Little string amplitudes (and the unreasonable effectiveness of 6D SYM)
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1
Center for Theoretical Physics, Massachusetts Institute of Technology
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Jefferson Physical Laboratory, Harvard University
We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in sixdimensional superYangMills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in N = 2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the 0 expansion. We find striking agreements with up to 2loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a Zk invariant point on the Coulomb branch. We comment on the issue of UV divergence at higher loop orders in the gauge theory and discuss the implication of

effectiveness of 6D
1 Introduction
Double scaled little string theory
The holographic description
SL(2)k/U(1)
SU(2)k as a Zk orbifold of U(1)k U(1)
2.6 Identifications among vertex operators in the coset theories
Massless string states
Spectral flow
Zk orbifold
SU(2)k/U(1) and N = 2 minimal model
Supersymmetric SU(2)3/U(1) and the compact boson
k = 2
k = 3
k = 4
k = 5
Nonnormalizable, delta function normalizable, and normalizable
pri
Correlators and amplitudes
Winding number conserving correlators
Bosonic SL(2)k+2 correlators and Liouville correlators
Scattering amplitude in the double scaled little string theory
Comparison to 6D superYangMills amplitudes
Structure of perturbative amplitudes
Evaluation of color factors and box integrals
A Normalizable vertex operators
A.1 NSsector
A.2 Rsector
B Threestate Potts model
There are two interesting maximally supersymmetric quantum field theories in six
dimensions: the (2, 0) superconformal field theory and the (1, 1) superYangMills theory. They
are believed to be the low energy limits of (2, 0) or (1, 1) little string theories (LST) [1
While the LST is a strongly coupled string theory that is difficult to get a handle
on, it admits a 1parameter deformation known as the double scaled little string theory
(DSLST) [7, 8], the parameter being an effective string coupling. In the weak coupling
limit, DSLST can be studied perturbatively. In particular, one can compute the spectrum
In this paper, we will focus on the (1, 1) theories. In this case, the LST reduces
to 6D SYM at low energies. More precisely, the interactions of massless modes of LST
are expected to be described by an effective theory that is 6D SYM deformed by higher
dimensional operators (such operators are highly constrained by supersymmetry and we will
C.1 Zamolodchikov recurrence formula for conformal blocks
C.2 Crossing symmetry
D 3loop UV finiteness of the fourpoint amplitude for the Cartan gluons 45
gs gY M mW
This can be understood by identifying the little string with instanton strings in the SYM.
The DSLST, on the other hand, is related to 6D SYM on its Coulomb branch. The relation
between the string coupling and the Coulomb branch parameter takes the form
Thus one may anticipate a relation between scattering amplitudes of massless modes in
the DSLST and those of the Cartan gluons in the 6D SYM of the schematic form
ADSLST (gs, 0, E) = ASY M (gY M , mW , E),
provided the identifications (1.1) and (1.2). Note however that the weak coupling limit of
DSLST corresponds to the regime far from the origin on the Coulomb branch. In the 6D
gauge theory, a priori one would expect higher dimensional operators to become
important in this limit, and it is not a priori clear whether the gluon scattering amplitudes in
the SYM, expanded in 1/m2W , should agree with that of the perturbative amplitudes of
massless modes of DSLST.
In fact, the 6D SYM diverges at threeloops, and one might be led to think that such
an agreement is impossible with just the SYM perturbative amplitudes (without including
any higher dimensional operators). However, a more careful inspection indicates that the
gauge theory is free of UV divergence at 3loops. This is because there are only two
candidate counter terms that are dimensional 10 operators allowed by sixteen supersymmetries;
one of them is a double trace operator known to be 1/4 BPS and finite in 6D SYM
according to [9], the other is a single trace operator that is nonBPS but vanishes when restricted
to the Cartan subalgebra of the SU(k). (We will explicitly verify the absence of this UV
divergence in appendix D.) This suggests that a certain nonrenormalization theorem is at
play, and one could hope that the 6D SYM by itself already captures some of the dynamics
of massless gluons on the Coulomb branch of the (1, 1) little string theory.
Remarkably, we find agreement between up to twoloop amplitudes of the massless
gluons on the Coulomb branch of the 6D SYM, expanded to leading order in 1/m2W , and
the tree level amplitude of the corresponding massless string modes in DSLST, expanded
the SU(k) SYM, based on previously known twoloop results derived from unitarity cut
methods [1015], and in the DSLST by a worldsheet computation that involves
expressing SL(2)/U(1) coset CFT correlators in terms of Liouville correlators, which are then
expressed as integrals of Virasoro conformal blocks. The computations on the two sides
utilize completely different techniques and the results are rather involved. Nonetheless,
we will evaluate the results numerically for k up to 5, and the answers on the two sides
strikingly agree.
While our result may be viewed as a strong check of the duality between DSLST and 6D
SYM, it also hints that the higher dimensional operators correcting the 6D SYM are under
control. Put differently, the perturbative 6D SYM seems to know a lot more about its UV
completion and particularly the instanton strings than one might have naively expected!
The paper is organized as the following. After reviewing the construction of DSLST
and its spectrum, we will study the tree level fourpoint amplitude of massless RR vertex
operators in six dimensions. In computing such amplitudes, we will need the singular limits
of certain correlators in the SL(2)/U(1) coset (cigar) CFT, which are further related to
Liouville correlators by Ribault and Teschners relations [16]. This will allow us to express
the DSLST amplitudes as integrals of Liouville/Virasoro conformal blocks.1 Order by order
in table 1 and the 6D SYM results are summarized in table 3.
These amplitudes are then compared with gluon scattering amplitudes of 6D SYM on
its Coulomb branch, expanded in 1/m2W to the leading order (if these amplitudes were the
full story, higher order terms in the 1/m2W expansion would be mapped to higher genus
1The integration of conformal blocks was circumvented in [17] as the authors of [17] were only concerned
blocks will be carried out.
double scaled little string amplitudes). The 1loop and 2loop amplitudes are perfectly
finite and can be obtained using unitarity cut method [1214], and the 3loop amplitudes
have been reduced to scalar integrals as well [18] (as already asserted, they are free of UV
divergences when the external lines are restricted to the Cartan subalgebra). An agreement
of the 1loop amplitude at order 1/m2W with the low energy limit of four gluon amplitude
involve highly nontrivial group theory factors). While the expressions on both sides are
in the threestate Potts model went into the DSLST amplitude computation!) We conclude
with a discussion on the implication of these results. Details on the numerical integration
of conformal blocks, based on Zamolodchikovs recursion relations [19, 20], are described
in appendix C.
Double scaled little string theory
In section 2.1, we review holographic descriptions of DSLST. In sections 2.22.6, we
construct the normalizable vertex operators of the supersymmetric
quantum numbers satisfying (2.56) subject to the identification (2.62) and (2.64). Our
(SL(2)k/U(1)
main result is in section 2.7, where we identity the massless bosonic vertex operators of
type IIA string theory on R1,5
(SL(2)k/U(1) SU(2)k/U(1))/Zk, which is the Tdual
description of IIB (1,1) DSLST. These massless string states correspond to the scalars and
gluons in the 6D SYM.
The holographic description
The physical definition of double scaled little string theory is the decoupled theory on
k NS5branes in type IIA or IIB string theory, spread out on a circle of radius r0 in a
transverse plane R
in string units. Here one may consider an energy scale of interactions comparable to the
string scale (without taking a low energy limit). The configuration or the decoupled theory
has a residual global symmetry Zk U(1).
In the case of IIB NS5branes, D1branes extending between NS5branes correspond
to Wbosons in the sixdimensional SU(k) gauge theory. They become tensionless in the
r0 0 limit, where we recover the strongly coupled (1,1) LST.
On the other hand, one may directly take the decoupling limit of the string worldsheet
theory in the NS5brane background. Recall that (1,1) LST is described by IIB string
theory on the linear dilaton background from the NS5brane near horizon geometry [3, 21].
Tdual to the SUL((21))k cigar CFT in IIA, similarly for (2,0) DSLST [7, 22, 23] (see also [24]).
This leads to the holographic description of DSLST, as IIA (in the (1, 1) case) or
type IIB (in the (2, 0) case) string theory with the target space given by [25]
in the NSR formalism. The Zk orbifolding acts simultaneously on the two (supersymmetric)
coset models.
The supersymmetric coset model SL(2)k/U(1) can be constructed from the bosonic
3 and the total bosonic U(1) current at level k (combining a U(1) from the
central charge is
i
The current 2 abcbc gives rise to an SL(2) current algebra at level 2. Altogether, we
have a level k SL(2) current
for the supersymmetric SL(2)k WZW model. This theory has N
= 1 superconformal
symmetry, with the supercurrent
G =
abajb 6 abc
as required for critical string theory.
SL(2)k/U(1)
One starts with the bosonic SL(2)k+2 WZW model, governed by the current algebra
ja(z)jb(0)
The antiholomorphic currents will be denoted as ja. The extension to supersymmetric
m = 9 +
3(k 2)
= 15,
Let us summarize the relations between the various currents in this construction. We
sum being the total SL(2)k current J a for the supersymmetric SL(2)k WZW model. The
J 3 =
j3 =
JR = i
XR,
iH =
x =
follows from (2.5) and (2.7) that the following relations hold among the bosonization scalars:
for JR to be well defined in the coset theory.
The NS superconformal primaries of the coset model can be constructed starting from
the primaries of the supersymmetric SL(2)k WZW model, which can be taken to be the
where we have defined
G =
JR =
j j1 ij2.
and charge m and m with respect to j3 and j3. The range of j and m, m will be discussed
later. The SL(2)/U(1) primary Vjs,ml,m is then obtained by factoring out a U(1) primary,2
js,lm,m = Vjs,ml,m eq k2 (mXm X ),
2As we are focusing on the massless string modes in this paper, we only need to take into account those
coset primaries that come directly from current primaries.
q k2 X, J3 = +
nience. The coset model primary Vjs,ml,m has conformal weights
j,m = j(j + 1) + m2
j,m = j(j + 1) + m2
respect to JR. Combining the above two observations, we obtain the Rcharges for the
coset primary Vjs,ml,m ,
R(Vjs,ml,m ) =
R(Vjs,ml,m ) =
Nonnormalizable, delta function normalizable, and normalizable
priDepending on the value of j, the primary operator Vjs,ml,m can either be nonnormalizable,
delta function normalizable, or normalizable along the radial direction of the cigar. The
nonnormalizable primaries correspond to generic real j, whereas the delta function
normomentum along the radial direction of the cigar). Note that the conformal weight formula
3This bound is slightly stronger than that imposed by the noghost theorem in string theory on
Vjs,ml,m = R(j, m, m ; k)Vslj1,m,m ,
R(j, m, m ; k) = (k)2j+1 (1
2j+1 )(j + m + 1)(j m + 1)(2j 1)
k
(1 + 2jk+1 )(m j)(m j)(2j + 1)
nonnegative. If n is negative we simply exchange the role of m and m in the formula for
the reflection coefficient. Using this reflection relation we can restrict the nonnormalizable
upper bound j < k1 to ensure the two point function of Vjs,ml,m is nonsingular [7, 8].3
2
The j3, j3 charges m and m are related to momentum and winding on the cigar (along
the circle direction),
SL(2) [28, 29].
m m = n Z,
m + m = kw kZ,
m =
m = n + wk
So far, for either the nonnormalizable or the delta function normalizable primary operators,
there are no constraining relations between m, m and j.
Importantly, there are also normalizable primary operators at special values of real
j [17, 27], corresponding to principal discrete series of SL(2) [29], namely
where m0 is given by
m0 =
(min{m, m },
min{m, m },
To be more precise, the normalizable vertex operators are the residue of Vj,m,m as
j j [27],
Vjno,mrm,m = Resjj Vj,m,m .
Only delta function normalizable primaries and normalizable primaries are being used
to construct the vertex operators of DSLST. The delta function normalizable primaries of
the SL(2)/U(1) will lead to a continuum of string modes that propagate down the cigar,
whereas the normalizable primaries will give rise to string modes localized at the tip of
the cigar, thus effectively living in six dimensions [7, 27]. The scattering amplitudes of the
latter is the subject of this paper.
The supersymmetric coset model SU(2)k/U(1) can be constructed similarly to the
SL(2)k/U(1) model. We will denote the primary operators and charges in the SU(2)k/U(1)
model with primes in order to distinguish them from those in the SL(2)k/U(1) model. Let
are (in this case there is no distinction between upper and lower indices)
ji0(z)jj0 (0)
Ji0 = ji0 2 ijk0j 0k.
The overall SU(2)k current of the supersymmetric WZW model is given by
Rcurrent J R0,
G0 =
J R0 = k2 j30 +
J30 = i
j30 = i
J R0 = i
X0,
r k 2
r k 2
XR0,
X0 =
r k 2 x0 +
XR0 =
r k 2
The NS superconformal primaries of the SU(2)k/U(1) coset model (which is the same
weights and Rcharges are
j0(j0 + 1) m02
R = k
For later application we will recall below two examples of supersymmetric SU(2)k/U(1)
coset models where the primary operators and correlators can be easily written down.
described as follows.
Supersymmetric SU(2)3/U(1) and the compact boson
with superconformal currents
G0(z) =
G0(z) =
From (2.22) and (2.23), we can read off the relations between the bosonization scalars
with dimension 32 . The coefficient
convention for G0(z).
The Virasoro primary operators are determined as usual
3n 23
n, w Z.
I dz znJ R0(z)(0) = 0,
zr+ 12 G0(z)(0) = 0, r > 0,
which is always true for the Virasoro primaries (2.30). In the NSsector, this implies that
momentum n and the winding number w,
G0+(z)G0(0) 3z3 +
3z3 +
2 T 0(z) +
J R0(0)
the identity operator. The former ones are
Moving on to the Rsector, the superconformal primary condition (2.31) implies that
n, w in (2.30):
1 3n +
1 3n 2 1,
2 1,
2 3n +
2 2
2 3n 2 2
corresponding to the Rsector primary operators
R : exp i 23 0(z) exp i 23 0(z) ,
exp i 23 0(z) exp i 23 0(z) ,
Spectral flow
, R = R = 6 ,
R = R = 2 .
another operator O
weight and Rcharge
= j(j + 1) + (m + )2
R =
On the other hand, in the SU(2)k/U(1) superconformal coset, the spectral flowed operator
R =
su, 12 are Rsector vertex operators.
Zk orbifold
As already mentioned, the worldsheet CFT in the holographic description of DSLST (either
type IIA or type IIB case) is
The Zk orbifold is inherited from the holographic dual of the (nondoublyscaled) LST,
with worldsheet description
stack of NS5branes. It is a standard fact that the supersymmetric SU(2)k WZW model
can be written as the Zk orbifold of the product of a supersymmetric U(1)k WZW model
and a supersymmetric coset model SU(2)k/U(1) [32],
SU(2)k =
Before proceeding to the Zk orbifolding in DSLST, let us recall how (2.41) works.
SU(2)k as a Zk orbifold of U(1)k
We will write down primary operators of the supersymmetric SU(2)k WZW model in the
language of
the unorbifolded U(1)k
SU(2)k /Zk. By comparing with the general primary operators in
SUU((12))k , we will be able to identify the action of the Zk orbifold.
the J30 charge,
j0,m0 ei0H0 = eiq k2 (m0+0)X0 Vjs0u,m,00 ,
language of U(1)k U(1)
SU(2)k /Zk, X0 is the (holomorphic part of the) compact boson at
operator (2.38) in the coset model SU(2)k/U(1), as can be checked by comparing the
q k2 X0. In the
weights and Rcharges on both sides.
To complete the identification of vertex operators on the two sides of (2.41), we need
to include the antiholomorphic part as well. The vertex operators coming from (2.42) are
On the other hand, if we were considering the unorbifolded U(1)k
operators would take the form
iq k2 (m0+0)X0iq k2 ( m0+0)X 0 Vjs0u,m,(0,0m,00).
iq k2 MX0iq k2 M X 0 Vjs0u,m,(0,0m,00),
(mod 1) respectively.
(2.43) would be reproduced from (2.44) with the identification
However, the quantization condition on M and M in the U(1)k are different from that
on m0, m 0 of vertex operators in the supersymmetric SU(2)k via this identification. The
condition M M Z translates into
SL(2)k/U(1) coset model. Hence we will use the same symbol H for the bosonization
J Rtot = iH + iH0,
Note that the orbifold projection demands that the total Zk charge vanishes, and this is in
particular obeyed by (2.45).
coset theory, on which the Zk orbifolding acts nontrivially. In preparation for the deformed
case, let us introduce some notations in the linear dilaton theory.
j0(j0 + 1)
eq k2 jeiH+i0H0 j0,m0 = eq k2 jeiH eiq k2 (m0+0)X0 Vjs0u,m,00 .
Now we will consider the deformation from (the internal part of) the worldsheet theory of
LST to that of DSLST [7, 8], namely
We would like to see how the vertex operator (2.51) is deformed. The weight and the
Rcharge (determined from (2.50)) of the vertex operator (2.51) are
with the quantum numbers satisfying
In the deformed theory, (2.51) maps to the following spectral flowed operator in
which indeed has the same weight and Rcharge (by (2.37) and (2.38)). In other words,
as well as the constraints (2.46) and (2.47), will survive the orbifold.
operators in the deformed theory
SU(2)k /Zk can be written as
Combining the holomorphic and antiholomorphic part, we see that a class of vertex
Identifications among vertex operators in the coset theories
both the SL(2)/U(1) and SU(2)/U(1) coset theories. These identifications can be traced
back to the ones in the bosonic coset theories. Let us start with the bosonic SU(2)kbos /U(1)
at level kbos, with primary operators Vjb0,oms 0 . The quantum number j0 lies in the range
0 j0
kb2os , whereas m0, unlike in the bosonic SU(2)kbos WZW model, is a priori
unconstrained. There is the following identification among primaries labeled by different
quantum numbers [17, 33]:5
Vjb0,oms 0 = V kbb2ooss j0, kb2os +m0
statement that m0m 0 (mod 1) is the charge with respect to the residual Zkbos action on
kbos
the coset theory.
j0,m0 ei0H0 = Vjs0u,m,00 eiq k2 (0+m0)X0
= Vjb0,oms 0 eiq k 22 m0x0 ei0H0 .
kmb0o2s only applies for m0 j0 which
labels the primaries obtained directly from factoring out U(1). Within this range, the identification is only
Vjb0,oms 0 = Vjs0u,m,00 ei0q kk2 XR0+im0 k(2k2) XR0
Vjs0u,m,00 = V ks2u2,0 j0,m0 k 22 eiq kk2 XR0 = V ksu2,01
SU(2)k/U(1) is not invariant under a shift on m0 alone. Rather, we have
to the Zk symmetry. We can write the above identifications in a more compact form,
Similarly, primary operators in the bosonic SL(2)k+2/U(1) are subject the following
identification6 [17, 34]
from which we obtain the identification for the supersymmetric SL(2)k/U(1) theory
2 j0,m0 k 22 = V ksu2,0+1
Vjb,mos = V kb2o2sj, k+22 +m,
Vjs,ml, = V ks2l,2+1j,m k+22 = V ksl,21
Note the sign difference in 1 and m
k+2 when compared with SU(2)k/U(1). Once
2
Massless string states
Now we discuss the construction of physical vertex operators in type IIA string theory on
SU(2)k /Zk, which is the Tdual description of IIB (1, 1) DSLST [7, 8].
U(1)
We will focus on the explicit description of massless string modes in the R1,5, localized at
the tip of the cigar. We will further restrict our attention to bosonic string modes, and
discuss the (NS,NS) sector and (R,R) sector separately.
the highest weight state of Dj to the lowest weight state of D+kbos j2
2
6We are considering here coset primaries that come directly from the lowest weight (resp. highest weight)
. The rest of the relations may be
thought of as extending the definition of Vjb, mos (see [34] for an explanation of the origin of this identification).
Consider the (NS,NS)sector vertex operators of the form [17, 27]
VNS = eeipX Vjs,ml,(,m,)Vj0,m0,m 0
J 3 =
in the (NS,NS)sector. Since VNS has no R1,5 spacetime index, it should be the vertex
operator for the sixdimensional scalar fields.
The mass shell condition is
2
= 0.
the Zk orbifold condition (2.57). The onshell condition for massless states then reduces to
Next let us examine the chiral GSO projection condition,
j0(j0 + 1) j(j + 1)
FL + R 1 2Z,
where FL is the holomorphic worldsheet fermion number of R1,5 (and similar for the
antiholomorphic sector). The total Rcharge can be computed using spectral flow (2.37)
R =
vertex operator of the form (2.66) that obeys GSO projection condition must satisfy (see
appendix A.1)
normalizable massless vertex operators in these two cases are
j =
, m =
j0 =
, m0 = 2 ,
for ` = 0, 1, , k 2,
( = 1, 0 = 0 or = 0, 0 = 1).
Note that m, m0 are both positive in these cases.
holomorphic part,
To summarize, the normalizable vertex operators from the internal CFT have, in their
2 , `+22 V `
2s,u,02` , V 2`s,l,`0+22 V 2`s,u2`,1,
2 , `+22 V `
The operators in (2.74) are not all independent, however. Recall that we have the
identifications (2.63) and (2.65), and therefore,
negative in these cases.
with the solutions
k 22 . Note that m, m0 are both
j = , m =
j0 = , m0 = ,
2 , `+22 V `
2s,u,02` = V ksl,20` , k2` V k2` , k2` ,
su,1
2 2 2
2 , `+22 V `
su,`1 = V ksl,21` , k2` V k2` , k2` .
su,0
2 , 2 2 2 2
In particular, the identification flips the sign of m and m0. This will be important when
we include the antiholomorphic part of the vertex operators.
Combining with the antiholomorphic part, normalizability requires in addition that
this case m < 0 but m > 0. On the other hand, from the identifications (2.75), some of
In the end, there are four inequivalent pairings between the holomorphic and
antiholomorphic quantum numbers that are allowed in the massless vertex operator
0 1
0 1 1
1 1 0
m, m , m0, m0 are all negative for the vertex operators pairing this way (see (2.74)), we can
The explicit forms of the four sets of normalizable vertex operators are
VNS1,` = eeipX V 2`s,l,(1,1)
su,(0,0) , VNS2,` = eeipX V 2`s,l,(0,0)
`+22 , `+22 V 2` , 2` , 2
su,(1, `1),
VNS3,` = eeipX V 2`s,l,(1`,+202), `+22 V 2` , 2` , 2
su,(0,1`), VNS4,` = eeipX V 2`s,l,(0`,+212), `+22 V 2` , 2` , 2
su,(1,0`),
or, equivalently, using the identifications (2.75), we can rewrite them as
VNS3,` = eeipX V 2`s,l,`(+202, ,`1+2)2 V `su`,(1`,0), VNS4,` = eeipX V 2`s,l,`(+221,,`0+2)2 V `su`,(0`,1),
+ +
2 , 2 , 2 2 , 2 , 2
VNSi, ` = VNSi, k2`,
i = 1, 2, 3, 4.
VNS1,0, VNS2,k2, or equivalently VNS1,k2, VN+S2,0, are in the untwisted sector, i.e. they
+
Let us compare this with the NS5brane or six dimensional gauge theory description.
We are at the point in the Coulomb branch moduli space where the k NS5branes are
spread on a circle in R
and at the same time rotates the circle of spread. The center of mass mode decouples,
are scalars that are linear combinations of collective coordinates of the NS5brane in the
R2 that contains the circle of spread, while Zej are scalars associated with the remaining
is a center of mass mode and decouples, thus absent from the DSLST spectrum. The
is absent from the spectrum. So indeed there is only a single massless complex scalar Z0
that is uncharged under the Zk symmetry.
In the (R,R)sector, we consider the vertex operators [17, 27]
VR = a,a e 2 2 eipX SaSea Vj,m,m
the U(1)k1 gauge bosons.
where Sa, Sea are the spin fields in the R1,5, and a and a are the indices in the 4 and 4 of
in the 15. In the massless case, VR will give the vertex operators for the field strength of
The onshell condition for the vertex operator (2.80) is
8
= 0. (2.81)
j0(j0 + 1) j(j + 1)
Its straightforward to derive that (see appendix A.2) physical vertex operators surviving
1
the GSO projection FL + R 2 2Z must have the following combinations of spectral flow
= 2 , 0 = 2 , and FL =
2 , 2 .
Let us first consider the case = 12 , 0 = 2 . We must have j = j0 and m = m0 1. The
1
solutions for the normalizable states are
or, equivalently,
The two are related by the reflection (2.86),
Note that m, m0 are both negative in this case.
solutions for the normalizable states are
Similarly, in the case = 2 , 0 = 12 , we must have j = j0 and m = m0 + 1. The
1
Note that m, m0 are both positive in this case.
are in fact identified by (2.63) and (2.65),
2 , `+22 V `
su,1/2 = V ksl,21`/,2k2` V k2` , k2` .
su,1/2
2 , 2` 2 2 2
Combining with the antiholomorphic part, as before, normalizability of the vertex
, 0 = 21 in the holomorphic
sector with = 2 , 0 = 12 in the antiholomorphic sector. In the end, the gauge boson
1
. The fact that m and m must take the
vertex operators are
VR,` = a,a e 2 2 eipX SaSea V `
2 , `+22 , `+22 V `
su,(1/2,1/2), ` = 0, 1, , k 2,
j =
, m =
` = 0, 1, , k 2
j =
, m =
` = 0, 1, , k 2,
j0 =
, m0 = 2 ,
j0 =
, m0 =
and Q = b + b1 = 5/2.
k = 4, ` = 0, 2 :
ADSLST 78.96 144.6 2 s12 + O(02s2).
4 ` ,
k = 4, ` = 0, 2 :
ADSLST
= 1.831.
due to the symmetry in s12, s13, and s14, so ADSLST is zero.
k = 5
The relevant operators in the supersymmetric SU(2)5/U(1) coset model for the gauge boson
vertex operators (2.87) and (2.88) are
Combining with the correlators in the R1,5 and SL(2)4/U(1) parts, we obtain the
following scattering amplitude,
Z dP
ADSLST
1 +
1 z + 1
1 z
C 3, 4, 2 iP F (1, 2, 3, 4; P ; z)2,
V0s,u0,,01/2,1/2(z, z) = V 32 , 32 , 2
3
V 1s,u1,1,/12,1/2(z, z) = V1s,u,1,1/12,1/2(z, z), = =
2 2 2
V1s,u1,,11/2,1/2(z, z) = V 12 , 12 , 2
su,1/2,1/2(z, z), = =
1
V 32s,u32,1,/322,1/2(z, z) = V0,0,0
su,1/2,1/2(z, z)
, R = R =
, R = R =
, R = R = 10
, R = R = 10
These four operators with dimension 3/40 can be realized as a free boson 0 and the
V0s,u0,,01/2,1/2(z, z) = V 32 , 32 , 2
su,1/2,1/2(z, z) = exp i
3
11The threestate Potts model is reviewed in appendix B.
0(z) exp i
i
0(z) exp i
i
0(z) exp i
= hV 3 3 3
= z 10 1 z 10
(z, z)V 3 3 3
(z, z)V 1 1 1
su,1/2,1/2
su,1/2,1/2
(, )i
= hV1,1,1
(0, 0)V1,1,1
(z, z)V1,1,1
su,1/2,1/2
(1, 1)V1,1,1
su,1/2,1/2
(, )i
(z, z)V0,0,0
(0, 0)V0,0,0
su,1/2,1/2
(1, 1)V0,0,0
su,1/2,1/2
(, )i
su,1/2,1/2
su,1/2,1/2
(, )i
and Q = b + b1 = 1/ 5 +
5. The amplitude is numerically computed to be
k = 5, ` = 0, 3 : ADSLST 110.4 264.5
iP
k = 5, ` = 0, 3 :
ADSLST
= 2.397.
(z, z) = V1,1,1
su,1/2,1/2
(z, z) = V 1
(z, z) = V0,0,0
su,1/2,1/2
su,1/2,1/2
1 1
= z 30 1 z 30 GPotts (z, z).
` = 0, 3
` = 1, 2
ADSLST
Z dP
ADSLST
Z dP
20 s13+ 175 GPotts (z, z)
iP
,
The fourpoint functions of the SU(2)5/U(1) coset model for each value of `, which
takes three possible values, 0, 1, 2, 3, are then
27.92 + 0 s12
51.28 60.11s12
78.96 144.6s12
157.9 + 0 s12
110.4 264.5s12
220.7 304.6s12
expansion, i.e. the coefficient of s12 divided by the s12independent term in the full amplitudes.
These ratios exactly match with the ratios of the 6D SU(k) SYM amplitudes listed in the last
appendix B. The amplitude is numerically computed to be
k = 5, ` = 1, 2 :
ADSLST
= 1.380.
The overall normalization of the amplitudes (which depends on k and `) have not been
lently, the expansion in the Mandelstam variables, of a particular amplitude are computed
unambiguously. These exactly match with the 1loop and 2loop 6D SU(k) SYM
ampliamplitudes to higher order, and they should be compared to higher than 2loop amplitudes
on the SYM side. We will comment on this later.
Comparison to 6D superYangMills amplitudes
The strong coupling limit of DSLST, namely LST, reduces to 6D (1, 1) SU(k) SYM in the
low energy limit. It is conceivable that the full dynamics of the massless degrees of freedom
at the origin of the Coulomb branch is described by a Wilsonian effective action, which
is that of SU(k) SYM deformed by an infinite set of higher dimensional operators, with a
12A fully supersymmetric regulator is assumed.
The DSLST corresponds to a point away from the origin on the Coulomb branch of this
theory, at which a Zk U(1) subgroup of the SO(4) Rsymmetry is preserved. In the gauge
A(E) = X(gY M E)2L+2AL(mW /E).
A priori, AL is not quite the same as the Lloop amplitude. As we expand the scalars
around their vevs, we obtain couplings that involve positive powers of gY M and
non
It seems reasonable to assume that the Wilsonian effective
Lagrangian. With such vertices, AL generally receives contributions from diagrams of no
The perturbative amplitude in DSLST, on the other hand, has the structure (after
A(E) = X(gY M mW )2h2Alhst
where h labels the genus (which is entirely unrelated to the loop order in the SYM theory).
also has an analytic expansion in 1/m2W at large mW , namely
While AL(mW /gY M ) is naturally defined by an analytic expansion in mW /gY M (as the
theory is free of infrared divergences), the duality with DSLST suggests that AL(mW /gY M )
AL(mW /E) = X
and in particular, we expect the tree level DSLST amplitude to agree with the 1/m2W part
of the gauge theory amplitude,
= X(gY M E)2L+4A(L1).
pendix D), the 3loop divergence is absent when the external legs are restricted to the
Cartan subalgebra. It turns out that the first UV divergence of the Cartan gluon
fourpoint amplitude arises at fourloop.13 In any case, one can ask whether the perturbative
to be yes, and in fact the finite 1loop amplitude in SYM, when expanded to first order
in 1/m2W , agrees with the leading low energy term in the tree level DSLST amplitude
13We thank the authors of [40] for pointing this out. The relevant fourloop divergence can be extracted
up to an overall multiplicative constant [17]. Since the 6D SYM by itself is UV finite at
2loop as well, one might suspect that A2(mW /E) is also given entirely by the 2loop 6D
Remarkably, we will find that this is indeed the case.
Structure of perturbative amplitudes
We will consider fourpoint amplitudes in SU(k) maximally supersymmetric YangMills
theory obtained from unitarity cut methods [1015]. While such amplitudes are mostly
studied in fourdimensional gauge theories, where the Lloop result can be expressed in
terms of the tree level amplitude together with scalar loop integrals, such formulae admit
straightforward generalizations to higher dimensions. It is known that up to 3loop order
dimensions can be extended to D dimensions by simply replacing the relevant scalar loop
integrals by the Ddimensional loop integrals [10, 11].
We will express the amplitudes in 6D SYM in terms of 6dimensional spinor helicity
are SU(2) SU(2) little group indices, and A, B are spinor indices of SO(6) or SO(5, 1)
Lorentz group. The amplitudes involving various particles in a supermultiplet will be
is convenient to express the latter in terms of the supermomenta
The colorordered fourpoint treelevel superamplitude can be written as
tree(1, 2, 3, 4) = s12s23
off. The delta function is the one in Grassmann variables. Explicitly, it can be expanded as
8(X qi) = 4(X qiA)4(X qiB)
(4!)2 i,j,k,`,m,n,r,s=1
ABCDqiAqjBqkC q`D EF GH qmE qnF qrGqsH .
respect to the SU(2) SU(2) Rsymmetry (not to be confused with the little group),
or rather, U(1) U(1) Cartan generators of the Rsymmetry group, these scalars have
transverse R
R2 of the 5brane are linear combinations of these two Cartan generators.
In this paper we are interested in the scattering amplitudes of the massless gluons in
due to the vanishing color factor. The nonvanishing loop amplitudes of the Cartan gluons
contain W bosons in the loops (as well as possibly massless gluon propagators at two loops
and higher).
The full 1loop amplitude in SYM is given by14 [12] (see also [13])
color factor associated with the box diagram. This relation holds in any D, which we now
need to replace I41loop(s12, s14) by15
scaled little string theory.
written in the form
In the last line we have expanded the result in 1/m2W . As explained, it is the order 1/m2W
result of the SYM amplitude that will be compared with the genus zero amplitude of double
In the end, the 1loop amplitude of Cartan gluons on the Coulomb branch can be
A1loop(1, 2, 3, 4) = i8(X qi)
C1234 + C1324 + C1243
summed over the species of W bosons if k > 2.
tree(1, 2, 3, 4) refers to the colorordered partial amplitude (the full amplitude is obtained by
summing over s12, s13, s14 channels), A
1loop(1, 2, 3, 4) and A
2loop(1, 2, 3, 4) are full amplitudes.
15The generalization to massive propagators in the loop is justified by consideration of unitarity cuts.
The full 2loop amplitude is given by the treelevel amplitude multiplied by 2loop
scalar integrals [14]
A2loop(1, 2, 3, 4) = s12s23Atree(1, 2, 3, 4)
2loop,P and Aabcd
ure 2. Once again, the propagators in the loops will be replaced by the appropriate massive
W boson or massless gluon propagators in the amplitudes on the Coulomb branch of the
The 3 and higherloop amplitudes generally contain logarithmic divergences. It is
likely that they still contain nontrivial information that captures the DSLST amplitudes
Evaluation of color factors and box integrals
need to impose the traceless condition by hand in this case, as the overall U(1) decouples
due to the interaction vertices). The external massless gluons will be labeled by vectors
~v1, , ~v4 in the Cartan subalgebra of su(k). The mass of the (ij)W boson is
mij = r0i j  = 2r0 sin
eter that will be related to the inverse string coupling of DSLST.
Expanding around the point in Coulomb branch with Zk symmetry, corresponding to
the NS5branes spreading out on the circle in a transverse R2, it is convenient to take ~va
to be Zk charge eigenstates,
vaj = (j1)na , j = 1, , k,
X na 0 mod k.
na of interest are16
for ` = 0, 1, , k 2.
As discussed before, the gluon vertex operator VR,` in DSLST has Zk momentum
(` + 1). Therefore, we see that in order to compare with the DSLST scattering amplitude
n1 = n2 = ` + 1, n3 = n4 = k (` + 1),
The 1loop amplitude, expanded to order 1/m2W , is of the form
A1loop =
i6=j
A(s12, s14) + O(mW4),
Plugging in the explicit expression for vai, we can further write
A(s12, s14) =
A1loop =
+ O(r04)
As was shown in [17], the sum collapses into a curiously simple answer,
A1loop =
2 A(s12, s14)min{na, k na} + O(r04).
Now consider the 2loop amplitude. In the planar case, let us label the W boson
running through vertices 1,2 by (ij), the W boson running through 3,4 by (`m), and the
where the scalar loop integral is
I2loop,P (mij , m`m, mnr) jnr`mi j`mnri
4
I2loop,P (mij , m`i, mj`) Y (vai va) Y (va` vai),
j
4
2 Y (vai va) Y (va` vam)
j
1 Z d6`1 d6`2
16We shift n3 and n4 by k for later convenience.
6.048
16.876
34.594
39.883
4.500
12.435
25.327
29.136
+ O(r04)
+ O(r04).
A1223l4oop,P
1/m2W expansion for fourgluon scattering in 6D SU(k) SYM. Here we choose the Zk charges for
The numbers are in units of s12s23A
This gives the colorweighted planar amplitude
Qa=1,2 e ikna (j`) sin( na(ij) ) Qa=3,4 sin( na(i`) )
k k
0 m,r=0
8k kX1 Z d6`1 d6`2
Qa=1,2 e ikna (rm) sin( nkam ) Qa=3,4 sin( nkar )
(2)6 (2)6 (`21 + sin2 km )3(`22 + sin2 kr )3((`1 + `2)2 + sin2 (mr) ) + O(r04).
k
(`21 + sin2 (ikj) )2(`22 + sin2 (ik`) )3((`1 + `2)2 + sin2 (jk`) )2
(2)6 (2)6 (`21 + sin2 km )2(`22 + sin2 kr )2((`1 + `2)2 + sin2 (m+r) )3
k
These convergent integrals and sums over color factors can be computed numerically.
The results for the planar and nonplanar contributions to the twoloop amplitude are
given in table 2, and the full twoloop amplitudes, whose expression is given by (4.11), are
listed in table 3. We see that the ratios listed in the last column remarkably match with
the ratios computed from DSLST that are listed in table 1, to the numerical precision of
the conformal block integration.
The tree DSLST amplitudes provides all order results gY2 M and first order in 1/m2W of
the UV completed 6D gauge theory on its Coulomb branch. While the agreement of 6D
A1loop
10.548
29.311
59.922
69.019
1.1720
1.8319
2.3969
1.3804
fourgluon scattering in 6D SU(k) SYM. Here we choose the Zk charges for the external gluons to
both in units of s12s23A
burning question is whether the SYM 3loop amplitude, which as discussed is finite when
the external lines are restricted to the Cartan subalgebra, agrees with the DSLST at
nextreduced to scalar integrals, it is merely a matter of evaluating these scalar integrals to
answer the question. We hope to report on the result in the near future.
One may also try to carry out the DSLST amplitude computation to higher genus,
and compare with the higher order terms in the 1/m2W expansion of the SYM amplitude
at each loop order. This is not easy as the relevant genus one fourpoint function in the
cigar coset CFT is not yet known, but would nonetheless be interesting.
From the point of view of the Abelian effective action on the Coulomb branch, the
2loop amplitude of order 1/m2W comes from the 1/4 BPS dimension 10 operator of the
of this term in the Coulomb effective action, with respect to higher dimensional nonBPS
operator corrections to the nonAbelian SYM theory. If so, then the 3loop test will be
particularly important, and an agreement with the DSLST tree amplitude at the next order
In any case, the big question here is, to what extent will the agreement between the
massless amplitudes of pure 6D SYM on the Coulomb branch and DSLST hold, and why
do they agree? It so happens that the Cartan gluon amplitude becomes divergent at
fourloop [40]. Therefore we will definitely see some nontrivial disagreement with the LST
known but the UV divergence with external legs in the Cartan subalgebra has yet to be
extracted [43].18 A priori, there could be all sorts of higher dimensional operators that
enter the Wilsonian effective action of the 6D gauge theory and correct the amplitudes
of the SYM theory itself. After all, we do expect the presence of the dimension 10
non17We thank Ofer Aharony for pointing this out.
18We thank the authors of [40] for explaining to us the results of [40, 43, 44].
BPS operator (see for instance [45]) as the counter term that cancels the general 3loop
divergence, even though this operator vanishes when the fields are restricted to the Cartan
subalgebra. A systematic investigation of the higher dimensional counter terms and their
effect on the Cartan gluon scattering amplitude is left to future work.
Finally, let us mention that the Wbosons in the 6D SYM are dual to D1branes
stretched between the NS5branes. The scatterings of strings with the D1branes
correspond to the scatterings of the Cartan gluons with the Wbosons, and also the scatterings
of the D1branes with themselves are dual to the scatterings of Wbosons. Some aspects
of open strings and Dbranes in DSLST are studied in [46] (also see [47] for the Dbranes
to the closed string twopoint amplitudes on a disc ending on stretched D1branes, and
compare with the scattering amplitudes of two Cartan gluons and two Wbosons.
Acknowledgments
We would like to thank Ofer Aharony, Clay Cordova, Lance Dixon, Thomas Dumitrescu,
Yutin Huang, Daniel Jafferis, Ingo Kirsch, SooJong Rey, David SimmonsDuffin, and
Andy Strominger for helpful conversations and correspondences at various stages of this
project. We would like to thank the Kavli Institute for Theoretical Physics and Aspen
Center for Physics during the course of this work. C.M.C. has been supported in part
by a KITP Graduate Fellowship. C.M.C. would like to thank the Physics Department
of National Taiwan University for hospitality during the final stage of the work. S.H.S.
is supported by the Kao Fellowship and the An Wang Fellowship at Harvard University.
X.Y. is supported by a Sloan Fellowship and a Simons Investigator Award from the
Simons Foundation. This work is also supported by NSF Award PHY0847457, and by the
Fundamental Laws Initiative Fund at Harvard University.
2(j0 j)(j0 + j + 1) = k(1 2 0 ).
Normalizable vertex operators
Meanwhile the GSO condition demands that
to solve is
and therefore,
For normalizable vertex operators we have
j0 j Z0.
m > j, j0 m0,
which implies that
This further implies,
which is equivalent to
This is impossible.
j j0 < m m0 m m0 =  0.
Assuming j > j0, we can combine the two equalities above to get
2j0 j > 2 + 02 1.
2 + 02 1 < 2(j j0) < 2 0.
Therefore the only normalizable solutions that survive the GSO projection satisfy
The onshell condition in the R sector is
and the GSO condition becomes
4(j0 j)(j0 + j + 1) = k(1 22 202),
Since we are looking at halfinteger spectral flows, this implies that
and therefore,
Then the massshell condition requires that
22 + 202,  02 4Z + 1.
j0 j Z0.
Assuming j > j0, we have the following inequality from the normalizability condition
which demands
This is impossible.
22 + 202 1 < 4(j j0) < 4 0,
4 0 22 + 202 1 + 8  02 + 7.
Their dimensions are
= , X = , Y = 3, = =
In addition to the scalar primaries, there are spin 1 primaries
Threestate Potts model
model. The scalar primary operators in this theory are
1, , X, Y, , , Z, Z.
and also spin 3 primaries,
is relevant to us,
The fusion rules of the primaries are given in [48]. Here, we only present the part that
can be written as
4 GPotts (z, z)
the conformal block.
4 GPotts(z, z)
F 7 (z)F 2 (z)
C0,3 F0(z)F3(z) C3,0
F3(z)F0(z) ,
4 GPotts(z, z)
F 7 (z)F 2 (z)
+ C0,3 F0(z)F3(z) + C3,0
F3(z)F0(z) ,
and the rest are found to be
1.09236,
Numerical methods
Zamolodchikov recurrence formula for conformal blocks
function of four scalars is expressed in terms of the threepoint functions and the conformal
where z is the cross ratio
q is defined by
z =
n=
F (i; z) = (16q)P 2 z Q42 1 2 (1 z) Q42 1 3
3(q)3Q24(1 +2 +3 +4 )H(i2; q),
K(1 z)
K(z) =
pt(1 t)(1 zt)
The product of (r, s) is taken over
and the product of (k, `) is taken over
Finally, the function H is determined by the following recurrence relation
m,n1
r = m + 1, m + 3, , m 1,
s = n + 1, n + 3, , n 1,
k = m + 1, m + 2, , m,
` = n + 1, n + 2, , n,
faster than the corresponding series in x; this can be seen, for example, for small values of
z if one notes q = 1z6 + O(z2).
then the matrix equation is
H(i2, 1,2 + 2q) = 1 + qq1RR,211+,,112
q2R2,1
1,2+22,1 H(i2, 1,2 + 2q)
to order qN .
We note some caveats in the implementation of this method. For special values of
the central charge, for example when c equals the central charge of a minimal model, or
therefore certain coefficients appearing in the recurrence relation diverge. Nonetheless, we
can deform the value of the central charge from c to c + , and as it must all the poles in
cancel. Therefore, with a small
and high enough numerical precision (high enough so
that the divergences cancel properly on the computer), we can still compute the conformal
blocks for these seemingly pathological values of the central charge.
Crossing symmetry
Consider the fourpoint function (C.1), and let us define
G(1, 2, 3, 4z)
which satisfies the following crossing relations
The complete set of transformations are
T 2 = S2 = 1, (T S)3 = 1.
, (1432)
z 1 , (1342)
ST S : z z 1
ST : z 1 z
, (1243)
, (1423)
T : z 1 z, (1324).
We can divide the complex plane into six fundamental regions:
I : Re z 2 , z 1 1,
II : z 1, z 1 1,
III : Re z 2 , z 1,
region I by the ST S, T S, T, ST, S transformations, respectively. An integral involving
conformal blocks over the entire complex plane can be rewritten as an integral over only
region I. This is useful for doing numerical integration because, first, region I is bounded,
and second, in this region q is bounded above by 0.0658287, which means that the
Zamolodchikov recurrence formula (C.6) converges very quickly.
amplitude for k = 2, 3, 4, 5 and ` = 0 as
d2zzs12 1 zs13 G(1, 2, 3, 4z)
1 zs12 (zs13 + zs14 )G(1, 3, 2, 4z) ,
and for k = 5, ` = 1 as
d2zz 31 s12 1 z 32 s13 G(1, 2, 3, 4z)GPotts (z)
2 2
(zs13 1 zs14 + zs14 1 zs13 )z 3 1 z 3 G(1, 4, 3, 2z)GPotts(z)
2 1
1 zs12 (zs13 + zs14 )z 3 1 z 3 G(1, 3, 2, 4z)GPotts (z) ,
3loop UV finiteness of the fourpoint amplitude for the Cartan gluons
In this appendix we verify that the 3loop 4point amplitudes of the U(1) Cartan gluons
(photon) in the SU(k) SYM is UV finite (see also [44]). This is indeed expected both
and from the inspection on the possible counter terms mentioned in the introduction.
The amplitude is reduced to scalar 3loop integrals, summarized in figure 2 of [18].
The four potentially logarithmic divergent diagrams are shown in figure 3 (ignoring the
signs on vertices for now). We will compute the divergent parts of these diagrams with
color factors included and show that they cancel among themselves.
Let us start with the SU(2) case where no actual calculation is needed to show the
cancellation of UV divergence. The key fact here is that since there is only one species of
6D SYM. The signs for the internal vertices denote the two index structures in the double line
notation; plus for the left vertex and minus for the right vertex in figure 4. We label the diagrams
following the notation in figure 2 of [18]. The above sign assignments together with the other four
of the scattering amplitudes of four Cartan gluons.
(k) = i
Cartan gluon, the amplitudes are invariant under permutation of all four external legs. It
follows that the logarithmic divergent part of the amplitude is proportional to (s12 + s13 +
Moving on to the general SU(k) 6D SYM, it suffices to show that the UV divergent
part is invariant under cyclic permutations on the external legs 2, 3, 4, from which it again
In the double line notation, each 3point vertex can be written as the difference of two
vertices shown in figure 4 with different index structures. Each diagram in figure 3 then
is easy to see that the only diagrams that give noncyclic invariant amplitudes are the four
The color factors for the four external Cartan gluons will be labeled by ~v1, , ~v4 in
the Cartan subalgebra of su(k) as in the previous section. The four diagrams in figure 3
only interested in the divergent part, we have set the external momenta to be zero.The number
indicates the propagator should be raised to the corresponding power.
can be expressed as19
3 2
1 Ilog
UVdivergent scalar integrals I1
log and I2log are defined in figure 5.
the vertices. The stands for the finite as well as the cyclic invariant terms. The two
Next, we need to sum over all the permutations on the external legs. After taking
the symmetry factors for each diagram appropriately, the noncyclic invariant part of the
divergent amplitude is proportional to
2N 2s12(v~1v~2)(v~3v~4)
1 Ilog +
1 Ilog
2I1log
2 2
2
3 2
1 Ilog
= 0. (D.2)
Note that we have grouped (v~1 v~3)(v~2 v~4) + (v~1 v~4)(v~2 v~3) with (v~1 v~2)(v~3 v~4) to the
In summary, in this appendix we have showed that the 3loop 4point amplitudes for
gluons in the Cartan subalgebra is free from divergence and we are then left with a finite
amplitude. The comparison with the DSLST amplitude at this order will be left for future
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