#### The \( T\overline{T} \) deformation at large central charge

HJE
The T T
Ofer Aharony 0 1
Talya Vaknin 0 1
0 Rehovot 7610001 , Israel
1 Department of Particle Physics and Astrophysics, Weizmann Institute of Science
We study Zamolodchikov's T T deformation of two dimensional quantum field theories in a 't Hooft-like limit, in which we scale the number of degrees of freedom c to infinity and the deformation parameter t to zero, keeping their product t · c fixed (more precisely, we keep energies and distances fixed in units of t · c). In this limit the Hagedorn temperature remains fixed, but other non-local aspects of the theory disappear. We show that in this limit correlation functions may be computed exactly, and they are local in space and polynomials in t. We compute explicitly the deformed three-point functions of the energy-momentum tensor for a T T -deformed conformal field theory.
1/N Expansion; Integrable Field Theories; Renormalization Group
1 Introduction 2 3 4
1
Introduction
The deformation and its effect on correlation functions
Correlation functions in the large c limit
Example: 3-point functions in the deformed CFT of a free scalar
all of our computations are valid for both signs of t). For this sign the energy spectrum
on a circle is real for large radii, but the ground state energy becomes complex for small
radii, corresponding to the Hagedorn instability. If we initially start from a conformal field
theory of central charge c, then this happens at a radius R obeying tc = − 32Rπ2 . This implies
that the deformed theories have a maximal temperature for which they are well-defined,
and that they may not be local theories at high energies. A specific interesting suggestion
is that [8] the deformation is equivalent to coupling the theory to Jackiw-Teitelboim [9, 10]
gravity, such that at high energies it is a gravitational theory with no local degrees of
freedom (see also [11]). For positive t there are states with complex energies at all radii
and the interpretation of these theories is not clear.
– 1 –
A recent study [12] showed that for some purposes the T T deformation has no local
effect, since it can be rewritten as a boundary term. This allowed the derivation of
differential equations for the t-dependence of the partition functions of T T deformed theories
on various manifolds. However, for the purpose of computing local correlation functions,
deformation terms proportional to the equations of motion, that were ignored in [12], must
be kept. We show here that local correlation functions in these theories do have a
nontrivial dependence on t, so that the deformation can be locally felt.1 In particular this is
true for correlators of the energy-momentum tensor, indicating that the theory is sensitive
to local changes in the background metric.
In this paper we analyze a specific limit in which correlation functions of T T -deformed
theories can be analyzed exactly. This is the large c limit — a limit of a large number
of degrees of freedom, analogous to the large N limit of non-Abelian gauge theories.2
We show that in this limit, when keeping fixed tc, and working to leading order in 1/c,
the correlation functions are actually polynomials in tc, and their large t (short distance)
behavior is controlled just by the 2-point functions of the undeformed theory. Note that
in this study we are analyzing energies that are fixed in units of 1/p|t|c; they can be large
in these units, but they are still much smaller than the scale 1/p|t|. For these energies
the correlation functions look like those of a standard local (but not scale-invariant) field
theory. Some non-local features of these theories, like their Hagedorn behavior, arise at
the scale 1/p|t|c, while others, like the change in the S-matrix, appear at the scale 1/p|t|.
We do not have anything to say here about the behavior at the scale 1/p|t|.
At large c it is also natural to study this theory using gauge/gravity duality [11, 13–15].
The same scaling of t with c is required there for the deformation to be seen in classical
gravity. At leading order in t and in 1/c it is a “double-trace deformation”, which corresponds
to changing the boundary conditions at infinity for the graviton in the dual gravitational
theory. By construction, this holographic picture with the boundary conditions as infinity
will agree with our computations at the leading order in 1/c. It was conjectured in [15]
(for the positive sign of t) that one can also go beyond this leading order and describe
the theory by putting a radial cutoff on the dual three-dimensional gravity theory, which
couples the field theory to two dimensional gravity. We will not discuss the holographic
picture here.
In this paper we only study correlation functions in infinite flat space; it would be
interesting to generalize our results to compact and/or curved spaces, and to study what
can be said there. It would also be interesting to consider the 1/c corrections to our results,
and to understand the transition between the behavior at energy scales of order 1/p|t|c
to the behavior at energies of order 1/p|t|.
Note that there is a closely related deformation analyzed in [
16–19
], for which the UV
completion is given by a “little string theory”. This deformation can also be studied in
the large c limit in which it has an interesting gravitational dual. In “little string theories”
1Similar computations were recently performed in [13].
2For conformal field theories this is the limit of a large central charge, but we use the same notation also
for more general quantum field theories with the property that all connected correlation functions of the
energy-momentum tensor scale as some constant c.
– 2 –
there can also be a separation of scales between the Hagedorn scale and the non-locality
scale, which is similar to what we find here. However, in this paper we do not discuss this
alternative deformation but rather the original one of [
1–3
].
2
The deformation and its effect on correlation functions
In this section we will show how to compute correlation functions of energy momentum
tensors in a quantum field theory (QFT) which is deformed by a “T T deformation”, first
introduced in [
1–3
]. This is a short-hand notation for a family of field theories parameterized
by a coupling contant t, which are perturbatively defined such that the change in the action
when infinitesimally increasing t by δt is proportional to det(T ),
S(t + δt) − S(t) = −8δt
Z ǫikǫjlTij (x)Tkl(x)d2x.
(2.1)
(2.2)
Here T is the energy-momentum tensor of the theory with parameter t. It depends
nontrivially on t, so that we cannot write the deformation in a simple way using the original
energy-momentum tensor (we will leave the dependence of T on t implicit for simplicity).
We assume that for every t the theory has a conserved symmetric energy-momentum tensor
which is a local operator; perturbatively in t this is definitely true. As shown in [
1
], this
particular combination of energy-momentum tensors has no singularity when the operators
are brought together, so that it is a well-defined operator.
We will work in infinite flat space in Euclidean signature, where the infinitesimal
deformation corresponds to inserting in the path integral
e8δt R ǫikǫjlTij(x)Tkl(x)d2x.
d
dt
Latin letters will denote flat-space coordinates, without distinguishing upper and lower
indices.
Computing correlation functions with this deformation to all orders in perturbation
theory in t is complicated in general. One complication is that it is not clear how
general local operators O(x) should depend on t, since operators can mix in a complicated
way under the deformation. For this reason, we will focus in this paper on correlation
functions of T itself, since its variation is fixed by (2.1) (consistent with its conservation
and symmetry). Consider an n-point correlation function of energy-momentum tensors,
hTm1n1 (x1) . . . Tmnnn (xn)i. The definition of the theory above implies that it obeys the
differential equation3
hTm1n1 (x1) · · · Tmnnn (xn)i = 8
d2x hTm1n1 (x1) · · · Tmnnn (xn)ǫikǫjlTij (x)Tkl(x)i
dt
Tm1n1 (x1) · · ·
dTmini (xi) · · · Tmnnn (xn) , (2.3)
Z
n
+ X
i=1
3We assume here for simplicity that the one-point function of det(T )(x), related [
1
] to the square of
the one-point function of Tmm(x), vanishes. On the plane this can always be achieved by a constant shift
(possibly depending on t) in Tmm(x).
– 3 –
with the initial condition at t = 0 given by the correlation function in the original,
undeformed theory. At the leading non-trivial order in t, a similar computation was performed
in [13] for a deformed conformal field theory. However, going to higher orders is difficult,
since at k’th order in perturbation theory, the deformation in the n-point function is related
to (n + 2k)-point functions of the original theory.
We will show in this section that the computation can be simplified by using Ward
identities, and then in the next section we will show that it simplifies even further in the
limit of a large number of degrees of freedom. We follow the notations of Cardy [12]
throughout this section.
We begin by performing a Hubbard-Stratonovich transformation with an auxiliary
symmetric tensor field hij to rewrite
e8δt R ǫikǫjlTijTkld2x ∝
[dh]e−(1/32δt) R ǫikǫjlhijhkld2x+R hijTijd2x.
Integrating out Φ yields an action quadratic in α,
Z
[dαi]e−(1/16δt) R αj∂i2αjd2x+R (αj∂jTii−2αj∂iTij+4δt(Tii)2)d2x.
– 4 –
A general symmetric tensor field may be decomposed locally as
hij = ∂iαj + ∂j αi + δij Φ,
where αi(x) is a vector field and Φ(x) a scalar field; globally there may be obstructions to
this, but in this paper we only work in flat space and hij is infinitesimal so we can ignore
them. Since hij in (2.4) is infinitesimal and couples to the energy-momentum tensor, we
can treat it as an infinitesimal change of the metric of the space that the field theory lives
on, and we can view (2.5) as its decomposition into an infinitesimal diffeomorphism αi(x),
and an infinitesimal deformation e
Φ of the conformal factor.
When writing the deformation in terms of this decomposition, the quadratic terms in
hij , multiplying (1/32δt) in (2.4), have the following contributions:
2ǫikǫjl(∂iαj )(∂lαk) + 2ǫikǫjl(∂iαj )(∂kαl),
and
4ǫikǫjl(∂iαj )(δklΦ) + ǫikǫjl(δij Φ)(δklΦ) = 4(∂kαk)Φ + 2Φ2.
The second term in (2.6) is a total derivative, so it can be turned into a boundary term,
and we will drop it since we are interested in flat space and the correlation functions we
are interested in decay fast enough such that it will not contribute. Unlike in [12], we will
keep the terms in the action that are proportional to the equations of motion, since they
will be important (without these terms the full deformation is a total derivative [12]). The
path integral over h now takes the form
[dΦ][dαi]e−(1/32δt) R (2ǫikǫjl(∂iαj)(∂lαk)+4(∂kαk)Φ+2Φ2)d2x+R (2∂iαjTij+ΦTii)d2x.
can use Ward identities to simplify the effect of the deformation, since pkTjk(p) vanishes
by the equations of motion, and its correlation functions are given by a sum of contact
terms with other operators:
pkhTjk(p)O1(p1) · · · On(pn)i = XhO1(p1) · · · (δj Oi)(pi + p) · · · On(pn)i,
(2.11)
j direction (for scalars this is just the derivative in the j direction, otherwise there are
additional terms).
We can use this to simplify the computation of n-point correlation
functions to all orders of perturbation theory.
Going back to our n-point correlation function hTm1n1 (x1) · · · Tmnnn (xn)i, we can now
rewrite our differential equation (2.3) for the momentum-space correlation function
(dropping the momentum-conservation delta function) as
d
dt
=16
w2 hTm1n1 (p1) · · · Tmnnn (pn)wkTjk(w) wk′ Tjk′ (−w) − wj Tll(−w) i
(2.12)
Tm1n1 (p1) · · ·
dTmini (pi) · · · Tmnnn (pn) .
dt
Using the Ward identity we can rewrite the first line on the right-hand side as the sum of
an n-point function of energy-momentum tensors and an (n + 1)-point function of
energymomentum tensors (using the fact that the variation of the energy-momentum tensor under
translations is the sum of derivatives of energy-momentum tensors). This allows us to write
the variation in the n-point correlator in terms of (n + 1)-point correlators at most. Similar
simplifications arise for correlation functions of other operators, but as mentioned above,
for them there is no preferred choice of their t-derivative, so we will not analyze them here.
3
Correlation functions in the large c limit
There is a special limit in which we can explicitly compute the correlation functions
discussed in the previous section, which is the limit of a large number of degrees of freedom.
For a conformal field theory (CFT) this is the large c limit, where c is the central charge,
but there is a similar limit for general quantum field theories (which we will still call the
large c limit). In the large c limit correlation functions of Tmn factorize. The connected
contribution to an n-point function of energy-momentum tensors is proportional to c at
large c, so that when we compute a general correlation function and look at the
contribution to it which is a product of k connected components, then this will scale as ck, and
– 5 –
the correlation function will be dominated by the contribution with the largest value of
k. Other contributions will be suppressed by powers of c. In other words, for even n at
leading order in c
hTm1n1 (x1) · · · Tmnnn (xn)i = hTm1n1 (x1)Tm2n2 (x2)i · · · hTmn−1nn−1 (xn−1)Tmnnn (xn)i+
other pairings.
(3.1)
The same scaling of correlators with c occurs for a more general class of operators called
“single-trace operators”. Their products (after removing singularities) are called
“multitrace operators”. The T T deformation operator is an example of a double-trace operator.
Suppose we now consider the variation (2.3) of a connected correlation function of n
energy-momentum tensors. In the first term on the right-hand side, the leading contribution
will come from products of two connected correlation functions, with the first one including
Tij (x) and some of the original n operators, and the other including Tkl(x) and the other
original operators. This will scale as one extra power of c compared to the connected
contribution to this first term, and also compared to the correlation function on the
lefthand side. Thus, if we want to have a finite large c limit, we need to scale t as 1/c, or
in other words take an ’t Hooft-like limit, in which we take c to infinity, keeping tc finite.
In this limit only the disconnected contributions to the first term on the right-hand side
in (2.3) will survive. Since t has dimensions of length squared, this means that we will be
computing correlation functions at distances of order p|t|c, or at energies of order 1/p|t|c,
in the deformed theory. We will assume this scaling in the rest of this paper.4
The change in the Lagrangian at first order in t is a double-trace operator, and thus the
change in Tmn at first order is in general a combination of a double-trace and a single-trace
operator. Plugging this back into the action, one can show that the terms of order tk in
the action are products of (k + 1) single-trace operators or less, and thus the same is true
also for the energy-momentum tensor. In other words,
∞
k=0
S = X tkS(k),
Tmn = X tkT m(kn),
∞
k=0
where S(k) and T m(kn) include products of up to (k + 1) single-trace operators.
In general we could have a one-point function hTmn(x)i = f (t)δmn, but in flat space
we can always subtract this away without affecting the conservation equation, so we will
assume that hTmn(x)i = 0. For two-point functions we then obtain from (2.12)
(3.2)
(3.3)
4With this scaling the full action of the deformed theory scales as c, suggesting that a saddle point
approximation may be used for computing correlation functions and other objects, but we will not use this here.
– 6 –
d
dt
=16
correlation function on this line is a one-point function which vanishes. The term with
dT /dt needs to be of order c2 to contribute in our large c limit, but it cannot factorize into
a product of two correlators, so its contribution is suppressed at least by a factor of c in
the limit that we are interested in. Thus, in the large c limit we obtain that
hTm1n1 (p1)Tm2n2 (p2)i = hTm1n1 (p1)Tm2n2 (p2)i|t=0,
and two-point functions are independent of t.5
Considering next 3-point functions, and using the Ward identities, we have
order (tc)nc2. Now this can happen, but only if we separate it into two correlators and have
no factors of t. Such a term can only come from having one single-trace operator in the
double-trace contribution to T m(1n) combining with Tm2n2 , and the other with Tm3n3 . So in
this term we get a product of two correlators of T with an operator that is independent of t,
and the same argument we used above implies that these 2-point functions are independent
of t. Thus, we find that the 3-point functions are linear in t, and their large t behavior
(for large c at fixed tc) is determined just by the 2-point functions in the original QFT
before the deformation. We will see an explicit example of this in the next section. Note
that since t is dimensionful, taking large t is the same as going to short distances or high
energies (but still of order p|t|c).
Similarly, if we consider the t-derivative of a connected 4-point function, we obtain on
the first line on the right-hand side a product of a 3-point function and a 2-point function,
which we already know to be linear in t. On the second line we can get a contribution from
the double-trace term in T m(1n), which gives a product of a 3-point function (which can be of
order t) with a 2-point function (that is independent of t). Or, we can get a contribution
from the triple-trace term in T m(2n), which gives t times a product of three 2-point functions
(which are independent of t). All other contributions are suppressed. So 4-point functions
are exactly quadratic in t in this limit. Similarly we find that, to all orders in perturbation
theory in t, and to leading order in 1/c, connected n-point functions are polynomials of
degree (n − 2) in t, and they scale as tn−2cn−1 at large t. Moreover, they are simply related
to the lower-point functions in the original QFT before the deformation; the leading large
t behavior is in fact determined just by 2-point functions in the original QFT. A similar
behavior arises also for correlation functions of other “single-trace operators”.
5This result was previously obtained by M. Porrati (D. Kutasov, private communication).
– 7 –
(3.4)
(3.5)
A large class of large c theories has a weakly coupled (and sometimes also weakly
curved) gravitational dual, by gauge/gravity duality (for CFTs this is given by a
gravitational theory on AdS3). In this language, the energy-momentum tensor is dual to a graviton
in the bulk, and the deformation (2.1) is given (in the large c limit with fixed tc), like other
“double-trace deformations” [20–22], by a change in the boundary condition for the
graviton (see [23, 24]). On general grounds it is clear that perturbatively computing correlation
functions in the gravitational theory precisely reproduces our analysis above, and this is
consistent with the explicit computation of some correlation functions in [13]. If one tries
to go away from the large c limit, the deformation on the gravitational side is more drastic
and the behavior near the boundary is significantly modified; it was suggested in [15] that
this corresponds to putting a finite cutoff in the gravitational theory, but there are many
ways of putting such a cutoff and it is not clear which, if any, of them corresponds to the
correct deformation. Since 1/c corrections correspond to loop corrections on the gravity
side, it is difficult to perform comparisons of the two sides away from the large c limit.
4
Example: 3-point functions in the deformed CFT of a free scalar
Using the methods described above, we can compute the three-point functions of
energymomentum tensors in a conformal field theory deformed by T T , in the large c limit. We use
the notations T = T++, T = T−− and Θ = T+− for the different components of the
energymomentum tensor, where in Euclidean signature x± = x1 ± ix2. Following equation (2.12),
the deformation will depend only on the two-point functions; however we need to keep
also the contact terms in the two-point functions, which have non-trivial effects since the
deformation is integrated. In momentum-space, the two-point functions including contact
terms (which are determined by conservation) are given by
3
hT (p)T (−p)i = − 1c2 pp−+ , hT (p)Θ(−p)i = 1c2 p2−,
hT (p)T (−p)i = − 1c2 pp−3+ , hT (p)T (−p)i = − 1c2 p+p−, hT (p)Θ(−p)i = 1c2 p2+,
hΘ(p)Θ(−p)i = − 1c2 p+p−.
There are a total of ten different three-point correlation functions. We will do a detailed
calculation for hT (p)T (q)Θ(−p−q)i and then state the results for the rest. Following (2.12)
we get
d
dt
hT (p)T (q)Θ(−p − q)i
=256
d2w
1
w−
+
+
+
Z
1
w+
1
w−
1
w+
hΘ(−p − q)Θ(−w)i w+hT (p)T (q)T (w)i + w−hT (p)T (q)Θ(w)i
hΘ(−p − q)Θ(−w)i w−hT (p)T (q)T (w)i + w+hT (p)T (q)Θ(w)i
hT (q)Θ(−w)i w+hT (p)Θ(−p − q)T (w)i + w−hT (p)Θ(−p − q)Θ(w)i
hT (q)Θ(−w)i (w−hT (p)Θ(−p − q)T (w)i + w+hT (p)Θ(−p − q)Θ(w)i)
(4.1)
– 8 –
+
+
+
w−
1
w+
dT (p)
dt
hT (p)Θ(−w)i w+hT (q)Θ(−p − q)T (w)i + w−hT (q)Θ(−p − q)Θ(w)i
hT (p)Θ(−w)i w−hT (q)Θ(−p − q)T (w)i + w+hT (q)Θ(−p − q)Θ(w)i
T (q)dΘ(−p − q)i + hT (p)
dΘ(−p − q)i + hT (p)T (q)
dT (q)
dt
dΘ(−p − q)
dt
For the first contributions we can apply the Ward identities (2.11), which for this case
take the form (see, for instance, [
25, 26
])
pn1 hTm1n1 (p)Tm2n2 (q)Tm3n3 (−p − q)i
= 2p(m3 hTn3)m1 (q)Tm2n2 (−q)i + 2p(m2 hTn2)m1 (−p − q)Tm3n3 (p + q)i
(4.2)
+ (p + q)m1 hTm2n2 (q)Tm3n3 (−q)i − qm1 hTm2n2 (−p − q)Tm3n3 (p + q)i,
with the brackets denoting symmetrization with a coefficient 21 . The contributions from
this part are universal, independent of the specific theory, and we find
HJEP05(218)6
16c2 p+ q− (p+q− − p−q+)2 + non-universal terms,
(4.3)
with the non-universal terms coming from the explicit derivatives dTmn/dt.
Similarly, for the other three-point functions, we get
hT (p)T (q)T (−p − q)i = −
hT (p)T (q)Θ(−p − q)i =
hT (p)Θ(q)Θ(−p − q)i =
hΘ(p)Θ(q)Θ(−p − q)i = −
hT (p)T (q)T (−p − q)i = −
hT (p)T (q)T (−p − q)i = −
hT (p)T (q)T (−p − q)i = −
hT (p)T (q)Θ(−p − q)i =
hT (p)Θ(q)Θ(−p − q)i =
9
p− q−
9 p−
64c2
9
9 p− + q− p− q−
8c2 p+ + q+ p+ q+ (p+q− − p−q+)2 + non-universal terms,
−16c2 p+ q+ (p+q− − p−q+)2 + non-universal terms,
40c2 p+ (p+q− − p−q+)2 + non-universal terms,
(p+q− − p−q+)2 + non-universal terms,
8c2 p+ q+ p− + q− (p+q− − p−q+)2 + non-universal terms,
9 p− q− p+ + q+
9 p+ q+ p− + q−
9 p+ q+ p+ + q+
8c2 p− q− p+ + q+ (p+q− − p−q+)2 + non-universal terms,
8c2 p− q− p− + q− (p+q− − p−q+)2 + non-universal terms,
9
p+ q+
9 p+
−16c2 p− q− (p+q− − p−q+)2 + non-universal terms,
40c2 p− (p−q+ − p+q−)2 + non-universal terms.
(4.4)
Many of these results are pure contact terms, or contact terms of two of the operators,
but we still write them since they will be important for going to higher-point correlation
functions.
– 9 –
In a given theory there will be specific expressions for dTmn/dt, and, as described above,
any “double-trace” terms that this includes will give a contribution to the large c 3-point
functions, which is a product of 2-point functions of the t = 0 theory. These contributions
are essential for the Ward identities to continue to hold after the deformation. In the case
of the free scalar, dTmn/dt was explicitly computed in [2], and it is easy to generalize this
to the case of N free scalars, with c = N . In that case each dTmn/dt is a “double-trace
operator”. The extra contributions to each correlation function are a sum of three products
of two 2-point functions, but the only non-contact term contribution in this case comes
from dΘ/dt ∝ T T .
Summing up the two types of contributions for the free scalar theory, the 3-point
functions that acquire a t-dependence that is not a contact term are, in position space6
d
dt
d
dt
d
dt
hT (x)T (y)Θ(0)i = 32 x4+y−4
,
hT (x)T (y)T (0)i =
hT (x)T (y)T (0)i =
c
2
128c2
128c2
3
3
1
1
(y− − x−)5x3+ +
(y+ − x+)5x3− +
1
1
(x− − y−)5y+3
(x+ − y+)5y−3
,
So the 3-point functions at separated points at finite t will include hT T T i and hT T T i that
take the same value as in the CFT (proportional to c), and the additional ones coming
from (4.5) that are proportional to tc2. At short distances (compared to p|t|c) the latter
will dominate. Our results show no sign of any non-locality in these theories, even though
the distances involved can be much smaller than the Hagedorn scale. In particular, they
are compatible with still having an operator product expansion for the local operators,
that includes new terms going as
T (x)Θ(0) ∼
T (x)T (0) ∼
tcT (0)
4
x+
,
tcx−T (0)
5
x+
Acknowledgments
We would like to thank S. Dubovsky, E. Gerchkovitz, A. Giveon, N. Itzhaki, O. Mamroud,
V. Narovlansky, A. Schwimmer, G. Torrents, R. Yacoby, S. Yankielowicz, A. B.
Zamolodchikov, and especially D. Kutasov, for many useful discussions. This work was supported
in part by the I-CORE program of the Planning and Budgeting Committee and the Israel
Science Foundation (grant number 1937/12), by an Israel Science Foundation center for
excellence grant, and by the Minerva foundation with funding from the Federal German
Ministry for Education and Research. OA is the Samuel Sebba Professorial Chair of Pure
and Applied Physics. TV is supported by the ERC STG grant 335182.
6These results agree with those of [13], even though the details of the computation are different.
tcT (0)
4
x−
,
tcx+T (0)
5
x−
(4.5)
(4.6)
T (x)Θ(0) ∼
T (x)T (0) ∼ tc
, T (x)T (0) ∼
∂−T (0)
3
x+
−
∂+T (0)
3
x−
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