PACS: 04.60.-m; 04.65.+e; 11.15.-q; 11.30.Ly

Our main goal in this work is to establish links between the (1+1)-matrix-brane system ¹ (see also Ref. 2) with qubit theory (see Ref. 3 and references therein) and non-compact Hopf maps ⁴. In such connections the (2+2)-signature plays a key role. It turns out that through the years the (2+2)-signature has become a very important notion in different scenarios of mathematics ^5-6 (see also Refs. 7 and 8) and physics. In fact, the (2+2)-signature emerges in several physical context, including self-dual gravity a la Plebanski (see Ref. 9 and references therein), consistent N=2 superstring theory ¹⁰-¹¹, N = (2,1)heterotic string ^12,13,14,15. Furthermore, it has been emphasized ^16-17 that Majorana-Weyl spinors exist in a spacetime of (2+2)-signature. Using the requirement of the SL(2,R)-group and Lorentz symmetries it has been proved ¹⁸ that (2+2)-target spacetime of a 0-brane is an exceptional signature. Moreover, the (2+2)-signature arises following an alternative idea to the notion of ‘worldsheets for worldsheets’ ¹⁹ or the 0-branes condensation ²⁰. Another recent motivation for a physical interest in the (2+2)-signature has emerged via the discovery ²¹ of hidden symmetries of the Nambu-Goto action. In fact, in this case the Nambu-Goto action in a (2+2)-target spacetime can be written in terms of a hyperdeterminant, revealing apparently new hidden symmetries of such an action.

In this work, we shall explore possible links between the (1+1)-matrix-brane system with qubit theory and non-compact Hopf maps. We start by reviewing the relation between Refs. 1 and 21, concerning the (2+2)-signature. For this purpose we shall consider both the (1+1)-matrix-brane theory and the hyperdeterminant structure approach, focusing on the (2+2)-signature.

First, consider the line element

Here, we shall assume that the indices μ,ν∈{1,2,3,4} and that the flat metric ημν=diag(1,1,-1,-1) determines the (2+2)-signature. Introducing the matrix

one finds that (1) can be written as

where a,b,m,n∈{1,2}. Moreover, by defining the alternative matrix

it is not difficult to show that (1) can also be written as

Here, ηab=diag(1,-1) and ηmn=diag(1,-1). This proves that the three line elements (1), (3) and (5) are equivalents. Thus, one can say that these equivalences provide an interesting connection between the signatures (1+1) and (2+2).

It turns out that such equivalences at the level of the line elements (1), (3) and (5) can be transferred to the matrix

In fact, (6) can be written in the following two equivalent forms

and

So, by introducing the quantity

it is straightforward to verify that

can be written as

Here, the notation Det (h _ab) means hyperdeterminant. Thus, one finds that the Nambu-Goto action

can also be written as

Actually, one has

Similarly, introducing the quantity

it is also straightforward to see that det(h _ab), given in (10), can also be written as

This means that the Nambu-Goto action (12) is also equivalent to

This shows that in (2+2)-dimensions S, S and S are equivalent actions. The interesting thing is that S reveals new hidden symmetries in the original Nambu-Goto action S²¹. Presumably, the same conclusion can be said in the case of the action S. (Details of the connections between the actions S, S and S can be found in Ref. 1.)

In order to related the previous discussion with qubit theory and Hopf maps it is convenient to introduce the mathematical notion of 2 by 2 real matrices M (2, R). It turns out that through the years the importance of M (2,R) has emerged in different scenarios of physics and mathematics, including Clifford algebras ^22-23, matroid theory ^24-25 (see also Refs. 26 to 32 and references therein), string theory ³³, 2d gravity ³⁴, 2t physics ³⁵, qubit theory (see Refs. 3 and references therein) among others. We argue that these connections may suggest that one may even consider the set M (2, R) as one of the underlaying structures of supersymmetry and M-theory ³⁶. This last observation is due in part to the fact that M (2, R) is linked to a 2-rank self-dual oriented matroid and to the fact that in both oriented matroid theory and M-theory the duality concept plays a fundamental role. Indeed, it has been proposed ²⁷ that oriented matroid theory may be considered as the underlying mathematical framework for M-theory.

Let us briefly recall some aspects of M (2, R). It is not difficult to see that any matrix

M=abcd

in M (2, R) (with diag(a,d) and antidiag(b,c)) can be written as

Here, δij=diag(1,1), εij=antidiag(1,-1), ηij=diag(1,-1) and λij=antidiag(1,1) are fundamental 2 by 2 matrices and the quantities x, y, r and s are related to the real quantities a, b, c and d by

Our first observation is that a complex number

where x and y are real numbers and i2=-1 , can also be written as ^37-38

In this case, the product of two complex numbers corresponds to the usual matrix dot product. According to (18), this is equivalent to set r = 0 and s = 0. So, from these simply observations one may conclude that the complex structure is contained in M (2,R). If instead, one sets y = 0 and r = 0 (or s = 0) in (18) one arrives to the so called split numbers ³⁹ (or semicomplex numbers (among other alternative names)). These kind of numbers shall play and important role below. But before, we use split algebra let just mention the following. Traditionally, one can not set x = 0 and y = 0 because the dot product of  ηik and λlj is not closed. In fact, one has

where the quantity δkl plays the role of the dot product. However, one may introduce a new kind of product (and therefore new kind of numbers which we shall call “niet” (from dutch word meaning “no”)) complexif instead of δkl one uses ηkl in such a way that the product combination of ηik and λlj is again closed. In fact, in this case one has

Denoting the matrices product with ηkl as a star ⋆ one sees that (23) becomes

Thus, one can show that all possible combinations of δ and ɛ with the dot product are equivalent to all possible combinations of ƞ and λ with the star product. Therefore, through the prescription

one discovers that the niet-complex algebra is isomorphic to the complex structure (see Ref. ⁴⁰ for more details). On the other hand the split number differ from the complex numbers in a number of facts. First while in the complex numbers ε2=-1 in the split numbers λ2=1. Furthermore, the fact that in the complex numbers one has zz*=x2+y2, in the case of split numbers one has ww*=x2-y2, where z*=x-iy and w*=x-jy, with j=λ. Of course, according to the Hurwitz theorem the split number structure does not form a division algebra. One can see this by assuming y=x and noting that in this case ww*=0. So, split numbers with y=x does not have inverse.

It is worth mentioning the following observations. It is known that the fundamental matrices given in (18) 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 in (18) and Kronecker products. It turns out that C (0,2) is isomorphic to the quaternionic algebra H. Thus, it is proved that all the other C (a, b)´ s can be constructed from the basic building blocks C (2,0), C (1,1) and C (0,2) (see Ref. ⁴¹ and references therein).

Let us now briefly describe the connection of the coordinates xij and ζam in M (2,R) with qubit theory. Let us first introduce the basis

A general qubit can be written as

For instance, a 3-qubit is expressed by

The central idea is to identify xij and ζam with 2-rebits which are the real version of the corresponding 2-qubits ψj1j2 (see Ref. ³ and references therein).

Let us now come back to consider again the three Nambu-Goto type actions S, S and S, given in (12), (13) and (17), respectively. First of all, the idea is to relate these actions with the Polyakov action

It is well known that this action is equivalent to the Nambu-Goto action (12) (and therefore to the other two actions 𝐒 and S). Let us recall how this is achieved. Making variations of (29) with respect to gab one obtains the expression

which can be used to substitute gab in (29) and in that way one obtains the Nambu-Goto action (12). One sees that in order to related (29) with the hyperdeterminant is enough to consider (7). In fact, in this case (29) becomes

which leads to (13). Of course, this is only true in 2+2-dimensions. Similarly, by writing (29) as

where (8) was used, one obtains (17), after variations of gab.

From the above observations one is tempting to raise the question: what could be the role of the matrices M (2,R) in the structure of M-theory? One knows that the duality concept is an essential aspect in M-theory. Similarly, duality is a central notion in oriented matroid theory. This is one of the reasons that oriented matroid theory has been proposed as the underlying mathematical structure of M-theory ²⁷. In this scenario one observe that M (2,R) describes a self-dual graphic oriented matroid and therefore is in agreement with both M(atroid) theory and M-theory. So, an audacious proposal could be that M (2,R) may be one of the essential building blocks of M-theory. This proposal is reinforce by the fact that M (2,R) is related to qubit theory via (2+2)-dimensions and to supersymmetry via the Clifford algebra.

By further research and in order to related the previous discussion with the qubit theory we shall consider the 2 + 10-dimensional spacetime. This signature has emerged as one of the most interesting possibilities for the understanding of both supergravity and super Yang-Mills theory in D = 11. What it is important for us is that the (2+10)-dimensional theory seems to be the natural background for the (2+2)-brane (see Refs. ¹, ¹⁰, ¹¹ and references therein). Thus, let us think in the possible transition

which, in principle, can be achieved by some kind of symmetry breaking applied to the full metric of the spacetime manifold M(2+10). It has been shown that the symmetry S L (2,R) makes the (2+2)-signature an exceptional one ¹⁸. On the other hand, the (0+8)-signature is Euclidean and in principle can be treated with the traditional methods such as the octonion algebraic approach. In pass, it is interesting to observe that octonion algebra is also exceptional in the sense of the celebrated Hurwitz theorem. Thus, one can say that both (2+2) and (0+8) are exceptional signatures. This means that the transition (33) is physically interesting.

Consider now the action of the (1+1)-matrix-brane in (2+10)-dimensional target spacetime background ¹,

where ην^σ^ is a flat metric and the indices ν^,σ^ now run from 1 to 12. Splitting the flat metric ην^σ^ according to the transition (2+10) → (2+2)+(0+8) one finds that (34) can be written as

where

and

Here, the indices A, B, run from 1 to 4 and A^,B^ run from 5 to 12. Using the change xA→xpq one can write S₁ in the form

while if one uses the change xA→ζpq one has

Now, both metrics gab and γmn ‘live’ in (1+1)-dimensions. So, according to our previous discussion (38) and (39) can be expressed in terms of the fundamental matrices δij,εij,λij and ηij which are elements of the basis of M (2,R). This means that we have proved that (2+2)-dimensional sector of M(2+10) can be connected with qubit theory via the elementary basis matrices δij,εij,λij and ηij.

It is interesting to observe that both actions S₁ and S1, given in (38) and (39) respectively, are double Weyl invariant in the sense that are invariants with respect to the transformations gab→efgab and mn→ehγmn, for arbitrary functions f and h. This is quite interesting because as it is known the Weyl invariance of the Polyakov action is linked to the critical dimensions of the target spacetime determined by the metric ηAB. If one adds to this observation the fact that the flat target metric ηAB in the action (36) is written in terms of either εckεdl or ηikηjl (according to (38) and (39), respectively), which are the qubit inspired metrics, one is tempted to conjecture a link between the critical dimensions and qubit theory in the (1+1)-matrix-brane theory.

Let us now discuss the (1+1)-matrix-brane theory from the perspective of split algebra. First, observe that if one has two split numbers dω1=dx1+jx3 and d ω2=dx2+jdx4 (remember; j=λ with j2=1) then one gets the invariant

which determines, in a natural way, a (2+2)-signature. This means that the Nambu-Goto action in (2+2)-dimensions can also be written as

or

where we wrote ω in terms of δij and λij. Similarly, in the case of (1+1)-matrix-brane system one must have

One of the reason to become interested in the structure (43) is because recently Hasebe ⁴ has introduced the mathematical concept of non-compact Hopf maps. In fact, in analogy to the Hopf maps (which play a key role in the paralellizabilty of spheres and division algebras ⁴²) S3→S1S2, S7→S3S4 and S15→S7S8 and using the split algebra Hasebe introduced the non-compact Hopf maps H2,1→H1,0H1,1, H4,3→H2,1H2,2 and H8,7→H4,3H4,4. Here, Hp,q denotes higher dimensional hyperboloids

where A = 1,…,p and A^=1,...,q+1. Indeed, in terms of the signature the non-compact Hopf maps may be also written as (2+2)→(1+1)(1+2), (4+4)→(2+2)(2+3) and (8+8)→(4+4)(4,5). So, the (2+2)-signature and the split algebra play a key role in these developments. Moreover, one may expect this approach to be useful in the context of M-theory since it has been shown ⁴³ that versions of M-theory lead to type IIA string theories in spacetime of signatures (0+10), (1+9), (2+8), (6+4) and (5+5), and to type IIB string theories of signatures (1+9), (3+7) and (5 +5). It turns out that these theories are linked by duality transformations. One notices that the (5+5)-signature is common to both type IIA strings and type IIB strings. So, one wonders whether Hasebe formalism and matrix-brane theory may also be related to the (5+5)-signature.

It is worth remarking that the split quaternions can also be related to the (2+2)-signature in a natural way. In fact, one may reveal split quaternionic structure in the action (43) by writing dp=dω1+idω2 and properly using the algebra between i, j, and k = ij, with k² = 1 (see the algebra (4) in Ref. 44). Considering such an algebra it is not difficult to show that dp can also be written as

where dz1=dx1+ix2 and dz2=dx3+idx4 and therefore one gets dpdp*= dz1dz1*-dz2dz2*. Thus, one finds that in this case the action (43) becomes

It turns out that one of the advantage of the formulation (46) is that may shed light on a possible route to supersymmetrize the (1+1)-matrix-brane theory via the proposed 2-spinors over the split quaternions structure ⁴⁴. This is particularly interesting because there exist already a formulation of the Dirac equation in terms of the split-quaternions. Moreover, the usual Dirac 4-spinor is replaced by a 2-spinor with split quaternionic components. In this framework, the SO(3, 2;R) symmetry of the Lorentz invariant scalar ψ ψ is manifest and therefore there exist a finite unitary representations of the Lorentz group over the split-quaternions (see Ref. ⁴⁴ for details).

Finally, since part of the motivation of considering non-compact Hopf maps it emerges from the concept of fuzzy spheres ⁴⁵ it may be interesting for further research to relate the (1+1)-matrix-brane with fuzzy geometry.