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

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

J.A. Nieto^I,*; E.A. León^II,†

^IFacultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, México

^IIDepartameto 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 i² = -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_ji and antisymmetric A_ij = -A_jiparts. In fact, we have

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^l-Γ_ik^l, one finds that the more general solution of (20) is

where Γ_kij = Γ_ki^lg_lj and T_kij = T_ki^lg_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 = ∂_iA_kj the expression (23) can be identified with the connection Γ_kij which presumably describes 2d gravity with torsion. Since A_ij can always be written as (18) we discover that (23) yields

Here, we used the notation ∂_kυ = 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_ij 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_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_ij looks like

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 { V¹,V²}, {V¹,V³}, { V²,V⁴} and { V³,V⁴} 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 { V¹,V²} → δ_ij, { V¹,V³} → η_ij, { V²,V⁴} →λ_ij and {V³,V⁴} → 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 q₁^a≡ q^a and q₂^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_ijΩ^ab+ε_ijη^ab in the form

with g'_ij = g_ijΩ. 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 x¹, x² are the elements of the main diagonal and the "time" coordinates x³, x⁴ 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_klη^ikη^jl and l^ij = l_klη^ikη^jl, where η^ij is the inverse flat 1+1 metric, and has the same components as η_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 x¹ = x, x² = y and y¹ = r, y² = 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_mⁱe_n^jη_ij via the zweibeins e_mⁱ. So, this motivate us to introduce the three-index curved metric

It seems very interesting to try to develop a gravitational theory based in g_mnλ, for at least two reasons. First, the η_ijk contains the four basic matrices (7), which we proved are linked to matroid theory. Therefore this establishes a bridge between matroids and g_mnλ. Thus, a gravitational theory based in g_mnλ 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.

(Received on 14 October, 2009)

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Publication Dates

• Publication in this collection
  03 Jan 2011
• Date of issue
  Dec 2010

History

• Received
  14 Oct 2009