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## Brazilian Journal of Physics

##
*Print version* ISSN 0103-9733

### Braz. J. Phys. vol.40 no.4 São Paulo Dec. 2010

#### https://doi.org/10.1590/S0103-97332010000400004

**2d Gravity with torsion, oriented matroids and 2+2 dimensions **

**J.A. Nieto ^{I,}^{*}; E.A. León^{II,}^{†}**

^{I}Facultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, México

^{II}Departameto de Investigación en Física de la Universidad de Sonora, 83000, Hermosillo Sonora , México

**ABSTRACT**

We find a link between oriented matroid theory and 2d gravity with torsion. Our considerations may be useful in the context of noncommutative phase space in a target spacetime of signature (2+2) and in a possible theory of gravity ramification.

**Keywords:** 2d-gravity, 2t physics, 2+2 dimensions.

As it is known, the theory of matroids is a fascinating topic in mathematics [1]. Why should not be also interesting in some scenarios of physics? We are convinced that matroid theory should be an essential part not only of physics in general, but also of M-theory. In fact, it seems that the duality concept that brought matroid theory from a matrix formalism in 1935, with the work of Whitney (see Ref. [2] and references therein), is closely related to the duality concept that brought M-theory from string theory in 1994 (see Refs. [3-11] for connections between matroids and various subjects of high energy physics and gravity). These observations are some of the main motivations for the proposal [12] of considering oriented matroid theory as the mathematical framework for M-theory. In this paper, we would like to report new progress in our quest of connecting matroid theory with different scenarios of high energy physics and gravity. Specifically, we find a connection between matroids and 2d gravity with torsion and 2+2 dimensions. In the route we find many new directions in which one can pursue further research, such as tame and wild ramification [13], nonsymmetric gravitational theory (see Ref. [14] and references therein) and Clifford algebras (see Ref. [15] and references therein). We believe that our results may be of particular interest not only for physicists but also for mathematicians.

In order to achieve our goal we first show that a 2×2-matrix function in two dimensions can be interpreted in terms of a metric associated with 2d gravity with torsion. Let us start by writing a complex number z in the traditional form [16]

where *x* and *y *are real numbers and *i*^{2} = -1. However, there exist another, less used, way to write a complex number, namely [17]

In this case the product of two complex numbers corresponds to the usual matrix product. These two representations of complex numbers can be linked by writing (2) as

Since one finds from (1) and (3) that the matrix can be identified with the imaginary unit *i*.

It turns out that the matrices and can be considered as two of the matrix bases of general real 2×2 matrices which we denote by M(2,R). In fact, any 2×2 matrix Ω over the real can be written as

where

Let us rewrite (4) in the form

where

Considering this notation, we find that (1) becomes

Comparing (6) and (8), we see that (8) can be obtained from (6) by setting *r *= 0 and *s* = 0. If *ad-bc* ≠ 0, that is if detΩ≠0, then the matrices in *M*(2,*R*) can be associated with the group *GL*(2,*R*). If we further require *ad-bc* = 1, then one gets the elements of the subgroup *SL*(2,*R*). It turns out that this subgroup is of special interest in 2t physics [18-20].

Now, consider the following four functions *F*(*x,y,r,s*),* G*(*x,y,r,s*),* H*(*x,y,r,s*) and *Q*(*x,y,r,s*), and construct the matrix

By setting

we get that γ can be written as

or

We can always decompose the matrix λ_{ij} in terms of its symmetric *g _{ij}* =

*g*and antisymmetric

_{ji}*A*= -

_{ij}*A*parts. In fact, we have

_{ji}From (11) or (12) we find that we can write *g _{ij}*(

*x,y,r,s*) in the form

while

An interesting possibility emerges by dimensional reduction of the variables *r *and *s*, that is by setting in (13) *r *= 0 and* s *= 0. We have

with

and

Of course, according to (8) the expressions (16), (17) and (18) can be associated with a complex structure. This observation can be clarified by using isothermal coordinates in which *w* = 0 and ξ = 0. In this case (16) is reduced to

where we wrote λ_{ij}(*x,y*)→ *f*_{ij}(*x,y*) in order to emphasize this reduction. In the traditional notation, (19) becomes *f*(*x,y*) = *u*(*x,y*)+*iv*(*x,y*). It turns out that the existence of isothermal coordinates is linked to the Cauchy-Riemann conditions for u and v, namely ∂*u*/∂*x* = ∂*v*/∂*y* and ∂*u*/∂*y* = -∂*v*/∂*x* [16].

One of the main reason for the above discussion comes from the question: is it possible to identify the symmetric matrix *g _{ij}*(

*x,y*) with 2d gravity? Assuming that this is the case the next question is then: what kind of gravitational theory describes λ

*(*

_{ij}*x,y*)? In what follows we shall show that λ

*(*

_{ij}*x,y*) can be identified not only with a nonsymmetric gravitational theory in two dimensions but also with 2d gravity with torsion. First, consider the covariant derivative of the metric tensor

Here, we assume that the symbols Γ_{ki}^{l} are not necessarily symmetric in the two indices k and i. In fact, if we define the torsion as T_{ki}* ^{l}* ≡ Γ

_{ki}

*-Γ*

^{l}*, one finds that the more general solution of (20) is*

_{ik}^{l}where Γ_{kij} = Γ_{ki}^{l}g_{lj} and *T _{kij} = T_{ki}^{l}g_{lj}.*

On the other hand, if we consider the expression

by substituting (16) into (22) one gets

Comparing (23) and (21) one sees that if one sets *T _{kji}* = ∂

_{i}

*A*the expression (23) can be identified with the connection Γ

_{kj}*which presumably describes 2d gravity with torsion. Since*

_{kij}*A*can always be written as (18) we discover that (23) yields

_{ij}Here, we used the notation ∂_{k}υ = υ* _{,k}*.

The curvature Riemann tensor can be defined as usual

The proposed gravitational theory, which may be interesting in string theory, can have a density Lagrangian* * of the form ~ ^{2}+ Λ [21], where L is a constant. In this context, we have proved that it makes sense to consider the nonsymmetric metric of the form (16)-(18) as a 2d gravity with torsion.

From the point of view of complex structure there are a number of interesting issues that arises from the above formalism. One may be interested, for instance, in considering the true degrees of freedom for the metric *g _{ij}*. In this case one may start with the Teichmuller space associated with the metric

*g*and then to determine the Moduli space of such a metric [22]. Another possibility is to consider similarities. In this case one may be interested to associate with the metric g

_{ij}_{ji}either the Riemann-Roch theorem [23] or the tame and wild ramification complex structure [13]. In the later case one may assume that the principal part of the metric

*g*looks like

_{ij}In this case the similarities can be identified with solitons associated with black holes. In this scenario our constructed route to 2d gravity with torsion provides a bridge which may bring many ideas from complex structure to 2d gravity with torsion and *vice versa*.

Let us now study some aspects of the above formalism from the point of view of matroid theory. Consider the matrix

with the index *A* taking values in the set

It turns out that the subsets { **V**^{1},**V**^{2}}, {**V**^{1},**V**^{3}}, { **V**^{2},**V**^{4}} and { **V**^{3},**V**^{4}} are bases over the real of the matrix (27). One can associate with these subsets the collection

which can be understood as a family of subsets of E. It is not difficult to show that the pair = (E,) is a 2-rank self-dual matroid. The fact that we can express* * in the matrix form (27) means that this matroid is representable (or realizable) [1]. Moreover, one can show that this matroid is graphic and orientable. In the later case the corresponding chirotope [1] is given by

Thus, we get, as nonvanishing elements of the chirotope χ* ^{AB}*, the combinations

The relation of this matroid structure with of our previous discussion comes from the identification { **V**^{1},**V**^{2}} → δ* _{ij}*, {

**V**

^{1},

**V**

^{3}} → η

*, {*

_{ij}**V**

^{2},

**V**

^{4}} →λ

_{ij}and {

**V**

^{3},

**V**

^{4}} → ε

_{ij}. The signs in (31) correspond to the determinants of the matrices δ

_{ij}, η

_{ij}, λ

_{ij}and ε

_{ij}, which can be calculated using (30). Therefore, what we have shown is that the bases of

*M*(2,

*R*) as given in (4) (or (7)) admit an oriented matroid interpretation. It may be of some interest to consider the weak mapping →

_{c}with

leading to the reduced pair _{c} = (*E*,_{c}). When the local structure is considered as in (14)-(18), one needs to rely in the matroid fiber bundle notion (see Refs. [24] and [25] and references therein). Therefore, we have found a link which connect matroid fiber bundle with 2d gravity with torsion.

It is worth mentioning the following observations. It is known that the fundamental matrices δ_{ij}, η_{ij}, λ_{ij} and ε_{ij} given in (7) not only form a basis for *M*(2,*R*) but also determine a basis for the Clifford algebras *C*(2,0) and *C*(1,1). In fact one has the isomorphisms *M*(2,*R*) ~ *C*(2,0) ~ *C*(1,1). Moreover, one can show that *C*(0,2) can be constructed using the fundamental matrices (7) and Kronecker products. It turns out that *C*(0,2) is isomorphic to the quaternion algebra *H*. Since all the others *C*(*a,b*)'s can be constructed from the building blocks *C*(2,0), *C*(1,1) and *C*(0,2), this means that our connection between oriented matroid theory and *M*(2,*R*) also establishes an interesting link with the Clifford algebra structure (see Ref. [15].and references therein).

Let us make some final remarks. The above links also apply to the subgroup *SL*(2,*R*) which is the main object in 2t physics. In this case it is known that noncommutative field theory of 2t physics [18-20] (see also Ref. [26]) contains a fundamental gauge symmetry principle based on the noncommutative group *U*_{*}(1,1). This approach originates from the observation that a world line theory admits a Lie algebra *sl*_{*}(2,*R*) gauge symmetry acting on phase space [18]. In this context, consider the coordinates *q* and *p* in the phase-space. The Poisson bracket

can be written as

where *q*_{1}^{a} ≡ *q*^{a} and *q*_{2}^{a} ≡ *p*^{a}, with a and b running from 1 to n. It worth mentioning that the expression (34) is very similar to the the definition of a chirotope (see Ref. [8] and references therein).

Recently, a generalization of (34) was proposed [27], namely

Here, Ω* ^{ab}* is skew-simplectic form defined in even dimensions. In particular, in four dimensions Ω

_{ab}can be chosen as

Here, by choosing η_{ab} = *diag*(-1,1,-1,1) we make contact with (2+2)-dimensions which is the minimal 2t physics theory (see Refs. [28-31]).

Let us write the factor in (35) *g _{ij}*Ω

^{ab}+ε

_{ij}η

*in the form*

^{ab}with **g**'_{ij} = *g _{ij}*Ω. We recognize in (37) the typical form (18) for a complex structure. This proves that oriented matroid theory is also connected not only with (2+2)-physics but also with noncommutative geometry.

An alternative connection with 2t physics can be obtained by considering the signature η* _{ab}* =

*diag*(1,1,-1,-1), and its associated metric:

In fact, by defining the matrix

it can be seen that (38) can be expressed as

where the indices* i,j,k,l *run from 1 to 2 as before, and η_{ij} stands for the third matrix defined in (7), namely η_{ij} ≡ . As before,noticing that in (38) the ßpatial" coordinates *x*^{1}, *x*^{2} are the elements of the main diagonal and the "time" coordinates *x*^{3}, *x*^{4} corresponds to the main skew diagonal in (39), *x ^{ij}* can be written in terms of the bases (7) as follows:

where we used the definitions

and considered the notation ε* ^{ij}* = ε

_{kl}η

*η*

^{ik}*and l*

^{jl}*= l*

^{ij}_{kl}η

*η*

^{ik}*, where η*

^{jl}*is the inverse flat 1+1 metric, and has the same components as η*

^{ij}_{ij}.

Finally, consider the three index object η* _{ijk}* with components

From these expressions and (7) it can be checked that η_{ijk} automatically satisfies

Therefore η_{ijk} has the remarkable property of containing all the matrices in (7). This means that an arbitrary matrix Ω_{ij} can be written as

where *x*^{1} =* x*, *x*^{2} = y and y^{1} = r, *y*^{2} = *s*. Here,* x,y,r *and *s* are defined in (5). Observe that η_{ijk} = η_{jik}, but η_{kij} ≠ η_{kji}. It is worth mentioning that a similar structure was proposed in Ref. [14] in the context of nonsymmetric gravity [32].

The inverse η^{ijk} of η* _{ijk}* can be defined by the relation

or

Explicity, we obtain the components

Traditionally, starting with a flat space described by the metric η* _{ij}* , one may introduce a curved metric

*g*=

_{mn}*e*η

_{m}^{i}e_{n}^{j}_{ij}via the

*zweibeins e*So, this motivate us to introduce the three-index curved metric

_{m}^{i}.It seems very interesting to try to develop a gravitational theory based in *g _{mn}*

_{λ}, for at least two reasons. First, the η

*contains the four basic matrices (7), which we proved are linked to matroid theory. Therefore this establishes a bridge between matroids and*

_{ijk}*g*

_{mn}_{λ}. Thus, a gravitational theory based in

*g*

_{mn}_{λ}may provide an alternative gravitoid theory (see Ref. [4]). Secondly, since the matrices (7) are also linked to Clifford algebras, such a gravitational theory may determine spin structures, which are necessary for supersymmetric scenarios. These and another related developments will be reported elsewhere [33].

**Acknowledgments**

J. A. Nieto would like to thank to M. C. Marín for helpful comments. This work was partially supported by PROFAPI 2007 and PIFI 3.3.

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(Received on 14 October, 2009)

* Electronic address: nieto@uas.uasnet.mx

† Electronic address: ealeon@posgrado.cifus.uson.mx