# The AdS 5 S 5 mirror model as a string

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2 DOUBLE WICK ROTATIONS OF LIGHT CONE GAUGE FIXED STRINGS We want to understand whether the double Wick rotated light cone worldsheet theory of a string on a given background can be realized by light cone gauge fixing a string on another background. At the bosonic level this works quite elegantly. In this letter we will consider d- dimensional backgrounds with coordinates {t, φ, x µ } and metric ds g mn dx m dx n = g tt dt + g φφ dφ + g µν dx µ dx ν, the metric components depending only on the transverse coordinates x α, and B fields that are nonzero only in the transverse directions. The string action is given by S = T dτdσ (g mn dx m dx n + B mn dx m dx n ), where T is the string tension. To fix a light cone gauge we introduce light cone coordinates [0] x + = t, x = φ t, and then fix a uniform light cone gauge (in the first order formalism) x + = τ, p + = 1. The bosonic action for a string on this background then takes the form ( S = T dτdσ 1 Y + B µν ẋ µ x ν), where Y = (ẋ µ x µ ) (ẋ µ ẋ µ g tt )(x νx ν + 1/g φφ ), and dots and primes refer to temporal and spatial derivatives on the worldsheet, having rescaled σ by T. We now observe that a double Wick rotation of the worldsheet coordinates τ i σ, σ i τ. gives an action of the same form, with g tt interchanged for 1/g φφ and B for B. This means that we can obtain the double Wick rotated worldsheet theory also directly by gauge fixing a string on a background with a metric with g tt and 1/g φφ interchanged and a B field with opposite sign. Note that we generically denote quantities in the double Wick rotated theory (the mirror theory) by tildes. Taking this construction and feeding it the metric of AdS 5 S 5 in global coordinates [1] ds = (1 + ρ )dt + dρ 1 + ρ + ρ dω 3 + (1 r )dφ + dr 1 r + r dω 3, we directly obtain ds = 1 1 r ( dt + dr ) + r dω 3 (1) ρ ( dφ + dρ ) + ρ dω 3, the metric that would result in the (bosonic) mirror theory. The transverse directions should not be affected by this transformation, but for the light cone directions this is more subtle and the φ direction need not be (taken) compact anymore. For the mirror version of AdS 5 S 5 we will take it noncompact in fact, upon considering the relation of our mirror space to the spaces appearing in the deformed sigma models of [17]. STRINGS ON (AdS 5 S 5 ) η The family of deformed sigma models of [17] can be labeled by a parameter κ [0, ), the undeformed AdS 5 S 5 string sigma model sitting at κ = 0. The corresponding metric is given by [] ds = f +(ρ) 1 f (κρ) dt + f + (ρ)f (κρ) dρ + ρ dθ ρ 3 + f (r) 1 f + (κr) dφ + f (r)f + (κr) dr + r dθ r 3, where f ± (x) = 1 ± x and dθ 3 is a particular κ- dependent deformation of the three-sphere metric in Hopf coordinates dθ ρ κ ρ 4 sin ζ (dζ + cos ζdψ1) + sin ζdψ, dθ r κ r 4 sin ξ (dξ + cos ξdχ 1) + sin ξdχ. The B field is given by ( ρ 4 sin ζ B = κ 1 + κ ρ 4 sin ζ dψ 1 dζ r 4 sin ξ ) 1 + κ r 4 sin ξ dχ 1 dξ, vanishing at κ = 0. parametrized as The tension can be conveniently T = g 1 + κ. The range of ρ is restricted to [0, 1/κ) to preserve the timelike nature of t, with a curvature singularity at ρ = 1/κ. At κ = 0 there is no singularity but rather the conformal boundary of anti-de Sitter space at ρ =. Now let us introduce rescaled coordinates t = κt, r = κρ, φ = κφ, ρ = κr,

3 3 and relabeled coordinates If we then also introduce ξ = ζ, ζ = ξ, χ i = ψ i, ψi = χ i. κ = 1/κ, the metric ds = κ ds becomes ds = f +( κ r) f ( r) d t 1 + f + ( κ r)f ( r) d r + r dθ r 3 + f ( κ ρ) f + ( ρ) d φ 1 + f ( κ ρ)f + ( ρ) d ρ + ρ dθ ρ 3, where now the dθ 3 factors contain tildes on κ and the angular variables. Up to the tildes and the factor of κ this is nothing but the deformed metric we started with, with g tt and 1/g φφ interchanged. Similarly, the B field precisely picks up a sign in addition to tildes and a factor of κ. This overall factor can now be naturally absorbed in the string tension. Identifying quantities with and without tildes puts us squarely in the situation of the previous section, proving that the deformed bosonic light cone theory at tension T (g) and deformation value κ is equal to the double Wick rotated theory at tension T ( κg) and deformation value κ. This is precisely the statement of mirror duality observed in [19], which we have now proven at the bosonic level. Just as the undeformed limit κ 0 gives the metric of AdS 5 S 5, the maximally deformed limit κ 0 gives precisely the mirror metric (1). From this perspective the mirror φ coordinate is naturally noncompact. Let us now discuss the mirror space in more detail. THE AdS 5 S 5 MIRROR BACKGROUND We obtained the mirror space (1) directly from AdS 5 S 5, but interestingly it is also closely related to ds 5 H 5. Namely, (by construction) our mirror space turns into ds 5 H 5 upon applying a timelike T-duality in t and a noncompact T-duality in φ. Indeed after a timelike T-duality the first line of eqn. (1) becomes (1 r )dt + dr 1 r + r dω 3, which is ds 5 in static coordinates. Similarly, T-duality in φ turns the second line into (1 + ρ )dφ + dρ 1 + ρ + ρ dω 3, which is H 5 in analogous coordinates. The timelike respectively noncompact nature of these T-dualities makes this a rather formal relation however. So while our original string lived on the maximally symmetric space AdS 5 S 5, its mirrored version does not quite live on the maximally symmetric space ds 5 H 5, but rather its doubly T-dual cousin (1). At this stage we should mention that based on simpler models [3] it was conjectured that the maximal deformation limit of the models of the previous section should correspond to ds 5 H 5 [17]. Indeed, this space was already extracted in a different κ limit combined with two spacelike T-dualities [4]. This limit requires taking one of the coordinates outside its natural range however, thereby changing the coordinate that is timelike, and as such does not appear to be smoothly connected to the general deformed geometry. Like ds 5 H 5, the resulting geometry is formally supported by an imaginary five-form flux. Our limit instead yields a geometry that is smoothly related to the generic case, and naturally results in a real worldsheet theory. We consider our doubly T-dual relation to ds 5 H 5, albeit part timelike and noncompact, close enough to the conjecture of [17]. Our mirror space is clearly related to the one obtained in [4] by one time and three spacelike T-dualities. Our mirror space is a product of two five-dimensional spaces, with curvature R = (4 1 r 1 r ) ( ) ρ 1 + ρ, showing a (naked) singularity at r = 1. This means that in strong contrast to AdS 5 S 5, the mirror space is singular. Sigma models on singular backgrounds are not necessarily ill-defined however, and the integrability of our string sigma model is actually a promising indication of good behaviour. A more pressing question is that of conformality of the sigma model at the quantum level. At one loop this is equivalent to the statement that our metric is part of a solution of supergravity, which we will now demonstrate. As our metric is T-dual to the metric of ds 5 H 5, it is natural to assume that the (type IIB) supergravity solution is supported by only a dilaton Φ and a self-dual five-form flux F, just as ds 5 H 5 formally is [5]. We then have to solve the equations of motion for the dilaton the metric 4 Φ 4( Φ) = R, R µν = µ ν Φ ! eφ F µρλσδ F ρλσδ ν, and the five-form ν ( gf ρλσδν ) = 0. The equation for the dilaton is solved by Φ = Φ 0 1 log(1 r )(1 + ρ ),

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