JHEP11(2024)134
Published for SISSA by
Springer
Received:
September 5, 2024
Accepted:
October 26, 2024
Published:
November 26, 2024
Angular fractals in thermal QFT
Nathan Benjamin
,
a,b
Jaeha Lee
,
a
Sridip Pal
,
a
David Simmons-Duffin
a
and Yixin Xu
a
a
Walter Burke Institute for Theoretical Physics, Caltech,
Pasadena, California 91125, U.S.A.
b
Department of Physics and Astronomy, University of Southern California,
Los Angeles, California 90089, U.S.A.
E-mail:
nathanbe@usc.edu
, jaeha@caltech.edu
, sridip@caltech.edu
,
dsd@caltech.edu
, yixinxu@caltech.edu
Abstract:
We show that thermal effective field theory controls the long-distance expansion
of the partition function of a
d
-dimensional QFT, with an insertion of any finite-order spatial
isometry. Consequently, the thermal partition function on a sphere displays a fractal-like
structure as a function of angular twist, reminiscent of the behavior of a modular form near
the real line. As an example application, we find that for CFTs, the effective free energy of
even-spin minus odd-spin operators at high temperature is smaller than the usual free energy
by a factor of
1
/
2
d
. Near certain rational angles, the partition function receives subleading
contributions from “Kaluza-Klein vortex defects” in the thermal EFT, which we classify.
We illustrate our results with examples in free and holographic theories, and also discuss
nonperturbative corrections from worldline instantons.
Keywords:
Effective Field Theories, Field Theory Hydrodynamics, Space-Time Symmetries,
Thermal Field Theory
ArXiv ePrint:
2405.17562
Open Access
,
©
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP11(2024)134
JHEP11(2024)134
Contents
1 Introduction
2
2 Folding and unfolding the partition function
4
2.1 Thermal effective action and finite velocities
4
2.2 Spin-refined partition functions: warm-up in 2d CFT
6
2.3 Folding and unfolding
8
2.4 The EFT bundle
10
3 Kaluza-Klein vortex defects
14
3.1 Background fields and EFT gauge
14
3.2 Effective action
18
3.3 Example: point-like vortex defects in 3d CFTs
19
3.4 Example: vortex defects in 4d CFTs
20
4 Fermionic theories
21
4.1 Review of 2d
22
4.2 Higher
d
24
5 Free theories
25
5.1 Massive free boson in 2d
25
5.2 3d CFTs
29
5.3 More examples with a defect action:
4
d CFTs
32
6 Topological terms: example in 2d CFT
33
7 Holographic theories
34
8 Journey to
β
= 0
37
9 Nonperturbative corrections
38
10 Discussion and future directions
41
A Qualitative picture of the 3d Ising partition function
43
B Review of plethystic sums
43
C More examples for 4d and 6d CFTs
49
D More on nonperturbative terms
51
D.1 Worldline path integral
51
D.2 A closer look at free field theory
54
D.3 Partition function from functional determinants
56
D.4 Free theories in odd dimension
58
– 1 –
JHEP11(2024)134
1 Introduction
Many aspects of conformal field theories (CFTs) are universal at high energies. A famous
example is Cardy’s formula, which states that the entropy of local operators at sufficiently
high energies takes a universal form in all unitary, compact 2d CFTs [
1
] (see [
2
–
4
] for a
precise formulation). Equivalently, the partition function of a 2d CFT
Tr
h
e
−
βH
+
iθJ
i
(1.1)
is universal in the high temperature regime
β
→
0
with
θ
∼
O
(
β
)
.
The derivation of Cardy’s formula uses invariance of the torus partition function under the
modular transformation
S
:
τ
7→−
1
/τ
. By instead using the full modular group
PSL
(2
,
Z
)
,
one finds similar universal behavior as
β
→
0
, near
any rational angle
θ
=
2
πp
q
, see e.g. [
5
].
This leads to universal “spin-refined” versions of the density of states. For example, in the
case
p
q
=
1
2
, the modular transformation
τ
7→
−
τ
2
τ
−
1
gives the universal behavior of
Tr
h
e
−
β
(
H
−
i
Ω
J
)
(
−
1)
J
i
= Tr
h
e
−
βH
+
iθJ
i
θ
=
π
+
β
Ω
,
(1.2)
in the regime
β
→
0
with
Ω
∼
O
(1)
. For any given 2d CFT, the logarithm of (
1.2) is
1
/
4
the logarithm of (
1.1 ) at high temperature, leading to a universal result for the difference
between densities of even- and odd-spin operators in 2d CFTs.
1
While modular invariance is not available on
S
d
−
1
×
S
1
in higher dimensions, higher
dimensional CFTs still display forms of universality at high energies, both in their density
of states [
12
–
18
], and OPE coefficients [
17
,
19
]. A central insight from [
14
,
20
,
21
] is that
the high temperature behavior of a CFT can be captured by a “thermal Effective Field
Theory (EFT)” that efficiently encodes the constraints of conformal symmetry and locality.
In [
15
–
18
], thermal EFT plays the role of a surrogate for the modular
S
-transformation (as
well as modular transformations on genus-2 surfaces).
In this work, we will be interested in “spin-refined” information about the CFT density
of states in general dimensions. In particular, we will study the partition function (
1.1) with
high temperature (
β
≪
1
) and finite angles
⃗
θ
. (In higher dimensions we promote
⃗
θ
and
⃗
J
to
vectors with
⌊
d/
2
⌋
components coming from the rank of
SO
(
d
)
.) The regime
⃗
θ
=
β
⃗
Ω
with
fixed
⃗
Ω
is captured by thermal EFT as discussed in [
14
,
16
,
17
,
20
,
21
]. However, when
⃗
θ
does not scale to zero as
β
→
0
, the naïve EFT description breaks down.
A simple example of a partition function with finite
⃗
θ
is ( 1.2): the relative density of
even-spin and odd-spin operators with respect to some particular Cartan generator
J
of
the rotation group. This observable is naïvely outside the regime of validity of the thermal
EFT, since
θ
remains finite as
β
→
0
.
More generally, we can consider a partition function that includes a rotation by finite
rational angles in each of the Cartan directions:
2
Tr
h
e
−
β
(
H
−
i
⃗
Ω
·
⃗
J
)
R
i
,
where
R
=
e
2
πi
p
1
q
1
J
1
+
···
+
p
n
q
n
J
n
.
(1.3)
1
Modular invariance on higher genus surfaces also leads to universal results for OPE coefficients in 2d
CFTs, as derived in [
6–10], and unified in [
11].
2
In parity-invariant theories, we can also include reflections.
– 2 –
JHEP11(2024)134
Figure 1.
A qualitative picture of
log
(
log
(
Z
))
in the 3d Ising CFT, where
Z
=
Tr
(
e
−
βH
+
iθJ
)
is the
S
2
×
S
1
partition function. To construct this picture, we took the leading terms in the EFT description
around each rational angle (up to denominator
15
), and combined them with a root-mean-square. We
give more detail in appendix
A .
Using a trick that was applied in [
22
] to study superconformal indices near roots of unity, we
will find a
different
EFT description for this partition function, in terms of the thermal EFT
on a background geometry with inverse temperature
qβ
and spatial manifold
S
d
−
1
/
Z
q
, where
q
=
lcm
(
q
1
,...,q
n
)
. This determines the small-
β
expansion of (
1.3) in terms of the usual
Wilson coefficients of thermal EFT, up to new subleading contributions from “Kaluza-Klein
vortices” that we classify. For example, the effective free energy density of (
1.3), coming from
the leading term in the thermal effective action, is smaller than the usual free energy density
by a factor of
1
/q
d
. In particular, the effective free energy density of even-spin minus odd-spin
operators described by (
1.2) is smaller by
1
/
2
d
. (This generalizes the factor of
1
/
4
in 2d.)
3
The EFT descriptions around each rational angle patch together to create fractal-like
behavior in the high-temperature partition function — see figure
1 for an illustration in the
3d Ising CFT. It is remarkable that effective field theory constrains the asymptotics of the
partition function in such an intricate way, even in higher dimensions.
Kaluza-Klein vortices appear whenever the rational rotation
R
does not act freely on
the sphere
S
d
−
1
. Each vortex creates a defect in the thermal EFT, whose action can be
written systematically in a derivative expansion in background fields. By contrast, when
R
generates a group that acts freely, no vortex defects are present, and the complete perturbative
expansion of (
1.3) in
β
is determined in terms of thermal EFT Wilson coefficients, with
no new undetermined parameters.
While most of our discussion and examples are focused on CFTs, our formalism also
applies to general QFTs. In particular, using thermal effective field theory, we derive a relation
between the partition function at temperature
T
with a discrete isometry of order
q
inserted,
3
Note that simply taking the density of states computed in [
17
] and inserting the phase
R
into the trace
will
not
give the correct answer to the partition function. For a demonstration of this in 2d, see appendix B
of [ 5].
– 3 –
JHEP11(2024)134
to the partition function with no insertion at temperature
T/q
, in the thermodynamic limit.
4
For example, we have
−
log Tr
H
(
M
L
)
h
e
−
βH
R
i
∼−
1
q
log Tr
H
(
M
L
)
h
e
−
qβH
i
+
topological
+
KK defects
(
as
L
→∞
)
.
(1.4)
Here,
M
L
is a spatial manifold of characteristic size
L
, with associated Hilbert space
H
(
M
L
)
,
R
is a discrete isometry of order
q
, and “
∼
” denotes agreement to all perturbative orders
in
1
/L
. The relation (
1.4) holds whenever the theory is gapped at inverse temperature
qβ
.
We write the most general relation in (
2.28), which we check in both massive and massless
examples. An interesting consequence of this simple formula is that twists by discrete
isometries can be sensitive to lower-temperature phases of the theory. For example, the
partition function of QCD at temperature
T >
Λ
QCD
, twisted by a discrete isometry with
order
q
, becomes sensitive to physics below the confinement scale when
T/q <
Λ
QCD
.
This universality of partition functions with spacetime symmetry insertions is in contrast
to the case for global symmetry insertions. The insertion of a global symmetry generator
operator is equivalent to turning on a new background field in the thermal EFT. The
dependence of the effective action on this background field introduces new Wilson coefficients
that are not necessarily related in a simple way to the Wilson coefficients without the global
symmetry background, see e.g. [
23
–
25
].
The paper is organized as follows. In section
2 , we present a derivation of our main
result: a systematic study of the high temperature expansion of the partition function of
any quantum field theory with the insertion of a discrete isometry. In section
3 , we look in
more detail at the Kaluza-Klein vortices that appear on
S
d
−
1
when the discrete isometry
(which is a rational rotation in this case) has fixed points. In section
4 , we discuss subtleties
that appear for fermionic theories. In section
5 , we give several examples in free theories
that illustrate our general results. In section
6 , we consider thermal effective actions with
topological terms. In section
7 , we apply our results to holographic CFTs. In section
8 , we
look at irrational
θ
. In section
9 , we discuss non-perturbative corrections in temperature.
Finally in section
10 , we conclude and discuss future directions.
2 Folding and unfolding the partition function
2.1 Thermal effective action and finite velocities
Equilibrium correlators of generic interacting QFTs at finite temperature are expected to have
a finite correlation length. Equivalently, the dimensional reduction of a generic interacting
QFT on a Euclidean circle is expected to be gapped. When this is the case, long-distance
finite-temperature observables of the QFT can be captured by a local “thermal effective
action” of background fields [
14
,
20
,
21
]. For example, consider the partition function of a
QFT
d
on
M
L
×
S
1
β
, where the spatial
d
−
1
-manifold
M
L
has size
L
. In the thermodynamic
4
We are extremely grateful to Luca Delacretaz for emphasizing the general QFT case to us.
– 4 –
JHEP11(2024)134
limit of large
L
, we have
Tr
H
(
M
L
)
[
e
−
βH
L
] =
Z
QFT
[
M
L
×
S
1
β
]
=
Z
gapped
[
M
L
]
∼
e
−
S
th
[
g,A,φ
]
+
nonperturbative in
1
/L
(
L
→∞
)
,
(2.1)
where
H
(
M
L
)
is the Hilbert space of states on
M
L
, and
H
L
is the Hamiltonian. Here,
the thermal effective action
S
th
depends on a
d
−
1
-dimensional metric
g
ij
, a Kaluza-Klein
gauge field
A
i
, and a dilaton
φ
, which can be obtained by placing the
d
-dimensional metric
in Kaluza-Klein (KK) form
G
μν
dx
μ
dx
ν
=
g
ij
(
⃗x
)
dx
i
dx
j
+
e
2
φ
(
⃗x
)
(
dτ
+
A
i
(
⃗x
))
2
,
(2.2)
where
τ
∼
τ
+
β
is a periodic coordinate along the thermal circle. The derivative expansion
for
S
th
becomes an expansion in inverse powers of the length
L
.
If the spatial manifold
M
L
possesses a continuous isometry
ξ
, then we can additionally
twist the partition function by the corresponding charge
Q
ξ
:
Tr
H
(
M
L
)
h
e
−
β
(
H
L
−
iαQ
ξ
)
i
.
(2.3)
Geometrically, this twist corresponds to a deformation of the background fields
g,A,φ
that
depends on
αξ
. In the thermodynamic limit, we can describe (
2.3) using the thermal effective
action, provided that the background fields
g,A,φ
remain finite as
L
→ ∞
. In particular,
the combination
αξ
must remain finite as
L
→∞
. The physical reason is that
iα
represents
the velocity of the system in the direction of
ξ
in the canonical ensemble. This velocity must
remain finite in order to have a good thermodynamic limit.
By contrast, suppose that
M
L
possesses a nontrivial discrete isometry
R
with finite
order
R
q
= 1
. If we twist the partition function by
R
,
Tr
H
(
M
L
)
h
e
−
βH
R
i
,
(2.4)
then physically this corresponds to a system whose “velocity” is of order
L
. The background
fields
g,A,φ
naïvely do not have a good thermodynamic limit, and we cannot apply the
thermal effective action in an obvious way.
2.1.1
Example: CFT partition function
An important example for us is the partition function of a CFT
d
on
S
d
−
1
×
S
1
β
. Conformal
invariance dictates that
Tr
h
e
−
β
(
H
−
i
⃗
Ω
·
⃗
J
)
i
= Tr
H
(
S
d
−
1
L
)
h
e
−
Lβ
(
H
L
−
i
⃗
Ω
·
⃗
J
L
)
i
.
(2.5)
On the left-hand side, we have the usual partition function of the CFT on a sphere of radius
1
.
On the right-hand side,
H
L
denotes the Hamiltonian on a sphere
S
d
−
1
L
of radius
L
, and
⃗
J
L
are
generators of isometries of the sphere, normalized so that the corresponding Killing vectors
are finite in the flat-space limit
L
→ ∞
. (For example, for a rotation of the sphere by an
angle
φ
, a Killing vector with a finite flat-space limit is
1
L
∂
∂φ
.)
– 5 –
JHEP11(2024)134
When
β
is small, we can set
L
=
O
(1
/β
)
on the right-hand side and try to apply the
thermal effective action (
2.1). We find that in order to have a good thermodynamic limit
as
β
→
0
, the angular potentials
⃗
Ω
must remain finite. Phrased in terms of the rotation
angle
⃗
θ
=
β
⃗
Ω
, we find that
⃗
θ
must scale to zero as
β
→
0
. Provided this is the case, the
1
/L
expansion of the thermal effective action gives an expansion in small
β
for the CFT
partition function.
We can also understand condition
⃗
θ
→
0
more explicitly from a direct computation
using the thermal effective action. In a CFT, the thermal effective action is constrained
by
d
-dimensional Weyl invariance. The most general coordinate- and Weyl-invariant action
takes the form
5
,
6
S
th
=
Z
d
d
−
1
⃗x
β
d
−
1
p
b
g
−
f
+
c
1
β
2
b
R
+
c
2
β
2
F
2
+
...
+
S
anom
.
(2.6)
Here
b
g
=
e
−
2
φ
g
,
b
R
is the Ricci scalar built from
b
g
,
F
2
is a Maxwell term, etc. The term
S
anom
accounts for Weyl anomalies (which are not important for the present discussion).
On the geometry
S
d
−
1
×
S
1
β
, we can easily determine
g,A,φ
and evaluate
S
th
[
17
]:
S
th
=
vol
S
d
−
1
Q
n
i
=1
(1 + Ω
2
i
)
"
−
fT
d
−
1
+ (
d
−
2)
(
d
−
1)
c
1
+
2
c
1
+
8
d
c
2
n
X
i
=1
Ω
2
i
!
T
d
−
3
+
...
#
.
(2.7)
We see that terms of order
T
d
−
1
−
k
=
β
k
−
d
+1
in the high-temperature expansion of
S
th
are multiplied by a polynomial in the angular potentials
Ω
i
of degree
k
(see e.g. examples
in [
26
]). Consequently,
θ
i
→
0
as
β
→
0
is necessary for the high-temperature expansion
to be well-behaved.
To summarize, the thermal effective action can describe “small” angles
θ
∼
β
Ω
, where
the angular velocity remains finite in the thermodynamic limit. However, results from the
thermal effective action like (
2.7) break down outside this regime. How can we access more
general angles?
2.2 Spin-refined partition functions: warm-up in 2d CFT
As a warm-up, in 2d CFT, we can compute partition functions at more general angles using
modular invariance. Let us review how this works and derive some example results. For
convenience, we write the partition function as:
Z
(
τ,
τ
) = Tr
h
e
2
πiτ
(
L
0
−
c
24
)
−
2
πi
τ
(
L
0
−
c
24
)
i
,
(2.8)
where
τ
=
iβ
2
π
+
θ
2
π
and
τ
=
τ
∗
.
7
The high temperature behavior of
Z
(
τ,
τ
)
at small angles
can be obtained by performing the modular transformation
τ
→ −
1
/τ
(similarly for
τ
)
5
For simplicity, here we assume that the theory is free of gravitational anomalies.
6
Note that [
17
] worked in conventions where
τ
has periodicity
1
, and
β
is absorbed into the field
φ
. In
this paper, we instead use conventions where
τ
has dimensionful periodicity
β
(later we will also have other
periodicities) so that explicit powers of
β
appear in the action (
2.6), as required by dimensional analysis. To
convert from the conventions of [
17
] to the conventions in this work, one shifts the dilaton by
φ
→
φ
+
log
β
.
7
Note that in this section,
τ
denotes the modular parameter of the torus, while in other sections
τ
denotes
Euclidean time. We hope this will not cause confusion.
– 6 –
JHEP11(2024)134
and approximating by the contribution of the vacuum state. The result agrees with the
thermal effective action:
Tr
h
e
−
β
(
H
−
i
Ω
J
)
i
∼
e
−
S
th
= exp
"
vol
S
1
(1 + Ω
2
)
f
β
#
= exp
"
4
π
2
β
(1 + Ω
2
)
c
12
#
(
CFT
2
)
,
(2.9)
where
f
=
2
πc
12
. Here, we assume
c
L
=
c
R
for simplicity. In this case, only the cosmological
constant term appears in the thermal effective action in 2d.
Now let us instead assume that
θ
2
π
is close to a nonzero rational angle
p
q
, so that
τ,
τ
are very close to
p
q
. Following [
5
], we can perform a different modular transformation to
map
(
τ,
τ
)
close to
±
i
∞
and approximate the partition function by the vacuum state in the
new channel. For example, let us study the partition function with an insertion of
(
−
1)
J
given in (
1.2). In this case, we have
τ
=
1
2
+
β
Ω
2
π
+
iβ
2
π
τ
=
1
2
+
β
Ω
2
π
−
iβ
2
π
, β
≪
1
,
Ω
∼
O
(1)
.
(2.10)
Modular invariance is the statement
Z
(
γ
◦
τ,γ
◦
τ
) =
Z
(
τ,
τ
)
, γ
∈
PSL(2
,
Z
)
.
(2.11)
An appropriate transformation in this case is
γ
=
±
−
1 0
2
−
1
!
∈
PSL(2
,
Z
)
,
(2.12)
which leads to
Tr
h
e
−
β
(
H
−
i
Ω
J
)
(
−
1)
J
i
= Tr
e
2
πi
e
τ
(
L
0
−
c
24
)
−
2
πi
e
τ
(
L
0
−
c
24
)
∼
exp
"
1
4
4
π
2
β
(1 + Ω
2
)
c
12
#
,
where
e
τ
=
−
1
2
+
πi
2
β
(1
−
i
Ω)
,
e
τ
=
e
τ
∗
.
(2.13)
On the right-hand side, we approximated the trace by the contribution of the vacuum state in
the
β
→
0
limit. We find that the partition function weighted by
(
−
1)
J
grows exponentially
in
1
/β
, with an exponent that is
1
/
4
of the un-weighted case (
2.9).
For a general angle
θ
2
π
close to
p
q
, we repeat the same logic above but with a more
complicated modular transformation, namely
γ
=
±
−
(
p
−
1
)
q
b
q
−
p
!
∈
PSL
(2
,
Z
)
,
(2.14)
where
(
p
−
1
)
q
is the inverse of
p
modulo
q
, and
b
is chosen so the matrix has determinant
1
. We get
Tr
h
e
−
β
(
H
−
i
Ω
J
)
e
2
πi
p
q
J
i
∼
exp
"
1
q
2
4
π
2
β
(1 + Ω
2
)
c
12
#
.
(2.15)
– 7 –
JHEP11(2024)134
In general, we find that the partition function of a 2d CFT weighted by
e
2
πi
p
q
J
grows
exponentially in
1
/β
, with an exponent that is
1
/q
2
of the un-weighted case (
2.9).
Because modular invariance is not available in higher dimensions, it will be useful to
rederive (
2.15) in a different way. We now describe two (related) approaches that can
generalize to higher dimensions.
2.3 Folding and unfolding
Thermal EFT naively breaks down in spin-refined partition functions like (
1.2) because the
large spacetime symmetry
(
−
1)
J
moves us outside the thermodynamic limit. One way to
recover an EFT description is to perform a change of coordinates that makes
(
−
1)
J
look
more like a global symmetry.
For example, consider a spin-refined partition function of a 2d QFT (not necessarily
conformal) on
S
1
L
×
S
1
β
,
Tr
h
e
−
βH
(
−
1)
J
i
,
(2.16)
where
(
−
1)
J
denotes a rotation of the spatial circle
S
1
L
by
π
. We can reinterpret
one
copy of
the QFT on
S
1
L
×
S
1
β
as
two
copies of the QFT on
(
S
1
L
/
Z
2
)
×
S
1
β
, with topological defects that
glue the two copies to each other, see the middle of figure
2 . In this picture, the operator
(
−
1)
J
becomes a topological defect that simply permutes the two copies of the QFT as
we move along the time direction. If we begin in one copy of the QFT and move by
β
in
Euclidean time, we pass once through the
(
−
1)
J
defect and go to the other copy. Moving by
β
again, we pass through the
(
−
1)
J
defect again and end up in the first copy. Thus, inserting
(
−
1)
J
into the partition function creates a new effective thermal circle of length
2
β
.
This reinterpretation of the path integral with a
(
−
1)
J
insertion is illustrated in figure
2 .
One wrinkle (that is clear in the figure) is that the effective thermal
S
1
2
β
is nontrivially
fibered over the spatial circle
S
1
L
/
Z
2
: when we go once around the new spatial circle, the
S
1
2
β
shifts by
β
.
So far, we have considered a rotation angle of
π
. However, it is straightforward to study
nearby rotation angles of the form
θ
=
π
+
β
Ω
. On the left-hand side of figure
2 , we simply
insert an additional topological operator along the spatial cycle that implements the small
rotation
e
iβ
Ω
J
. Following the manipulations in the figure, we end up with a product of two
such operators on the new spatial cycle
S
1
L
/
Z
2
, which together implement a rotation of
2
β
Ω
.
The advantage of this rewriting of the path integral is that we can now smoothly take
the thermodynamic limit
L
→∞
and use the thermal effective action. The effective inverse
temperature is
2
β
, the rotation angle is
2
β
Ω
, and the effective spatial cycle is
S
1
L
/
Z
2
.
In fact, the above construction is straightforward to generalize to twists by any ra-
tional angle:
Tr
h
e
−
β
(
H
−
i
Ω
J
)+2
πi
p
q
J
i
.
(2.17)
We interpret (
2.17) as the partition function of
q
copies of the QFT on the space
S
1
L
/
Z
q
, with
appropriate topological defects that glue the copies together. The operator
e
2
πi
p
q
J
becomes a
topological defect that permutes the copies of the QFT as we move around the Euclidean
– 8 –
JHEP11(2024)134
L
β
2
β
L/
2
=
⇒
=
⇒
Figure 2. Left:
The torus partition function with spatial cycle of length
L
, inverse temperature
β
, and an insertion of
(
−
1)
J
. The
(
−
1)
J
insertion means we must glue the top and bottom of the
figure with a half shift around the spatial circle. We split the figure into a left and right half using
the trivial defect (vertical dashed line), and for convenience we color the right half grey.
Middle:
Placing the black and grey rectangles on top of each other, we can interpret this same observable
as the partition function of two copies of the QFT (black and grey) on an
(
L/
2)
×
β
rectangle, with
boundary conditions inherited from the left figure.
Right:
Finally, we can re-stack the two copies
of the QFT, resulting in a single copy of the QFT with a new spatial circle of length
L/
2
and an
effective thermal circle of length
2
β
. Note that the effective thermal circle is nontrivially fibered over
the new spatial circle.
time circle. This creates an effective thermal circle
S
1
qβ
, which is fibered over
S
1
L
/
Z
q
. We
can now apply thermal EFT on
S
1
L
/
Z
q
.
2.3.1
Example: 2d CFT
As an example application, we can recover our previous answer for the spin-refined partition
function of a 2d CFT. For a twist by
(
−
1)
J
, we find
Tr
h
e
−
β
(
H
−
i
Ω
J
)
(
−
1)
J
i
∼
e
−
S
th
[
S
1
/
Z
2
×
S
1
2
β
]
= exp
"
2
πc
12
vol(
S
1
/
Z
2
)
2
β
(1 + Ω
2
)
#
= exp
"
1
4
4
π
2
β
(1 + Ω
2
)
c
12
#
.
(2.18)
In the action, we obtain one factor of
1
2
from the smaller spatial cycle
S
1
/
Z
2
, and another
factor of
1
2
from the larger thermal circle, resulting in an overalll factor of
1
4
that agrees
with the result from modular invariance (
2.13).
8
More generally, for a twist by
2
πp
q
, the thermal effective action gives
Tr
e
−
β
(
H
−
i
Ω
J
)
e
2
πip
q
J
∼
e
−
S
th
[
S
1
/
Z
q
×
S
1
qβ
]
= exp
"
2
πc
12
vol(
S
1
/
Z
q
)
qβ
(1 + Ω
2
)
#
= exp
"
1
q
2
4
π
2
β
(1 + Ω
2
)
c
12
#
.
(2.19)
8
The fact that the thermal circle is nontrivially fibered plays no role here because the thermal effective
action is the integral of a local coordinate-invariant quantity that does not detect global features of the
bundle. In a theory with a gravitational anomaly, the thermal effective action would contain an additional
1
-dimensional Chern-Simons for the Kaluza-Klein gauge field, which can detect the nontrivial topological
structure of the thermal circle bundle, see section
6 . The nontrivial topology also enters into nonperturbative
corrections, see section
9 .
– 9 –
JHEP11(2024)134
We find that the effective free energy at high temperature for the spin-refined partition
function (
2.17) is down by a factor of
q
2
, in agreement with (
2.15). Note that the precise
permutation of the copies of the CFT implemented by
e
2
πi
p
q
J
depends on
p
, but the length
of the resulting thermal circle does not. Consequently the partition function is independent
of
p
, up to nonperturbative corrections as
β
→
0
.
9
2.3.2
Higher dimensions
The above construction works for
d >
2
as well, and on more general geometries. Consider a
QFT
d
on any
(
d
−
1)
-dimensional spatial manifold
M
L
with a discrete isometry
R
of finite
order
R
q
= 1
. Again, we can reinterpret one copy of the QFT on
M
L
×
S
1
β
as
q
copies of
the QFT on
(
M
L
/
Z
q
)
×
S
1
β
, with topological defects that glue the copies to each other. In
this picture,
R
is represented as a topological defect that simply permutes the
q
copies of the
QFT as we move along the time direction, creating an effective inverse temperature
qβ
.
2.4 The EFT bundle
Before exploring further consequences of this idea, it will be helpful to adopt a more abstract,
geometrical perspective on this construction. Consider again a
d
-dimensional QFT with
spatial manifold
M
L
. Given an isometry
U
∈
Iso
(
M
L
)
, the partition function twisted by
U
,
10
Tr
H
(
M
L
)
h
e
−
βH
U
i
,
(2.20)
is computed by the path integral of the CFT on the mapping torus
M
β,U
≡
(
M
L
×
R
)
/
Z
,
(2.21)
where
Z
=
⟨
h
⟩
is generated by
h
:
M
L
×
R
→M
L
×
R
,
h
: (
⃗x,τ
)
7→
(
U⃗x,τ
+
β
)
,
(2.22)
where
⃗x
is a coordinate on
M
L
.
Now let us specialize to
U
=
R
, where
R
has order
q
. In this case, the
q
-th power of
h
acts very simply: it leaves
M
L
invariant, and shifts
τ
by
qβ
:
h
q
: (
⃗x,τ
)
7→
(
⃗x,τ
+
qβ
)
.
(2.23)
Consequently, it is useful to decompose
Z
∼
=
q
Z
×
Z
q
=
⟨
h
q
⟩×
(
⟨
h
⟩
/
⟨
h
q
⟩
)
, and obtain the
mapping torus
M
β,R
via two successive quotients. We first quotient by
q
Z
∼
=
⟨
h
q
⟩
(which
turns
R
into
S
1
qβ
), and then quotient by
Z
q
=
⟨
h
⟩
/
⟨
h
q
⟩
:
M
β,R
= ((
M
L
×
R
)
/q
Z
)
/
Z
q
= (
M
L
×
S
1
qβ
)
/
Z
q
.
(2.24)
9
There is
p
-dependence if the theory is fermionic (see section
4 ) or has a gravitational anomaly (see
section
6 ).
10
Here, we abuse notation and write
U
for both the isometry and the operator implementing its action on
the Hilbert space
H
(
M
L
)
.
– 10 –
JHEP11(2024)134
The quotient
(
M
L
×
S
1
qβ
)
/
Z
q
on the right-hand side of (
2.24) can be viewed as a bundle
in two different ways. Firstly, it is a
M
L
-bundle over
S
1
qβ
/
Z
q
∼
=
S
1
β
. This is the usual point
of view of the trace as a spatial manifold evolving over Euclidean time
β
. However, we
can alternatively view
(
M
L
×
S
1
qβ
)
/
Z
q
as an
S
1
qβ
bundle over
M
L
/
Z
q
. We call this latter
description the “EFT bundle.” In section
2.3 , the EFT bundle was a nontrivial
S
1
qβ
bundle
over
S
1
L
/
Z
q
. As we saw, the virtue of the EFT bundle is that the thermodynamic limit
L
→ ∞
is straightforward: we can dimensionally reduce along the effective thermal circle
S
1
qβ
without leaving the thermodynamic limit. The theory is then described by thermal EFT
with effective inverse temperature
qβ
and spatial cycle
M
L
/
Z
q
.
Suppose for the moment that the action of
R
on
M
L
is free, so that
M
L
/
Z
q
is smooth.
(This is the case, for example, for a rational rotation of the spatial circle in 2d.) For any
term in the thermal effective action that is the integral of a local density, the effect of the
quotient by
Z
q
is simply to multiply its contribution by
1
/q
. Thus, we conclude
−
log Tr
h
e
−
βH
R
i
∼−
1
q
log Tr
h
e
−
qβH
i
+
topological
(
if the
R
action is free
)
.
(2.25)
Here, “
∼
” denotes agreement to all perturbative orders in the
1
/L
expansion. The term
“topological” indicates potential contributions from a finite number of terms capable of
detecting the topology of the EFT bundle, which cannot be written as the integral of a local
gauge/coordinate-invariant density. We discuss such terms in section
6 .
Let us pause to note that the result (
2.25) really only requires that the theory be gapped
at inverse temperature
qβ
(not necessarily at inverse temperature
β
), since we only use
locality of the thermal effective action on the right-hand side.
2.4.1
Adding “small” isometries
Just as before, we can also consider inserting into the trace an additional “small” isometry
U
=
e
iβ
(
αQ
ξ
)
, where
ξ
is a Killing vector on
M
L
,
Q
ξ
is its corresponding charge, and
α
is
the corresponding thermodynamic potential. We will be mainly interested in the case where
U
commutes with the discrete isometry
R
, so we assume this henceforth. The insertion
of
U
can be thought of as a topological defect that wraps
M
L
. Consequently, the defect
wraps
q
times around the base of the EFT bundle
M
L
/
Z
q
, resulting in an effective rotation
U
q
. We conclude that
−
log Tr [
gR
]
∼−
1
q
log Tr [
g
q
] +
topological
(
if the
R
action is free
)
,
(2.26)
where
g
=
e
−
βH
U
.
In fact, this argument applies to any global symmetry element
V
as well, so (
2.26) holds
when
g
is multiplied by a global symmetry group element:
g
=
e
−
βH
UV
. We can think of
V
as implementing a nontrivial flat connection for a background gauge field coupled to the
global symmetry. In this case, the “topological” terms in (
2.26) could include contributions
from nontrivial topology of this connection.
– 11 –
JHEP11(2024)134
We can also understand the insertion of “small” isometries geometrically. Again, the idea
is to view the mapping torus
M
β,UR
as the result of two successive quotients
M
β,UR
= ((
M
L
×
R
)
/
⟨
h
q
⟩
)
/
Z
q
=
M
qβ,U
q
/
Z
q
,
where
h
: (
⃗x,τ
)
7→
(
UR⃗x,τ
+
β
)
,
(2.27)
where
Z
q
=
⟨
h
⟩
/
⟨
h
q
⟩
, and we have used
(
UR
)
q
=
U
q
. On the right-hand side, we have
the mapping torus
M
qβ,U
q
which is described by the thermal effective action at inverse
temperature
qβ
, with small isometries
U
q
turned on. The effect of the
Z
q
quotient is to
multiply the contribution of any integral of a local density by
1
/q
. This again leads to (
2.26).
11
The work [
22
] uses similar ideas to characterize superconformal indices of 4d CFTs near
roots of unity. Our novel contribution is to apply these ideas in not-necessarily-supersymmetric,
not-necessarily-conformal theories, on general spatial geometries, and also to describe the
effects of Kaluza-Klein vortices (see below), which do not appear in superconformal indices.
2.4.2
Non-free actions and Kaluza-Klein vortex defects
What happens if the action of
R
is not free? For example, in a 3d QFT on
S
2
×
S
1
β
, the
action of
(
−
1)
J
(where
J
is the Cartan generator of the rotation group) has fixed points
at the north and south poles of
S
2
. In this case, the EFT bundle degenerates at the fixed
loci of nontrivial elements of
Z
q
, namely
R,...,R
q
−
1
∈
Z
q
. After dimensional reduction,
these degeneration loci becomes defects
D
i
(with
i
labelling the set of defects) in the
d
−
1
dimensional thermal effective theory. We call them “Kaluza-Klein vortex defects” because
the KK gauge field
A
has nontrivial holonomy around them, as we explain in section
3.1 .
Each defect
D
i
contributes to the partition function a coordinate-invariant effective
action
S
D
i
of the background fields
g,A,φ
in the infinitesimal neighborhood of
D
i
. We then
have the more general result
−
log Tr [
gR
]
∼−
1
q
log Tr [
g
q
] +
topological
+
X
D
i
S
D
i
.
(2.28)
We conjecture that for generic interacting QFTs, the KK vortex defects will be gapped. (In
fact, in this work, we will study several examples of free theories where the appropriate
defects are still gapped.) In this case, each
S
D
i
will be a local functional of
g,A,φ
.
In CFTs, the defect actions
S
D
i
are additionally constrained by Weyl-invariance, just
like the bulk terms in the thermal effective action. We will determine the explicit form of
S
D
:=
P
D
i
S
D
i
in CFTs later in section
3 . For now, we simply note that the leading term in
the derivative expansion of
S
D
i
in a CFT is a cosmological constant localized on
D
i
:
12
S
D
i
=
a
D
i
Z
D
i
d
n
i
y
(
qβ
)
n
i
q
b
g
|
D
i
+
higher derivatives
.
(2.29)
Here, we assume that
D
i
is
n
i
-dimensional,
y
are coordinates on the defect, and
b
g
|
D
i
denotes
the pullback of
b
g
=
e
−
2
φ
g
to
D
i
. This term behaves like
β
−
n
i
as
β
→
0
. In the case
n
i
= 0
, i.e.
11
When
U
and
R
don’t commute, the same logic works but we have
M
qβ,
(
UR
)
q
/
Z
q
on the right-hand side
of ( 2.27). We can still use thermal EFT, since
(
UR
)
q
is
O
(
β
)
close to the identity.
12
S
D
i
itself can also have topological terms; if the topological term has no derivatives (i.e. the Wilson line
of the KK photon), it will contribute at the same order in
β
as the defect cosmological constant.
– 12 –