1. Introduction

Some years ago, Duff^{1} discovers hidden new symmetries in the Nambu-Goto action^{2}^{-}^{3}. It turns out that the key mathematical tool in such a discovery is the Cayley hyperdeterminant^{4}. In this pioneer work, however, the target space-time turns out to have an associated (2+2)-signature, corresponding to two time and two space dimensions. It was proved in Ref. 5 and 6 that the Duff’s formalism can also be generalized to (4+4)-dimensions and (8+8)-dimensions. Here, we shall prove that if one introduces a Wick rotations for various coordinates then one can actually extend the Duff’s procedure to any signature in 4-dimensions. Moreover, we also prove that our method can be extended to curved space-time in ( 2 2𝑛 + 2 2𝑛 )-dimensions and ( 2 2𝑛+1 + 2 2𝑛+1 )-dimensions.

There are a number of physical reasons to be interested on these developments, but perhaps the most important is that eventually our work may be useful on a possible generalization of the remarkable correspondence between black-holes and quantum information theory (see Refs.^{7}^{-}^{10} and references therein).

2. Mathematical development

Let us start recalling the Duff’s approach on the relation between the Nambu-Goto action and the (2+2)-signature. Consider the Nambu-Goto action^{2}^{,}^{3},

Here, the space-time coordinates 𝑥 𝜇 are real function of two parameters (𝜏,𝜎)= 𝜉 𝑎 and 𝜂 𝜇𝜈 is a flat metric, determining the signature of the target space-time. Moreover, the parameter 𝜖 takes the values +1 or −1, depending whether the signature of 𝜂 𝜇𝜈 is Euclidean or Lorenziana, respectively.

It turns out that by introducing the world-sheet metric 𝑔 𝑎𝑏 one can prove that (1) is equivalent to the action^{11} (see also Ref.^{12} and references therein)

which is, of course, the Polyakov action (see Ref. 12 and references therein). In fact, from the expression

obtained by varying the action (2) with respect to 𝑔 𝑎𝑏 , it is straightforward to show that from (2) one obtains (1) and vise versa. Hence, the actions (1) and (2) are equivalents.

It is convenient to define the induced world-sheet metric

Using this definition, the Nambu-Goto action (1) becomes

It is not difficult to see that in (2+2)-dimensions the expression (4) can be written as

where 𝑥 𝑖𝑗 denotes a the 2×2- matrix

It is important to observe that (7) corresponds to the set 𝑀(2,𝑅) of any 2×2-matrix. In fact, by introducing the fundamental base matrices

one observes that (7) can be rewritten as the linear combination

Let us now introduce the expression

If one uses (4) one gets

However, if one considers (6) one obtains

where 𝒟𝑒𝑡( ℎ 𝑎𝑏 ) denotes the Cayley hyperdeterminant of ℎ 𝑎𝑏 , namely

Of course, (11) and (12) imply that

In turn, (14) means that in (2+2)-dimensions the Nambu-Goto action (5) can also be written as

Note that, since in this case one is considering the (2+2)-signature one must set 𝜖=+1 in (5).

In (4+4)-dimensions the key formula (6) can be generalized as

While in (8+8)-dimensions one has

(see Refs. 5 and 6 for details). So by considering the real variables 𝑥 𝑖 1 ... 𝑖 𝑛 and properly considering the matrices 𝜀 𝑖𝑗 and 𝜂 𝑖𝑗 the previous formalism can be generalized to higher dimensions. Of course, in such cases the Cayley hyperdeterminant 𝒟𝑒𝑡( ℎ 𝑎𝑏 ) must be modified accordingly.

Observing (7) one wonders whether one can consider in (6) other signatures in 4-dimensions besides the (2+2)-signature. It is not difficult to see that using the Wick rotation in any of the coordinates 𝑥 1 , 𝑥 2 , 𝑥 3 or 𝑥 4 one can modify the signature. For instance, one can achieve the (1+3)-signature if one uses the prescription 𝑥 2 →𝑖 𝑥 2 in (6). This method lead us inevitable to generalize our method to a complex structure. One simple introduce the complex matrix

where the variables 𝑧 1 , 𝑧 2 , 𝑧 3 and 𝑧 4 are complex numbers. The expression (6) is generalized accordingly as^{13}

Thus, in this case, the Cayley hyperdeterminant becomes

and consequently the Nambu-Goto action must be written using (20). Of course, the Nambu-Goto action, or the Polyakov action, must be real and therefore one must choose any of the coordinates 𝑧 1 , 𝑧 2 , 𝑧 3 and 𝑧 4 in (20) either as pure real or pure imaginary.

Similarly, the generalization to a complex structure can be made by introducing the complex variables 𝑧 𝑖 1 ... 𝑖 𝑛 and writing

or

depending whether the signature is ( 2 2𝑛 + 2 2𝑛 ) or ( 2 2𝑛+1 + 2 2𝑛+1 ), respectively.

One can further generalize our procedure by considering a target curved space-time. For this purpose let us introduce the curved space-time metric

Here, 𝑒 𝜇 𝐴 denotes a vielbein field and 𝜂 𝐴𝐵 is a flat metric. The Polyakov action in a curved target space-time becomes

Using (23), one sees that this action can be written as

So, by defining the quantity

the action in (25) reads as

Hence, in a target space-time of (2+2)-dimensions one can write (27) in the form

where

Here, we considered the fact that one can always write

Observe that in this development one can consider a generalization of (4) namely

and therefore in (2+2)-dimensions this expression becomes

while in (4+4)-dimensions and (8+8)-dimensions one obtains

and

respectively.

At this stage, it is evident that if one wants to generalize the procedure to any signature in a curved space-time one simply substitute in the action (27) either

or

depending whether the signature is ( 2 2𝑛 + 2 2𝑛 ) or ( 2 2𝑛+1 + 2 2𝑛+1 ), respectively. Here, we used the prescription 𝐸 𝑎 𝑖 1 ... 𝑖 𝑛 → ℰ 𝑎 𝑖 1 ... 𝑖 𝑛 , with ℰ 𝑎 𝑖 1 ... 𝑖 𝑛 a complex function.

In order to include 𝑝-branes in our formalism, one notes that the expression (35) and (36) can still be used. In such a case, one allows the indice 𝑎 in (35) and (36) to run from 0 to 𝑝. Braking such kind of indices as 𝑎=( 𝑎 1 , 𝑎 2 ) for a 3-brane, as 𝑎=( 𝑎 1 , 𝑎 2 , 𝑎 3 ), for a 5-brane and so on one observes that (35) and (36) can be written as

or

respectively. The analogue of Cayley hyperdeterminant in this case will be

and therefore the corresponding Nambu-Goto action becomes

Conclusions and comments

We have generalized the Duff’s procedure concerning the combination of the Nambu-Goto action and the Cayley hyperdeterminant in target space-time of (2+2)-dimensions. Such a generalization first corresponds to a curved worlds with ( 2 2𝑛 + 2 2?? )-signature or ( 2 2𝑛+1 + 2 2𝑛+1 )-signature. Using complex structure we may be able to extend the procedure to any signature. Further, we generalize the method to 𝑝-branes.

It turns out that these generalization may be useful in a number of physical scenario beyond string theory and 𝑝-branes. In fact, since the quantity 𝑧 𝑗 1 ... 𝑗 𝑛 can be identified with a 𝑛-complex rebit one may be interested in the route leading to oriented matroid theory [14] (see also Ref. 15 and 16). In this direction, using the phirotope concept (see Ref. 17 and references therein), which is a complex generalization of the concept of chirotope in oriented matroid theory, a link between super 𝑝-branes and qubits (in this context) has already been established [17]. Thus, it may be interesting for further developments to explore the connection between the results of the present work and supersymmetry via the Grassmann-Plücker relations (see Refs. 8 and 9 and references therein). It is worth mentioning that such relations are natural mathematical notions in information theory linked to 𝑛-qubit entanglement. Indeed, in such a case, the Hilbert space can be broken in the form 𝐶 2𝑛 = 𝐶 𝐿 ⊗ 𝐶 𝑙 with 𝐿=2𝑛−1 and 𝑙=2. This allows a geometric interpretation in terms of the complex Grassmannian variety 𝐺𝑟(𝐿,𝑙) of 2-planes in 𝐶 2𝑛 via the Plücker embedding. In this context, the Plücker coordinates of Grassmannians 𝐺𝑟(𝐿,𝑙) are natural invariants of the theory (see Ref. 9 for details). However, it has been mentioned in Ref. 18, and proved in Refs. 19 and 20, that for normalized qubits the complex 1-qubit, 2-qubit and the 3-qubit are deeply related to division algebras via the Hopf maps, 𝑆 3 → 𝑆 1 𝑆 2 , 𝑆 7 → 𝑆 3 𝑆 4 and 𝑆 15 → 𝑆 7 𝑆 8 , respectively. In order to clarify the possible application of these observations in the context of our formalism let us consider the general complex state ∣𝜓⟩∈ 𝐶 2𝑛 ,

where | 𝑖 1 𝑖 2 ... 𝑖 𝑛 ⟩=| 𝑖 1 ⟩⊗| 𝑖 2 ⟩⊗...⊗| 𝑖 𝑛 ⟩ correspond to a standard basis of the 𝑛-qubit, and 𝐶 𝑖 1 … 𝑖 𝑛 is a complex quantity which real and imaginary parts can be identified in terms of two rebits ( 𝑎 𝑖 1 … 𝑖 𝑛 and 𝑏 𝑖 1 … 𝑖 𝑛 ) in the form 𝐶 𝑖 1 … 𝑖 𝑛 = 𝑎 𝑖 1 … 𝑖 𝑛 +𝑖 𝑏 𝑖 1 … 𝑖 𝑛 . It is interesting to make the following observations. First, one finds that a 3-rebit and 4-rebit have 8 and 16 real degrees of freedom, respectively. Thus, one learns that the 4-rebit can be associated with the 16 degrees of freedom of a 3-qubit. It turns out that this is the kind of embedding discussed in Ref. 9. Second, one may expect that the quantum development of the Nambu-Goto action in n-dimensions leads to consider the 16-dimensions of target space-time as the maximum dimension required by division algebras via the Hopf map 𝑆 15 → 𝑆 7 𝑆 8 . Finally, the question arises whether in our generalized formalism one may also find hidden symmetries of the Nambu-Goto action in the sense of Ref. 1. In (2+2)-dimensions the hyperdeterminant turns out to be invariant under

Here, the first 𝑆𝐿(2,𝑅) is a global subgroup of the world-sheet diffeomorphism. The second two factors are spacetime Lorentz in (2+2)-dimensions, namely 𝑆𝑝𝑖𝑛(2,2)≅𝑆𝐿(2,𝑅)×𝑆𝐿(2,𝑅). By complexifying the 𝑥 𝜇 one may take different real forms, 𝑆𝑝𝑖𝑛(2,2)≅𝑆𝐿(2,𝑅)×𝑆𝐿(2,𝑅), 𝑆𝑝𝑖𝑛(1,3)≅𝑆𝐿(2,𝐶), 𝑆𝑝𝑖𝑛(4)≅𝑆𝑈(2)×𝑆𝑈(2) to obtain various signatures. However, only in (2+2)-dimensions one has the three factors 𝑆𝐿(2,𝑅) in the same footing and hence additional 𝑆 3 . In the case of (4+4)-dimensions one may consider the chain of maximal embeddings and branches,

However, these subgroups are not full symmetry groups and therefore it is difficult to reveal hidden discrete symmetries of the Nambu-Goto action in this case. In other cases the analysis seems even more difficult, but this motivate us to explore in more detail these developments.