Tensionless strings, correspondence with SO(D,D) sigma model

String theory with perimeter action is tensionless by its geometrical nature and has pure massless spectrum of higher spin gauge particles. I demonstrate that liner transformation of the world-sheet fields defines a map to the SO(D,D) sigma model equipped by additional Abelian constraint, which breaks SO(D,D) to a diagonal SO(1,D-1). The effective tension is equal to the square of the dimensional coupling constant of the perimeter action. This correspondence allows to view the perimeter action as a"square root"of the Nambu-Goto area action. The aforementioned map between tensionless strings and SO(D,D) sigma model allows to introduce the vertex operators in full analogy with the standard string theory and to confirm the form of the vertex operators introduced earlier.

It is generally expected that high energy limit, or what is equivalent the tensionless limit α ′ → ∞, of string theory should have massless spectrum M 2 N = (N − 1)/α ′ → 0 and should recover a genuine symmetries of the theory [1,2,3]. Of course this observation ignores the importance of the high genus G diagrams, the contribution of which A G ≃ exp{−α ′ s/(G + 1)} is exponentially large compared to the tree level diagram [1,2,3]. The ratio of the corresponding scattering amplitudes behaves as A G+1 /A G ≃ exp{α ′ s/G 2 } and makes any perturbative statement unreliable and requires therefore nonperturbative treatment of the problem [4,5,6,7,8] 2 .
The tensionless model with perimeter action suggested in [17,18,19] does not appear as a α ′ → ∞ limit of the standard string theory, as one could probably think, but has a tensionless character by its geometrical nature [17]. Therefore it remains mainly unclear at the moment how these two models are connected. However the perimeter model shares many properties with the area strings in the sense that it has world-sheet conformal invariance, contains the corresponding Virasoro algebra, which is extended by additional Abelian generators. This makes mathematics used in the perimeter model very close to the standard string theory and allows to compute its massless spectrum, critical dimension D c = 13 [18,19] and to construct an appropriate vertex operators [21,22].
Comparing literally the spectrum of these two models one can see that instead of usual exponential growing of states, in the perimeter case we have only linear growing. In this respect the number of states in the perimeter model is much less compared with the standard string theory and is larger compared with the field theory models of the Yang-Mills type. From this point of view it is therefore much closer to the quantum field theory rather than to the standard string theory. At the same time its formulation and the symmetry structure is more string-theoretical. Perhaps there should be strong nonperturbative rearrangement of the spectrum in the limit α ′ → ∞ before the spectrum of the area and the perimeter strings can become close to each other.
Our aim here is to give a partial answer to these questions. As we shell see the liner transformation of the world-sheet fields defines a map to the SO(D,D) σ-model equipped by an additional Abelian constraint, which breaks SO(D,D) to a diagonal SO(1,D-1). The effective string tension is equal to the square of the dimensional coupling constant m of the perimeter action This relation allows to view the perimeter action as a "square root" of the Nambu-Goto area action m = 1/2α ′ . The mass-shell quantization condition of the SO(13, 13) σ- the value of the first Casimir operator K 2 of the Poincaré algebra in 26-dimensions, is translated through the dictionary into the quantization condition for the square W = w 2 D−3 of the Pauli-Lubanski form w D−3 of the Poincaré algebra in 13-dimensions because as we shall see (26) K 2 | in 26−dim. = 2m (k · π)| in 13−dim. The aforementioned correspondence allows to introduce the vertex operators in full analogy with the standard string theory and to confirm the form of the vertex operators introduced earlier in [21,22].
The perimeter string model was suggested in [17] and describes random surfaces embedded in D-dimensional space-time with the following action Laplace operator and m has dimension of mass. There is no Nambu-Goto area term in this action. The action has dimension of length L and the dimensional coupling constant is m. Multiplying and dividing the Lagrangian by the square root (∆(h)X µ ) 2 one can represent it in the σ-model form [18] where the operator Π µ is We shall consider the model B, in which two field variables X µ and h αβ are independent. The classical equation is and world-sheet energy-momentum tensor The operator Π is a space-like vector, The energy momentum tensor is conserved ∇ a T ab = 0 and is traceless h ab T ab = 0, thus we have two-dimensional world-sheet conformal field theory with the central charge c = 2D [18]. We have equation of motion (4) together with the primary constraint equations (5) and (6), the secondary constraints have the form Θ 1,0 = Π∂ + Π, Θ 0,1 = Π∂ − Π, Θ 1,1 = ∂ + Π∂ − Π [18] . The equivalent form of the action (3) is where the Π µ field is now an independent variable and the ω αβ are a Lagrange multipliers. The system of equations which follows fromŚ is equivalent to the original equations (4) and (6) and the corresponding new energy momentum tensorT αβ acquires an additional term which depends only on the field Π where h abT ab = 0. The central charge c = 2D of the Virasoro algebra remains untouched and demonstrates the absence of additional contributions to the central charge due to the primary and secondary constraint (6) (see also [24] for alternative calculation).
Correspondence with the SO(D,D) σ-model . Let us introduce the new variables as follow 1 then the action (3) will take the form If one considers the 2D dimensional target space with the combined coordinates and fully symmetric Lorenzian signature space-time metric with D pluses and D minuses then the action (11) will have formal interpretation in terms of σ-model being defined on a 2D dimensional target space with the symmetry group SO(D, D) From this expression of the action we can deduce that the effective string tension T ef f is equal to the square of the mass m The last relations allow to view the tensionless string theory, which is defined by the perimeter action (3), as a "square root of the Nambu-Goto area action" m = 1/2α ′ . This interpretation has deep geometrical origin because in some sense the perimeter L, which was defined for the two-dimensional surfaces in (3), can be consider as a square root of the surface area. This intuitive interpretation can be made more precise if one recalls Zenodor-Minkowski isoperimetric inequality [25,26], which tells that L 2 ≥ 4πS, with the equality taking place only for a sphere.
The crucial constraint (6) The Abelian constraint (15) can be considered as a "compactification" to a hyperboloid manifold H D . The X µ and Π µ fields (10) are actually light cone coordinates on M D,D . The energy momentum tensor (5) will take the form It is therefore clear that we should have 2D c = 26 and recover the previous result [18] D c = 13.
It is important to get better idea about the algebra (19). The transformation (10) naturally leads to the oscillators and brings the algebra (19) to the form This is a standard algebra of the oscillators with the following signature In terms of the above oscillators the "target space" coordinates (10) Φ M = (Φ µ 1 1 , Φ µ 2 2 ) have the form: The above SO(D,D) σ-model interpretation of the tensionless string theory allows to introduce the vertex operators in the full analogy with the standard string theory case. Indeed the vertex operator for the ground state has the form: where K M = (k µ 1 1 , k µ 2 2 ), Φ M = (Φ µ 1 1 , Φ µ 2 2 ) and has conformal dimension equal to the square of the momentum K M Therefore substituting the expressions for the field Φ M = (Φ µ 1 1 , Φ µ 2 2 ) in terms of the original world-sheet fields X µ and Π µ (10) we shall get where the momenta k and π are: We can now translate the conformal dimension ∆ of the ground state vertex operator V K into the language of our momenta k and π This clearly confirms the form of the vertex operator and its conformal dimension obtained earlier in [21]. Indeed the general form of the vertex operators suggested in [21] is given by the formula the conformal spin should be equal to zero, therefore n 1 + ... + n j =ñ 1 + ... +ñ j = N.
In conclusion I would like to thank Luis Alvarez-Gaume, Ignatios Antoniadis, Ioannis Bakas, Lars Brink and Kumar Narain for stimulating discussions and CERN Theory Division for hospitality.