#### Semiclassical spectrum for BMN string in Sch 5 × S 5

HJE
S5
Hao Ouyang 0 1 2 4 5 6
0 School of Physical Sciences
1 19B Yuquan Road , Beijing 100049 , P. R. China
2 Chinese Academy of Sciences
3 . We compute the
4 Institute of High Energy Physics and Theoretical Physics Center for Science Facilities
5 19A Yuquan Road Beijing 100049 , P. R. China
6 University of Chinese Academy of Sciences
We investigate the algebraic curve for string in Sch5 semiclassical spectrum for BMN string in Sch5 S5 from the algebraic curve. We compare our results with the anomalous dimensions in sl(2) sector of the null dipole deformation of N = 4 super Yang-Mills theory.
Gauge-gravity correspondence; Integrable Field Theories
Semi-classical quantization of the BMN string
Fluctuation energies of excitations
One-loop shift
Comparison between the string and the gauge theory results
Conclusion and discussion
3
4
5
1
2.1
2.2
2.3
3.1
3.2
1 Introduction 2
S5 is related to the spectrum of scaling dimensions
in planar N = 4 supersymmetric Yang-Mills theory via the AdS/CFT duality [1{3].
Integrability on both sides of the duality helps us dramatically
nding 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 in nitely long composite operators
corresponds to Hamiltonian of integrable spin chain [5]. This implies that the spectrum can
be solved e ciently by the Bethe ansatz.
On the string side, classical integrability of superstrings in AdS5
S5 follows from the
existence of an in nite number of conserved charges [6] generated by the monodromy matrix
of the Lax connection. Algebraic curve for classical solution of superstring in AdS5
11] can be obtained from the Lax connection. It plays an important role in studying the
S5 [7{
semiclassical strings in AdS5
S5.
In recent years, much attention has been enjoyed by the integrable deformations of
AdS/CFT. One intriguing example is the Schrodinger spacetime [12{14]. Schrodinger
spacetime can be obtained from AdS background by an appropriate TsT
(T-duality-shiftT-duality) transformation [15] or null Melvin twist and has been shown to be classically
integrable [16{18]. String theories in Schrodinger spacetime is dual to null dipole deformed
eld theories [19] (see also [20{22]). It is interesting to study the spectrum on both sides
of the Schrodinger/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
{ 1 {
twist [24, 25] in the spin chain picture. The traditional Bethe ansatz is inapplicable due to
the absence of a vacuum state. One-loop 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 Schrodinger/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 AdS5/CFT4 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 uctuation
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 Sch5 S5 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 uctuation 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
2.1
Algebraic curve for strings in Sch5
S5
Sch5
S5 from TsT transformation
Schrodinger spacetime can be constructed by applying a TsT transformations to the AdS
background [12{14, 19]. In this paper we are interested in a particular case of Sch5
obtained by acting a TsT transformation on AdS5
solution of type IIB supergravity
S5.1 We begin with the AdS5
The ve-form eld strength is given by
1A more general class of Schrodinger deformations of AdS5
X5 has been studied in [27].
{ 2 {
S5
S5
(2.1)
(2.2)
(2.3)
(2.4)
subgroup and has two more generators corresponding to a non-relativistic scale
transformation D^ + M^ +
and a special conformal transformation K^ . The energy of the string
is de ned as the global charge associated with the symmetry (P^+ + K^ )=p2 which is
related to the non-relativistic scale transformation D^ + 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
d
dx
= d ~ +
= dx~ +
dx+
z2 ;
d ~ +
dx+
z2
+ A :
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
We perform a TsT transformation to this geometry. We make a rst T-duality along ,
followed a shift x
! x
After this TsT transformation, the solution reads
, and then apply a second T-duality along
coordinate.
+ A ) ;
d
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
HJEP12(07)6
x (2 )
x (0) = LJ;
is the deformation parameter in the dual eld theory and
are global charges associated with symmetries J^ and P^ , and p
= R2= 0 is the square
root of 't Hooft coupling.
with
where
with
Integrability of the string sigma model is preserved by TsT transformation, so strings in
Schrodinger spacetime is integrable [16{18] (see also [28{42]).
We now construct the Lax connection for IIB superstring in Sch5
S5. The type
IIB superstring in AdS5
S5 can be described by the a sigma-model in supercoset space
of the super group SU(2; 2j4) over SO(4; 1)
coordinates, we choose the coset representative as
SO(5) [43]. To describe strings in Poincare
g (x ; z; ; i; i; ) = B (x ; z; ; i; i) eF ( );
B(x ; z; ; i; i) = eix+P^++ix P^ +ix1P^1+ix2P^2 e i log(z)D^ ei J^B1( i; i);
and
represents the fermionic coordinates. We use a matrix representation such that
^
P
D^ =
i
2
04 4 CC ;
1
C
C
C
A
We can decompose the current J into
J = J
(0) + J
(
1
) + J
(2) + J
(3);
where
(J (i)) = inJ . Then the equation of motion of string in AdS5
S5 is equivalent to
the conservation of the Noether current
Lax connection L(x) for the superstring in AdS5
S5 has been derived in [6]
If the current satis es the equation of motion, the Lax connection is at
Using this at connection, one can de ne the monodromy matrix
dL
L ^ L = 0:
T (x) = P exp
d L (x) :
Z 2
0
The eigenvalues of T (x) do not depend on
and generate an in nite number of conserved
quantities. The current components J
do not have an explicit dependence on x
and .
and thus quasi-momenta for strings in Sch5
S5 background.
Then the Lax connection L in the undeformed case can be used to derive a Lax connection
The quasi-momenta pi(x) are functions de ned from the eigenvalues
nep^1; ep^2; ep^3; ep^4 ep~1; ep~2; ep~3; ep~4o
j
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 ! 1, the expansion of the Lax connection is
L =
g 1dg
g 1 kg + O
2
x
1
x2 :
Expanding the monodromy matrix, we get ~
T
g( ; 0)T g( ; 0) 1
= g( ; 0)g( ; 2 ) 1 1
= eiL(P J^ JP^ ) 1
0
0
0
p~4
BB p~1 C
BB p^4 CCC = BB
B
B
0
B
B
B
B
B
B
B
B
B
B
q LJP
p x
0
0
O(
1
)
+ S
+ S
O(
1
)
x BB +J1 + J2
B +J1
J2 + J3 CC
J1 + J2 + J3 CA
J1
J2
J3
where Q satis es Q^4^1 =
2 ip2P . Behavior of the quasi-momenta at x ! 1 reads
J3 CCC + O x 2 ;
(2.29)
(2.31)
(2.32)
(2.33)
(2.34)
where S and Ji are spins of the string. The quasi-momenta p^1 and p^4 are connected by a
square root cut which has a branch point at in nity. The quasi-momenta satisfy
I
1
dx p~21 + p~22 + p~23 + p~24
p^
2
1
p^
2
2
p^
2
3
p^
24 = 0;
(2.30)
so we still have the constraint between length and lling fractions (see [9, 11]).
2.3
BMN string
in [23]
We now exemplify the discussion above with BMN string solution in Sch5
S5 presented
i = i = x2 = x1 = 0;
x
+ =
z =
r
tan( )
p
p
2
2m
;
sec( ):
x
=
=
m
!
+ ! ;
+
tan( )
2m
;
Virasoro constraint gives
The conserved global charges are
2m2
2 + !2 = 0:
= p
;
P
M =
p
m;
J = p
!;
where we denote
as the global energy of the string associated with the non-relativistic
scale transformation D^ + M^ + . Then the classical energy of the BMN string is given by
= pJ 2 + 2M 2:
{ 6 {
The quasi-momenta of the BMN string are
with
x x
(x) = p x2
x
2
1
:
x
1
1 f
p
+
i
pj = 0; x 2 Cij:
{ 7 {
We nd an expected square root cut [csc 2 ; 1] connecting p^1 and p^4.
3
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
is the energy of the excitations. To
compute
, we add a perturbation
p(x) to p(x) associated with the classical solution.
The perturbation p(x) has a single pole at xn. The position xn is determined by
pi xinj
pj xinj = 2 n:
The residues at the poles are
resx=xinj p^k = ( ik
jk)
xinj Nnij; resx=xinj p~k = ( jk
ik)
xinj Nnij;
where i < j and Nnij is the excitation number for excitation with polarizations (ij) and
mode number n and the function
(x) is de ned as
the residues at x =
1 are synchronized as
p^1; p^2; p^3; p^4jp~1; p~2; p~3; p~4 =
;
; ;
j ;
; ;
g + : : :
For each cut Cij connecting pi(x) and pj(x) the perturbation p(x) satis es
where the superscript
denotes above and below the cut.
(2.35)
(2.36)
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
where ij are the o -shell uctuation energies.
The asymptotic behaviour of the perturbation p(x) is
p~4
BB p~1 CC = p4
+ N 2^3^ + N 2^4^ + N 2^3~ + N 2^4~ 1
N 2^3^
N 1^3^
N 1~3^
N 2~3^ C
N 1~3~
N 2~3~
N 1~4~
N 2~4~
N 1~3^
N 2~3^
N 1~4^
N 2~4^
+N 1~3~ + N 2~3~ + N 1^3~ + N 2^3~
+N 1~4~ + N 2~4~ + N 1^4~ + N 2^4~
C
C
C
C
C
A
CC + O x 2 :
The order O(x1=2) terms in the p^1 and p^4 are determined by the constraint (2.30). Since
the classical solution already obey the constraint (2.30), one can show these lling fractions
Nnij satisfy
X nNnij = 0:
n;ij
using Riemann bilinear identity (see eqs. (3.38) and (3.44) in [11]).
We now determine the most general form of the perturbation
p(x) for the BMN
string. When small poles are added to
p^2(x) with a square root cut, the branch point
x0
p
1=px
x0 near the branch
are rational functions of x and K(x) = p
p
x x
point x0 = sin 2 . The most general form for p^2(x) is f (x) + g(x)=K(x) where f and g
x0. From (3.6) and inversion symmetry
we get
0
B
CC :
C
One can obtained all other o -shell uctuation energies from the knowledge of 2~~3 and
^2^3 alone using the e cient method provided in [45]. From inversion symmetry we have
~1~4(y) =
~2~3(1=y) +
~2~3(0);
~1~4(y) =
~2~3(1=y)
2:
the quasi-momenta for BMN string in Sch5
stant terms:
p^1 =
p~1 =
p^4;
p~4 + 2
So the o -shell uctuation energies satis es
sin ;
ij (y) =
1
2
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
S5 are pairwise symmetric up to
con
Fluctuation energies of excitations
^^
We rst consider the excitation 23. We have p~ = 0, for i = 1; 2; 3; 4, and therefore
i
f (x) = 0. We take the following ansatz
g(x) =
X
n
x
n
x
K
x
n
x
n
N
n
+
2
p
+
4
p
X
n
N
n
:
and the level matching condition
n
;
(x) =
2
2x + 2xK(x)
x
2
1
;
nN
n
= 0:
3 ~~
We next consider the S excitation 23. We start with the following ansatz
Large x asymptotic of p~ and p~ give
1 4
Large x asymptotic of p^ give
1
The lling fractions satisfy we can solve these equations and obtain
X
n
=
X
n
nN
n
= 0;
N
n
~~ ~~
23 23
x
n
;
{ 9 {
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.26)
(3.27)
2
p
1
1
+
2
2
+
x
=
=
X
n
2
X
n
X
n
n
X
1
K(
1
)+
2
K(
1
)+
=
1
(cos
sin )
2
(cos +sin )+
= 0:
(3.25)
g(x) =
1
x
K(
1
)
1
2
K(
1
)
x + 1
+
2
p
p~ =
2
p~ =
3
1
1
+
2
x + 1
X
n
x
n
x
n
N
n
N
n
;
1
:
x
1
x
x
n
N
n
x
x
n
:
2
;
2
p
(x) =
sin 2
where
(x) =
(x) =
(x) =
(x) =
(x) =
(x) =
We now solve the pole position x . We choose the solution x
n
> 1 for small LM to be
physical poles. We have
n
2
p
p
2 L M
n
x
x
^^
14
n
^^
13
n
x
n
x
~~
14
n
x
~^
23
n
x
^~
23
n
x
^~
24
n
x
^~
14
n
x
^~
13
n
=
=
=
=
x
~^
24
n
= x
~^
14
n
=
Finally using (3.12) and (3.15) we nd all the o -shell uctuation energies
(LM
p
16
p
p
2
p
+ n
2
n
2
LM )
n
2
2
J + n( n
p
p
2LM
1
J + O (
1
) ;
LM ) +
n
p
J
;
p
p
(LM + n)
(LM + 4 n)
2
J + n(LM + n) + J
(3.29)
connecting p^1 and p^4 and become x4^nj (xin1^).
The exact expressions of x^in^j are very complex, so we only consider the leading order terms
Pluging xn into the o -shell uctuation energies, in the large J limit we get the on-shell
uctuation frequencies
The one loop shift is equal to one half of the graded sum of all uctuation mode frequencies.
Using zeta function regularization, we have
(n + q)2 + pn = q2 + ( 2; 1 + q) + ( 2; 1
q) = 0:
(3.32)
Therefore when we compute the one loop shift energy at order J 2, only the contribution
from
^1^4 is nontrivial. Then we sum over the energies of the sl(2) modes to get the
1 loop =
+
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 ! 1 and
Type IIB superstring in Sch5
=J 2 kept xed and small (see e.g. [26, 46{48]).
S5 is dual to null dipole deformed N = 4 super Yang-Mills.
The sl(2) sector nontrivially a ected 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
t(u)Q(u) = (u + i=2)J Q(u + i) + (u
i=2)J Q(u
i);
and
The 1-loop energy is given by
t(u) = 2uJ + LM J uJ 1 + : : : :
(
1
) =
i
Q(u + i=2)
Q(u
i=2) u=0
:
1
k=0
Q(u) = X pk+m(u)LkM k;
(4.1)
(4.2)
(4.3)
(4.4)
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 pk+m is a polynomial in u of degree k + m. The small LM expansion of Q can
be interpreted as a function with a
nite number of zeros near the Bethe roots in the
undeformed limit and an in nite number of zeros of order L 1M 1. In the string picture
the zeros of order L 1M 1 correspond to the cut connecting p^1 and p^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:
Q0 = 1
LM u + L2M 2
+ O L3M 3 ;
Q1 = (u
un) +
LM
2J u2 + 2J u2n + 4u2n + 1
2(J + 2)
J u2
In the undeformed case, Q0 and Q1 correspond to the ground state and one particle state
Then the energies of the ground state and one particle state are
(01) =
(11) =
2(J + 2) (4u2n + 1)
+
L2M 2 4(J (J (J + 10) + 20) + 24)u2n + (J
6)(J + 2)2
8 2(J + 2)3(J + 3) (4u2n + 1)
L3M 3 48J 3u3n + 4(J + 2)2(J (4J + 7) + 12)un
3 2(J + 2)5(J + 3)(J + 4) (4u2n + 1)
c4 = 96J 3 J 5 + 36J 4
92J 3
1808J 2
4656J
8 2J
L2M 2
8 2J
+
+
+ O(J 3);
n
2
8 2J 2
L4M 4 2n2 +6
96J 2 ( 4n2)
!
the undeformed case, the energy of an excited state above ground state energy is
(01) = X Nn
n
n
2
L4M 4
16 ( 4J 2n2) + O L5M 5
+ O(J 3); (4.12)
where Nn is excitation number for mode number n, and we assume that the total
momentum is zero.
To compare the spectral curve result with the spin chain result, we expand
for small LM and obtain the energy shift of the sl(2) sector
=
X Nn1^4^ ^1^4(x^1n4^) =
X Nn1^4^
n
n
n
2
where we have used the level matching condition. The result agrees with (4.12).
The energy of the spin chain ground state to order L6M 6 has been computed in [23]
8 2J
L2M 2
L4M 4
+
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).
5
Conclusion and discussion
In this paper we study the algebraic curve for superstring in Sch5
S5 and its application
to the spectral problem. The asymptotic properties of the quasi-momenta for strings in
Sch5
S5 are nontrivial. The point at in nity 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 eld results obtained
by Baxter equation. We provide a detailed test of the Schrodinger/dipole CFT duality.
Our results encourage further exploration of integrability in Schrodinger/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 eld theory side to get a precise match with string theory predictions.
One can also study the three dimensional counterpart of Sch5, the warped AdS3. We hope
that integrability would be a powerful tool for the spectral problem of warped AdS3/dipole
CFT duality [49].
Acknowledgments
The author would like to thank Nikolay Bobev, Hui-Huang Chen, Fedor Levkovich-Maslyuk,
Zhibin Li, Wei Song, Jun-Bao Wu and Konstantin Zarembo for very helpful discussions
and comments. This work is supported in part by the National Natural Science Foundation
of China (Grant No. 11575202).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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