{'fraction_non_alphanumeric': 0.05457878294993933, 'fraction_numerical': 0.008855497895851083, 'mean_word_length': 4.075874721530599, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 16, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A-infinity structure on simplicial complexes

A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as anticipated, it is a part of non-trivial A-infinity structure, involving a chain of poly-linear operations, constrained by nilpotency relation: (d + \wedge + m + ...)^n = 0 with n=2.


Introduction
In the space of ordinary differential forms two operations are naturally defined: linear exterior derivative d and bilinear exterior multiplication ∧. They satisfy a triple of quadratic relations: i.e. nilpotency of d : which can be rewritten as a single nilpotency constraint Eq. (1) is often used as a basic relation in construction of topological field theories [1]. A natural deformation of (1) in BRST-BV formalism [2] is the A ∞ -structure [3], which can be symbolically written as and involves an infinite chain of k-linear operations with all natural k. Exterior derivative d = m (1) and exterior product ∧ = m (2) are just the first two members of this chain. More accurately, (2) is an infinite collection of quadratic constraints: k+l=n+1 m (k) m (l) = 0 n = 1, 2, 3, . . . i.e.
It is a long-standing claim [4] that the A ∞ -structure is a natural property of discrete de Rham complex, i.e. arises in the study of finite-difference forms on simplicial complexes [5], in particular, on triangulations of continuous manifolds. Despite certain efforts and progress in investigation of this idea ( [6], [7]) it does not attract attention that it deserves and explicit realization is still lacking. Once established, such construction should possess further generalization to higher-order nilpotents, in the spirit of non-linear algebra [8]. Such generalizations are expected to provide new tools for a variety of applications, from topology to strings. It is the goal of this paper to present explicit formulas about the forms on simplicial complexes, operations on the space of these forms and relations between these operations. Such down-to-earth description of the lowest components of the A ∞ -structure shows what explicit realization means and can hopefully stimulate the search for equally explicit formulas for its cubic and higher-order generalizations.

Basic example in dimension one
We begin with a one-dimensional example of discrete forms and operations. Consider a one-dimensional discrete space (a graph). Such a space contains points (vertices) and links between some of them. Function f i is defined on its vertices i, while 1-form ψ ij is defined on links ij.

Differential d
Discrete derivative of a function df is a 1-form: for a given link ij, connecting the points i and j, it is a difference between the values of f at the two ends of a link: In one dimension, differential of a 1-form is zero, and so the d 2 = 0 identity is trivial.

Product ∧
The product of two functions is the point-wise product: A natural definition for a product of a function f and a 1-form ψ is again an 1-form, In one dimension, a product of two 1-forms is zero.

The Leibnitz rule d ∧ + ∧ d = 0
Operations d and ∧ satisfy the usual Leibnitz rule The calculation is straightforward (f and g are functions) : If we define differential not only on functions, but also on pairs of functions as then the Leibnitz rule (10) can be represented in three different ways: and, finally In this paper, we often use this kind of notation. We call this form of A ∞ -relations brief, in contrast with the detailed form (10). The brief form, implied by the BRST-BV formalism, is extremely useful, both from practical and theoretical points of view. First of all, higher A ∞ -relations become more and more complicated, and one needs the brief form to handle them in a simple and transparent way. Second, in the brief form the A ∞ structure admits a simple interpretation as deformation of d by ∧, see s.7.7 below.

Non-associativity ∧ 2 = 0
The discrete product is not obligatory associative. While (f ∧ ψ) ∧ g = f ∧ (ψ ∧ g), for the same triple in another order (f ∧ g) ∧ ψ = f ∧ (g ∧ ψ). Indeed, is not vanishing unless df = 0 or dg = 0. This is a fact of great importance: in general, a discrete product satisfying the Leibnitz rule fails to be associative. If we define the product ∧ not only on pairs of forms, but also on triples (converting them into pairs): In this notation non-associativity simply means that ∧ 2 = 0. This is just another example of the brief form of A ∞ relations: simplification occurs when we define operations on a larger set of arguments.

Higher operation m
Consider a 3-linear operation m which converts a triple of a function f and two 1-forms ψ and χ into a 1-form The following (non)associativity relations hold: Complemented by m(f, g, ψ) i = m(f, ψ, g) i = 0, they provide examples of the first non-trivial relation (6) of the A ∞ structure. This relation states that for any forms a, b, c Degree of a function |f | = 0, while for a 1-form |ψ| = 1. In full detail eq.(19) states that This example is simple: in one dimension only 0-forms and 1-forms are present. However, it is informative: already at this level discrete non-associativity can be observed, together with the higher A ∞ operation m = m (3) . Non-associativity is not an isolated phenomenon, it is an element of non-trivial A ∞ structure of discrete algebra of operations. Non-associativity is fully described by the operation m (3) : instead of ∧ 2 = 0 a more complicated identity ∧ 2 = −dm − md holds. The rest of the paper is devoted to generalization of this example to arbitrary forms on arbitrary simplicial complexes.

Logic and plan of the presentation
We proceed in four steps.

Simplicial complexes, forms and operations
First, in section 4 we introduce the three "starting" operations m (1) = d, its conjugate ∂ = d † and m (2) = ∧, which act on the space Ω * (M ) of forms on an ordered simplicial complex M . Without ordering, M can be considered as equilateral triangulation of a manifold (perhaps, singular) or simply as triangular lattice in arbitrary dimension. Some of the definitions in s.4 will be given in a form, which allows and suggests further generalizations, especially to higher nilpotent operators d n : In the main part of this paper we consider only d = d 2 , details of the n > 2 case will be presented elsewhere [9].

Formulation of the problem
Second, in section 5 we turn to the A ∞ structure. In this paper we do not discuss its origins and hidden meanings, even cohomological and BRST-BV aspects of the story will not be considered. Instead our goal is to construct operations m (p) on simplicial complex in the most explicit way, and (2) is used as defining property of these operations, not as a theorem derived for some given "natural" operations. In order to use (3) in this quality, we rewrite it as a sequence of recurrent relations: (3) ∧ and so on. Now the problem is formulated, and the first thing to do is to check its consistency: for (22) to have a solution m (p) the r.h.s. should satisfy This consistency condition is considered in section 5.4. If (23) is true, solution to (22) can exist, but it does not need to be unique -and it actually is ambiguous. We consider two solutions: one naive, but somewhat distractive, in s.6, another nice and fully satisfactory, but somewhat sophisticated, in s.7. These will be our steps three and four.

Naive solution with the help of K-operator
At step three, in section 6, we resolve eq.(22) with the help of the d −1 -operator K [10], Such operator exists if d is cohomologically trivial. Even for topologically trivial simplicial complexes there could be a problem with this at the level of 0-forms (since zeroth homology does not vanish) and we use a special trick -postulate -to cure this problem and make all formulas work universally on entire Ω * . K operator can be constructed from arbitrary operator α, provided associated "Laplace operator" ∆ α ≡ dα + αd is invertible: In most applications one take α = ∂, so that ∆ ∂ = dd + + d + d or simple ∆ is the ordinary Laplace operator.
Given K ∂ , one can immediately resolve (22): and this is our "naive" solution. It is indeed a solution: as requested by (22). For a given simplicial complex M eq. (27) is an absolutely explicit formula. Moreover for a finite complex with finite number #(M ) of points any operator is represented by a finite-size matrix, and therefore satisfies a finite-degree algebraic equation. It follows that inverse ∆ −1 is actually a finite-degree polynomial of ∆. Eq.(27) provides quite reasonable formulas for a complex M which consists of a single simplex, they are especially nice because on a simplex the Laplace operator trivializes, ∆| simplex = multiplication by #(points) However, in general ∆ is given by a highly non-local expression that depends in sophisticated way on simplicial complex M : a minor modification of M , say, addition of an extra simplex at the boundary, changes all the formulas. Thus, despite (27) formally resolves our problem, this is not the desired solution; we want another one, which is local: does not depend on the structure of M far away from the place where m (p ) is evaluated. Such local solution exists, but its construction is somewhat involved and includes a number of new notions and ideas.

Local solution and the local K-operator
This construction of local operations m (p ) in section 7 is our step four.
However, (30) is not much better than (27), because it is still not local. The real advantage of (30) is that it has straightforward local modifications. On the space Ω ⊗p * with p ≥ 2 one acquires a new possibility: one can localize operation ∂, which was totally non-local on Ω * . In section 7.
acting from Ω ⊗p * (M ) to Ω * (M ), is our desired local expression for m (p) . In section 7.5 we justify the validity of this formula.
In addition to being local, eq.(33) appears to be simple and nice. This is because of distinguished properties of the localized Laplace operator ∆ loc , considered in section 7.2. This operator is actually acting within particular simplexes, ∆ loc = ∆ loc | simplex , but it does not coincide with ∆| simplex and does not satisfy (29) and even (31). Still, it is a remarkably simple operator: its eigenvalues, though not all coincident as for (29), are subsequent integers and inverse operator is easy to construct. This makes (33) a very explicit formula, for every particular complex it can be effectively handled by MAPLE or Mathematica. It goes without saying that eqs.(17) and (18) are immediately reproduced by (33). Higher operations are described in section 7.7.
Moreover, as explained in s.7.7, the entire A ∞ structure can be considered as a ∧-induced conjugation of the nilpotent operator d -and this interpretation survives generalization to higher nilpotents d n .
Finally, in the Appendix section 9 we present a number of explicit calculations with discrete forms. These include examples concerning discrete differential, discrete wedge multiplication, higher operations, Laplace operators and more.

Discrete space
The elementary sets σ are called simplexes. Complex is thought to be a "discrete space" -a number of vertices (0-simplexes), with some pairs connected by links (1-simplexes), some triples -by triangles (2-simplexes), some quadruples -by tetrahedra (3-simplexes) and so on. Of course, the simplest thing to do is to connect everything: consider a set of N ≥ 1 points together with all its subsets. This is a closed simplex. For example, a simple disc is a closed simplex, while its boundary is not, because the triple {1, 2, 3} is not connected.
Examples of simplicial complexes are provided by triangulations of continuous manifolds. Note, however, that these should be triangulations, not just arbitrary discretizations. For example, rectangular lattices with hypercubic, rather than tetrahedra (simplicial) sites are not simplicial complexes according to our definition (because diagonal of a square is not a link). They can be easily completed to triangulation by adding diagonals (this, however, can be done in different ways, since there are two possible choices of diagonal in each square; independence of the choice of triangulation in this context is often called flip symmetry).
We also consider an ordering of vertices in M . Linear ordering is a function ord(·) that labels vertices x with integer (and even natural) numbers ord(x). Given any simplex σ and vertex x ∈ σ, we define to be the number of vertices y ∈ σ with ord(y) ≤ ord(x). For a generic sub-simplex σ ′ ∈ σ Here |σ| denotes the dimension of σ. The number β σ ′ →σ = (−1) ord(σ ′ →σ) = ±1 defines the parity of embedding of a subset σ ′ ⊂ σ into an ordered set σ. It will be used below to specify coefficients in the definitions of various operations.

Discrete forms
Given simplicial complex M , a discrete p-form is a function defined on its p-simplexes. 0-forms are functions on vertices, while 1-forms are defined on links, 2-forms -on triangles, etc. That is a discrete (finite-difference) analogue of smooth forms, because a smooth p-form is a function on p-dimensional submanifolds. Its value on a particular submanifold is equal to the integral over this submanifold.
Simplexes can be formally added and multiplied by a number. In topology, such linear combinations of simplexes are usually called chains, and linear functions on chains are called co-chains. It is natural to use co-chains to represent discrete forms. However, a discrete form ω can be treated also a chain: This should not cause any confusion: since a finite-dimensional linear space is isomorphic to its dual space, both points of view are possible. Co-chain description was adopted in s.2, while most of calculations below are done with chains, but one can always pass to the dual picture.
The wide-spread approach to discrete forms is the Whitney-form approximation ( [6], [7]). In this approach discrete forms are represented by some collection of smooth forms (Whitney forms) and discrete operations (derivative, wedge product, etc) are naturally constructed from their smooth counterparts. In this paper we study the subject from essentially different point of view -we consider discrete forms per se, as linear combinations of simplexes, without a reference to any smooth approximation. Consequently, we use only linear algebra to handle the discrete forms and operations.

Operations
Given a simplicial complex, denote through Ω * = ⊕ ∞ m=0 Ω m the linear space of all discrete forms on it. Ω * is naturally Z-graded into 0-forms Ω 0 , 1-forms Ω 1 and so on. The space Ω * ⊗ Ω * is a space of pairs of forms, Ω * ⊗ Ω * ⊗ Ω * is a space of triples, and so on. The basis in Ω ⊗p * is formed by all possible p-uples of simplexes: for example, Ω * ⊗ Ω * is spanned by pairs of simplexes.
An operation with forms is any linear map (tensor) and duality are immediately present. In addition we define a property, inherited from the structure of simplicial complex: namely, locality.

Locality
Given p simplexes σ 1 , σ 2 , . . . , σ p , we define their local envelope to be a minimal simplex which contains all of them. If such a simplex does not exist, the local envelope is ∅ (for example, ∪(∂D) = ∅). For any simplex σ, denote through Ω ⊗p * (σ) a subspace spanned by p-uples with local envelope equal to σ. Then the whole space Ω ⊗p * is decomposed into subspaces: An operation O is said to be local if it maps Ω ⊗p * into Ω ⊗q * and never decreases the local envelope: In a similar way, a co-local operation O : Ω ⊗p * → Ω ⊗q * is the one that never increases the envelope: Operation that is both local and co-local, is called strictly local. A strictly local operation O : Ω ⊗p * → Ω ⊗q * maps Ω ⊗p (σ) into Ω ⊗q * (σ) and hence preserves the local envelope. Obviously, a composition of (co-)local operations is (co-)local. Duality converts a local operation into co-local and vice versa.
The meaning of locality becomes clear in the dual picture -consider discrete forms as functions on simplexes. Each operation is fully specified, if values of its image on all simplexes are known. However, this value on simplex σ can depend on the values of operation's arguments on larger domain of our simplicial complex which lie beyond σ. It is exactly the case when we call the operation non-local.
Localisation. It is especially important that a non-local operation O still can contain a nonzero local part, because its value O(e) where e ∈ Ω ⊗p * (σ) can have a nonzero projection onto the local subspace Ω ⊗q * (σ ′ ⊃ σ). The same happens to co-local part and to strictly local part. A generic operation O is a sum of three parts:

Discrete d and ∧
In order to discretise differential forms, especially important are the analogues of smooth exterior derivative d and exterior wedge product ∧. We define them on any ordered simplicial complex. From now on Discrete de Rham differential d : Ω * → Ω * is defined as follows: Here σ is a simplex in M , while x -a vertex in M . Operator d has a dual (conjugate) operator ∂, the co-differential: Discrete Stokes theorem dω(σ) = ω(∂σ) is nothing but the duality between d and ∂.
In other words, there is a natural pairing between chains and forms: < σ, ω >=< ω, σ >= ω(σ) Operators d and ∂ are conjugate w.r.t this pairing: Differential maps a p-form into a (p + 1)-form, and co-differential converts a p-form into a (p − 1)-form. From the dual point of view, differential adds one vertex to a simplex in all possible ways, with appropriate signs, and co-differential removes one vertex from a simplex in all possible ways, with appropriate signs.
Operations d and ∂ can be found in any combinatorial topology textbook (see, for example, [5]), and have the well-known properties. First, these operators are nilpotent: This property is most widely used in combinatorial topology: homology groups of d ("simplicial homologies") are applied to the study of topological properties of simplicial complexes, in particular, of triangulated manifolds. Second, the anti-commutator is a natural discrete Laplace operator. Its properties are related to topology of the complex M via Hodge theory. If M is just a closed simplex, then △ is very simple: it is a scalar operator, multiplying any form by the number of vertices in the simplex, i.e. does not depend on the choice of the form. If M is not exactly a simplex, but still has topology of a disc, then △ is non-degenerate. Finally, if M has non-trivial topology, then △ becomes degenerate, its kernel consists of harmonic p-forms and represents the p-th homology of M .
Discrete wedge product ∧ : Ω * ⊗ Ω * → Ω * is defined as follows: Here σ and σ ′ are simplexes in M, intersecting at a point: dim(σ ∩ σ ′ ) = 0, otherwise wedge product is zero. This product is not as popular in literature as the discrete de Rham differential, however, it is indeed the correct discrete realization of exterior product. First, as a consequence of the intersection rule, it converts a p-form and a q-form into a (p + q)-form. Second, as a consequence of the choice of signs β, it is skew-symmetric: Finally, it satisfies a graded Leibnitz rule with the differential d: Associativity is the only property of smooth exterior product that is broken. However, we do know that instead discrete exterior product is a part of non-trivial A ∞ structure. This structure is most natural at the level of simplicial complex, and associativity arises only in the continuum limit.
The following definition of A ∞ structure or algebra is most widely used [3].
A ∞ structure on a Z-graded linear space Ω * is a countable collection of multilinear maps The signs factors are defined by where |ω| is the degree of a form ω ∈ Ω * . Degree of a p-linear map X is defined as Explicitly, the first three identities read Bilinear map m (2) : Ω * × Ω * → Ω * is usually thought to be a product on Ω * , while operator m (1) : Ω * → Ω * is thought to be a differential. The first A ∞ -identity implies that m (1) is square-zero, the second one implies that m (1) is a graded derivation of m (2) (Leibnitz rule) and the third one implies that the product m (2) is associative up to the higher operation m (3) . Such abstract A ∞ structure has a lot of realizations, one of the most interesting arises in quantum field theory.
However, the physical systems are usually infinite-dimensional. The finite-dimensional A ∞ structure on simplicial complexes is more simple and transparent. In this case, the graded linear space Ω * is the space of discrete forms on a complex, operator m (1) has a meaning of discrete de Rham derivative and m (2) is naturally identified with discrete wedge product.

The brief/nilpotent form
In section 2, we introduced a brief formulation of the A ∞ -structure -a simplification that happens if we define m (p) as a linear operation acting not only on Ω ⊗p * , but on arbitrary space Ω ⊗q * with arbitrary q: The latter action is called a "lifting rule" and is defined for q ≥ p as follows: where ξ-function in the sign factors is given by the same (48). Actually, in this way we lift our operations to the total linear Fock-like space (we put Ω ⊗0 * = C). This space is also called a tensor algebra over Ω * . Note that Ω * * is doubly graded: there is a conventional degree-grading inherited from Ω * plus a tensor-grading: into forms, pairs of forms, triples of forms and so on. By means of lifting, operations m (p) (previously the poly-linear maps) act as linear operators in this space. Operators are much easier to handle, as compared to the multilinear maps. In these terms the system of constraints (47) looks substantially better: Eq.(52) is a brief form of (47).
Explicitly, these constraints read (m (1) ) 2 = 0 and so on. They can be nicely unified into a single (tensor-)graded constraint: In other words, the A ∞ structure on simplicial complex is nothing but a single operator D : Ω * * → Ω * * , satisfying a single nilpotency constraint The peculiarity is that the linear space Ω * * is graded, and the single relation (56) -simple and natural at the level of the whole tensor space -gives rise to sophisticated and numerous constraints (47) at the level of graded components. Brief formulation removes the grading and reveals the underlying nilpotent structure.

Solving A ∞ relations for m (p)
In what follows we consider only the special case of m (1) = d and m (2) = ∧ defined in 4.5. This restriction is implied by consideration of continuum limit: for high-resolution triangulations of smooth manifolds, we want the A ∞ structure to reproduce the smooth algebra (1) with increasing precision. In other words, in this paper we look only at A ∞ structures with the correct continuum limit. This natural constraint is very strong: it fixes m (1) = d and m (2) = ∧ unambiguously. These and only these two discrete operations reproduce the smooth operations d and ∧ in the continuum limit, while all the higher operations m (p) with p ≥ 3 vanish.
With this choice of m (1) and m (2) the two first A ∞ -equations (53) are satisfied: The higher A ∞ equations (52) have a recursive structure: where M (p ) is the right hand side

K-operator method
On each step of recursion we need to solve an equation where operation m (p) is the only unknown quantity and M (p ) at the right hand side is already known. Such equations can be solved by generic "K-operator" method [10], which can be roughly described as finding the inverse operator for differential. Consider generic equation dx + xd = y and any operator K, satisfying Then both x = yK and x = Ky are solutions, provided only dy = yd holds, which is the consistency condition. They are indeed solutions: We call them the right-hand and left-hand solutions respectively. In our problem, x = m (p) and y = M (p ) .
To begin with, consistency condition has to be checked. For p = 3 consistency d∧ 2 = ∧ 2 d follows from d ∧ + ∧ d = 0. For generic p ≥ 3 the check is straightforward: The right-hand solution of (57) is m (p ) = M (p ) K and the left-hand solution is m (p ) = KM (p ) . Note that the freedom is not exhausted by the single choice between the right and left solutions: it is possible to independently choose either one or another at each step of recursion. If we select only right-hand solutions each time, a pure right A ∞ structure arises: In what follows only this pure right-hand construction will be considered. A pure left-hand A ∞ structure can be constructed in a similar way, however, in section 7 we explain that only the right-hand construction can produce a local A ∞ structure. If we are not interested in locality, then both left or right choices can be made on equal footing. It is not necessary but often convenient to consider nilpotent K-operators, satisfying Such K-operators provide especially simple A ∞ structures: In this case, the whole series D = m (1) + m (2) + m (3) + . . . can be collected into a compact expression. Indeed,

Making use of 1/(1 +
. ., it is further simplified: It turns out that the A ∞ nilpotent operator D is a conjugation of the bare nilpotent operator d: D = U dU −1 . The transformation map U is equal to in this particular case (it would be U = 1+K∧ if we choose the pure left-hand solutions). Of course, conjugation preserves nilpotency:

The first (naive) solution
There is a natural way to construct a K-operator from any operator α : Ω * * → Ω * * . Introduce The same is true also forK α = △ −1 α α. Of course, △ α is rarely invertible. For arbitrary α this operator does not respect grading and △ α acts from a sub-space of a given grading into another. Even if the sub-space remains intact, there can be zero modes.
The simplest nontrivial operator α one can choose is α = ∂, so that ∆ ∂ = d∂ + ∂d = ∆ is the ordinary Laplace operator, which respects all gradings. If M is just a closed simplex, then △ is most simple: it is a scalar operator, multiplying any form by the number of vertices in the simplex. Therefore α = ∂ is especially convenient for calculations on a simplex. However, it is also of use on generic simplicial complexes.
If we choose α = ∂, then, since ∂ 2 = 0, K = K ∂ = ∂ · △ −1 = △ −1 · ∂ and occasionally Therefore, according to (62), we obtain the following (left-hand) A ∞ structure: Note that this solution is as explicit as a formula can be! The differentials d, ∂ and the wedge multiplication ∧ are defined in section 4.5 in pure combinatorial terms, and such operations m (p) can be calculated easily on various simplicial complexes.

The second (local) solution
The solution (67) is just an illustration of how the method works: we pick a K-operator and immediately obtain a solution. Another K-operator would produce another A ∞ structure. Actually, the K-operator method is capable of solving more complicated problems. It can be used to construct not just any but specific A ∞ structures, with given properties, for example, locality. In this section we are going to construct a local A ∞ structure. An A ∞ structure is said to be local, if all its operations m (p) : Ω ⊗p * → Ω * are local. This is a strong additional requirement.
Local A ∞ structure can be built with the help of a local K-operator. Note that neither ∂ nor K ∂ are local, and the solution (67) is not local as well. To construct a local solution, let us consider some important properties of the discrete differential.

Localisation of d and ∂
• In section 4.4 we have outlined that any operation can be decomposed into three operations with definite locality properties. This decomposition can be applied to operations d and ∂: • We are especially interested in describing the strictly local parts [d] and [∂]. First of all, on the space of forms Ω * the local envelope is trivial: for any simplex σ, ∪(σ) = σ. Differential d(σ) increases σ and the local envelope by adding a vertex from beyond σ. Consequently, the strictly local part of d vanishes. The same is true for ∂: it decreases the local envelope by removing a vertex. Therefore we obtain with p ≥ 2. Operations d and ∂ are defined on these spaces via lifting (50). For example, on the space of pairs Ω * ⊗ Ω * the differential act on the first simplex, and then on the second one: There can be vertices beyond a that belong to b -adding such a vertex to a does not change the envelope. This is what allows d(a ⊗ b) to contain a strictly local (envelope-preserving) part. There can be vertices that belong to both a and b, removing such a vertex from a does not change the envelope. Therefore ∂(a ⊗ b) contains a strictly local (envelope-preserving) part. 1 Consider the following minimal example: the complex is a 1 The term {1} ⊗ {1} is non-local in this case, because its local envelope {1} does not contain {1, 2}, which is the local envelope of the argument. One can see, that localization is nothing but omitting non-local terms.
• Differential d is a local operation on the space Ω * of single forms. However, on the whole Fock space Ω * * it contains a non-local part. The non-local part of d (operation d − ) is not identically zero for a very special reason: because the value d(σ 1 , σ 2 , . . . , σ p ) can have empty envelope, even if the argument σ 1 , σ 2 , . . . , σ p had envelope σ = ∅. • In other words, the image of operation d − always has empty envelope: Due to this fact, the following equality holds: • Finally, let us prove an important identity We are going to use this identity later. The proof is quite trivial, if the complex M is a simplex. In this case, according to (29), the discrete Laplase operator d∂ + ∂d is a scalar operator (proportional to unity) on the space of forms. According to the lifting rule (31), it is also scalar on the whole Fock space. It is clear that ∆ + should vanish then, because any scalar operator is strictly local.
If the complex M is not a simplex exactly, select any p-uple of simplexes e with local envelope equal to σ.

Operator △ loc = [d][∂] + [∂][d]
In what follows the central role will be played by local Laplace operator On any simplicial complex, irrespective of its topology, this operator has interesting spectral properties. Denote through Ω ⊗p * (σ, τ ) the subspace, spanned by collections of p simplexes with given local envelope σ and set of free points τ ⊂ σ. It is convenient to study the properties of △ loc on these subspaces separately, because the operator does not mix them. The properties of △ loc depend on the numbers n = |σ| and k = |τ |. Let us describe them in detail.

The eigenvectors
The eigenvectors are given by the following explicit construction. Suppose first that all the points are free, i.e. n = k. In the simplest case p = p, k = 1, n = 1 the space consists of p-uples with only one point which is itself a free point. Then in the natural basis  it is straightforward to prove that the local Laplace operator is represented by the p × p matrix where I is the unity matrix. The matrix (72) has two eigenvalues, 0 and p, with the multiplicities 1 and p − 1 respectively. Denote The eigenvector |0 = E · (1, 1, 1, 1 . . .) corresponds to λ = 0, while |1 , . . . , |p − 1 -a possible choice is . . . correspond to λ = p.
Using the eigenvectors with one free point, it is possible to construct the generic eigenvectors, for k ≥ 1 and with some non-free points e = σ 1 ⊗ . . . ⊗ σ p , k(e) = 0. The explicit construction is | n 1 , n 2 , . . . , n k = e ⋆ |n 1 ⋆ |n 2 ⋆ |n 3 ⋆ . . . ⋆ |n k The ⋆ operation here attaches one collection of simplexes to another: so that △ loc (e 1 ⋆ e 2 ) = (△ loc e 1 ) ⋆ e 2 + e 1 ⋆ (△ loc e 2 ) ∀e 1 , e 2 This construction explains all the spectral properties. In every particular case, all of them can be easily checked by using only combinatorial definition of d, ∂ and localization [·]. However, some technical details of a generic proof (mainly concerning the sign factors χ) are still missing. This sign χ depends on various factors including the number and configuration of non-free points. We have not found yet any convenient formula to express it.

A local K-operator
Given a strictly local operation [∂], it is possible to construct a K-operator following the procedure from section 6. This operator satisfies the main K-operator identity (60), however, it is still not local, because d itself has a non-local part (see s.7.1). To solve the problem, we introduce [K] is a solution, we use the property ∪(d − (e)) = ∅ (see s.7.1.). This property implies that composition of ǫ with any strictly local operation vanishes. In particular, all operations M (p ) are strictly local on Ω ⊗p * . The following cancellation of non-locality happens: Hence ǫ does not affect the results: One can see that M (p ) [K] is indeed a solution: This cancelation of non-local contributions is one of the main reasons why the right-hand K-operator construction (see 5.4) is used. Another reason is that operators [∂] and [K] vanish on the space Ω * of single forms, so that in the left-hand construction they would be useless:

Inversion of △ loc
Operator [K] includes the inverse of local Laplace operator △ loc . We know everything about eigenvectors and eigenvalues of △ loc , therefore it is easily invertible: on each subspace Ω ⊗p * (σ, τ ), where n = |σ| and k = |τ | the identity holds, because it is the spectral equation for a finite-size matrix △ loc . It follows that Coefficients α are expressed through the eigenvalues λ by Vieta formula: where S i denotes the i-th elementary symmetric polinomial. This makes the calculation of inverse Laplace operator straightforward. However, evaluation of subsequent powers △ k loc becomes more and more complicated when the number of free points is large. In order to simplify calculations, one can make use of the eigenvectors and multiplicities and derive an explicit formula for the action of △ −1 loc on any p-uple of simplexes e, in terms of n(e), k(e) and p. We present this formula in the Appendix.

A local A ∞ structure
The K-operator [K] is nilpotent: it satisfies [K] 2 = 0. Therefore, according to (62) we obtain an A ∞ structure This A ∞ structure is local (it is made of local operations only) and has a correct continuum limit (due to correct choice of the first discrete operations d and ∧). Like (67) eq.(79) is an absolutely explicit formula. In the Appendix below we calculate a few first operations m (p ) with the help of (79). In the simplest cases this calculation can be even done by hand. In generic case, starting actually from p = 4, this becomes a complicated calculation and computer facilities can be used. However, this matters only if one is interested in explicit expressions for operations. From pure theoretical point of view most important is that we obtained a nilpotent A ∞ operator As already mentioned in s.5.2, this operator is not a homogenious element w.r.t. the tensor-grading. It can be decomposed into the graded components: These graded components represent particular A ∞ operations.

Conclusion
The local A ∞ structure (79) is a particular example of a wide class of local A ∞ structures. In other words, equation (22) has (infinitely) many solutions and even more, infinitely many local solutions. A natural claim is that the A ∞ structures are in one-to-one correspondence with nilpotent operators on the Fock space Ω * * . Higher nilpotent analogues of A ∞ structure should exist, being the ∧ n -induced deformations of the d n operators. The relevant operators D n on the Fock space Ω * * satisfy D n n = 0 For explicit description of the local A (n) ∞ structures see [9].

Appendix
In this section we present a number of explicit calculations with discrete forms. They should be regarded as examples and illustrations.

Simplicial complexes
One-dimensional complexes: A discrete form f ∈ Ω * (M ) is a linear combination of these simplexes: Differential d acts on Ω * (M ) as follows: It is easy to check that d 2 = 0 is true.
The dual operator ∂ acts on Ω * (M ) as follows: It may be convenient to represent the linear operators with matrices. Obviously, the matrix of ∂ is equal to the transposed matrix of d.
It is easy to check that the Laplace operator △ = d∂ + ∂d = 3 · id.
The first non-trivial products are Localisation means omitting the terms of the right hand side, which have local envelope not equal to the local envelope of argument:  Construction (75) explains all these results. Compare cases 1 and 2 to observe the non-trivial χ sign.

Discrete quantum field theory
There is a straightforward physical interpretation of the construction. The local Laplace operator △ loc is a Hamiltonian operator for a discrete quantum field theory, defined on a Fock phase space which is constructed from a one-particle phase space Ω * . The one-particle states are described by forms on a simplicial complex (wave functions). The free points correspond to excitations. Note that the Hamiltonian △ loc arises naturally from the local differentials on complex. It is essentially discrete and vanishes in the continuum limit. It would be interesting to investigate the field-theoretical properties of this theory, compared to smooth analogues.

Inverse of Local Laplace Operator
Inversion of local Laplace operator is necessary to construct the solution (79). We present a combinatorial rule of action of inverse operator △ −1 loc on any p-uple of simplexes e: The result is equal to a sum over all rearrangements of free points into p groups (simplexes). Non-free points remain on their places. Each rearrangement is taken with a weight where F 0 = p k n(n−1)(n−2)...(n−k), Q a b = C a b p a (k−a)! and j is the number of free points in the rearrangement that has been moved to another simplex. Also, each rearrangement should be taken with a sign (−1) s1 (−1) s2 . (−1) s1 is the parity of permutation of free points, associated with the rearrangement. (−1) s2 counts the parity of jumps of free points over the non-free points. This rule certainly provides the same results as the main spectral formula (78), but it is more convenient for calculations, for reasons pointed out in s.7.6.

Discrete forms and operations: co-chain description
In the co-chain picture a discrete p-form is a function on p-simplexes, which can be represented as function f (i, j, . . .) of p + 1 vertexes i, j, . . . Note that formulas below do not depend on the simplicial complex at all. This is a nice property of local operations in the co-chain description.
Second, we present the local solution m (3) , obtained with the help of (79). The numeric factors (like 7/1296) typically arise from the inversion of local Laplace operator.