Green-Schwarz, Nambu-Goto Actions, and Cayley's Hyperdeterminant

It has been recently shown that Nambu-Goto action can be re-expressed in terms of Cayley's hyperdeterminant with the manifest SL(2,R) X SL(2,R) X SL(2,R) symmetry. In the present paper, we show that the same feature is shared by Green-Schwarz sigma-model for N=2 superstring whose target space-time is D=2+2. When its zweibein field is eliminated from the action, it contains the Nambu-Goto action which is nothing but the square root of Cayley's hyperdeterminant of the pull-back in superspace \sqrt{\hyperdet(\Pi_{i \a\Dot\a})} manifestly invariant under SL(2,R) X SL(2,R) X SL(2,R). The target space-time D=2+2 can accommodate self-dual supersymmetric Yang-Mills theory. Our action has also fermionic kappa-symmetry, satisfying the criterion for its light-cone equivalence to Neveu-Schwarz-Ramond formulation for N=2 superstring.


Introduction
Cayley's hyperdeterminant [1], initially an object of mathematical curiosity, has found its way in many applications to physics [2]. For instance, it has been used in the discussions of quantum information theory [3] [4], and the entropy of the STU black hole [5] [6] in fourdimensional string theory [7].
However, the NSR formulation [16][17] has a drawback for rewriting it purely in terms of a determinant, due to the presence of fermionic superpartners on the 2D world-sheet. On the other hand, it is well known that a GS formulation [12] without explicit world-sheet supersymmetry is classically equivalent to a NSR formulation [11] on the light-cone, when the former has fermionic κ -symmetry [20] [15]. From this viewpoint, a GS σ -model formulation in [14] of N = 2 superstring [16][17] [13] seems more advantageous, despite the temporary sacrifice of world-sheet supersymmetry. However, even the GS formulation [14] itself has an obstruction, because obviously the kinetic term in the GS action is not of the NG-type equivalent to a Cayley's hyperdeterminant.
In this paper, we overcome this obstruction, by eliminating the zweibein (or 2D metric) via its field equation which is not algebraic. Despite the non-algebraic field equation, such an elimination is possible, just as a NG action [9][10] is obtained from a Polyakov action [21].
Similar formulations are known to be possible for Type I, heterotic, or Type II superstring theories, but here we need to deal with N = 2 superstring [16] with the target space-time 3) The N = 2 here implies the number of world-sheet supersymmetries in the Neveu-Schwarz-Ramond (NSR) formulation [11]. Its corresponding Green-Schwarz (GS) formulation [12][13] [14] might be also called 'N = 2' GS superstring in the present paper. Needless to say, the number of world-sheet supersymmetries should not be confused with that of space-time supersymmetries, such as N = 1 for Type I superstring, or N = 2 for Type IIA or IIB superstring [15]. D = 2 + 2 instead of 10D. We show that the same global [SL(2, IR)] 3 symmetry [8] is inherent also in N = 2 GS action in [14] with N = (1, 1) supersymmetry in D = 2 + 2 as the special case of [13], when the zweibein field is eliminated from the original action, reexpressed in terms of NG-type determinant form.
As is widely recognized, the quantum-level equivalence of NG action [9][10] to Polyakov action [21] has not been well established even nowadays [22]. As such, we do not claim the quantum equivalence of our formulation to the conventional N = 2 NSR superstring [16] [17] or even to N = 2 GS string [13] itself. In this paper, we point out only the existence of fermionic κ -symmetry and the manifest global [SL(2, IR)] 3 symmetry with Cayley's hyperdeterminant as classical-level symmetries, after the elimination of 2D metric from the classical GS action [14] of N = 2 superstring [16] [17].
Then it is natural to 'supersymmetrize' this conjecture [24], such that all the supersymmetric integrable models in D ≤ 3 are generated by SDSYM in D = 2 + 2 [18] [19], and thereby the importance of N = 2 GS σ -model in [14] is also re-emphasized.
A is the superspace pull-back, Γ ij is a product of such pull-backs:  [25] for D = (2, 2; 2, 2) target superspace [19] [14]. Its explicit form is We use the underlined Greek indices: α ≡ (α, . for the pair of fermionic indices, where α, β, ··· = 1, 2 are for chiral coordinates, and . α, 2 are for anti-chiral coordinates [19]. The indices µ, ν, ··· = 1, 2, 3, 4 are for curved fermionic coordinates. Similarly to the superspace for the Minkowski space-time with the signature (+, −, −, −) [25], a bosonic index is equivalent to a pair of fermionic indices, e.g., [19]. Relevantly, the only non-vanishing supertorsion components are [19][14] (2.5) The antisymmetric tensor superfield B AB has the superfield strength (2.7) In our formulation, the lagrangian (2.1a) needs the 'square root' of the matrix Γ ij , analogous to the zweibein e i (j) as the 'square root' of the 2D metric g ij , defined by In terms of light-cone coordinates, this implies formally the Virasoro conditions [27] The only caveat here is that our γ i (j) is not exactly the zweibein e i (j) , but it differs only by certain factor, as we will see in (4.6).
The result (2.10) is not against the original results in NG formulation [9][10]. At first glance, since the NG action has no metric, it seems that Virasoro condition [27] will not follow, unless a 2D metric is introduced as in Polyakov formulation [21]. However, it has been explicitly shown that the Virasoro conditions follow as first-order constraints, when canonical quantization is performed [10]. Naturally, this quantum-level result is already reflected at the classical level, i.e., the Virasoro condition (2.10) follows, when the ij indices on Γ ij ≡ Π i a Π ja are converted into 'local Lorentz indices' by using the γ's in (2.8).
Most importantly, Det (Π iα . α ) in (2.1b) is a Cayley's hyperdeterminant [1][8], related to the ordinary determinant in (2.1a) by The In a sense, this invariance is trivial, because SL(2, IR) ⊂ GL(2, IR), where the latter is the 2D general covariance group. The latter is implied by the definition of Γ ij ≡ Π i a Π ja and γ (i) j in (2.8). Eventually, we have δ p Π (i) A = 0, while L WZNW is also invariant, thanks to δ p Π (i) A = 0. This concludes The second and third factor groups in SL(2, IR) × SL(2, IR) × SL(2, IR) act on the fermionic coordinates α and . α in D = (2, 2; 2, 2), which need an additional care. We first need the alternative expression of L WZNW by the use of Vainberg construction [28] [29]: We need this alternative expression, because superfield strength G ABC is less ambiguous than its potential superfield B AB avoiding the subtlety with the indices α and .
It was pointed out in ref. [8] that 'hidden' discrete symmetry also exists in NG-action under the interchange of the three indices for [SL(2, IR)] 3 . In our system, however, this hidden triality seems absent. This can be seen in (2.1b), where the Cayley's hyperdeterminant or L NG indeed possesses the discrete symmetry for the three indices i α .
α, while it is lost in L WZNW . This is because the mixture of Π iα . α and Π i α or Π i . α via the non-zero components of B AB breaks the exchange symmetry among i α . α, unlike Cayley's hyperdeterminant.

Fermionic Invariance of our Action
We now discuss our fermionic κ -invariance. Our action (2.1) is invariant under The κ − α is the parameter for our fermionic symmetry transformation, just as in the conventional Green-Schwarz superstring [12] [20]. Since Z M is the only fundamental field in our formulation, (3.1c) is the necessary condition of (3.1a) and (3.1b).
We can confirm δ κ I = 0 easily, once we know the intermediate results: By using the relationships, such as Thus the fermionic κ -invariance δ κ I = 0 works also in our formulation, despite the absence of the 2D metric or zweibein. The existence of fermionic κ -symmetry also guarantees the light-cone equivalence of our system to the conventional N = 2 GS superstring [14].

Derivation of Lagrangian and Fermionic Symmetry
In this section, we start with the conventional GS σ -model action [14] for N = 2 superstring [16] [17], and derive our lagrangian (2.1) with the fermionic transformation rule (3.1).
This procedure provides an additional justification for our formulation.
The N = 2 GS action I GS ≡ d 2 σ L GS [14] which is light-cone equivalent to N = 2 NSR superstring [16][17] has the lagrangian where λ has only the negative component: Only in this section, the local Lorentz indices are related to curved ones through the zweibein as in Π (i) instead of γ i (j) in the last section. In the routine confirmation of δ λ L GS = 0, we see its parallel structures to δ κ L = 0.
We next derive our lagrangians L NG and L WZNW from L GS in (4.1). To this end, we first get the 2D metric field equation from I GS 7) As is well-known in string σ -models, this field equation is not algebraic for g ij , because the r.h.s. of (4.3) again contains g ij via the factor Ω. Nevertheless, we can formally delete the 6) We use the parameter λ instead of κ due to a slight difference of λ from our κ (Cf. eq. (4.8)).

7)
We use the symbol .
= for a field equation to be distinguished from an algebraic one.
metric from the original lagrangian, using a procedure similar to getting NG string [9][10] from Polyakov string [21], or NG action out of Type II superstring action [12], as Thus the metric disappears completely from the resulting lagrangian, leaving only √ −Γ which is nothing but L NG in (2.1). As for L WZNW , since this term is metric-independent, this is exactly the same as the second term of (4.1).
We now derive our fermionic transformation rule (3.1) from (4.2). For this purpose, we establish the on-shell relationships between e i (j) and our newly-defined γ i (j) . By taking the 'square root' of (4.3a), we get the e i (j) -field equation expressed in terms of the Π's, that we call f i (j) which coincides with e i (j) only on-shell: Note that the f 's is proportional to the γ's by a factor of Ω/2, as understood by the use of (4.3), (4.5) and (2.8): Recall that the factor Ω contains the 2D metric or zweibein which might be problematic in our formulation, while γ i (j) , γ (i) j are expressed only in terms of the Π i A 's. Fortunately, we will see that Ω disappears in the end result.
Our fermionic transformation rule (3.1a) is now obtained from (4.2a), as where λ and κ are proportional to each other by Such a re-scaling is always possible, due to the arbitrariness of the parameter λ or κ.
The complete disappearance of Ω in our transformation rule (3.1) is desirable, because Ω itself contains the metric that is not given in a closed algebraic form in terms of Π i A . If there were Ω involved in our transformation rule (3.1), it would pose a problem due to the metric g ij in Ω. To put it differently, our action (2.1) and its fermionic symmetry (3.1) are expressed only in terms of the fundamental superfield Z M via Π i A with no involvement of g ij , e i (j) or Ω, thus indicating the total consistency of our system. This concludes the justification of our fermionic κ -transformation rule (3.1), based on the N = 2 GS σ -model [14] light-cone equivalent to N = 2 NSR superstring [16][17].

Concluding Remarks
In this paper, we have shown that after the elimination of the 2D metric at the classical level, the NG-action part I NG of GS σ -model action [14] for N = 2 superstring [16] [17] is entirely expressed as the square root of a  [14]. The hidden discrete symmetry pointed out in [8], however, seems absent in N = 2 string [17][19] [14] due to the WZNW-term L WZNW .
We have also shown that our action (2.1) has the classical invariance under our fermionic κ -symmetry (3.1), despite the elimination of zweibein or 2D metric. Compared with the original I GS [14], our action has even simpler structure, because of the absence of the 2D metric or zweibein. Due to its fermionic κ -symmetry, we can also regard that our system is classically equivalent to NSR N = 2 superstring [16] [17], or N = 2 GS superstring [13]. As an important by-product, we have confirmed that the Virasoro condition (2.10) are inherent even in the NG reformulation of N = 2 GS string [14] at the classical level. This is also consistent with the original result that Virasoro condition is inherent in NG string [9][10].
One of the important aspects is that our action (2.1) and the fermionic transformation rule (3.1) involve neither the 2D metric g ij , the zweibein e i (j) , nor the factor Ω containing these fields. This indicates the total consistency of our formulation, purely in terms of superspace coordinates Z M as the fundamental independent field variables.
In this paper, we have seen that neither the 2D metric g ij nor the zweibein e i (j) , but the superspace pull-back Π iα . α is playing a key role for the manifest symmetry [SL(2, IR)] 3 acting on the three indices iα .
α. In particular, the combination Γ ij ≡ Π i a Π ja plays a role of 'effective metric' on the 2D world-sheet. This suggests that our field variables Z M alone are more suitable for discussing the global [SL(2, IR)] 3 symmetry of N = 2 superstring [16][17] [14].
It seems to be a common feature in supersymmetric theories that certain non-manifest symmetry becomes more manifest only after certain fields are eliminated from an original lagrangian. For example, in N = 1 local supersymmetry in 4D, it is well-known that the σ -model Kähler structure shows up, only after all the auxiliary fields in chiral multiplets are eliminated [31]. This viewpoint justifies to use a NG-formulation with the 2D metric eliminated, instead of the original N = 2 GS formulation [13] We are grateful to W. Siegel and the referee for noticing mistakes in an earlier version of this paper.