Semiclassical spectrum for BMN string in $Sch_5\times S^5$

We investigate the algebraic curve for string in $Sch_5\times S^5$. We compute the semiclassical spectrum for BMN string in $Sch_5\times S^5$ from the algebraic curve. We compare our results with the anomalous dimensions in $sl(2)$ sector of the null dipole deformation of $\mathcal{N} = 4$ super Yang-Mills theory.


Introduction
Spectrum of superstrings in AdS 5 × S 5 is related to the spectrum of scaling dimensions in planar N = 4 supersymmetric Yang-Mills theory via the AdS/CFT duality [1][2][3]. Integrability on both sides of the duality helps us dramatically finding and understanding the AdS/CFT spectrum (For a big review, see [4]). In the N = 4 supersymmetric Yang-Mills theory, the planar anomalous dimension matrix of infinitely long composite operators corresponds to Hamiltonian of integrable spin chain [5]. This implies that the spectrum can be solved efficiently by the Bethe ansatz.
On the string side, classical integrability of superstrings in AdS 5 × S 5 follows from the existence of an infinite number of conserved charges [6] generated by the monodromy matrix of the Lax connection. Algebraic curve for classical solution of superstring in AdS 5 × S 5 [7][8][9][10][11] can be obtained from the Lax connection. It plays an important role in studying the semiclassical strings in AdS 5 × S 5 .
In recent years, much attention has been enjoyed by the integrable deformations of AdS/CFT. One intriguing example is the Schrödinger spacetime [12][13][14]. Schrödinger spacetime can be obtained from AdS background by an appropriate TsT (T-duality-shift-T-duality) transformation [15] or null Melvin twist and has been shown to be classically integrable [16][17][18]. String theories in Schrödinger spacetime is dual to null dipole deformed field theories [19] (see also [20][21][22]). It is interesting to study the spectrum on both sides of the Schrödinger/dipole CFT duality with the methods of integrability.
Integrability in null dipole deformed N = 4 super Yang-Mills was discussed in detail in [23]. The dipole deformation can be described as a Jordan cell Drinfeld-Reshetikhin twist [24,25] in the spin chain picture. The traditional Bethe ansatz is inapplicable due to the absence of a vacuum state. Oneloop spectrum of the nontrivial twisted sl(2) sector was instead obtained from the Baxter equation. In the large J limit, the anomalous dimension of the ground state perfectly matches the classical energy of the BMN string at order J −1 .
The purpose of this paper is to study the Schrödinger/dipole CFT duality by comparing semiclassical spectrum around classical string solutions to anomalous dimension of operators in the sl(2) sector at order J −2 in the large J limit. One reason to study the order J −2 terms is that in the well studied AdS 5 /CFT 4 correspondence, the gauge theory and string results match at order J −2 in the BMN limit [26]. One can expect that the null dipole deformation preserves this matching. Another reason is that at order J −2 we should consider one-loop quantum string theory corrections to the string energy, while the previous test at order J −1 involve purely bosonic classical string energies. We compute fluctuation energies of the excitations and the one-loop shift of the ground sate energy from algebraic curve. We show that semiclassical spectrum around the BMN string solution perfectly matches the spin chain prediction. This paper is organized as follows. In section 2 we discuss the Sch 5 × S 5 background and TsT transformation in detail. We discuss the algebraic curve for strings in this background and obtain the quasi-momenta for the BMN string. In section 3, we review the algebraic curve method for computing the fluctuation energies around classical string solutions. Then we compute the semiclassical spectrum for the BMN strings. In section 4, we compare string theory results obtained in section 3 with the 1-loop spectrum in the sl(2) sector of the null dipole deformation of N = 4 super Yang-Mills theory.
2 Algebraic curve for strings in Sch 5 × S 5 2.1 Sch 5 × S 5 from TsT transformation Schrödinger spacetime can be constructed by applying a TsT transformations to the AdS background [12][13][14]19]. In this paper we are interested in a particular case of Sch 5 × S 5 obtained by acting a TsT transformation on AdS 5 × S 5 1 . We begin with the AdS 5 × S 5 solution of type IIB supergravity 3) The five-form field strength is given by We perform a TsT transformation to this geometry. We make a first T-duality along ψ, followed a shift x − → x − − µψ, and then apply a second T-duality along ψ coordinate. After this TsT transformation, the solution reads The five-form F 5 is invariant under the transformation. The TsT translation preserves the symmetries that commute withĴ andP − . The symmetries of SO(4, 2) that commute withP − generate the Schrödinger group. The Schrödinger group contains Galilean group as a subgroup and has two more generators corresponding to a non-relativistic scale transformationD +M +− and a special conformal transformationK − . The energy of the string is defined as the global charge associated with the symmetry (P + +K − )/ √ 2 which is related to the non-relativistic scale transformationD +M +− by a similarity transformation. Holography enable one to compute the non-relativistic conformal dimensions of operators at strong coupling as the energies of strings.
The relations between the original and dual coordinates are Here we make a slight abuse of notation that we use the same symbols for forms on the target space and their pull-back to the worldsheet. We consider the closed strings on the deformed background. The dual coordinates satisfy periodic boundary conditions. The original coordinates have the following twisted boundary conditions where L = 2πµ/ √ λ is the deformation parameter in the dual field theory and are global charges associated with symmetriesĴ andP − , and √ λ = R 2 /α ′ is the square root of 't Hooft coupling.
We now construct the Lax connection for IIB superstring in Sch 5 ×S 5 . The type IIB superstring in AdS 5 ×S 5 can be described by the a sigma-model in supercoset space of the super group SU (2, 2|4) over SO(4, 1) × SO(5) [43]. To describe strings in Poincaré coordinates, we choose the coset representative as and Θ represents the fermionic coordinates. We use a matrix representation such that The current associated with g can be defined as where The Lie superalgebra su(2, 2|4) has a Z 4 grading structure associated with a Z 4 automorphism Ω. The automorphism Ω in this matrix representation is defined by We can decompose the current J into where Ω(J (i) ) = i n J . Then the equation of motion of string in AdS 5 × S 5 is equivalent to the conservation of the Noether current Lax connection L(x) for the superstring in AdS 5 × S 5 has been derived in [6] If the current satisfies the equation of motion, the Lax connection is flat Using this flat connection, one can define the monodromy matrix The eigenvalues of T (x) do not depend on τ and generate an infinite number of conserved quantities. The current components J α do not have an explicit dependence on x − and ψ. Then the Lax connection L in the undeformed case can be used to derive a Lax connection and thus quasi-momenta for strings in Sch 5 × S 5 background. The quasi-momenta p i (x) are functions defined from the eigenvalues of the monodromy matrix T (x). They are generating functions of conserved physical quantities. For instance, we can read the conserved global charges from the behavior at large x. Large x asymptotic properties of the quasi-momenta for strings with twisted boundary condition are more complex than those for close strings. Below we analysis the asymptotic behavior of the quasi-momenta. At x → ∞, the expansion of the Lax connection is Expanding the monodromy matrix, we get In our representation, it takes the form so we still have the constraint between length and filling fractions (see [9] [11]).

BMN string
We now exemplify the discussion above with BMN string solution in Sch 5 × S 5 presented in [23] (2.31) Virasoro constraint gives µ 2 m 2 − κ 2 + ω 2 = 0. (2.32) The conserved global charges are where we denote ∆ as the global energy of the string associated with the non-relativistic scale trans-formationD +M +− . Then the classical energy of the BMN string is given by The quasi-momenta of the BMN string arê We find an expected square root cut [csc 2γ, ∞] connectingp 1 andp 4 .

Semi-classical quantization of the BMN string
A powerful method for computing the semiclassical spectrum around string solutions is proposed in [44]. Here we begin with a review of this method for the reader's convenience. The semiclassical spectrum around the BMN string is given by where ∆ cl + ∆ 1−loop is the ground state energy, and δ∆ is the energy of the excitations. To compute δ∆, we add a perturbation δp(x) to p(x) associated with the classical solution. The perturbation δp(x) has a single pole at x n . The position x n is determined by The residues at the poles are where i < j and N ij n is the excitation number for excitation with polarizations (ij) and mode number n and the function α(x) is defined as the residues at x = ±1 are synchronized as δ{p 1 ,p 2 ,p 3 ,p 4 |p 1 ,p 2 ,p 3 ,p 4 } = 1 x ± 1 δ{α ± , α ± , β ± , β ± |α ± , α ± , β ± , β ± } + ...

(3.5)
For each cut C ij connecting p i (x) and p j (x) the perturbation δp(x) satisfies where the superscript ± denotes above and below the cut. The poles result in an energy shift where Ω ij are the off-shell fluctuation energies.
(3.28) Finally using (3.12) and (3.15) we find all the off-shell fluctuation energies (3.29) We now solve the pole position x n . We choose the solution |x n | > 1 for small LM to be physical poles. We have (3.30) The exact expressions of xîĵ n are very complex, so we only consider the leading order terms in the large J expansion. When LM = 0, a finite number of x1 j n (x i4 n ) will enter the cut connectingp 1 andp 4 and become x4 j n (x i1 n ). Pluging x n into the off-shell fluctuation energies, in the large J limit we get the on-shell fluctuation frequencies Ω23(x23 n ) = λ(πn 2 − LM n) (3.31) Then we obtained the energy shift δ∆ given by (3.7).

One-loop shift
The one loop shift is equal to one half of the graded sum of all fluctuation mode frequencies. Using zeta function regularization, we have n∈Z (n + q) 2 + pn = q 2 + ζ(−2, 1 + q) + ζ(−2, 1 − q) = 0. (3.32) Therefore when we compute the one loop shift energy at order J −2 , only the contribution from Ω14 is nontrivial. Then we sum over the energies of the sl(2) modes to get the one-loop shift: (3.33)

Comparison between the string and the gauge theory results
Comparison between the results obtained in the gauge theory and string theory is possible in the large spin regime with J → ∞ and λ/J 2 kept fixed and small (see e.g. [26,[46][47][48]). Type IIB superstring in Sch 5 × S 5 is dual to null dipole deformed N = 4 super Yang-Mills. The sl(2) sector nontrivially affected by the deformation has been studied in [23]. The one-loop spectrum of the sl(2) sector can be obtained by Baxter equation. It is proposed in [23] that the Baxter equation takes the same form as in the undeformed case and The 1-loop energy is given by We now solve the Baxter equation in the expansion in LM . At each order in LM , the Q-function is simply a polynomial. We write the ansatz where p k+m is a polynomial in u of degree k + m. The small LM expansion of Q can be interpreted as a function with a finite number of zeros near the Bethe roots in the undeformed limit and an infinite number of zeros of order L −1 M −1 . In the string picture the zeros of order L −1 M −1 correspond to the cut connectingp 1 andp 4 . Substituting the above ansatz into the Baxter equation, we can determine Q(u) up to multiplication by a function in LM . We consider the following two solutions: Q 1 = (u − u n ) + LM −2Ju 2 + 2Ju 2 n + 4u 2 n + 1 2(J + 2) In the undeformed case, Q 0 and Q 1 correspond to the ground state and one particle state respectively.
To compare the spectral curve result with the spin chain result, we expand Ω14(x14 n ) for small LM and obtain the energy shift of the sl (2)  (4.14) The order J −1 term matches the classical quantity ∆ cl − J, and the order J −2 terms perfectly match the one-loop shift ∆ 1−loop given in (3.33).

Conclusion and discussion
In this paper we study the algebraic curve for superstring in Sch 5 × S 5 and its application to the spectral problem. The asymptotic properties of the quasi-momenta for strings in Sch 5 × S 5 are nontrivial. The point at infinity is a branch point of a cut connecting two Riemann sheets. We compute the semiclassical spectrum of the BMN string. Remarkably, we show that in the large J limit the string results match the gauge field results obtained by Baxter equation. We provide a detailed test of the Schrödinger/dipole CFT duality. Our results encourage further exploration of integrability in Schrödinger/dipole CFT duality. It would be nice to derive the full quantum spectral curve of null dipole deformed N = 4 super Yang-Mills theory, because the quasi-momenta are related to the quantum spectral curve in the strong coupling limit. It is also worth trying to obtain higher-order corrections on the field theory side to get a precise match with string theory predictions. One can also study the three dimensional counterpart of Sch 5 , the warped AdS 3 . We hope that integrability would be a powerful tool for the spectral problem of warped AdS 3 /dipole CFT duality [49].