Thermal one-point functions and single-valued
polylogarithms: a spurious remark or
something deeper?
Based on T.P & A. Stergiou 1806.02340 (PRL) and T.P. 2105.03530 (PLB)

Tassos Petkou
September 2, 2021
Division of Theoretical Physics
School of Physics
Aristotle University of Thessaloniki
GreeceTable of contents
1. Introduction and Motivation
2. The conformal OPE in nontrivial geometries
3. The OPE inversion formula
4. Thermal one-point functions from the effective action (thermal freeenergy)5. Thermal one-point functions as single-valued polylogarithms.
6. Outlook
7. Extra material

1Introduction and Motivation

CFTs in nontrivial geometries: generalities
There is a deep connection between the RG and the couplingof CFTs to nontrivial background geometries.
Conformal transformations of d-dimensional flat space (i.e. withmetric ηµν ) are coordinate reparametrizations x µ 7→ x 0µ (x) that
preserve the norm of vectors up to a scale factor.
x µ 7→ x 0µ : ds 2 = ηµν dx µ dx ν 7→ ds 02 = ηµν dx 0µ dx 0ν ≡ Ω2 (x)ds 2∂x 0µ ∂x 0ν
= Ω2 (x)ηρσ∂x ρ ∂x σ• Hence, conformal transformations are equivalent to a local rescalingof the flat metric ηµν 7→ Ω2 (x)ηµν i.e.
– either we work with the coordinates x 0µ and metric ηµν ,– or we use the original coordinates x µ and the metric Ω2 (x)ηµν .
• In d = 2 x 0µ (x) are general analytic functions of x µ .
• In d > 2 x 0µ (x) are at most quadratic in x µ .
⇒ ηµν

2

CFTs in nontrivial geometries: generalities
CFTs are those QFTs that have a number (sometime finite in d = 2,certainly infinite in d > 2) of quasiprimary local operators O(x) that
under conformal transformations behave asx µ 7→ x 0µ : O(x) 7→ O0 (x 0 ) = Ω∆ (x)O(x 0 ) ≡ Ω∆ (x)O(x)
η

η

η

Ω2 η

where ∆ is the scaling dimension. (I considered scalars for simplicity.)• The conformal Ward identities express the invariance of correlation
functions under conformal transformationsx µ 7→ x 0µ : hO(x)...i 7→hO0 (x 0 )...i = Ω∆ (x)hO(x 0 )...i
η

η

η

∆

≡ Ω (x)hO(x)...iΩ2 η≡ hO(x)...iη• The simplest example: the 2pt function of a scalar field φ(x) withdimension ∆. Under the scale transformation x µ → x 0µ = λx µ forwhich Ω = λ we have1
1
hφ(x1 )φ(x2 )i = 2∆ → hφ0 (x10 )φ0 (x20 )i = λ2∆ 0 2∆ ≡ hφ(x1 )φ(x2 )i(x12 )x12

3

CFTs in nontrivial geometries: generalities
CFT correlators in the nontrivial - but conformally flat - geometry Ω2 ηµν are fully determined by the flat space correlators.
• Consider the 2-pt function of scalars in flat space. Going to sphericalcoordinates ~x → (r , Ωd−1 ) with ds 2 = dr 2 + r 2 dΩ2d−1 one hashO(~x1 )O(~x2 )i =

1
2∆
~x12

→

→ hO(r1 , θ)O(r2 , 0)i =

1
1
= 2
2
2∆
(~r1 − ~r2 )(r1 + r2 − 2r1 r2 cos θ)∆• If there was a conformal transformation of flat space with metric ηµν
to the metric Ω2 (x)ηµν , the Ward identity would give[Ω(r1 , θ)Ω(r2 , 0)]−∆ hO(r1 , θ)O(r2 , 0)i = hO(r1 , θ)O(r2 , 0)i
η

Ω2 η

4

CFTs in nontrivial geometries: example in d = 2• In d = 2 all analytic transformations are conformal transformations
since for ds 2 = dx 2 + dy 2 = dzd z̄ , z = x + iy , z̄ = x − iy andunder the general analytic transformations z 7→ z 0 = f (z),z̄ 7→ z̄ 0 = f¯(z̄) we haveds 2 7→ ds 02 = ∂z f (z)∂z̄ f¯(z̄)dzd z̄ ≡ Ω2 (z, z̄)ds 2• Choosing f (z) = L ln(z/L) and f¯(z̄) = L ln(z̄/L) we findds 2 7→ ds 02 =

L2
L2
L2 2
2
2
dzd z̄ = 2
[dx
+
dy
]
≡
[dr + r 2 dθ2 ]z z̄
x + y2
r2• Now, we observe the the metric shown in the last equality is actually
the reparametrization of the metric on R × S 1 i.e. underρr = Le L ⇒ ds 2 ≡L2 2[dr + r 2 dθ2 ] = dρ2 + L2 dθ2r2
5

CFTs in nontrivial geometries: example in d = 2• So we havehO(r1 , θ)O(r2 , 0)iΩ2 η≡ hO(ρ1 , θ)O(ρ2 , 0)i

R×S 1

1
1
= 2∆
∆
ρ
−ρ
L
2 cosh 1 L 2 − 2 cos θ• The metric reparametrization above generalises for all d > 2 asρr = Le L ⇒ ds 2 ≡L2 2[dr + r 2 dΩ2d−1 ] = dρ2 + L2 dΩ2d−1
r2

However, there is no conformal transformation of flat d-dimensional
space with Ω2 (r , Ωd−1 ) = L2 /r 2 , so CFT correlation functions on
R × S d−1 cannot be determined by those on Rd .

6

CFTs in nontrivial geometries: Weyl invariance• One possibility is to consider Weyl invariant QFTs. These are QFTs
that have operators O(x) with a definite behaviour under general
local Weyl rescalings of the flat metricηµν 7→ Ω2 (x)ηµν ⇒ O(x) 7→ O0 (x) = Ω∆ (x)O(x)• The above behaviour is independent of the spin of the operator O(x).
This leads to the corresponding Ward identities expressing Weyl
invariance of correlation functions asΩ∆ (x)hO(x)...iΩ2 η≡ hO(x)...iη• So, for Weyl invariant theories[Ω(r1 , θ)Ω(r2 , 0)]−∆ hO(r1 , θ)O(r2 , 0)i = hO(r1 , θ)O(r2 , 0)iη≡ hO(ρ1 , θ)O(ρ2 , 0)i

R×S d−1

=

Ω2 η

=

1
1
L2∆ 2 cosh ρ1 −ρ2 − 2 cos θ∆L7

CFTs in nontrivial geometries: Weyl invariance• Weyl invariance connects background effective actions with the RG.
Consider the renormalised effective action Wr [gµν , J] of a theory
coupled to gµν (x) and a scalar source J(x) with Weyl-weight (i.e.
scaling dimension) d − ∆. For constant Weyl rescalings we haveδg µν (x) = 2g µν (x) , δJ(x) = (d − ∆)J(x)hence we have the following local form if the RG flow equation



Z
√
∂
δ
d
δ
µ Wr = µ+ d d x g 2g µν µν + (d − ∆)J(x)Wr = 0
dµ
∂µ
δg
δJ(x)
µ is the RG mass scale, absent in CFTs except in the presence of
anomalies.
• Conformal invariance in flat space implies Weyl invariance in a generalbackground (??) [e.g. M. Luty et. al. (2017)].
8

CFTs in nontrivial geometries: recap
In two-dimensional CFTs correlators in the thermal geometry
R × SL1 are fully determined by those on R2 .
In d > 2 it appears that only for Weyl invariant theories
correlators in R × SLd−1 are fully determined by those on Rd .
However, the thermal geometries SL1 × Rd−1 , although conformally flat, are not related neither by a conformal transformation
nor by a Weyl rescaling to Rd for d > 2.
We generally need additional data to describe CFTs in thermal
geometries for d > 2. What are these additional data, andhow are they related to the RG properties of QFTs?
9

The conformal OPE in nontrivial
geometries

The conformal OPE in nontrivial geometries: generalities• The conformal OPE is the statement that quasiprimary operatorsform a complete basis for operator products in a CFT i.e. for scalars
φ(x1 )φ(x2 ) =

1

1+
2∆

x12

X

1

Os

2(∆− ∆2s + 2s )x12[x12 · Os (x2 )]where [x12 · Os (x2 )] denotes the spin-s, dimension-∆s contribution
with all its descendants.
• Using the OPE for correlators in nontrivial geometries we could
evaluate them if we knew the 1-pt functions hOs (x)i for the relevant
quasiprimary operators.
• Nevertheless, for 1-pt functions we generically havehO(x)iΩ2 ηµν = [Ω(x)]−∆ hO(x)iηµν = 0 for O(x) 6= 1
and we need to be careful.
10The conformal OPE in nontrivial geometries: d = 2 example• Since since all 1-pt functions vanish on Rd , namely hO(x)i = 0 andη

in d = 2 the plane is conformally related to the thermal geometry it
would appear that all 1-pt functions of quasiprimary operators vanish.
• Nevertheless, in d = 2 there exist operators transforming anomalously
i.e. the energy momentum tensor
T (z) → T 0 (z 0 ) = [f 0 (z)]2 T (z 0 )+f 000 f 0 − 23 f 002
c{f (z), z}1 , {f , z} =12
f 02

• Using the above one obtains
c 1
24 L2
• We conclude that thermal correlation functions in d = 2 do receive
contributions from nontrivial 1pt functions of non-quasiprimary
operators i.e. from conformal anomalies. Setting u = ρ cos φ andLθ = ρ sin φ the thermal 2-pt function becomes [J. Cardy (1986)]1∆φ ρ 2
cos
2φ
+
·
·
·
hφ(ρ, φ)φ(0, 0)i = 2∆φ 1 −12 L2ρhT (z)iR×Sβ1 = −11The conformal OPE in nontrivial geometries: Sβ1 × Rd−1
• In the Sβ1 × Rd−1 geometry the 1-pt functions of scalar quasiprimaries
can only depend on a single parameter asbO
hO(x)iSβ1 ×Rd−1 = hO(0)iSβ1 ×Rd−1 = ∆Oβ• For SO(d) irreducible tensors we havehTµν... (0)iSβ1 ×Rd−1 =bT(êµ êν ... − traces)
β ∆Twhere x µ = (τ, x) are coordinates on Sβ1 × Rd−1 with periodτ ∼ τ + β, r = |x| and θ ∈ [0, π] is a polar angle when Rd−1 iswritten in spherical coordinates. êµ are unit vectors in the τ -direction.
• Then, the thermal two-point function takes the generic formhφ(x)φ(0)iβ ≡ g (r , cos θ) =X
Os ∆Os νr
Cs (cos θ)
aOs
β
r 2∆φ
12

The conformal OPE in nontrivial geometries: Sβ1 × Rd−1

• Csν (cos θ) are Gegenbauer polynomials with ν = d/2 − 1.
• The coefficients aOs are given by
aOs =

s!
2s (ν)

gφφOs bOs
COss

with COs and gφφOs the corresponding 2-pt and 3-pt function
coefficients, and (a)n the Pochhammer symbol.
• The unit operator 1 is the unique operator with dimension zero, andhere
22∆φ −d Γ(∆φ )a1 = d d
π 2 Γ( 2 − ∆φ )so that the momentum-space two-point function is unit-normalized.

13

The conformal OPE in nontrivial geometries: d = 2 example• The thermal 2-pt function gives (there is an extra factor of 2 in thenormalization of the d = 2 Gegenbauers)aT =g T bT∆φ=−CT
12• Using the Ward identity we then find
gT =

d∆φ
CT
⇒ bT = −d −1
24

confirming that the e.m. tensor coefficient CT coincides with theconformal anomaly c in d = 2.
• There is yet another coefficient c̃ related to thermal 1-pt functionshTτ τ iR×S d−1 = −(d − 1)[fβ − f∞ ] =
β

bT
ζ(d)
= −2(d − 1) d c̃βdβwhere fβ is the free energy density. For d = 2 we see that c̃ ∼ c.
14

The conformal OPE in nontrivial geometries: recap

The conformal OPE can be used to study correlators in nontrivial geometries.
For general d > 2 we need additional data, in the form of 1-ptfunctions, to determine the correlators.
Nevertheless, in d = 2 it appears as if the thermal correlator
is fully determined by the plane result. This is a consequenceof the fact that the only nonzero 1-pt functions are those ofanomalously transforming operators, and they depend on thecentral charge. The latter cancels in the OPE.

15

The OPE inversion formula

The OPE inversion formula: the Euclidean formula• Further information regarding the thermal 2-pt function can beobtained using an OPE inversion formula. [L. Iliesiu et. al. 1802.10266 (JHEP)]• Complexifying ∆ one defines the spectral function a(∆, s) via
g (r , cos θ) =X I −+i∞ d∆
s

−−i∞

2πi

a(∆, s)Csν (cos θ)
r 2∆φ −∆

whose poles at ∆ = ∆Os with residues −aOs yield the physicalspectrum.
• Assuming that the physical poles lie on the right of the imaginary axis
one can close the contour clockwise for r < 1 (we set β = 1 fromnow on) if a(∆, s) does not grow exponentially at infinity.

16

The OPE inversion formula: the Euclidean formula• We can then use the orthogonality of Gegenbauer polynomials to
project on a spin-s state and then integrate with a suitable power inthe region of convergence r ∈ [0, 1] to obtain a(∆, s) as
1a(∆, s) =
Ns,ν

Z 1

dr

Z 1

0

r ∆−2∆φ +1

−1

where
x = cos θ , Ns,ν =1

dx (1 − x 2 )ν− 2 Csν (x)g (r , x)21−2ν πΓ(s + 2ν)(s + ν)Γ(s + 1)Γ2 (ν)• This is termed Euclidean inversion formula.

17

The OPE inversion formula: the Lorenzian formula• Writing x = cos θ = (w + 1/w )/2 with w = e iθ one can transformthe Euclidean formula into a contour integral over the unit circle inthe complex-w plane.
• To exploit further the analytic structure of the 2-pt function
g (r , cos θ) one would like to allow w to explore the full complex
plane. This can be done by a suitable complexification of the
Euclidean variables r , θ, defining z = rw and z̄ = r /w which are now
independent real variables.
• As a function of w , i.e. in the w -plane, g (r , w ) has the cuts(−∞, −1/r ), (−r , 0), (0, r ) and (1/r , ∞). One also has to assume
that it does not grow faster than w s0 (resp. 1/w s0 ) for large (resp.
small) w for some constant s0 > 0.

18

The OPE inversion formula: the Lorenzian formula• Moreover, one needs to use the analytic extension of the Gegenbauer
polynomials to the whole complex plane as [M. Costa et. al. 1209.4355 (JHEP)]
Csν (w ) =Γ(s + 2ν)(Fs (1/w )e iνπ + Fs (w )e −iνπ )Γ(ν)Γ(s + ν + 1)

where
Fs (w ) = w s+2ν 2 F1 (s + 2ν, ν; s + ν + 1; w 2 )19

The OPE inversion formula: the Lorenzian formula• Then, the integral giving a(∆, s) will receive contributions from thediscontinuities across the cuts of g (r , w ) as well as from the arcs at
infinity. The final result isa(∆, s) = aDisc (∆, s) + θ(s0 − s) aarcs (∆, s)
where
Z 1
aDisc (∆, s) = Ks

d z̄
z̄

Z ∞

∆
dz h
(z z̄)∆φ − 2 −ν
z
0
1
r !
i
z̄
2ν
Disc[g (z, z̄)]× (z − z̄) Fszwith

Γ(s + 1)Γ(ν)
4πΓ(s + ν)
• The discontinuity relevant for the evaluation of the above integral is
the one across the cut (1/r , ∞), as all others are related to it.
Ks = (1 + (−1)s )20

The OPE inversion formula: explicit example
Inversion formulae (i.e. spectral decompositions) are most useful when there is an independent evaluation of the correlation
functions.
• For bosons (scalars) the simplest ansatz is to consider the
momentum-space thermal 2-pt function
G (d) (ωn , p) =1

2 , ωn = 2πn , n = 0, ±1, ±2, . . . ,ωn2 + p2 + mth• This is motivated by known work on thermal field theory which showsthat fields develop generically a thermal mass mth at finite
temperature.
• We are actually asking whether the simple ansatz above can define athermal CFT. We make no reference to a Lagrangian, although it is
known that the 2-pt function can be obtained, for example, in thelarge-N limit of the O(N) model [T. P. et. al. hep-th/9803149 (PLB) ].
21

The OPE inversion formula: explicit example• In arbitrary-d the above 2-pt function can be Fourier-transformed todG (d) (τ, x) =

 −1
∞ 
X
mth 2
K d −1 (mth |Xn |) , Xn = (τ − n, x)d2(2π) 2 n=−∞ |Xn |1

where Kα (x) is the modified Bessel function of the second kind.
p• Defining z = τ + i|x| we find |Xn | = (n − z)(n − z̄).
• We focus on odd d = 2k + 1, k = 1, 2, . . . , and in that case we maywrite
G (2k+1) (τ, x) =∞X1
2k+1 π kn=− inf ty

withLk,p =

k−1
k−1
X
mth
Lk,p
−mth |Xn |e|Xn |k(m|Xn |)pth
p=0(k − 1 + p)!
2p p!(k − 1 − p)!
22

The OPE inversion formula: explicit example• The latter coefficients also appear in the Bessel polynomials
r
n
X
2 1/x
p
e Kn+ 12 (1/x)
yn (x) =Ln+1,p x =πx
p=0• The relevant discontinuity Disc(G (d) ) follows simply fromunderstanding the discontinuity of the function
k−1
√ X
ak−1
Lk,p
√
f (k) (x) = √ k e −a x( x)(a x)pp=0

across the cut due to the square-root branch point at x = 0.
• Assuming that the cut goes from x = 0 to x = ∞ we find that

√
√
2  1
Disc(f (k) (x)) = k−1 √ Uk (x) cos(a −x) + Vk (x) sin(a −x)
x
−x
√
√ 
1
Uk (x) = 2 θk−1 ( x) + θk−1 (− x)√√ 1Vk (x) = √ θk−1 ( x) − θk−1 (− x)2 xwith θn (x) = x n yn (1/x) the so-called reverse Bessel polynomials.

23

The OPE inversion formula: explicit example• We can now calculate the spectral function a(∆, s). For the
discontinuity part we findΓ(k − 12 )1
(k)aDisc,0 (∆, s) = (1 + (−1)s ) 2s+k
2s! Γ(k + s − 12 )
k−1+s
X

×

n=0

where Liα (z) =

P∞

n=1 z

2n+1 (2(k − 1 + s) − n)! n
mth Li2k−1+s−n (e −mth )n! (k − 1 + s − n)!
n/nα is the polylogarithm.

• The result above follows just from the leading term in a z̄-expansionof the inversion formula. It gives the contributions of higher-spin
conserved currents with ∆ = d − 2 + s.
(k)

(k)

• Subleading terms in the z̄-expansion, denoted as aDisc,1 , aDisc,2 , . . . . ,would give the contributions of higher-twist operators.
24Gap equations from the inversion formula: bosonic case(d)

• The arc part aarcs (∆, s) is nonzero only for s = 0. We find
(d)
aarcs
(∆, 0) = ∆  ∆ − d + 2
∆
Γ −
m
Γ
−
√
d−5
th
2
2
2∆− 2 π1

• Notice that for mth = 0 only the ∆ = 0 term survives giving thecontribution of the identity operator. This, along with the(k)corresponding mth = 0 contributions from aDisc (∆, s), yield thespectrum of generalized free CFTs.
• When mth 6= 0 and for ∆ > 0 the above yields contributions of an
infinite tower of scalar operators with ∆ = 2m, m = 1, 2, . . . , as wellas contributions with ∆ = d − 2 + 2l, l = 0, 1, 2, . . . .
• The former correspond to operators of the form σ m , m = 1, 2, . . . ,where σ is the shadow of φ2 .

25

The conformal OPE in nontrivial geometries: recap
We have seen that thermal one-point functions represent additional data needed to describe CFTs in nontrivial (i.e. thermal)
geometries.
If one knew the thermal 2pt function one could read thethermal one-point functions of all operators in the spectrum
using an conformal inversion formula of the OPE. Alternatively,
knowing the thermal one-point functions one might reconstruct
the thermal 2pt functions.
An important remaining question is: what are the physicalparameters that determine the thermal one-point functions andhow are they related to the RG propeprties of the CFT??

26

Thermal one-point functions
from the effective action(thermal free energy)

Thermal one-point functions from the free energy
A few words about systems at imaginary chemical potential• The canonical partition function of a system at finite temperature
T = 1/β with a global U(1) charge operator Q̂ having integer
eigenvalues is
Z 2π
Z 2π
dθ iθQ h −β Ĥ−iθQ̂ idθ iθQθZc (β, Q) =
e
Tr e
=
e
Zgc (β, µ = −i ) ,
2π
2π
β
0
0
where Zgc (β, µ) is the grand canonical partition function withimaginary chemical potential µ.
• The latter function exhibits certain periodicity properties wrt θ. For
example in SU(N) gauge theories with the fermion number operator
Q̂ is a multiple of N in the confining phase and Zgc (β, µ) will be
periodic with a θ-period 2π/N. However, in a high temperaturedeconfining phase one should find a 2π θ-period i.e. the breaking of
ZN symmetry signals a deconfining transition [A. Roberge & N Weiss (1986)]
27

Thermal one-point functions from the free energy• Consider the Euclidean action action for a massive complex scalar
field φ(x) in odd d-dimensions in the presence of an imaginary
chemical potential (or equivalently in the presence of the temporalcomponent of real gauge potential)
Z β
SE (β; m, µ) =Zdτ

~ 2 + m2 |φ|2 .
d d−1 ~x |(∂τ − iµ)φ|2 + |∂φ|0

• To evaluate the grand canonical partition function and thecorresponding free energy (grand canonical potential) of the theory
one may cure the short distance singularity and other calibration
issues by subtracting the zero temperature mass and chemical
potential results to obtainZ
1
Zgc (β; m, µ) ≡(Dφ̄)(Dφ)e −SE = e −βFgc (β;m,µ) .
Z (0; 0, 0)
28

Thermal one-point functions from the free energy• By a simple scaling argument the grand canonical free energy isusually written as
Fgc (β; m, µ) =Vd−1
Cd (βm, βµ) ,βdwith Vd−1 the spatial volume.
• The dimensionless function Cd has been extensively studied over theyears e.g. [A. Castro-Neto et. al. (1992), T. Appelquist et. al. (1999), A. Leclair (2005)] as a measureof the degrees of freedom along the RG, although it does not appear
to satisfy the requirements of a c-theorem [S. Giombi et. al. (2011)].
• Here I will point out some properties of Cd as a function of theparameters m, µ and notably also d. These properties are nicely
presented using the set of complex variablesz = e −βm−iβµ , z = e −βm+iβµ ⇒ Cd (βm, βµ) ≡ Cd (z, z) .
29

Thermal one-point functions from the free energy• The calculation of Fgc can be done along the lines described in the
appendix of [T.P. et. al. (2018)] and yields

Cd (z, z) = −Kd lnd |z| −
where
Kd =


Sd−1 ¯d (z, z) ,
z)
+
i
i
(z,
d
(2π)d−1πSd12π d/2,S=.
dd(2π)d sin(πd/2)
Γ(d/2)• For d ≥ 3 the integral id (z, z) is given byZ z
id (z, z) =
0

dw
w



1 z
ln w − ln
2 z

# d−3
 "
2
2
1 z
2
ln w − ln− ln |z|ln(1−w )2 z

and i¯d (z, z) is obtained from above by exchanging z ↔ z. For d = 1
we have i1 (z) = − ln(1 − z) and i¯1 (z) = − ln(1 − z).
30Thermal one-point functions from the free energy• For odd d the calculation of Cd can be reduced to a finite series of
iterating integrals and then to a finite double series using results such
as

k
Z z
k
X
dw
1 z
(−1)`+1 k! `ln w − lnln(1 − w ) =ln |z|Lik+2−` (z) ,
w
2 z
`!
0
`=0

where Lin (z) are the usual polylogarithms.
• The result can be written as
id (z, z) + i¯d (z, z) = −


Γ d+1
2
Id+2 (z, z) ,
d −1

with
d−3
2
X
(−1)n (d − 3 − n)! 2n lnn |z|
Id (z, z) =[Lid−2−n (z) + Lid−2−n (z)] .
d−3
n!
−
n
!
2
n=0

31

Thermal one-point functions from the free energy• In the simple free field theory studied here m2 and µ parametrize thedeformations of the free Hamiltonian by the operators O = |φ|2 and←→Q = i φ̄ ∂τ φ, the latter being the charge density defined from the←→U(1) current Jµ = i φ̄ ∂µ φ.
• Evidently, the normalized thermal averages (integrated thermal
one-point functions) of the above deformations are obtained as
moments of the free energy. Assuming they are uniform i.e.
RhO(x)i = βVd−1 hOi one obtains

hOid =
hQid =1
β d−2

D̂Cd (z, z) , D̂ =1(z∂z + z∂z )2 ln |z|

1
L̂Cd (z, z) , L̂ = (z∂z − z∂z )β d−1• Namely, hOid and hQid are the responses of Cd along the radial andangular directions in the two-dimensional space of massive and U(1)deformations of the free CFTs.

32

Thermal one-point functions from the free energy• To further unveil the physical content of the above it is instructive toconsider the case d = 1. This corresponds to a system of two
noninteracting harmonic oscillators with frequency ω ≡ m. The
twisted partition function of the oscillators is given by
Z1 = TrH1,2 e −β Ĥ+iβµQ̂ .
• The Hamiltonian and the twist operator are respectively
Ĥ =

2
X
p̂ 2
i

i=1

2

+

m2 xi2, Q̂i = âi† âi , Q̂ = Q̂1 − Q̂2 ,2

The trace in (1) is taken over the tensor product Hilbert space
H1,2 ≈ {|n1 i ⊗ |n2 i}, n1 , n2 = 0, 1, 2, ...

33

Thermal one-point functions from the free energy• Before I proceed with the trivial calculation, let me write the partition
function in the slightly unconventional way as follows
2

Z1 = TrH1,2 e −β(Ĥ0 +m Ô)+iβµN̂ , Ô =1 2(x̂ + x̂22 )2 i

and Ĥ0 = (p̂12 + p̂22 )/2 is the free Hamiltonian or equivalently in thiscase the kinetic energy.
• Calculating the partition function one easily obtains
C1 (z, z) = − ln |z| + ln(1 − z) + ln(1 − z) ,This matches the result for the free energy above for d = 1.

34

Thermal one-point functions from the free energy• The thermal averages of the operators Ô and Q̂ are given by



β
1
1
2
hÔi1 = −1 + |z|+2 ln |z|z(1 − z) z(1 − z)11
hQ̂i1 = |z|2−z(1 − z) z(1 − z)
• Also recall that in our simple model there is a virial theorem at work
relating the thermal averages of the operator Ô and the HamiltonanĤ as2ω 2 hÔi1 = hĤi ,and therefore it is not surprising that the operator O is related to theradial Hamiltonian in general d-dimensions.

35

Thermal one-point functions from the free energy• The generic formulae above yield an intimate connection among
theories in different dimensions. The crucial point is the following
result which can be proven by a direct calculation
D̂Cd (z, z) = −1
Cd−2 (z, z) ,4π

for d = 1, 3, 5, .. with the boundary condition C−1 (z, z) = −4πhOi1 .
• Then from the above one obtains
Cd (z, z) = −4πβ d hOid+2
which shows that the free energy of the d-dimensional theory is
proportional to the one-point function of the operator O ind + 2-dimensions. One also finds
hOid = −4πβ 2 D̂hOid+2 , hQid = −4πβ 2 D̂hQid+2 ,36

Thermal one-point functions from the OPE: test

• In the case of a real scalar field studied in [T.P & A. Stergiou (2018)], φ2 withdimension ∆φ2 = d − 2 was the only relevant operator in the OPE
φ × φ which was not a shadow operator, and there were no currents
with odd spin. In the case of the complex scalar field studied here
there is another relevant operator which is not a shadow operator; thisis the s = 1 current Jµ with dimension ∆J = d − 1. The contributionof the above two operators in the thermal two-point function is(d)g (r , cos θ) =(d)(d)
gφφφ2 (d)gφφJ (d)aI+[shadows]
+
b
+
r
cos
θ
b + ···(d) φ2
(d) J
r d−2
C 2
C
φ

J

• The OPE coefficients depend on the thermal mass and the imaginary
chemical potential, namely bOs = bOs (z, z).

37

Thermal one-point functions from the OPE: test

• The operators φ2 and J0 correspond to the operators O and Q whose
thermal one-point functions are given by the general formulae given
previously.
• If one wanted the have a conformal OPE, then the arbitrary scaleparameter introduced in the theory by bφ2 should not be there.
Requiring its vanishing is the condition (gap equation) suggested in[T.P & A. Stergiou (2018)] that determines the critical values of the thermal
mass and chemical potential. For µ = 0 this condition gives a class ofalgebraic equations for high order polylogarithms and it explains thecritical behaviour of bosonic systems in generic even dimensions. (seeextra material).

38

Thermal one-point functions from the OPE: test
• The thermal two-point function of the free theory is

ν
∞
X
1
m
inµ
g (r , cos θ; m, µ) =eKν (m|Xn,ζ |) ,d|Xn,ζ |(2π) 2 n=−∞pwith Xn,ζ = (n − ζ)(n − ζ̄), and ζ = re iθ .
• The n = 0 term gives the usual zero temperature result for themassive two-point function, while the Bessel functions havepolynomial expansions for d odd.
• In d > 2 the zero temperature contribution has a divergent m → 0
expansion where two cut-off independent terms stand out: the term
giving the massless two-point function and the r -independent term
proportional to md−2 . In the OPE expansion these correspond to thecontribution of the unit and |φ|2 operators respectively.
• Notice that the contributions from shadow scalar operators comefrom the process of the cut-off subtraction and renormalization of thetheory - which I assume that it can somehow be done and it does notaffect the results.

39

Thermal one-point functions from the OPE: test
• After some algebra I find# "Γ 1 − d2
Γ d−1
gφφφ2
d−2
2
√
m
+ Id (z, z̄) ,bφ2 = aICφ2
Γ(d − 2)2 π

1 Γ d−1gφφJ2bJ = − aI
Id (z, z̄)CJ
2 Γ(d − 2)
where aI =pπ

12 (2π) d2

1
2

d−3
2

Γ(d−2).
Γ( d−1
2 )• The functions Id (z, z̄) was given above and Id (z, z) isd−12X(−1)n (d − 1 − n)! 2n lnn |z|
Id (z, z̄) =[Lid−1−n (z) − Lid−1−n (z)] .
d−1n!
2 −n !
n=040

Thermal one-point functions assingle-valued polylogarithms.

Thermal one-point functions as single-valued polylogarithms• The results for the free energy and the thermal one-point functions
are particular cases of the single-valued polylogarithms Pw (z, z)
constructed by Brown [F. C. Brown (2004), O. Schnetz (2013)]. Explicitly one observes
that for d ≥ 1

d+1

Id (z, z) = (−1) 2 Γ



d +1
2



Pwd (z, z) , wd ≡ 0{nd } 10{nd } ,with nd = (d − 3)/2.
• The index wd denotes a ”word” in the ”two-letter alphabet” {0, 1}
and one normalises such that P∅ = 1 and P0{n} = 2n lnn |z|/n!.
Notice that P1 (z, z) = ln(1 − z) + ln(1 − z). It can be shown thatthe single-valued polylogarithm Pwd (z, z) satisfies for d ≥ 1
z∂z P0wd 0 (z, z) = P0wd (z, z) , z∂z P0wd 0 (z, z) = Pwd 0 (z, z) ,which are the form of Knizhnik-Zamolodhikov equations.
41

Thermal one-point functions as single-valued polylogarithms• Actually, one can show that the above yields the second orderdifferential equations studied e.g. in [J. Drummond (2012)]. The KZ-likeequations then imply
L̂P0wd 0 (z, z) = P0wd (z, z) − Pwd 0 (z, z)
1
2
D̂P0wd 0 (z, z) =Pwd (z, z) =P0wd−2 0 (z, z) ,nd + 1d +1
which are effectively a different form of the relationships given for thefree energy and the thermal one-point functions.

42

OutlookOutlook• It is well-known that single-valued polylogarithms are closely related
to Feynman graphs in gauge and string theories. The observation
made in this work seems to connect two apparently different physicalquantities; thermal one-point functions in odd-dimensional theories
and multiloop graphs in N = 4 SYM in d = 4.
• Contrasting these two manifestations of single-valued polylogarithms
one notices that in the first case the variables z and z parametrise
relevant deformations of free CFTs while in the second caseparametrize spacetime points.
• Moreover, in the first case the order of the polylogarithms is related
to spacetime dimension while in the latter case to the number ofloops in the graphs.

43

Outlook• It would be nice to understand further the implications, if any, of ourobservation. As a hint, one may consider interpreting the charge
one-point functions as higher-dimensional generalizations of the twist
one-point function operator in the system of two harmonic oscillators.
For example in d = 3 we find [T.P. et. al. (2018)]
4πβ 2 hQi3 = −4iD(d) , D(z) = =Li2 (z) + ln |z|Arg(1 − z) ,with D(z) the famous Bloch-Wigner function (single-valued
dilogarithm).

44

Outlook• D(z) appears in the simplest tree graph of N = 4 SYM
1
π2

d 4x

Z

(x − x1 )2 (x − x2 )2 (x − x3 )2 (x − x4 )2=

14i
2 x 2 z − z̄ D(z)x1423where the change of variables is
u=

2 2
2
x34
x12
x 2 x34v= z z̄
, v = 12= (1 − z)(1 − z̄) ,
2
2
2
2
u
x13 x24x14 x23• What is perhaps less known is that the integral above can beobtained as a limit that corresponds to a particular kind of twisting.
Namely, using a result given [T.P. (1994)] one obtains after some algebra
h
i
(4)
(4)
lim Γ() G2− (v , Y ) − v  G2+ (v , Y ) =→04i
D(z) ,z − z̄

(d)where G∆ (v , u) is the standard d-dimensional conformal block of ascalar operator with dimension ∆.

45

Outlook• The important point, first stressed in [T.P. (1994)] and explicitly
demonstrated in [F. Dolan & H. Osborn (2000)] is that the  → 0 divergencecancels out in leaving a finite result. Hence, the first nontrivial laddergraph gives a twisting of shadow scalar conformal blocks. Perhaps
this interpretation generalises to higher loops.
• One may also suspect the relevance of our results to recent works onlarge-charge expansions where imaginary chemical potentials play animportant role [L. Alvarez-Gaume et. al. (2019]. In another direction, it would beinteresting to study the manifestation of the shuffle algebra of thepolylogarithms in the context of thermal field theories.

46

Extra material

Gap equations and vacuum structure of CFTs
• .. continuing the discussion of the gap equation: Notice that φ2operator enters the spectrum both from the discontinuity part as well
as from the arc part of the spectral function.
• If we demand the absence of this operator from the spectrum thetwo above contributions must cancel each other. This gives rise to acondition that determines mth , namely

k−1 n+1
X
2(2(k − 1) − n)!
n=0

n!

1
2k−1
n
mth
Li2k−1−n (e −mth ) = − √ mthΓ(−k+ 21 )(k − 1 − n)!
2 π• This is the so-called gap equation and it is here presented for any
d = 2k + 1, k = 1, 2, . . . .
47

Gap equations and vacuum structure of CFTs

• The subleading terms in the give higher poles at∆ = d − 2 + 2l, l = 1, 2, . . . ,. These correspond to scalar operatorsof the form φ ∂ 2l φ.
• Such operators also arise from subleading terms in the z̄ expansion ofof the discontinuity parts of the spectral function., namely from(k)(k)
aDisc,1 , aDisc,2 , . . . . These operators should also disappear from thespectrum when the gap equation is satisfied.
• Although we have verified this in a couple of cases, we do not have ageneral proof as yet.

48

Gap equations from the inversion formula: bosonic case

• The arc contribution of the identity operator provides a quickconsistency check of our computations. Since the identity operator
has ∆ = 0 we see that the pole associated with it appears due toΓ(− ∆2 ).
• For the residue find
d−3

2 2
(d)
Res (aarcs(∆, 0)) = − √ Γ( d2 − 1)∆=0π
• This exactly reproduces the correct normalization of the identity
operator (in our conventions).

49

Gap equations and vacuum structure of CFTs
• The Bloch-Wigner function D(z) ≡ D2 (z) gives the volume of ideal
tetrahedra in Euclidean hyperbolic space H3 whose four vertices lie in∂H3 at the points 0, 1, ∞, and z (z is a dimensionless cross ratiohere). These tetrahedra are the building blocks for general hyperbolic
manifolds - the volume of the latter arises as the sum of ideal
tetrahedra after a suitable triangulation [e.g. Zagier].
• It is known [Witten (98), Gukov (03)] that complex SL(2, C) Chern-Simons
theory with purely imaginary level corresponds, at least semiclassically, to Euclidean three-dimensional gravity with negative
cosmological constant. We have shown in [T.P. et. al. (18)] that hQi3 arises
as a purely imaginary Chern-Simmons level in the 3d fermionic model
coupled to background monopole operator. This way we may be able
to understand the presence of hyperbolic volumes in the gap
equations and eventually in the free energy of our models.
50

Gap equations and vacuum structure of CFTs
• It is also possible to study finite-temperature fermionic 2-pt functions
using the inversion formula. The simplest case to consider is the
singlet projection of the two-point functions of Dirac fermions ψi (x),ψ̄i (x) in odd dimensions, ∆Õ νX
s C (cos θ)rshψi (x)ψ̄i (0)iβ ≡ g̃ (r , cos θ) =
ãÕs
β
r 2∆ψ
Õs 6=1d−1with ∆ψ = ∆φ + 1/2 and i, j = 1, 2, . . . , 2 2 spinor indices.
• This vanishes at zero temperature which is is a manifestation of thefact that the unit operator is absent in the finite-temperature OPE.
• The corresponding unit-normalized momentum-space 2-pt function isG̃ (d) (ωn , p) =m̃th2
ωn2 + p2 + m̃thwhere the fermionic Matsubara frequencies are ωn = 2π(n + 1/2),
n = 0, ±1, ±2, . . . .
51

Gap equations and vacuum structure of CFTs
• The fermionic propagator vanishes for m̃th = 0 so we will only
consider m̃th =6 0 in the fermionic case from now on. The calculations
follow closely the bosonic case e.g. it is known that fermionicMatsubara sums reduce to a linear combination of bosonic ones.
• We then notice that by virtue of the relationship ∆ψ = ∆φ + 1/2, thefermionic formulas can all be obtained from the bosonic ones by thesimple shift ∆ → ∆ − 1.
• The arc contributions in the fermionic case are thus given by(d)ãarcs(∆, 0) = −1∆− d−322

 ∆ − 1  ∆ − d + 1∆−1
Γ −√ m̃th Γ −22π• This gives operators of dimension ∆ = 2m + 1 and ∆ = d − 1 + 2m,
m = 0, 1, 2, . . . .

52

Gap equations and vacuum structure of CFTs

• The former are contributions that do not arise from the discontinuity
part, having the form σ̃ m with σ̃ the shadow field of ψ̄ψ. Note that,as expected, there is no contribution from the unit operator.
• The latter provide contributions from operators of the form ψ̄∂ 2m ψthat coincide with those coming from the discontinuity.
• The fermionic gap equation is the condition for the cancellation of
the latter operators from the spectrum and it reads

k−1 n+1
X(2(k − 1) − n)!
2
n=0

n!

(k − 1 − n)!

1
n+1
2k
m̃th
Li2k−1−n (−e −m̃th ) = − √ m̃thΓ(−k+ 21 )2 π

53Gap equations and vacuum structure of CFTs

The Lorentzian inversion formula together with an ansatz forthe form of the thermal 2-t function can be used to bootstrap
bosonic and fermionic CFTs in arbitrary odd-d dimensions.
The nontrivial dynamics corresponds to a rearrangement of theoperator spectrum. The gap equation arises as the condition
that certain classes of operators drop out from the spectrumof the nontrivial CFT.
The resulting picture for the operator spectrum corresponds
to the well-known large-N CFTs that arise from a generalised
Hubbard-Stratonovich transformation (see later).

54

Further lessons from the gap equation: solutions• The bosonic gap equation in d = 3 reads−mth = 2 log(1 − e −mth )with the well-known solution (related to the ”golden mean”)
(d=3)

mth

√

= 2 log( 1+2 5 ) ≈ 0.96242• In d = 5 the bosonic gap equation becomes3−16 mth= Li3 (e −mth ) + mth Li2 (e −mth )This has a complex conjugate pair of solutions given numerically by
(d=5)

mth

≈ 1.17431 ± 1.19808i

55

Gap equations and vacuum structure of CFTs

• In fact, we find that for d = 3, 7, 11, . . . the bosonic gap equation has
a unique real solution for mth and complex solutions that come inconjugate pairs - except for d = 3 where there are no complex
solutions. i.e. in d = 7 we find a real and a pair of complex conjugate
solutions.
• For d = 5, 9, 13, . . . we do not find any real solutions, and thegap equation only has pairs of complex conjugate solutions. I.e.
d = 5 we only find the solutions above, while in d = 9 we find four
complex conjugate pairs of solutions. Notice also that mth = 0 isnever a solution of the bosonic gap equations.

56

Gap equations and vacuum structure of CFTs
• The fermionic gap equations in d = 3, 5 are given respectively by2−m̃th= 2m̃th log(1 + e −m̃th ) ,42− 16 m̃th= m̃th Li3 (−e −m̃th ) + m̃th
Li2 (−e −m̃th )• For d = 3 and m̃th 6= 0 The fermionic gap equation has only a pair of(d=3)
complex conjugate imaginary solutions m̃th
= ±2πi/3. For d = 5
it has a pair of opposite real solutions, as well as a pair of complex
conjugate imaginary ones which can be found numerically.
• This pattern continues to higher dimensions, namely for
d = 7, 11, 15, . . . there is no real solution to the corresponding
fermionic gap equation, while for d = 9, 13, 17, . . . there isalways a pair of opposite real solutions and an increasing
number of complex conjugate ones.
57

Gap equations and vacuum structure of CFTs
The above pattern for the solutions of bosonic and fermionicgap equations for all odd-d fits nicely with a renormalizationgroup understanding of universality classes of scalars andfermions in general dimensions.
• In the bosonic case the standard lore is that the large-N universality
class for scalars in d = 2k + 1, k = 1, 2, . . . , is accessible via the εexpansion starting from d = 2k + 2.
• The Hubbard–Stratonovich transformation introduces a field σ via theclassically marginal interaction σφ2 . σ has dimension ∆σ = 2 in all d,and the scalars φ can be integrated out resulting in an effective
potential of the general formdVeff (σ) ∼ Trd log(−∂ 2 + σ) + g∗ σ 2 + · · ·with g∗ some critical dimensionless coupling.
58

Gap equations and vacuum structure of CFTs

• For general d the effective potential can also receive contributions
dfrom terms involving derivatives of σ, but the term σ 2 is universal.
• Performing the Trd log calculation in d − ε one finds that fordd = 4, 8, 12, . . . there is a resulting contribution of the form σ 2 log σ 2 ,which is positive and dominates for large σ. Thus, besides various
possible local minima, the effective potential has a global minimum.
d• On the other hand, for d = 6, 10, 14, . . . the term σ 2 leads to an
unbounded potential, and hence to the absence of a global
minimum, regardless of the sign of the Trd log contribution. This
matches exactly the pattern we see for mth : a real mth implies a
global minimum, while a complex mth signals unstable local
extrema with nonzero decay width.

59

Gap equations and vacuum structure of CFTs

40
30
20
10

-3

-2

1

-1

2

3

-10
-20
-30

60

Gap equations and vacuum structure of CFTs
• In the fermionic case our results are consistent with large-N
universality classes in d = 2k + 1, k = 1, 2, . . . that are accessible via
the ε expansion starting from a generalization of theGross–Neveu–Yukawa model to d = 2k + 2 [P. Zinn-Justin NPB B367 (1991)]• The corresponding Hubbard–Stratonovich transformation introducesσ̃ via the classically marginal interaction σ̃ ψ̄ψ. Here σ̃ has dimension∆σ̃ = 1 in all d, and one gets an effective potential of the form the
Trd log term enters with the opposite sign
Veff (σ̃) ∼ −Trd log(∂/ + σ̃) + g̃∗ σ̃ d + · · ·• σ̃ d gives always a bounded from below contribution (recall d is even).
However, the Trd log term changes sing as d − ε: for d = 4, 8, 12, . . .
it gives a negative contribution that dominates at infinity leading toan unstable vacuum structure, while for d = 6, 10, 14, . . . it gives a
positive contribution that guarantees the presence of a global
minimum. In either case there can be a number of unstable extrema.
This matches the obtained pattern for the m̃th .

61

Gap equations and vacuum structure of CFTs

3.5
3.0
2.5
2.0
1.5
1.0
0.5

-1.0

-0.5

0.5

1.0

62