A 4D-2D equivalence for large-N Yang-Mills theoryGökçe Başar,1, ∗ Aleksey Cherman,2, † Keith R. Dienes,3, 1, ‡ and David A. McGady4, 5, §1
2

Department of Physics, University of Maryland, College Park, MD 20742 USA
Department of Physics, University of Minnesota, Minneapolis, MN 55455 USA
3
Department of Physics, University of Arizona, Tucson, AZ 85721 USA
4
Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8586, Japan
5
Department of Physics, Princeton University, Princeton NJ 08544 USA

General string-theoretic considerations suggest that four-dimensional large-N gauge theories
should have dual descriptions in terms of two-dimensional conformal field theories. However, fornon-supersymmetric confining theories such as pure Yang-Mills theory, a long-standing challenge
has been to explicitly show that such dual descriptions actually exist. In this paper, we consider
the large-N limit of four-dimensional pure Yang-Mills theory compactified on a three-sphere in thesolvable limit where the sphere radius is small compared to the strong length scale, and demonstratethat the confined-phase spectrum of this gauge theory coincides with the spectrum of an irrational
two-dimensional conformal field theory.

Introduction. Confining gauge theories in the largeN limit are believed to have dual descriptions as weaklycoupled string theories [1]. Since string theories have 2Dworldsheet conformal field theory (CFT) descriptions, itis therefore expected that confining 4D gauge theories
may have alternative descriptions based on 2D CFTs.
However, for non-supersymmetric quantum field theories(QFTs) such as Yang-Mills (YM) theory, no concrete relation between large-N confining theories and 2D CFTs
has ever been found.
In this paper we tackle this problem by studying thelarge-N limit of 4D pure SU (N ) YM theory, formulated
at temperature T = β −1 and compactified on a threesphere S 3 of radius R. One can thus view the theory3
× Sβ1 with Euclidean metric signature.
as living on SR
The virtues of this setting are two-fold. First, thanks toasymptotic freedom, if we take ΛR  1 where Λ is the
YM strong scale, then the ’t Hooft coupling λ ≡ g 2 N
becomes small — i.e. λ(1/R) → 0. As a result, thetheory becomes solvable for any temperature β ∼ N 0 .
Second, it is known [2] that large-N YM theory stays in
the confined phase when β/R & 1, even when λ → 0. In
this context “confinement” means that the system has
an unbroken center symmetry and that its free energy
scales as N 0 . As sketched in Fig. 1, it is plausible thatthe physics of YM theory is smooth as a function of ΛR.
Thus, the ΛR  1 regime of the large-N confined phaserepresents a particularly tractable 4D starting point inour search for a dual 2D description.
Rather than attempt a string-theory construction of a
2D dual for large-N YM theory, we shall instead analyze
the confined-phase spectrum of YM theory in the solvable
ΛR  1 limit. Remarkably, we find that a surprisingly
simple 2D CFT description emerges. Thus, in this limit,
we conclude that the large-N confined-phase spectrum of
4D YM theory coincides with the spectrum of a 2D CFT.
Specifically, recall that the spectrum of a QFT is encoded in its thermal partition function. We take 4D YM
theory to be minimally coupled to the S 3 metric, so that

β/R

Confined(1)β/R ∼ 1/(Λ R)
Deconfined

0

ΛR

1

FIG. 1. A conjectured phase diagram for large-N YM theory
on S 3 × S 1 . In the analytically tractable regime ΛR  1, thedeconfinement transition occurs at β ∼ R, while for ΛR  1,lattice studies have shown that it occurs at β ∼ 1/Λ. This
sketch illustrates the natural conjecture that these two limiting cases are smoothly connected. The results of this paper
apply in the ΛR → 0 region indicated by the blue line.

the Kaluza-Klein energies on the three-sphere are given
by En = n/R in the λ → 0 limit [2]. The partition
function then takes the form
ZYM (β/R) =∞X

dn e−βEn =

n=0

∞
X

dn q n(1)

n=0where q = e−β/R and dn counts the number of states withenergy En . Our main result, then, will be the demonstration that ZYM coincides with a chiral partition functionof a 2D CFT:ZYM (τ ) = Z2D (τ ) .

(2)

In writing Eq. (2), we have analytically continued q toe2πiτ with τ ∈ H, the complex upper half-plane. Thus
Im τ = β/(2πR). It can also be shown that Re τ =µI β/(2π), where µI ∈ R is an imaginary chemical potential for a combination of the S 3 angular momenta.

2
The 4D partition function. We begin by briefly
explaining the computation of ZYM , leaving a moreleisurely exposition to Ref. [3]. To calculate the 4D partition function ZYM (τ ), we take the large-N limit withΛ held fixed, which means taking the continuum limit
after the large-N limit. Working on S 3 × S 1 , we allow an imaginary chemical potential µI for the Cartan
Q3L + Q3R charges associated to the two SU (2) angularmomenta on S 3 , which has the isometry group SO(4) 'SU (2)L × SU (2)R , and assume that β, R, µI ∼ N 0 . We
will not consider states with energies & N because theylie beyond our UV cutoff. As is typical in studies oflarge-N theories, we shall work with the U (N ) version
of YM theory rather than the SU (N ) version [4]. WhenΛR → 0, the microscopic degrees of freedom of YM theory reduce to an infinite collection of color-adjoint-valued
harmonic oscillators. These oscillators are counted by themassless-vector partition function, which can be written
as zv (τ ) = (6q 2 − 2q 3 )/(1 − q)3 . The physical states
are then determined by imposing the color Gauss law.
In the λ = 0 confined phase, the physical single-particle
states can be identified with single-trace operators, andtheir energies are proportional to their scaling dimensions. The counting problem for these states, and also forthe multi-particle states, has been solved [2, 5], and theresulting grand-canonical confined-phase partition function is given by
ZYM (τ ) =∞Y(1 − q n )31=
n
1 − zv (q ) n=1 1 − 3q n − 3q 2n + q 3nn=1∞
Y

= 1 + 6q 2 + 16q 3 + 72q 4 + ...

(3)

As expected in any confining large-N theory, we findthat the dn grow exponentially for large n. Thus, there
are Hagedorn singularities in ZYM (τ ). In Eq. (3), we
Cnfind√ dn ∼ e and En ∼ n for large n, with C ≡ log(2 +√3) ≈ 1.317. This√ contrasts with the behaviors dn ∼e n and En ∼ n that would arise for a string theory
with a flat target space. Of course, we are not in flatspace: the spacetime curvature is ∼ 1/R, which is of thesame scale as the effective string tension α0 ∼ 1/R2 that
follows from our spectrum. The scaling properties of dn
in Eq. (3) imply that the leading Hagedorn singularity of
ZYM (β, µI ) is at βH /R = C, µI = 0. Consequently, atµI = 0, there must be a phase transition to a deconfinedphase at βH . This is discussed in detail in Refs. [2, 6].
Modular symmetries. We now observe that the denominator in Eq. (3) can be factorized with roots thatare inverses of each other:
1 − 3q n − 3q 2n + q 3n = (1 + q n )(1 − q n z)(1 − q n /z) (4)√where z = 2 + 3. This pivotal algebraic observation
was first made in Ref. [7] in the context of uncovering asubtle “temperature-reflection” symmetry for ZYM . For
our purposes, however, the key point is that this allows

ZYM to be written as
ZYM =

∞
Y
n=1

(1 − q n )3
(1 + q n )(1 − q n z)(1 − q n z −1 ).

(5)This observation is very important because the structure of Eq. (5) matches the structure of the productrepresentations of the Dedekind η-function and generalized Jacobi ϑ-functions. (In the related context ofadjoint QCD, this was also noted in Ref. [8].) Specifically, the Dedekind Q
η-function has the product represen∞tation η(τ ) = q 1/24 n=1 (1 − q n ), while the generalized
αP2
ϑ-function ϑ β (τ ) ≡ n∈Z q (n+α) /2 e2πinβ has a product representation of the form
∞ 
Y
α
α2 /2
(1 − q n )ϑ β (τ ) = q
n=1

× (1 + q1
n− 2 +α 2iπβe)(1 + q1
n− 2 −α −2iπβ

e


) .
(6)

Under the S : τ → −1/τ and T : τ → τ + 1 generators
Z), we find η(−1/τ ) =√ of the modular group SL(2,−i τ η(τ ) and η(τ + 1) = eiπ/12 η(τ ), while
√
 
 
S : ϑ αβ (−1/τ ) = −i τ e−2πiαβ ϑ −βα (τ ) ,
α


α
iπα2
T : ϑ β (τ + 1) = e
ϑ β + α + 1/2 (τ ) .
(7)
Given these definitions, the structure of Eq. (5) allows
us to rewrite the 4D partition function ZYM as a finite
product of Dedekind η-functions and Jacobi ϑ-functions:!s√− 2e−iπb η(τ )2 η(τ )
3
 1/2 (8)
ZYM (τ ) = η(τ )ϑ2 (τ )
ϑ b + 1/2 (τ ) where b = i log(z)/2π ≈ 0.21i, where ϑ2 (τ ) ≡ ϑ 1/2(τ ),0
and where the identity 2η(2τ )2 = η(τ )ϑ2 (τ ) has beenused in passing from Eq. (5) to Eq. (8). The fact thatb is imaginary is the reason the degeneracy factors dn in
Eq. (3) grow as dn ∼ eCn . The expression in Eq. (8) —and our interpretation of this expression in terms of specific 2D CFTs, as discussed below — are the key resultsof our paper, with many striking consequences.
Modularity versus dimensionality. The first interesting implication of Eq. (8) becomes apparent upon
realizing that it is extremely unusual for the partition
function of a 4D theory to be expressible as a finite product of modular functions, as in Eq. (8). The large-|τ |behavior of a modular function is tied, through the S
modular transformation, to its behavior near |τ | = 0.
For example, the Dedekind η-function has the large-|τ |expansion η(τ ) = q 1/24 (1 − q + ...); the S transformation then requires this function
to behave at small |τ | as√η(τ ) ∼ exp[−iπ/(12τ )]/ −iτ . Similar statements can bemade for the ϑ-functions. Thus, if a partition function

3
can be written as a finite product of modular η-functionsand ϑ-functions, then it must have the leading behaviorlim
lim log Zmodular (τ ) = σR/β(9)arg τ →π/2 |τ |→0

for a constant σ. This amounts to the statement thatlog Zmodular ∼ T as T → ∞. This is indeed the expectedbehavior for a 2D QFT. However, it is certainly not the
expected behavior for a 4D QFT, for which we generically
expectlog Z4D ∼ T 3

as T → ∞ .

(10)

In this sense, 4D QFTs whose partition functions can be
written in terms of modular functions behave as if they
were 2D QFTs, since they follow Eq. (9) rather thanEq. (10).
pβ
1 + (µI R)2 and arg τ =In our case, |τ | = 2πR−1cot (µI R). If we were to reverse the order of limits
on the left side of Eq. (9) and take the T → ∞ limit
with µI = 0, pure YM theory would follow the scaling in
Eq. (10). Such a limit cannot be studied from Eq. (3) due
to the Hagedorn singularities, and the physics is governedby the deconfined phase. But with the order of limits indicated in Eq. (9), which amount to taking β/R → 0
before µI R → 0, Eq. (9) holds for pure YM theory. Notethat in other theories such as adjoint QCD with periodic
boundary conditions for fermions, the Hagedorn singularities do not lie along arg τ = π/2; the two limits thencommute [8] and these theories exhibit 2D behavior in
the sense of Eq. (9) irrespective of the order of limits.
Vacuum energy. Another major consequence ofEq. (8) is that the modular properties of the η- and ϑfunctions fix the vacuum energy EYM of our large-N YM
theory to be zero.
To see this, we first recall that if we write the qseries expansionof a modular function f (τ ) in the formP∞
f = q ∆ n=0 an q n , then ∆ can be thought of as the2D vacuum energy. Its value is fixed by the modular
properties of f and tied to the values of an . Were one
to abitrarily shift ∆ → ∆ + c, the modular propertiesof f (τ ) would be ruined because the S-transformation
would map q c = e(2πiτ )c to e(−2πi/τ )c , thereby preventing q c f (τ ) from transforming as a modular function.
Next, we observe that the vacuum  energy associatedto the η-function is 1/24, while ϑ ab has vacuum energy
a2 /2. Summing the vacuum energies of the individual
modular functions in Eq. (8) we obtain a striking result:
EYM = 0 .

(11)

Indeed, this is the only value consistent with the qexpansion for ZYM given in Eq. (3), provided that EYM
is calculated in a renormalization scheme which is consistent with the modular properties of ZYM made evident
in Eq. (8). This value, EYM = 0, coincides with the result implied by T-reflection symmetry [7], and also agrees

with a direct evaluation of the sum over the confinedphase spectrum of finite-temperature large-N YM theory
compactified on S 3 , as performed in Ref. [9].
CFT interpretation. The striking modular structure of Eq. (8) suggests that the spectrum of our 4D YM
theory coincides with that of a chiral (e.g., left- or rightmoving) 2D CFT. This motivates the central question
we shall now explore for the rest of this paper: what isthe 2D CFT which gives rise to Eq. (8), and thus gives a
2D description of 4D YM theory in the large-N limit?
Unfortunately, we will not be able to give a complete
answer to this question. The reason ultimately has to dowith the fact that many distinct CFTs can have coincident spectra without being equivalent. They may differ,
for example, in their correlation functions. In general, themost important aspects of a given 2D CFT are governedby its central charge (conformal anomaly) c and its spectrum of operator conformal dimensions hi , i = 1, ..., n,where n is the number of so-called “primary” fields in
the CFT. Along with the explicit traces over states,
knowledge of c and the hi ’s goes a long way in nailing
down relevant aspects of the CFT such as its selection
rules and correlation functions. But partition functions(eff)
are only sensitive to the combinations hi≡ hi − c/24,
rather than the values of c and hi individually. Consequently, without additional assumptions about the CFT
in question (such as the assumption of unitarity, which
would additionally tell us that min {hi } = 0), this represents a fundamental limitation on our ability to specify
a unique CFT.
We will therefore answer a different but related question: do there exist any 2D CFTs to which our large-N
YM theory is isospectral ? Remarkably, we shall showthat at least one such 2D CFT indeed exists. To see this,
we first recall that a free c = 1 scalar CFT has a leftmoving spectrum whose trace is given by 1/η(τ ), while
the Z2 orbifold of this CFT has a chiral sector whose
trace is (2η(τ )/ϑ2 (τ ))1/2 . Furthermore, the direct product of two copies of the c = −26 bc ghost CFT has aleft-moving spectrum whose trace is given by η(τ )4 . Perhaps the most challenging to interpret is the remaining
factor in Eq. (8), specifically√− 2e−iπb η(τ )


.
(12)
ϑ b +1/21/2 (τ )However, this can be identified as the trace of the chiral(e.g., left-moving) states in the vacuum sector of the c =2 bosonic βγ ghost CFT recently explored in Ref. [10].
This is a logarithmic CFT [11], and it has a U (1) conserved charge. Thus the vacuum-sector chiral partition
function of the c = 2 βγ CFT depends on the choice of acomplex fugacity z = e+µβ . To match with our expres√sions for YM theory, we set µβ = 2πi b = − log(2 + 3).
Putting this together, we therefore conclude that theexpression in Eq. (8) can be viewed as the trace over4
the chiral spectrum of a theory which is the direct product of five known CFTs, one of which is irrational. This
then justifies the central claim of this paper in Eq. (2):there is indeed an irrational 2D CFT which is isospectral
to the finite-temperature large-N 4D YM compactified
on S 3 in the ΛR → 0 limit. Aside from explaining ourobservations concerning EYM and the small-|τ | behavior of ZYM , this fact has an intriguing further implication. Two-dimensional CFTs have infinite-dimensional
symmetries, which always include the Virasoro symmetry. Eq. (2) then suggests that large-N YM theory has ahidden Virasoro symmetry. It would be very interesting
to demonstrate this explicitly within YM theory.
Primary operator spectrum. We now collect information concerning the spectrum of conformal dimensions(eff)hicorresponding to the primary fields of this tensorproduct CFT. Our approach proceeds by determining
the diagonal modular-invariant associated with the expression in Eq. (8), and then computing the eigenvalues(eff)of the modular T operator to extract hi .
We begin by defining the quantities
!1/2
√
η(τ )
− 2 e−iπbn η(τ )4
 + 1/2, (13)
Tm,n ≡ϑ mb
ϑ PP(m)/2
nb + 1/2 (τ )(n)/2 (τ )where {m, n} are relatively prime integers (a relationshipwhich we shall henceforth denote m ⊥ n), and P (k) ≡1k
2 (1 + (−1) ), k ∈ Z. Thus P (k) = 0, 1 for odd or even k,
respectively. The set {Tm,n } is a basis for a vector spaceover the field C with two key properties: it contains the“seed term” in Eq. (8), and it is the minimal set which is
closed under the action of the SL(2, Z) modular group.
The first property follows by noting that T0,1 (τ ) coincides with Eq. (8). The verification of the second property proceeds in two steps. First, it can be shown√ that, up
to overall phases and extraneous factors of −iτ , the S
and T modular transformations map Tm,n to T−n,m andTm,n+m , respectively. Second, we observe that if {m, n}
are relatively prime, then {−n, m} and {m, n + m} arealso relatively prime. Since all modular transformations
can be generated by sequences of S and T , it then follows
that the full modular “orbit” of our seed term T0,1 is contained within the set of coprime integers {m, n}. Indeed,
it is also possible to demonstrate [3] that the modular
orbit actually covers all coprimes.
As a result, the minimal “diagonal” modular-invariant
generated from Eq. (8) is given by
X
2
Zdiagonal = (Im τ )3/2
Tm,n .
(14)

FIG. 2. The numerical values of Eq. (14) with |m|, |n| ≤ 10,
plotted within the unit-q disk.

for an isolated set of points corresponding to the Hagedorn singularities. The numerical values of Zdiagonal on
the interior of the unit-q disk are shown in Fig. 2.
In order to extract the spectrum of effective conformal(eff)dimensions hi , we now rewrite Zdiagonal in a basis of
eigenfunctions of the modular T : τ → τ +1 operator. We
do this because such eigenfunctions χ(τ ) will have eigen(eff)values exp[2πihi ] under T , allowing us to read off the(eff)values of hi(mod 1). Fortunately, constructing eigenfunctions of the T -operator from linear combinations of
the Tm,n ’s in Eq. (13) is relatively straightforward. Since
Tm,n (τ + 1) = eπi{[1−P (m)]/8+m |b| } Tm,n+m (τ ) , (15)2

we see that any linear combination which includes Tm,nmust also include Tm,n+m , Tm,n+2m , and indeed all
Tm,n+km where k ∈ Z. Our T -invariant linear combinations can therefore be indexed by an arbitrary integerm and a second integer ` ⊥ m obeying 0 ≤ ` < m.
Hence T -eigenfunctions can be constructed analogously
to Bloch eigenfunctions, by summing over all components
Tm,`+km with k ∈ Z with a Bloch phase α ∈ [0, 1) ⊂ R:
X
χm,`,α =e2πiαk Tm,`+mk .
(16)
k∈Z

It then follows that
(eff)

m⊥n

The appearance of the factor of (Im τ )3/2 is standard
when combining holomorphic and anti-holomorphic components, such as our Tm,n factors, each of which has modular weight k = 3/2. It also ensures that Zdiagonal is fully
modular-invariant. Moreover, it can be verified numerically that the infinite sum in Eq. (14) converges except2χm,`,α (τ + 1) = e2πihm,`,α χm,`,α (τ ) ,

(17)



1 1 − P (m)+ m2 |b|2 − α .
2
8

(18)

where
(eff)

hm,`,α =One might wonder whether {χm,`,α } is the complete
set of T -eigenfunctions. However, we have verified this by5checking that summing over χm,`,α reproduces Eq. (14):X X Z 1
3/2
dα |χm,`,α |2 . (19)
Zdiagonal = (Im τ )m∈Z 0≤`<m`⊥m

0This confirms that Eq. (18) is the desired set of effective conformal dimensions (mod 1) of the primary operators in our CFT. The fact that these dimensions dependon α — a continuous real variable — confirms that we are
dealing with an irrational CFT [12]. Our observations
are consistent with the 2D logarithmic CFT interpretation discussed above, since it is known that logarithmic
CFTs typically have a continuously infinite number ofprimary operators [13].
Outlook. We have presented evidence that the confined phase of finite-temperature 4D non-supersymmetriclarge-N pure Yang-Mills theory compactified on a threesphere of radius R is isospectral to an irrational 2D CFT
in the ΛR → 0 limit. This gives credence to the hope andexpectation that non-supersymmetric large-N confining
gauge theories are dual to 2D CFTs. Moreover, as weshall demonstrate in a separate paper [3], modularity inthe sense of Eq. (8) turns out to be a generic property oflarge-N confined-phase gauge theories with adjoint massless matter in the λ → 0 limit. In Ref. [3] we shall also
show that this structure is present in the large-N limitof the N = 4 superconformal index.
Our results suggest a large number of interesting topics for future research. For example, it is important tounderstand whether our large-N 4D-2D spectral equivalence extends to correlation functions, and to explore
how it is related to other known 4D-2D relations, such as
those discussed in Refs. [14]. It would also be interesting
to understand the origin of Eq. (8) within string theory,
perhaps by making contact with the ideas in Refs. [15].
Given recent progress in the understanding of the bulk
duals of 2D CFTs (see, e.g., Ref. [16]), it is tempting towonder whether our results may help to uncover the bulk
dual of YM theory and of other non-supersymmetric 4D
adjoint-matter theories. It would also be interesting tounderstand whether the continuous spectrum of primary
operators in the 2D theory suggested by our analysis has
an interpretation in 4D YM theory. Finally, there remains the very important question of determining how
our 4D-2D relation might evolve for λ > 0.
Acknowledgements. We are grateful to O. Aharony,
D. Berenstein, S. Caron-Huot, S. Cremonesi, N. Dorey,
G. Dunne, D. Hofman, J. Kaplan, Z. Komargodski,
J. McGreevy, D. O’Connor, E. Poppitz, S. J. Rey, D. Ridout, D. Tong, M. Unsal, S. Wood, and M. Yamazaki for
helpful discussions at various stages during the gestationof this work. AC thanks the Instituto de Fisica Teoretica of UAM and DAMTP of Cambridge University for
hospitality, and DAM thanks the Institute for Advanced
Study and the Niels Bohr International Academy for hospitality and support during the completion of this work.

GB and AC thank the Department of Energy for support under Grants DE-FG02-93ER-40762 and DE-FG0294ER40823 respectively. The research of KRD was supported in part by the Department of Energy under Grant
DE-FG02-13ER-41976 and by the National Science Foundation through its employee IR/D program. The opinions and conclusions expressed herein are those of theauthors, and do not represent any funding agencies.

∗

gbasar@umd.edu
acherman@umn.edu‡
dienes@email.arizona.edu§
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