We revisit the study of a 2D quantum field theory in the hydrodynamic regime
and develop a formalism based on Euclidean one-loop partition functions that is
suitable to analyze transport properties due to gauge and gravitational
anomalies. To do so, we generalize the method of a modified Dirac operator
developed for zero-temperature anomalies to finite temperature, chemical
potentials and rotations.Comment: 5 page

Ng, Gim Seng

Surówka, Piotr

English

arXiv

AbstractWe revisit the study of a 2D quantum field theory in the hydrodynamic regime and develop a formalism based on Euclidean one-loop partition functions that is suitable to analyze transport properties due to gauge and gravitational anomalies. To do so, we generalize the method of a modified Dirac operator developed for zero-temperature anomalies to finite temperature, chemical potentials and rotations

Elsevier - Publisher Connector 

One-loop effective actions and 2D hydrodynamics with anomalies 

Physics Letters B 746 (2015) 281–284Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
One-loop effective actions and 2D hydrodynamics with anomalies
Gim Seng Ng a, Piotr Surówka b,∗
a Departments of Physics, McGill University, Montreal, H3A 2T8, Canada
b Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received 10 February 2015
Accepted 7 May 2015
Available online 12 May 2015
Editor: L. Alvarez-Gaumé
We revisit the study of a 2D quantum ﬁeld theory in the hydrodynamic regime and develop a formalism 
based on Euclidean one-loop partition functions that is suitable to analyze transport properties due to 
gauge and gravitational anomalies. To do so, we generalize the method of a modiﬁed Dirac operator 
developed for zero-temperature anomalies to ﬁnite temperature, chemical potentials and rotations.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.1. Introduction
Quantum anomalies play an important role in the hydrody-
namic regime, leading to new transport properties [1,2]. This may 
have consequences in various physical systems such as heavy-ion 
collisions, cosmology, and condensed matter systems [3]. Therefore 
a detailed theoretical understanding of anomalies in relativistic ﬂu-
ids is necessary and has led to numerous recent developments in 
the ﬁeld. Much of the work has been based on the consistency of 
hydrodynamic expansions, including second law of thermodynam-
ics, or semiclassical kinetic theory analysis [4–10]. In addition, the 
necessity to include anomalies in ﬂuid dynamics is supported by 
AdS/CFT arguments [11–14].
A natural framework to study anomalies at zero temperature 
utilizes heat kernels or zeta-function regularizations [15,16]. The 
advantage of these techniques comes from the fact that they can 
be used to describe different types of anomalies, including contin-
uous and discrete anomalies. However, previous attempts [17,18]
to capture anomalous contribution at ﬁnite temperature (but with-
out rotations) did not give any non-zero result. Using hydrody-
namic reasoning and state/operator counting in free massless the-
ories, however, we now know that in order to get a non-zero 
contribution we need to consider rotating ﬂuids [4,5].
The aim of this paper is to show that in the Euclidean path-
integral method the spectrum of an appropriate “chiral” operator 
[19] at ﬁnite temperature and density for rotating ﬂuids indeed 
gives the required chiral-half of the results. This paper concen-
trates on a (1 + 1)-dimensional rotating ﬂuid (by which we mean 
a Lorentz-boosted ﬂuid), though we expect and hope to apply the 
* Corresponding author.
E-mail address: surowka@physics.harvard.edu (P. Surówka).http://dx.doi.org/10.1016/j.physletb.2015.05.011
0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article
SCOAP3.method to higher-dimensional ﬂuids in the presence of magnetic 
ﬁelds.
Let us consider a Dirac-type operator D on a Euclidean 
torus T 2, upon which the Euclidean path-integral is used to com-
pute the canonical partition function at ﬁnite temperature β , 
chemical potential μ and rotation . The one-loop approximation 
to the partition function is:
Z [β,μ,] ≈ e−Scl det[D] , (1)
where the classical contribution is denoted as e−Scl . Our aim is to 
calculate the one-loop effective action:
W [β,μ,] ≡ ln Z [β,μ,] + Scl ≈ lndet[D]. (2)
It describes quantum effects in the presence of background ﬁelds 
in the one-loop approximation of quantum ﬁeld theories.
In this Letter, we show how to isolate the chiral-contribution 
to the thermodynamics of a (1 + 1)-dimensional ﬁeld theory with 
global anomalies living on the manifold T 2(β,) with coordinates 
(t, x). The thermal circle’s periodicity is (t, x) ∼ (t + β, x + β)
while the compact spatial circle’s periodicity is given by (t, x) ∼
(t, x + 1). The complex parameter of the torus is then τ ≡ τ1 +
iτ2 ≡ β + iβ/(2π R), where R is the radius of spatial circle. We 
are interested in studying the one-loop effective action W [β, μ, ]
in the high temperature limit where β  1 and β/R  1 or, 
equivalently, τ → 0. A method developed to account for the con-
tribution due to chiral and Lorenz anomalies at ﬁnite temperature 
and the interpretation in terms of anomalous hydrodynamics is the 
central result of this Letter.
2. 2D free Dirac fermion
To warm up and provide a starting point for the subsequent 
inclusion of anomalies, let us ﬁrst review the computation of the  under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by 
282 G.S. Ng, P. Surówka / Physics Letters B 746 (2015) 281–284partition function of a non-anomalous, massless, free Dirac opera-
tor D = (β/(2π))(i/D). It can be obtained through
det[D] =
√
det[D2] . (3)
One trick to evaluate det[D2] is to ﬁnd Weyl spinors ψ±n (i.e. 
5ψ
±
n = ±ψ±n ) which furthermore satisfy
Dψ±n = λ±n ψ∓n , (4)
such that ψ±n diagonalizes D
2:
D2ψ±n = λ2nψ±n , λ2n ≡ λ+n λ−n . (5)
As a result we obtain
det[D2] =
∏
n
λ+n λ
−
n , (6)
and thus
det[D] =
√∏
n
(λ+n λ
−
n ) . (7)
In this work, we will always impose the (A, A)-boundary con-
dition (where A means anti-periodic while the ﬁrst (second) slot 
indicates the thermal (spatial) circle):
ψ±n (t, x) = −ψ±n (t, x+ 1) = −ψ±n (t + β, x+ β) . (8)
A basis for such Weyl spinors satisfying Eq. (4) and Eq. (8) is pro-
vided by
ψ±n (t, x)
∝ exp [+2π i(n1 + 1/2)t/β + 2π i(n2 + 1/2)(x− t)]u± ,
(9)
where 5u± = ±u± and n ∈ Z2. In our conventions,
i/D =
(
0 −∂t + i(2π R)−1∂x
∂t + i(2π R)−1∂x 0
)
. (10)
Applying (β/(2π))i/D to ψ±n , we obtain
λ+n = i
[(
n1 + 1
2
)
−
(
n2 + 1
2
)
τ¯
]
, λ−n = (λ+n )∗ , (11)
λ+n λ
−
n =
∣∣∣∣
(
n1 + 1
2
)
−
(
n2 + 1
2
)
τ
∣∣∣∣
2
. (12)
In evaluating the inﬁnite product of the eigenvalues, we use the 
identity
∞∏
n1=−∞
(
n1 + 1
2
+ a
)
= −2i cos (πa) (13)
and the Hurwitz zeta function regularization of[ ∞∏
n=0
q
− 12
(
n+ 12±a
)]
= q− 12 ζ(−1,1/2±a) = q−1/48+a2/4. (14)
The result for the one-loop effective action is
W [τ , τ¯ ] = 1
2
log
∣∣∣∣∣q−1/24
∞∏
n=1
(
1+ qn− 12
)2∣∣∣∣∣
2
= 1 log
∣∣∣∣ θ(τ )
∣∣∣∣
2
, q ≡ e2π iτ , (15)
2 η(τ )where the η and θ functions are deﬁned as
η(τ ) ≡ q1/24
∞∏
n=1
(
1− qn) ,
θ(τ ) ≡ q−1/24η(τ )
∞∏
n=1
(
1+ qn−1/2
)2
. (16)
Using the modular properties of these functions
θ(τ )
η(τ )
= θ(−1/τ )
η(−1/τ ) , (17)
and their asymptotic expansions, we obtain as τ → 0
lim
τ→0
θ(τ )
η(τ )
= exp
[
+2π i 1
24τ
]
(18)
implying the well-known Cardy’s formula for a 2D massless Dirac 
fermion
lim
τ→0W [τ , τ¯ ] = +2π i
(1/2)
24τ
− 2π i (1/2)
24τ¯
(19)
consistent with the fact that a 2D massless Dirac fermion is equiv-
alent to one left moving Weyl fermion and one right moving Weyl 
fermion with central charges cL = cR = 1/2. We note that although 
in some sense a Dirac fermion is equivalent to two Weyl fermions, 
the chiral contribution (i.e. contributions proportional to cL − cR ) 
cannot be captured by taking an appropriate half of Dirac’s an-
swer [5]. This is related to the fact that the anomalous contribution 
to the effective action is purely imaginary [19], while the above 
answer is purely real. A way to overcome this issue and to isolate 
the Weyl fermion’s contribution in the Euclidean path integral has 
been proposed by [19] and will be the focus of the next sections.
3. Chiral chemical potential
In this section, we will show how to appropriately calculate 
a determinant which captures the contribution of a single Weyl 
fermion. Let us ﬁrst introduce a non-trivial chemical potential. It 
can be done either by modifying the Dirac operator in a chiral 
way or equivalently by imposing boundary conditions for the basis 
spinors differently depending on their chirality:
ψ±n (t, x) = −ψ±n (t, x+ 1) = −eiβμ∓ψ±n (t + β, x+ β). (20)
Then, it is easy to see that Eq. (12) will be modiﬁed in the follow-
ing way:
λ+n λ
−
n =
[(
n1 + 1
2
+ ν−
)
−
(
n2 + 1
2
)
τ¯
]
×
[(
n1 + 1
2
+ ν+
)
−
(
n2 + 1
2
)
τ
]
, (21)
where iβμ± ≡ 2π iν± and ν± ’s are real. Similar computations as 
before now yield
W [τ ,ν+; τ¯ , ν−] = 1
2
log
[
θ(ν+, τ )
η(τ )
θ(ν−, τ )
η(τ )
]
, (22)
where z± = e2π iν± and
θ(ν±, τ ) ≡
∞∏
n=1
(1− qn)(1+ z±qn−1/2)(1+ z−1± qn−1/2) . (23)
Note that θ(τ ) = θ(0, τ ). The θ(ν, τ ) function has the following 
modular transformation:
G.S. Ng, P. Surówka / Physics Letters B 746 (2015) 281–284 283θ(ν, τ )
η(τ )
= e−π iν2/τ θ(ν/τ ,−1/τ )
η(−1/τ ) , (24)
which has the small τ behavior of the form
lim
τ→0
θ(ν, τ )
η(τ )
= exp
[
2π i
(
1
24τ
− ν
2
2τ
)]
. (25)
This gives
lim
τ→0W [τ ,ν+; τ¯ , ν−]
= 2π i
[
(1/2)
24τ
− (1/4)ν
2+
τ
]
− 2π i
[
(1/2)
24τ¯
− (1/4)ν
2−
τ¯
]
,
(26)
which reproduces Cardy’s formula (with chemical potentials) with 
central charges cL = cR = 1/2 and Kac–Moody levels kL = kR =
1/4.
Tuning ν− = 0 and ν+ = 0 (or vice versa), we could isolate 
the chiral-half of the answer, which probes the kL or kR part in-
dependently. The above result is not accidental but follows from 
the identiﬁcation of chiral determinants with a determinant of a 
modiﬁed Dirac-type operator which acts on Dirac spinors [19]. The 
equivalence between the “chiral boundary conditions” introduced 
above and the modiﬁed Dirac-type operator method of [19] will 
be explained below.
In the construction of [19], one naively starts by trying to study 
the eigenvalue problem of the operator
i/D+ = iμ
(
∂μ + Aμ
)
P+, P± ≡ 1
2
(1± 5) , (27)
and identiﬁes the path-integral over a single Weyl mode as the de-
terminant of such an operator. However, since this operator maps 
the P+ = 1 subspace to the P+ = 0 subspace, the eigenvalue prob-
lem for i/D+ is not well-deﬁned. Instead, one studies the modiﬁed 
operator
Dˆ ≡ iμ (∂μ + AμP+)= i
(
0 /∂−
/D+ 0
)
, (28)
where the background ﬁeld Aμ is coupled only to the chiral half 
of the operator. Since Dˆ acts on Dirac fermions, it has twice the 
number of degrees of freedom as the operator of Weyl fermions. 
But this doubling would affect only the overall normalization and 
not the effective action (as a functional of the background ﬁeld).
Furthermore, following [19], the det[Dˆ] can be computed by 
realizing that
det[Dˆ] = det[(i/∂−)(i/D+)] . (29)
If we found Dirac spinors ψ+n such that
(i/∂−)(i/D+)ψ+n = λ2nψ+n , (30)
then
det(Dˆ) =
∏
n
λ2n . (31)
A particular convenient choice will be to ﬁnd Weyl spinors ψ±n
with the properties that
i/D+ψ+n = λ+n ψ−n , i/∂−ψ−n = λ−n ψ+n , (32)
such that λ2n = λ+n λ−n . The determinant now reads1
1 We note that a different choice of basis may change the λ−n and λ+n (as deﬁned 
in Eq. (32)) but the products λ−n λ+n remain unchanged [20], i.e.det(Dˆ) =
∏
n
(λ+n λ
−
n ) . (34)
Let us now compute the determinant of Dˆ . Restricting to 
Aadxa = A0dt , we have
Dˆ =
(
0 −(∂t + A0) + i(2π R)−1∂x
∂t + i(2π R)−1∂x 0
)
. (35)
Thus, the effect of introducing the chiral coupling to the gauge 
ﬁeld is equivalent to shifting the “upper-right” part of the operator 
by ∂t → ∂t + A0. This implies that the λ−n will get shifted appro-
priately while λ+n remains unchanged:
λ+n = i
[(
n1 + 1
2
)
−
(
n2 + 1
2
)
τ¯
]
, (36)
λ−n = −i
[(
n1 + 1
2
+ ν+
)
−
(
n2 + 1
2
)
τ
]
, (37)
once we have identiﬁed
A0 = iμ+ = i ν+
Rτ2
. (38)
Note that since Aadxa = A0dt is deﬁned such that t is the Eu-
clidean time, the A0 is purely imaginary for real μ+ .
The above computation shows that the chiral coupling is im-
plemented by sending n1 → n1 + ν+ in λ−n . This is equivalent to 
not modifying the operator D but instead changing the boundary 
conditions for the ψ+n modes according to Eq. (20) with μ− = 0
but μ+ = ν+/(Rτ2). This argument demonstrates that one can in-
terchangeably implement chiral couplings either by: i) imposing 
chiral boundary conditions while using D or ii) using the chiral 
operator Dˆ instead of D together with non-chiral boundary condi-
tions. We will use this property to calculate the chiral contribution 
of rotation to our system.
4. Chiral coupling to rotation
Motivated by the equivalence presented in the previous sec-
tion, we shall now investigate what happens in the case of zero 
chemical potentials, but instead we impose the “chiral boundary 
condition” for the rotation. More precisely, we require
ψ±n (t, x) = −ψ±n (t, x+ 1)
ψ−n (t, x) = −ψ−n (t + β, x+ β)
ψ+n (t, x) = −ψ+n (t + β, x) , (39)
i.e. the “twist” induced by the rotation is imposed only on one 
of the Weyl spinors. Similar to the computations in the chemi-
cal potential case in the previous section, it is straightforward to 
show that the above “chiral boundary condition” for the rotation 
is equivalent to the modiﬁcation of the operator to “chiral-couple” 
rotation to the vielbeins. At zero temperature, such modiﬁcations 
of the operator or boundary conditions exactly reproduce what one 
expects from Lorentz anomalies. This was proven in [16]. At ﬁnite 
temperature and rotations, we will now show that rotations can be 
incorporated in a similar manner and the outcome agrees with the 
standard Cardy’s formula.
In analogy with the previous sections we calculate appropriate 
eigenvalues
λ2n = λ+n λ−n (33)
is independent of the choice of basis for the ψ± .n
284 G.S. Ng, P. Surówka / Physics Letters B 746 (2015) 281–284λ+n = i
[(
n1 + 1
2
)
−
(
n2 + 1
2
)
κ¯
]
, (40)
λ−n = −i
[(
n1 + 1
2
)
−
(
n2 + 1
2
)
τ
]
, (41)
where κ ≡ iβ/(2π R), κ2 = β/(2π R) and τ ≡ β + iβ/(2π R). The 
rest of the computation proceeds as before:
det[Dˆ] = θ(τ )
η(τ )
θ(κ)
η(κ)
⇒ W [τ , τ¯ ] = 1
2
log
[
θ(τ )
η(τ )
θ(κ)
η(κ)
]
, (42)
where q ≡ e2π iτ and p ≡ e2π iκ . Using the standard modular trans-
formations and the asymptotic expansions of the theta and eta 
functions, we obtain
lim
β→0W [τ ,κ] =
1
2
[
2π i
(
1
24τ
)
− 2π i
(
1
24κ
)]
. (43)
To isolate the chiral or anomalous contributions, we should now 
take the imaginary part of the W [19]. Thus, we deﬁne
Wanom[β,] ≡ Im [W [β,]] (44)
leading to
lim
β→0Wanom[T ,] =
1
T
[
(2π R)(2π R)
(1− (2π R)2)
]
2π
[
(1/2)
24
T 2
]
,
(45)
which is just the imaginary part of the Cardy’s formula with 
cR = 1/2 and cL = 0. We see that the imaginary part of det[Dˆ]
automatically computes the anomalous contribution to the effec-
tive action (and hence all the thermodynamical quantities) in the 
high-temperature limit.
5. Conclusions
In this Letter, we developed a method to deal with ﬁnite-
temperature anomalies using one-loop determinants of an appro-
priate operator in two dimensions. This method can be generalized 
to higher dimensions and has many potential applications. It would 
be interesting to investigate its analogue in odd dimensional ﬁeld 
theories in the study of discrete anomalies at ﬁnite temperature. 
Another interesting direction would be to isolate the contribu-
tions of anomalies to entanglement entropy with this technique. 
Finally we believe that the generalization to all even dimensions 
can provide yet another perspective on the anomaly polynomial “replacement rule”, which captures anomalous contributions to the 
partition function at ﬁnite T and μ [4–6,8,7].
Acknowledgements
We thank T. Dumitrescu, G. Dunne, K. Jensen and R. Lo-
ganayagam for useful discussions and correspondence. P.S. is grate-
ful to the Mainz Institute for Theoretical Physics (MITP) and Cal-
tech for their hospitality and partial support during the completion 
of this work. G.S.N. was supported by an NSERC Discovery Grant. 
P.S. was supported by a Marie Curie International Outgoing Fellow-
ship, grant number PIOF-GA-2011-300528.
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We revisit the study of a 2D quantum field theory in the hydrodynamic regime and develop a formalism based on Euclidean one-loop partition functions that is suitable to analyze transport properties due to gauge and gravitational anomalies. To do so, we generalize the method of a modified Dirac operator developed for zero-temperature anomalies to finite temperature, chemical potentials and rotations

Gim Seng Ng

Piotr Surówka

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Physics Letters B

One-loop effective actions and 2D hydrodynamics with anomalies

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One-loop effective actions and 2D hydrodynamics with anomalies

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