# Topology at the Planck Length

###### Abstract

A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a ‘smooth’ change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting geometries. Applications are considered to the theory of -branes within the context of the proposed (atrix) theory.

PACS 02.40.-k, 04.20.Cv

LPTHE Orsay 97/34

## 1 Motivation and notation

Since the early efforts of Wheeler [36] in this direction it has often been speculated that the topology of space might not be a well defined dynamical invariant. There can of course be no smooth time evolution of a space of one topology into that of another; a classical space cannot change topology without the formation of a singularity. However the ‘true’ description of space and of space-time should reasonably include quantum fluctuations and it is possible that a quantum space-time exists which seems in a quasi-classical approximation to evolve from a space of one topology to that of another, the ‘exact’ quantum space-time being a sum over many topologies.

Noncommutative geometry furnishes a possible alternative mathematical language in which one can also discuss this question. A change in topology is possible simply because even classically the topology of a manifold is not a well defined quantity at all length scales. A change in space topology can occur if the dynamical evolution of space-time is such that the space enters a regime in which its description requires the use of noncommutative geometry. This could be expected to occur near a classical singularity. We are able to treat this problem only in 2+1 dimensions where we can identify space with a smooth compact surface of genus . Even here we can give no reasonable field equations which would dynamically implement the change of space topology which we consider.

In Section 2 we propose a definition of a fuzzy surface of ill-defined topology. In Section 3 we discuss the differential structures of fuzzy surfaces and we describe in detail an explicit example of topology change from the 2-sphere to the 2-torus through a sequence of such fuzzy surfaces. We speculate on the analogous transition between two compact surfaces of arbitrary genus. Finally in Section 5 we discuss our results in light of the recent fuzzy description of -branes.

When using in the noncommutative context a word which is usually defined only for an ordinary differential manifold we enclose it in quotation marks if there is a chance of ambiguity.

## 2 Topological fuzzy surfaces

The general definition of a fuzzy surface has been given elsewhere [27, 29]. Points are replaced by elementary cells (Planck cells) of a quantum of area. If the ‘surface’ is to be in some sense compact then there can only be a finite number of cells and the structure algebra must be of finite dimension. It is usually taken to be the algebra of complex matrices. The ‘topology’ is encoded in a filtration of the algebra which we shall introduce for each genus . The differential structure is encoded in the differential calculus over the algebra. We shall suppose that there exist fuzzy versions of compact surfaces of arbitrary genus although we know of explicit constructions only in the particular cases and .

Let be an element of . For each we shall introduce a norm If the sequence has a limit for some value of then we consider to be a matrix approximation to a function on . The limiting procedure is rather obscure and we shall consider it only on the algebra of polynomials in a set of basic matrices, the ‘coordinates’. The choice of this set will define the filtration and hence the value of . Since is a simple -algebra a morphism is necessarily of the form for some hermitian . In the commutative limit these maps tend to symplectomorphisms of into itself [9, 23, 15, 17, 7, 26]. A general ‘coordinate transformation’ would be a map of the form which respects the algebraic structure only in the commutative limit. If such a map is singular in the commutative limit then the resulting transformation is a change of topology.

### 2.1 Genus zero

Let be the cartesian coordinates of and the euclidean metric. The ordinary round sphere of radius is defined by the constraint . The algebra of smooth functions on is a completion of the quotient of the algebra of polynomials in the by the ideal generated by those which contain as a factor.

A fuzzy version of the sphere [5, 9, 23, 15, 17, 7, 27] is constructed using an -dimensional irreducible representation of the Lie algebra of the group . We let , for , be the generators and we raise and lower indices using the Killing metric . We introduce a macroscopic length scale , the radius of the sphere, and a microscopic area scale which are related, for large , by the equation

The integer counts the number of elementary cells of area . The Casimir relation is written as and the commutation relations of the ‘coordinates’ are given by

We shall consider the length scale as fixed and so in the limit as . We can identify therefore

Any matrix can be written as a polynomial in the ,

where the are completely symmetric and trace-free. We can associate to the function . Set . We have defined then a vector-space map

This map cannot of course respect the product structures of the respective algebras but if and are two polynomials of order less than some integer then one can show that

For each integer introduce the vector space of symmetric polynomials of order in the . Obviously

The filtration of the algebra which defines the sphere is given by the and is a filtration of the polynomials of order on the sphere.

We define the norm of an element as

In particular we find that

We introduce the norm of an element as

Then if we have

The norm of a generic element of grows as .

### 2.2 Genus one

Let be again a length scale and consider the torus defined to be the subset of with coordinates subject to the conditions . Consider the two functions

The algebra of smooth functions on is a completion of the algebra of polynomials in and .

A fuzzy version of the torus was constructed by Weyl [35], Schwinger [33] and others [16, 1, 2] to describe a finite version of quantum mechanics. One introduces elements and which satisfy the Weyl relation

as well as the constraints

The algebra generated by and is isomorphic then to the matrix algebra . Define the area parameter by the relation

This is the same as (2.1.1) if one replaces the area of the sphere of radius by the area of the torus. It is worth noticing that when described by the algebra the torus has ‘cells’; each observable can take possible values. It is therefore to be compared with an approximation on a lattice with

as unit of length.

An explicit form for and can be easily found [33]. There is an orthonormal basis , , of such that and are given by

and such that the cyclicity condition

holds. One can introduce matrices and defined by the relations

In the basis it is obvious that one can choose such that

There is also an orthonormal basis in which the is diagonal. The two bases are related by the ‘Fourier transformation’ [33]

If we define, for arbitrary ,

then we can write

The Fourier transformation is unitary because of the relations and , . A short calculation yields the relation

In the limit we must have and therefore . We recover then the commutation relation

which is equivalent to the Weyl relation (2.2.2).

Because as we can identify

Introduce . Any matrix can be written as a polynomial in the :

where the are completely symmetric. We can associate to the function . Set . We have defined then a vector-space map

As above, if and are two polynomials of order less than some integer then one can show that

For each integer introduce the vector space of symmetric polynomials of order in the . Obviously

The filtration of the algebra which defines the torus is given by the and is a filtration of the polynomials of order on the torus.

We define again the norm of an element by (2.1.7). In particular we find that

We define the norm of an element as

Then if we have

The norm of a generic element of grows as .

### 2.3 Higher genera

We conjecture that the construction of Section 2.1 and Section 2.2 can be extended to arbitrary genus. Let be a surface of genus and choose generators of which define in the limit coordinates on . There might be a large number of the which satisfy relations. For each integer introduce the vector space of symmetric polynomials of order in the such that

The filtration of the algebra which defines is given then by the .

We define again the norm of an element as (2.1.7). We introduce the norm of an element as

Then, if we should have

The norm of a generic element of grows as . We shall return to this in Section 3.3.

### 2.4 Continuous transitions

From the point of view of noncommutative geometry a transition is possible between space-times of different topology simply because an individual space-time is never completely in a ‘pure’ topological state. As long as is not equal to zero the correct description of every surface is given in terms of a filtration of the matrix algebra for some (very large) integer . A transition occurs when one filtration becomes more appropriate than another. Below we shall introduce differential calculi on and we shall be in a position to speak of the noncommutative analog of a smooth scalar field. A generic such field on a surface of genus must have finite action and every other action must be ‘almost always’ infinite. If during the time evolution the action changes so that has finite action for the genus then this means that the surface has evolved towards a different topology.

The difference in topology between the sphere and the torus is expressed in a discontinuity in the functions . These discontinuities will not show up in the norm (2.1.7) we have put on . They do show up however if we use the action as norm since it contains derivatives. We shall discuss in the following section how a topological transition can be induced using the partition function after we have introduced differential calculi and the associated scalar-field actions.

## 3 Smooth fuzzy surfaces

Every surface can of course be endowed with a differential structure and the associated de Rham calculus of differential forms. To speak of a smooth fuzzy surface we must be able to define a differential calculus on each fuzzy which in some sense has the de Rham calculus as a limit. Since the de Rham calculus is based on the derivations of the algebra of functions it is natural to require that for each the differential calculus over be based on derivations. This idea was first suggested by Dubois-Violette [12] and developed by Dubois-Violette et al. [13]. We shall use a modified version proposed later by Dimakis [11]. Since we shall restrict our considerations here to scalar fields and shall not therefore need explicitly the differential calculus we shall not enter into the details of its construction. Some more details will be given where necessary in Section 4. We recall that a classical scalar field defined on a fuzzy surface of any genus is an element of . The form of the matrix determines the genus of the surface on which it is to be considered an approximation to a regular function.

For each we shall define a differential structure over in order to be able to speak of the noncommutative analogue of a smooth scalar field. We can then define an action which tends to the action of a complex classical field on . Let be an element of which tends to a function on . Then we have

We shall use the action to define a Sobolov-like norm on the matrices and a Sobolov norm on the limit functions. We shall return to this in Section 3.4.

### 3.1 Genus zero

The derivations

satisfy the commutation relations

In the commutative limit these derivations tend towards vector fields on the sphere defined by the action of the Lie algebra of . The relation defines the space of vector fields on the sphere as a (projective) submodule of the free -module generated by the .

We choose the differential calculus defined in terms of the , that is, with the 1-forms defined by the relation

Since the sphere is not parallelizable the differential calculus must be defined on a parallelizable bundle over it. The details of this have been described elsewhere [26, 21, 8]. It is important only to recall the existence of a special basis or frame which is dual to the derivations and which commutes with the elements of the algebra.

We define the action of the matrix on the surface as the trace

where the laplacian is the covariant laplacian with respect to the geometry we have put on the sphere and is an arbitrary (positive) potential function. The normalization has been chosen so that

where is the usual action of the classical complex scalar field . Obviously we shall have

for almost all elements .

It is of interest to note that because of the identity

it is possible to write the action (3.1.4) without explicitly using the derivations. The 2-form is defined using a straightforward generalization of the standard duality in forms which relies on the existence of the preferred frame.

### 3.2 Genus one

The vector fields

form a basis of the free -module of vector fields on the torus. Their action on the generators is given by

and of course they commute:

The dual de Rham 1-forms are given by

Because the torus does not have as large an invariance group as the sphere it is more difficult to find a differential calculus over which tends to the de Rham calculus. This fact leads us to believe that the introduction of appropriate noncommutative differential calculi over fuzzy surfaces of higher genera will be a delicate matter.

Were it not for the extra constraints (2.2.3) which distinguish the ‘quantum’ torus from the ‘quantum’ plane we could have used the ‘quantum’ analog of (3.2.1) and introduced a differential calculus based on the outer derivations defined by

If we extend formally the algebra and admit hermitian elements and then these derivations become inner and can be written, using the relation (2.2.4) as

The associated frame is formally identical to (3.2.2):

It is easy to see [11] that the associated differential calculus admits a flat metric-compatible torsion-free linear connection.

But the above derivations are not compatible with the constraints (2.2.3). With these constraints the algebra is a matrix algebra and all derivations must be inner. This leads to problems. It is of course in itself not surprising to encounter a situation where ‘quantization’ is inconsistent with certain constraints; this feature of quantum mechanics was known to Dirac. Using the representations of Section 2.2 the commutation relations

are easily derived. We have here introduced the projectors

As every element of the algebra they can be expressed as polynomials in the generators:

It follows that the action of the derivations (3.2.4) on the generators of the algebra is given by

The highly singular projector term on the right-hand side of each of these equations is due to the constraints (2.2.3). It is because of these terms that we find and as we must.

The 1-forms dual to the derivations (3.2.4) are given by

If we compare (3.2.10) with (3.2.2) we see that the could in a weak way be considered to tend to the . The problem of the singular limit of the differential calculus is hidden however in the differentials ; the differential calculus based on the derivations (3.2.4) does not tend to the de Rham differential calculus on the torus.

It was of course not necessary to use a differential calculus based on derivations and one can introduce many another differential calculi over the ‘quantum’ torus. There are in fact many which can be constructed [11] based on derivations but which are not real. It is easy to see however that whatever the definition of and the 1-forms (3.2.5) cannot commute with the elements of the algebra and that the resulting differential calculus will not have them as a preferred frame. Also to define the action we will have to be able to define a Laplace operator using the derivations.

To construct the torus we identified the points with and with . In the ‘quantized’ version this becomes the cyclicity condition (2.2.6) which gives rise to the singular projector terms in the derivations (3.2.9). One can eliminate them by a procedure which is equivalent to folding, so to speak, the torus at or . For this we suppose that is even in the formulae of Section 2.2 and we consider the possibility of a differential calculus based on the derivations of the form (3.2.4) with and replaced respectively by and defined by

We shall suppose that so that we have

The matrices and are not defined then uniquely by the Formulae (2.2.7). This fact is related to the fact that only by using additional topological conditions was von Neumann able to deduce the uniqueness of the representation of the Heisenberg commutation relations. For a discussion of this and an introduction to the problems connected with the quantization of the torus as a classical phase space as well as reference to the previous literature on the subject we refer to the lecture by Emch [14] or to the recent article by Narnhofer [30].

If we choose

and introduce the ‘step functions’

we find

The commutation relations (3.2.14) are almost as singular as (3.2.6). The presence however of the extra factors permits us to ‘renormalize’ the and define

We find then in the limit

We introduce the step functions

We can claim then that in a weak way

and comparing (3.2.1) with (3.2.16) we find that

The limit of the derivations are vector fields on the torus which form a basis of but which are not continuous along the lines , .

We have not succeeded in finding real derivations of which tend to real smooth vector fields on the tours. The limit is a rather singular limit and it need not be true that an arbitrary vector field on the torus is the limit of a derivation. We constructed the algebra using generators and relations. This is the noncommutative version of the method of defining a curved manifold by an embedding in a higher-dimensional flat euclidean space. This procedure works well for the sphere but the flat torus possesses no such embedding. We refer to the book by Thorpe [34] for a discussion of this point.

The 1-forms dual to the derivations (3.2.16) are given by

These are almost as singular as the limit of the expressions given by (3.2.10). It is important however for us to have the derivations to define the Laplace operator.

We define the action of the matrix on the surface as the trace

where the laplacian is the covariant laplacian with respect to the geometry we have put on the torus and is an arbitrary (positive) potential function. The normalization has been chosen so that

where is the usual action of the classical complex scalar field . From (3.2.16) we find

and similarly for .

Obviously we shall have

for almost all elements . As an example consider the ‘coordinate’ on the sphere. With the conventions we have been using one finds the expression

The numerical factor in this expression is valid only for large values of . Since

there follows then the estimate

At least one of the ‘coordinates’ of the fuzzy sphere becomes singular then when considered as an element of the fuzzy torus.

### 3.3 Higher genera

An introduction to general Riemann surfaces can be found for example in the lecture notes by Schlichenmaier [32]. The algebra of functions on each surface has been ‘quantized’ using general -algebras [24, 25]. We conjecture in fact that this can be done using matrix algebras and that differential calculi can be constructed over which tend in some way to the de Rham differential calculus of for each genus . The construction of Berezin [5] as well as the fact that each can be endowed with a metric of constant Gaussian curvature is some encouragement. If the differential calculus is based on derivations then one can define a Laplace operator and an action

with

where is the usual action of the classical complex scalar field on the Riemann surface .

### 3.4 Smooth transitions

A generic classical field on a surface of genus must have finite action and every other action must be infinite. If during the time evolution the action changes so that has finite action for some other genus then this means that the surface has evolved towards a different topology. To describe a topological transition from the sphere to the torus one introduces a ‘temperature’ and an action such that for and for . The transition will be of first order. It can be made to be of infinite order by choosing for and for and choosing as action a smooth functional

in the region . One task of a noncommutative version of gravity would be to motivate this ad hoc change of action functional, to calculate, that is, the function .

The partition function for a complex scalar field over a surface of genus is given by

The matrix approximation [27] is given by

where the path integral is now a well-defined integration over matrices. We suppose that the ‘real’ value of is ‘large’ but not infinite, given by (2.1.1) or (2.2.4). We can then claim that the expression (3.4.2) is the ‘correct’ one and (3.4.1) is the approximation. For the contributions from almost all those matrices which approximate functions on the torus (and other genera) are suppressed since . On the other hand for the contributions from almost all those matrices which approximate functions on the sphere (and other genera) are suppressed since .

## 4 -branes

Matrices can also be used to give a finite ‘fuzzy’ description of the space complementary to a Dirichlet -brane, a description which will allow one perhaps to include the reasonable property that points should be intrinsically ‘fuzzy’ at the Planck scale. Strings naturally play a special role here since they have a world surface of dimension two and an arbitrary matrix can always be written as a polynomial in two given matrices. We refer to the literature for a description of Dirichlet branes in general [31, 6, 10] and within the context of (atrix)-theory [4, 18, 22, 3]. The action of the matrix description of the complementary space is conjectured [9] to be associated to the action in the infinite-momentum frame of a super-membrane of dimension . Since quite generally the compactified factors of the surfaces normal to the -branes are of the Planck scale we conclude from the arguments of the previous sections that they have ill-defined topology and that a matrix description will include a sum over many topologies.

We consider a -dimensional manifold with a Kaluza-Klein reduction to a Dirichlet brane of dimension . The is known as a -brane. The manifold is therefore a bundle over with fibre an -dimensional manifold . We shall suppose for simplicity that the fibration is trivial, , and that all manifolds are parallelizable. We shall suppose also, as is usual in Kaluza-Klein theory, that is space-like. Let Greek indices take the values 1 to , Latin indices the values to and Latin indices the values to . We introduce a moving frame on with a moving frame on . Consider now an electromagnetic field on and write the field strength as

Then the electromagnetic action in takes the form

Although we have argued elsewhere [27] that the entire should be described by a noncommutative algebra we shall suppose here that the -brane can be described by an ordinary smooth manifold and that only the need be ‘quantized’. This means that the algebra of (smooth) functions on is replaced by a finite-dimensional matrix algebra, the algebra of matrices, a procedure which is analogous to the quantization of a compact phase space [5], for example spin. It means also that the algebra of de Rham differential forms on must be replaced by a differential calculus over . This entire procedure has just been described for in Section 4. For we refer to Grosse et al. [20] and for general to Madore [28]. In the case we have seen that only genus zero and one can be considered with any success.

The components of the moving frame introduced on are to be replaced by a noncommutative equivalent such that in some sense we have

We have seen in Section 4 how difficult this limit is to define even for . We shall use the same symbol to denote a de Rham form and the equivalent fuzzy form. Let be an element of . Then in typical Kaluza-Klein fashion we can write as the sum of a ‘horizontal’ term in and a ‘vertical’ term in . More details of this can be found elsewhere [27].

### 4.1 Curved complements

The case in which the space complementary to a -brane is curved and compact is the easiest to treat conceptually from a fuzzy point of view. It contains at least a simple generic example, which has been worked out in detail and although we formulate them more generally, most of the following calculations have been shown to be valid only in this one example. The case has however the drawback in that, being curved, the models have no immediate supersymmetric extension. A rudimentary version of ‘noncommutative supergravity’ would have to be developed for this purpose. This would involve introducing besides the noncommuting bosonic generators a set of non-anticommuting fermionic generators and defining a linear connection on the entire structure. This has yet to be done.

We identify the gauge transformations on as the unitary elements of . The way in which can be identified with the local transformations in the commutative limit has been explained by Grosse & Madore [19] in the case and genus zero. A gauge transformation is therefore given by

with an element of , the group of local gauge transformations on .

Now has a preferred 1-form which is invariant [13] under a gauge transformation:

We have here decomposed . Choose an integer and an anti-hermitian basis of the Lie algebra of . Restrict to those values such that has an irreducible representation of dimension and restrict the to be an orbit of the adjoint representation of . For example if then can take any values and which is the dimension of minus the number of Casimir operators. If then which is again the dimension of minus the number of Casimir operators. The manifold is a manifold embedded in and defined by the cubic Casimir operator of . If we introduce the derivations and the 1-forms dual to them then . (In the limit (4.1.1) is singular [26] in the case and genus zero; this is because the sphere is not parallelizable and in the commutative limit the must be defined on a parallelizable bundle over it) We can decompose then

where is the difference between two connections and so transforms under the adjoint representation of . We write . Then a straightforward calculation leads to the identities

for the ‘fuzzy’ and mixed components of the electromagnetic field strength. The structure constants are defined with respect to the basis of .

A rather dubious mathemetical argument leads, at least in the case and genus zero [19], to the limit

The is given by

where

is the potential. We see that it can be thought of as the field strength in the fuzzy directions.

If we consider the as ‘coordinates’ on the fuzzy version of then is given by , which is a stable zero of the potential (4.1.8). Another obvious stable zero is given by . There are in general other stable zeros, the number of which increases with . For example, in the case of the 2-sphere there are in all (the partition function) zeros of . One can think of this as meaning that there are possible ‘positions’ for the -branes and they are all stable with the same potential energy. Energy is required however to transit from one state to another. In a complicated way the number of massless modes increases as one ‘approaches’ the vacuum . By this we mean that when there is a multiplet of massless modes, when there is only a multiplet and the number of massless modes in the vacuua between these two extremes depends on the characteristics of the vacuum.

The are curved ‘manifolds’ in general and endowed with a linear connection. The covariant derivatives of in the directions normal to are given by

It vanishes therefore on the stable vacuum given by but not on the others. Except for the ‘curvature term’ in the expression of the ‘vertical’ components of the Yang-Mills field strength the action (4.1.7) is identical to the bosonic part of the one which has been proposed in (atrix) theory, To see if it is possible to obtain exactly the (atrix)-theory action we turn our attention to flat complements .

### 4.2 Flat complements

The simplest example of a compact manifold which could admit a flat metric is the -torus. We have not succeeded in treating this case for general values of but it would seem from our considerations of Section 3.2 that it is very difficult if not impossible to define a differential calculus on a noncommutative version of the 2-torus which tends smoothly to the de Rham differential calculus and which admits a flat metric. The case of a flat is paradoxally more difficult to treat from a fuzzy point of view than the curved one.

## Acknowledgment

The authors would like to thank A. Kehagias and S. Theisen for enlightening conversations. One of the authors (JM) would like to thank J. Wess for his hospitality at the Ludwig-Maxmillian Universität, München and the other (LAS) would like to thank CNPq, Brazil for financial support.

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