphically they correspond to s-, t-, and u-channels in string diagrams satisfying also this kind of fusion rules. The basic condition would be that same amplitude results irrespective of the choice made. The duality conditions guarantee associativity in the formation of the n-point amplitudes without any further assumptions. The reason is that the writing explicitly the expression for a particular bracketing of n-point function always leads to some bracketing of one particular 4-point function and if duality conditions hold true, the associativity holds true in general. To be precise, in quantum theory associativity must hold true only in projective sense, that is only modulo a phase factor. - This framework encourages category theoretic approach. Besides different bracketing there are different permutations of the vertices of the triangle. These permutations can induce a phase factor to the amplitude so that braid group representations are enough. If one has representation for the basic braiding operation as a quantum phase q=exp(i2p/N) , the phase factors relating different bracketings reduce to a product of these phase factors since (AB)C is obtained from A(BC) by a cyclic permutation involving to permutations represented as a braiding. Yang-Baxter equations express the reduction of associator to braidings. In the general category theoretical setting associators and braidings correspond to natural isomorphisms leaving category theoretical structure invariant.
- By combining the duality rules with the condition that 4-point amplitude vanishes, when any two points co-incide, one obtains from s
_{k}=s_{l}and s_{m}=s_{n}the condition stating that the integral of U^{2}(A_{klm})f^{2}(x_{kmr}) over the third point s_{r}vanishes. This requires that the phase factor U is non-trivial so that Q must be non-vanishing if one accepts the identification of the phase factor as Bohm-Aharonov phase. - Braiding operation gives naturally rise to a quantum phase. Braiding operation maps A
_{klm}to A_{klm}-4p since oriented triangles are in question and braiding changes orientation of the original triangle and maps the triangle to its complement. If the f is proportional to the exponent exp(-A_{klm}Q), braiding operation induces a complex phase factor q=exp(-i 4pQ). - For half-integer values of Q the algebra is commutative. For Q = M/N, where M and N have no common factors, only braided commutativity holds true for N ³ 3 just as for quantum groups characterizing also Jones inclusions of HFFs. For N=4 anti-commutativity and associativity hold true. Charge fractionization would correspond to non-trivial braiding and presumably to non-standard values of Planck constant and coverings of M
^{4}or CP_{2}depending on whether S^{2}corresponds to a sphere of light-cone boundary or homologically trivial geodesic sphere of CP_{2}.

** 4. Finite-dimensional version of the fusion algebra**

Algebraic discretization due to a finite measurement resolution is an essential part of quantum TGD. In this kind of situation the symplectic fields would be defined in a discrete set of N points of S^{2}: natural candidates are subsets of points of p-adic variants of S^{2}. Rational variant of S^{2} has as its points points for which trigonometric functions of q and f have rational values and there exists an entire hierarchy of algebraic extensions. The interpretation for the resulting breaking of the rotational symmetry would be a geometric correlate for the choice of quantization axes in quantum measurement and the book like structure of the imbedding space would be direct correlate for this symmetry breaking. This approach gives strong support for the category theory inspired philosophy in which the symplectic triangles are dictated by fusion rules.

**4.1 General observations about the finite-dimensional fusion algebra**

- In this kind of situation one has an algebraic structure with a finite number of field values with integration over intermediate points in fusion rules replaced with a sum. The most natural option is that the sum is over all points involved. Associativity conditions reduce in this case to conditions for a finite set of structure constants vanishing when two indices are identical. The number M(N) of non-vanishing structure constants is obtained from the recursion formula M(N) = (N-1)M(N-1)+ (N-2)M(N-2)+...+ 3M(3) = NM(N-1), M(3)=1 given M(4)=4, M(5)=20, M(6)=120,... With a proper choice of the set of points associativity might be achieved. The structure constants are necessarily complex so that also the complex conjugate of the algebra makes sense.
- These algebras resemble nilpotent algebras (x
^{n}=0 for some n) and Grassmann algebras (x^{2}=0 always) in the sense that also the products of the generating elements satisfy x^{2}=0 as one can find by using duality conditions on the square of a product x=yz of two generating elements. Also the products of more than N generating elements necessary vanish by braided commutativity so that nilpotency holds true. The interpretation in terms of measurement resolution is that partonic states and vertices can involve at most N fermions in this measurement resolution. Elements anti-commute for q=-1 and commute for q=1 and the possibility to express the product of two generating elements as a sum of generating elements distinguishes these algebras from Grassman algebras. For q=-1 these algebras resemble Lie-algebras with the difference that associativity holds true in this particular case. - I have not been able to find whether this kind of hierarchy of algebras corresponds to some well-known algebraic structure with commutativity and associativity possibly replaced with their braided counterparts. Certainly these algebras would be category theoretical generalization of ordinary algebras for which commutativity and associativity hold true in strict sense.
- One could forget the representation of structure constants in terms of triangles and think these algebras as abstract algebras. The defining equations are x
_{i}^{2}=0 for generators plus braided commutativity and associativity. Probably there exists solutions to these conditions. One can also hope that one can construct braided algebras from commutative and associative algebras allowing matrix representations. Note that the solution the conditions allow scalings of form C_{klm}® l_{k}l_{l}l_{m}C_{klm}as symmetries.

** 4.2 Formulation and explicit solution of duality conditions in terms of inner product**

Duality conditions can be formulated in terms of an inner product in the function space associated with N points and this allows to find explicit solutions to the conditions.

- The idea is to interpret the structure constants C
_{klm}as wave functions C_{kl}in a discrete space consisting of N points with the standard inner productáC _{kl}, C_{mn}ñ =

å

rC _{klr}C*_{mnr}. - The associativity conditions for a trivial braiding can be written in terms of the inner product as
áC _{kl}, C*_{mn}ñ = áC_{km}, C*_{ln}ñ = áC_{kn}, C*_{ml}ñ. - Irrespective of whether the braiding is trivial or not, one obtains for k=m the orthogonality conditions
áC _{kl}, C*_{kn}ñ = 0 .For each k one has basis of N-1 wave functions labeled by l ¹ k, and the conditions state that the elements of basis and conjugate basis are orthogonal so that conjugate basis is the dual of the basis. The condition that complex conjugation maps basis to a dual basis is very special and is expected to determine the structure constants highly uniquely.

- One can also find explicit solutions to the conditions. The most obvious trial is based on orthogonality of function basis of circle providing representation for Z
_{N-2}and is following:C _{klm}= E_{klm}×exp(if_{k}+f_{l}+f_{m}),f _{m}= n(m)2p/N-2 .Here E

_{klm}is non-vanishing only if the indices have different values. The ansatz reduces the conditions to the formå

_{r}E_{klr}E_{mnr}exp(i2f_{r}) = å_{r}E_{kmr}E_{lnr}exp(i2f_{r}) = å_{r}E_{knr}E_{mlr}exp(i2f_{r}) .In the case of braiding one can allow overall phase factors. Orthogonality conditions reduce to å

_{r}E_{klr}E_{knr}exp(i2f_{r}) = 0 . If the integers n(m), m k, l span the range (0,N-3) ortogonality conditions are satisfied if one has E_klr=1 when the indices are different. This guarantees also duality conditions since the inner products involving k,l,m,n reduce to the same expression å_{r ¹ k,l,m,n}exp(i2f_{r}) . - For a more general choice of phases the coefficients must have values differing from unity and it is not clear whether the duality conditions can be satisfied in this case.

** 4.3 Do fusion algebras form little disk operad?**

The improvement of measurement resolution means that one adds further points to an existing set of points defining a discrete fusion algebra so that a small disk surrounding a point is replaced with a little disk containing several points. Hence the hierarchy of fusion algebras might be regarded as a realization of a little disk operad and there would be a hierarchy of homomorphisms of fusion algebras induced by the improvements of measurement resolution. The inclusion homomorphism should map the algebra elements of the added points to the algebra element at the center of the little disk.

A more precise prescription goes as follows.

- The replacement of a point with a collection of points in the little disk around it replaces the original algebra element f
_{k0}by a number of new algebra elements f_{K}besides already existing elements f_{k}and brings in new structure constants C_{KLM}, C_{KLk}for k ¹ k_{0}, and C_{Klm}. - The notion of improved measurement resolution allows to conclude
C _{KLk}=0, k ¹ k_{0},C _{Klm}=C_{k0lm}. - In the homomorphism of new algebra to the original one the new algebra elements and their products should be mapped as follows:
f _{K}® f_{k0},f _{K}f_{L}® f_{k0}^{2}=0 ,f _{K}f_{l}® f_{k0}f_{l}.Expressing the products in terms of structure constants gives the conditions

å

MC _{KLM}=0 ,

å

rC _{Klr}=

å

rC _{k0lr}=0 .The general ansatz for the structure constants based on roots of unity guarantees that the conditions hold true.

- Note that the resulting algebra is more general than that given by the basic ansatz since the improvement of the measurement resolution at a given point can correspond to different value of N as that for the original algebra given by the basic ansatz. Therefore the original ansatz gives only the basic building bricks of more general fusion algebras. By repeated local improvements of the measurement resolution one obtains an infinite hierarchy of algebras labeled by trees in which each improvement of measurement resolution means the splitting of the branch with arbitrary number N of branches. The number of improvements of the measurement resolution defining the height of the tree is one invariant of these algebras. The fusion algebra operad has a fractal structure since each point can be replaced by any fusion algebra.

** 4.4 How to construct geometric representations of the discrete fusion algebra?**

Assuming that solutions to the fusion conditions are found, one could try to find whether they allow geometric representations. Here the category theoretical philosophy shows its power.

- Geometric representations for C
_{klm}would result as functions f(A_{klm}) of the symplectic area for the symplectic triangles assignable to a set of N points of S^{2}. - If the symplectic triangles can be chosen freely apart from the area constraint as the category theoretic philosophy implies, it should be relatively easy to check whether the fusion conditions can be satisfied. The phases of C
_{klm}dictate the areas A_{klm}rather uniquely if one uses Bohm-Aharonov ansatz for a fixed the value of Q. The selection of the points s_{k}would be rather free for phases near unity since the area of the symplectic triangle associated with a given triplet of points can be made arbitrarily small. Only for the phases far from unity the points s_{k}cannot be too close to each other unless Q is very large. The freedom to chose the points rather freely conforms with the general view about the finite measurement resolution as the origin of discretization. - The remaining conditions are on the moduli |f(A
_{klm})|. In the discrete situation it is rather easy to satisfy the conditions just by fixing the values of f for the particular triangles involved: |f(A_{klm})| = |C_{klm}|. For the exact solution to the fusion conditions |f(A_{klm})|=1 holds true. - Constraints on the functional form of |f(A
_{klm})| for a fixed value of Q can be deduced from the correlation between the modulus and phase of C_{klm}without any reference to geometric representations. For the exact solution of fusion conditions there is no correlation. - If the phase of C
_{klm}has A_{klm}as its argument, the decomposition of the phase factor to a sum of phase factors means that the A_{klm}is sum of contributions labeled by the vertices. Also the symplectic area defined as a magnetic flux over the triangle is expressible as sum of the quantities ∫ A_{μ}dx^{μ}associated with the edges of the triangle. These fluxes should correspond to the fluxes assigned to the vertices deduced from the phase factors of Y(s_{k}). The fact that vertices are ordered suggest that the phase of Y(s_{j}) fixes the value of ∫ A_{μ}dx^{μ}for an edge of the triangle starting from s_{k}and ending to the next vertex in the ordering. One must find edges giving a closed triangle and this should be possible. The option for which edges correspond to geodesics or to solutions of equations of motion for a charged particle in magnetic field is not flexible enough to achieve this purpose. - The quantization of the phase angles as multiples of 2π/(N-2) in the case of N-dimensional fusion algebra has a beautiful geometric correlate as a quantization of symplecto-magnetic fluxes identifiable as symplectic areas of triangles defining solid angles as multiples of 2π/(N-2). The generalization of the fusion algebra to the p-adic case exists if one allows algebraic extensions containing the phase factors involved. This requires the allowance of phase factors exp(i2π/p), p a prime dividing N-2. Only the exponents exp(i∫ A
_{μ}dx^{μ})= exp(in2π/(N-2)) exist p-adically. The p-adic counterpart of the curve defining the edge of triangle exists if the curve can be defined purely algebraically (say as a solution of polynomial equations with rational coefficients) so that p-adic variant of the curve satisfies same equations.

** 4.5 Does a generalization to the continuous case exist?**

One can consider an approximate generalization of the explicit construction for the discrete version of the fusion algebra by the effective replacement of points s_{k} with small disks which are not allowed to intersect. This would mean that the counterpart E(s_{k},s_{l},s_{m}) vanishes whenever the distance between two arguments is below a cutoff a small radius d. Puncturing corresponds physically to the cutoff implied by the finite measurement resolution.

- The ansatz for C
_{klm}is obtained by a direct generalization of the finite-dimensional ansatz:C _{klm}= k_{sk,sl,sm}Y(s_{k})Y(s_{l})Y(s_{m}) .where k

_{sk,sl,sm}vanishes whenever the distance of any two arguments is below the cutoff distance and is otherwise equal to 1. - Orthogonality conditions read as
Y(s _{k})Y(s_{l})ó

õk _{sk,sl,sr}k_{sk,sn,sr}Y^{2}(s_{m})dm(s_{r})= Y(s _{k})Y(s_{l})ó

õ

S^{2}(s_{k},sl,s_{n})Y ^{2}(s_{r})dm(s_{r})=0.The resulting condition reads as

ó

õ

S^{2}(s_{k},sl,s_{n})Y ^{2}(s_{r})dm(s_{r})=0This condition holds true for any pair s

_{k},s_{l}and this might lead to difficulties. - The general duality conditions are formally satisfied since the expression for all fusion products reduces to
Y(s _{k})Y(s_{l})Y(s_{m})Y(s_{n})X ,X = ó

õ

S^{2}k _{sk,sl,sm,sn}Y(s_{r})dm(s_{r})= ó

õ

S^{2}(s_{k},s_{l},s_{m},s_{n})Y(s _{m})dm(s_{r})=- ó

õ

D^{2}(s_{i})Y ^{2}(s_{r})dm(s_{r}) , i = k,l,s,m .These conditions state that the integral of Y

^{2}any disk of fixed radius d is same apart from phase factor: same result follows also from the orthogonality condition. This condition might be difficult to satisfy exactly and the notion of finite measurement resolution might be needed. For instance, it might be necessary to restrict the consideration to a discrete lattice of points which would lead back to a discretized version of algebra.

The article Category Theory and Quantum TGD gives a summary of the most recent ideas about applications of category theory in TGD framework. See also the new chapter Category Theory and TGD of "Towards S-matrix".

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