Dissipative hydrodynamics in superspace

0.49 MB·

by  Kristan Jensen

Prepared for submission to JHEP
7 Dissipative hydrodynamics in superspace
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2 Kristan Jensen,a Natalia Pinzani-Fokeeva,b,c and Amos Yaromb

] aDepartment of Physics and Astronomy, San Francisco State University, San Francisco, CA 94132,
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t USA
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p bDepartment of Physics, Technion, Haifa 32000, Israel
e cDepartment of Mathematics, University of Haifa, Haifa 31905, Israel
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[ E-mail: [email protected], [email protected],

1 [email protected]
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Abstract: We construct a Schwinger-Keldysh effective field theory for relativistic hy-
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4 drodynamics using a superspace formalism. Superspace allows us to efficiently impose the
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symmetries of the problem and to obtain a simple expression for the effective action. We
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. show that the theory we obtain is compatible with the Kubo-Martin-Schwinger condition,
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0 which in turn implies that Green’s functions obey the fluctuation-dissipation theorem. Our
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approach complements existing formulations found in the literature.
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Contents
1 Introduction 1
2 Symmetries 4
2.1 Doubled symmetries 7
2.2 Topological Schwinger-Keldysh symmetry 10
2.3 The reality condition 15
2.4 Kubo-Martin-Schwinger symmetry 17
3 Ward identities and degrees of freedom 31
4 Constructing the effective action 34
4.1 The structure of the action 36
4.2 The derivative expansion 38
4.3 The fluctuation-dissipation relation 42
5 Discussion and outlook 43
5.1 Comparison with previous work 43
5.2 Outlook 45
1 Introduction
The study of fluid mechanics dates back to ancient Greece and the works of Archimedes.
Sincethen, hydrodynamicshasundergonecountless transformations andmodificationsbefore
settling into its modern form. Yet, while the dynamics of fluids are prevalent and common,
a full understanding of fluid dynamics is still lacking in many respects.
From a field theoretic viewpoint, relativistic fluid dynamics is a low-energy effective
descriptionintermsofconstitutiverelationsforthestresstensorandotherconservedcurrents.
Theconstitutive relations allow oneto solve theassociated conservation equations andobtain
the universal behavior of fully retarded, thermal correlation functions. Up until the works
of [1–3], see also [4, 5], there was no theory from which one could consistently evaluate
symmetric, advanced and other correlation functions associated with the dynamics of the
fluid. More simply,fluiddynamicsdidnotfollow fromanaction principle. Itwas inthissense
that hydrodynamics was incomplete. The current treatise merges the somewhat orthogonal
constructions of [2] and [1, 3]. We will elaborate on the differences between our formalism
and that in the literature when appropriate.
Thecurrentworkandthatof[1–3]donotstandbythemselves. Avariationalprinciplefor
dissipationless fluiddynamicswasformulated duringthelastcenturyinthecontext ofgeneral
– 1 –
relativity, seee.g., [6]oralso[7,8]. Thisvariational principlewasrecently revisitedandrecast
in modern language in [9–11]. Contemporaneously with these developments, the authors
of [12, 13] argued that hydrodynamics simplifies dramatically in hydrostatic equilibrium.
Moreover, in that limit, the constitutive relations can be obtained from a local generating
function. While these approaches shed light on the structure of a possible effective action for
hydrodynamics and offer an alternative to some phenomenological approaches [14, 15], they
are lacking in several aspects. Apart from failing to capture dissipation, they do not account
for all possible non-dissipative transport phenomena. (For instance, they fail to account for
Hall viscosity [16–18].) Yet another modern approach to obtain an action for hydrodynamics
involves adding the effects of stochastic noise [19]. Other attempts include [20–23]. Most
recently, the authors of [17, 18, 22, 24] have advocated that the Schwinger-Keldysh formalism
is the natural setting to write an effective action for dissipative fluid dynamics.
The Schwinger-Keldysh formalism [25, 26] was developed around the middle of the past
century in order to obtain a generating function for connected correlators in a state described
by a density matrix ρ in the far past. Recall that vacuum correlation functions can be
−∞
computed by an appropriate variation of the vacuum generating functional with respect to
sources,withonesuchsourceforeachoperator. TheSchwinger-Keldyshgeneratingfunctional
naturally has two sources associated with each operator. This feature allows one to not only
compute the retarded correlation functions in the state ρ , but also partially symmetric
−∞
and advanced ones.
Let us review the definition and attributes of the Schwinger-Keldysh partition function.
We begin with a generic quantum field theory which, in the infinite past, is in a mixed state
characterized by a density matrix ρ (which is not necessarily normalized). TheSchwinger-
−∞
Keldysh partition function is given by
†
Z[A ,A ] Tr U [A ]ρ U [A ] . (1.1)
1 2 ≡ 1 1 −∞ 2 2
(cid:16) (cid:17)
HereU [A ]isthetime-evolution operator,evolvingstatesfromtheinfinitepasttotheinfinite
1 1
future. It is a functional of external sources which we schematically denote as A . The time
1
evolution operator U [A ] is similarly defined, and is distinct from U via its dependence on
2 2 1
A .
2
For a quantum field theory with a Lagrangian description, the Schwinger-Keldysh parti-
tion function may be written as a functional integral. If we denote the fundamental fields of
the theory by φ and the action by S[φ] then,
Z[A ,A ] = [dφ ][dφ ]exp i S[φ ] S[φ ]+ ddx (O [φ ]A O [φ ]A ) . (1.2)
1 2 1 2 1 2 1 1 1 2 2 2
− −
Z (cid:20) (cid:18) Z (cid:19)(cid:21)
Here O is the operator conjugate to A, and the fields φ and φ satisfy boundary conditions
1 2
in the infinite past and future. In the past, the boundary conditions depend on ρ . In the
−∞
future, they are identified, lim φ (t) = lim φ (t). We also require that the sources
t→∞ 1 t→∞ 2
asymptote to the same values in the past and future, lim A (t) = lim A (t) and
t→∞ 1 t→∞ 2
lim A (t)= lim A (t). See e.g. [2, 27] for a modern discussion.
t→−∞ 1 t→−∞ 2
– 2 –
Equation (1.2) gives an ultraviolet description of the Schwinger-Keldysh partition func-
tion for theories with a Lagrangian description. Due to the universality of hydrodynamics it
is expected that when the initial state is thermal, i.e. ρ = e−bH (or its variants with a
−∞
chemicalpotential), theinfraredbehaviorofZ willalsobeuniversal. Moreprecisely,following
the usual logic of Wilsonian effective field theory, we may write
Z[A , A ] = [dξ ][dξ ]eiSeff[ξ1,ξ2;A1,A2], (1.3)
1 2 1 2
Z
when the sources A and A vary over arbitrarily long scales. Here S [ξ , ξ ;A ,A ] is a
1 2 eff 1 2 1 2
low-energy Schwinger-Keldysh effective action with (doubled) infrared degrees of freedom ξ
1
and ξ . We expect that S is universal, and further that it may be viewed as an effective
2 eff
action for hydrodynamics.
In order to obtain an expression for S in terms of the infrared degrees of freedom
eff
ξ and ξ one follows the standard path taken when constructing effective actions. Namely,
1 2
one identifies the infrared degrees of freedom of the theory, the fundamental symmetries
associated with their dynamics, and then constructs the most general action compatible with
those symmetries. In the context of Schwinger-Keldysh actions for thermal states, this line
of research was pioneered in [1–3].
Both [2]and [1,3]have established thatS possessesanilpotentsymmetry transforma-
eff
tion reminiscent of supersymmetry. This is not the first time that such a nilpotent symmetry
appears in the context of dissipative dynamics, the canonical example being the Langevin
equation, c.f., [28] or [29] and references therein. The authors of [1, 3] have proposed an a’
priori superspace construction which is naturally associated with the formalism developed in
[17, 18] in order to capture the symmetries of S . The authors of [2] have implemented the
eff
symmetries associated with S in a more direct manner and observed an “emergent” su-
eff
peralgebra reminiscent of supersymmetry. In this work we offer a hybrid construction where
we build an effective action S using a super-Lagrangian from the ground up.
eff
In section 2 we provide a comprehensive discussion of the symmetries of the Schwinger-
Keldysh generating function, the resulting supersymmetry algebra, and how to implement
this symmetry in an effective action. In section 3 we elaborate on the low-energy degrees of
freedomoftheinfraredtheory. Insection4weconsideraconfigurationwithfixedtemperature
and velocity, such that the only dynamical field is the chemical potential. In this probe limit
we demonstrate that the action principle developed is compatible with known constitutive
relations and with thefluctuation-dissipation theorem. While our formalism is similar to that
of [1, 3], the physical reasoning is comparable to that of [2]. Our resulting action differs from
that of both groups. We discuss the differences and similarities of the different approaches in
section 5 where we also provide an outlook.
While this work was being completed we became aware of [30] by Gao and Liu, which
has overlap with this work, and of [4, 5] which has overlap with some parts of section 2.
– 3 –
2 Symmetries
Themainchallengeinconstructinganyeffectivetheoryistoidentifyitsdegreesoffreedomand
itsymmetries. Inwhatfollows wewillstudyexact symmetries associated withtheSchwinger-
Keldysh partition function (1.1) and an initial thermal state. Being exact symmetries, they
must be imposed on the Wilsonian effective theory at any energy scale. In what follows we
will discuss in detail how to impose these symmetries on the Schwinger-Keldysh low-energy
effective action.
Our exposition largely draws on results obtained in the recent work of Crossley, Glorioso
and Liu [2], henceforth CGL, and Haehl, Loganayagam and Rangamani [1, 3] (see also [4,
5]), henceforth HLR. These two approaches to construct the Schwinger-Keldysh effective
action are similar but not quite the same. In a sense, our work provides a distillation of the
superspace formulation of HLR [1, 3] with the approach of CGL [2]. In section 5 we discuss
similarities and differences between the two approaches and ours.
Recall that the Schwinger-Keldysh generating function for a generic initial state ρ is
−∞
given by (1.1) or by (1.2) when a Lagrangian description of the ultraviolet theory is available.
In this work we primarily study the case when the initial state is thermal, e.g., ρ =
−∞
exp( bH). In this instance the boundary conditions in the past are implemented by an
−
additional segment in the integration contour along imaginary time,
Z[A ,A ]= [dφ ][dφ ][dφ ]exp i S[φ ,A ] S[φ ,A ] exp S [φ ,A ] , (2.1)
1 2 1 2 E 1 1 2 2 E E E
− −
Z h i h i
(cid:0) (cid:1)
where S is the Euclideanized action, and the A are the time-independent sources that
E E
characterizetheinitialstate. Forexample,theinitialstatemaybeathermalstateonR Sd−1,
×
and the radius of the Sd−1 would be just such a source. The A are related to the sources
E
A and A in the far past by lim A (t) = lim A (t) = A .
1 2 t→−∞ 1 t→−∞ 2 E
†
Recalling the definition of the partition function Z[A ,A ] = Tr(U [A ]ρ U [A ]), we
1 2 1 1 −∞ 2 2
see that variations of the Schwinger-Keldysh generating functional, W = ilnZ, lead to
−
connected correlation functions of the form
ρ
−∞
Tr O(τ )...O(τ ) O(t )...O(t ) , (2.2)
1 m n 1
Tr(ρ )T T
−∞
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
e
where and denote the time-ordering and anti-time-ordering operators respectively, and
T T
O is the operator conjugate to A. The first string of operators on the right-hand side of (2.2)
e †
comes from the variation of U and the second from the variation of U .
2 1
The chief virtue of the Schwinger-Keldysh formalism is that it computes correlation
functions with a wide family of operator orderings. For instance, the symmetric, retarded,
– 4 –
and advanced two-point functions of O all follow from variations of W,
1 ρ δ2W
−∞
G (t , t ) = Tr O(t ),O(t ) = ,
sym 1 2 1 2
2 (cid:18)Tr(ρ−∞){ }(cid:19) δAa(t1)δAa(t2)(cid:12)Aa=Ar=0
ρ δ2W (cid:12)
−∞
G (t , t ) = iθ(t t )Tr [O(t ), O(t )] = i (cid:12) , (2.3)
ret 1 2 1 2 1 2
− (cid:18)Tr(ρ−∞) (cid:19) δAa(t1)δAr(t2)(cid:12)Aa=Ar=0
ρ δ2W (cid:12)
−∞
G (t , t ) = iθ(t t )Tr [O(t ), O(t )] = i (cid:12) ,
adv 1 2 2 1 1 2
− − (cid:18)Tr(ρ−∞) (cid:19) δAr(t1)δAa(t2)(cid:12)Aa=Ar=0
(cid:12)
wherewehavegonetotheso-called r/abasisanddefinedtheaverageanddiffe(cid:12)rencequantities
1 1
A = (A +A ) , A = A A , O = (O +O ) , O = O O . (2.4)
r 1 2 a 1 2 r 1 2 a 1 2
2 − 2 −
In the r/a basis, r-type operators are conjugate to a-type sources and vice versa,
ddx(O A O A ) = ddx(O A +O A ) . (2.5)
1 1 2 2 r a a r
−
Z Z
For these reasons we will use, e.g.,
δ3W
G = . (2.6)
raa
δAaδArδAr Aa=Ar=0
(cid:12)
(cid:12)
In this notation, (cid:12)
G = iG , G = iG , G =G . (2.7)
ret ra adv ar sym rr
We refer the reader to [2] for a modern discussion.
We note in passing that equation (2.2) makes it clear that not all operator orderings can
beobtainedfromtheSchwinger-Keldyshpartitionfunction,includingtheout-of-time-ordered
four-point functions which diagnose the onset of chaos [31, 32].
After gaining some familiarity with the Schwinger-Keldysh partition function (1.1) we
will, in the remainder of this section, discuss some of its symmetries and the expected in-
frared degrees of freedom required to describe hydrodynamic behavior. We will focus on four
symmetries of the partition function which are independent of the dynamics of the micro-
scopic theory and are generated as a result of the special structure of the Schwinger-Keldysh
partition function or, in one instance, are a marked feature of thermal states [2, 3]. These
four symmetries are:
1. Doubled symmetries. Thefunctional integral representation (1.2) makes it clear that, in
the absence of gravitational anomalies, Z has a doubled reparameterization invariance.
Thefirstweight in theSchwinger-Keldysh functional integral, involving the1 fields, can
be written with any choice of coordinates, and so can the second. This is true for any
initial state. Similarly, if the microscopic theory has a flavor symmetry group G, then
Z is invariant under a doubled flavor gauge invariance, whereby the 1 and 2 weights in
the Schwinger-Keldysh functional integral may be expressed in different flavor gauges.
– 5 –
2. Topological Schwinger-Keldysh symmetry. Consider the Schwinger-Keldysh parition
function (1.1). If we align the sources of the partition function such that A = A
1 2
then unitarity and cyclicity of the trace imply that
Z[A = A = A] = Tr U[A]ρ U†[A] =Tr(ρ ). (2.8)
1 2 −∞ −∞
(cid:16) (cid:17)
A normalized Z[A = A = A] is independent of the sources A. Otherwise, it may
1 2
depend on the values of the sources in the initial state A through ρ . Going to
E −∞
the r/a basis (2.4), equation (2.8) implies that when a-type sources are set to zero, all
variations with respect to the r-type sources at times t > must vanish. Thus, in
−∞
particular,
G = 0. (2.9)
aa...a
We conclude that the Schwinger-Keldysh partition function becomes topological when
A = A .
1 2
3. Reality and positivity. As emphasized by [2–4], the complex conjugate of Z is given by
Z[A ,A ]∗ = Tr U [A∗]ρ U†[A∗] = Z[A∗,A∗] (2.10)
1 2 2 2 −∞ 1 1 2 1
(cid:16) (cid:17)
for any Hermitian initial state and complexified sources. In terms of the generating
functional W = ilnZ the condition (2.10) amounts to
−
W[A ,A ]∗ = W[A∗,A∗]. (2.11)
1 2 − 2 1
Equation(2.11)istheSchwinger-Keldyshanalogueoftheusualstatementthatunitarity
impliesthattheWilsonianeffectiveactionisreal,andforthisreasonwecallthisareality
condition. However, as (2.11) allows for the effective action S to have an imaginary
eff
part, we will also restrict the imaginary part of S for the functional integral to
eff
converge.
4. KMS symmetry. The partition function possesses an additional symmetry when the
initial state is thermal. In the absence of conserved charges, an initial thermal state
has the form ρ = e−bH. This is the time evolution operator in imaginary time,
−∞
translating t t ib. Thus,
→ −
Z[A (t ),A (t )] = Tr U†[A (t )]e−bHU [A (t ib)] , (2.12)
1 1 2 2 2 2 2 1 1 1−
(cid:16) (cid:17)
which may also be generalized to initial states at nonzero chemical potential. Equation
(2.12) leads to the usual statement of the Kubo-Martin-Schwinger (KMS) condition for
thermal correlation functions [33–35]. Following [2], (2.12) together with CPT invari-
ance leads to a non-local Z symmetry of the partition function given by (2.91).
2
In the remainder of this Section we discuss each of these symmetries in detail. In Section
2.1 we describe how the doubling of sources in the Schwinger-Keldysh functional integral
– 6 –
naturally leads to a sigma model-like theory. In Section 2.2 we argue, following [1–5], that
every Schwinger-Keldysh theory posseses a topological sector which can be made manifest
using superspace techniques. Then in Section 2.3 we examine how the reality condition
restricts various expressions which can appear in the effective action. Finally in Section 2.4
we focus on thermal states and the KMS symmetry.
2.1 Doubled symmetries
In writing the r and a-type sources as in (2.4) we have glanced over a subtle point, which
we have not seen discussed elsewhere in the literature. In order to construct the r and a-
type combinations, we need to compare the 1 and 2-type operators and sources at the same
point. But,inprinciple, wecouldusedifferentcoordinates x andx whengivingafunctional
1 2
integral description of the time-evolution operators U and U . Or, to make the issue more
1 2
severe, suppose that the source we turn on is an external metric, viz. Z[g (x ),g (x )].
1µν 1 2ρλ 2
In this case, U is the time-evolution operator on a spacetime which differs from the
1 1
M
spacetime on which U evolves time. In order to construct the r and a-type operators
2 2
M
one needs a method by which a point x on can be compared with a point x on .
1 1 2 2
M M
In order to resolve the issue raised in the previous paragraph, we require that and
1
M
are diffeomorphic to each other; a diffeomorphism from to associates a point x
2 1 2 2
M M M
in with a point x in . This is a global restriction on the sources appearing in the
2 1 1
M M
Schwinger-Keldysh partition function. Equivalently, there exists an “auxiliary spacetime” M,
which is diffeomorphic to and , and we can use any diffeomorphism from M to
1 2 1
M M M
and to form average and difference combinations on M. At first sight, it might seem ill-
2
M
advised to introduce yet another spacetime, especially if we do not have to. However, doing
so has the advantage that it allows us to treat the 1 and 2 fields on an equal footing, which
will prove useful soon. In what follows we will call M a “worldvolume” and and the
1 2
M M
“target spaces,” in analogy with a sigma model.
Let σi denote worldvolume coordinates. Then the diffeomorphisms from M to and
1
M
are locally represented by maps xµ(σi) and xµ(σi), which we can use to pull back the
M2 1 2
metrics on and as
1 2
M M
g (σ) = g (x (σ))∂ xµ∂ xν, g (σ) = g (x (σ))∂ xµ∂ xν. (2.13)
1ij 1µν 1 i 1 j 1 2ij 2µν 2 i 2 j 2
This allows us to properly define the average and difference metrics
1
g (σ) = (g (σ)+g (σ)) , g (σ) = g (σ) g (σ). (2.14)
rij 1ij 2ij aij 1ij 2ij
2 −
The Schwinger-Keldysh partition function can then be rewritten in terms of g , g , and
aij rij
x (σ) and x (σ),
1 2
Z[g (x ),g (x )] = Z[g (σ),g (σ);x (σ),x (σ)]. (2.15)
1µν 1 2ρλ 2 rij akl 1 2
The functional variations of Z with respect to g and g yield correlation functions of
aij rkl
ij
operators which could be called the “average” and “difference” stress-energy tensors T and
r
Tkl respectively.
a
– 7 –
There is a similar story when the microscopic theory has a flavor symmetry group G,
in which case we can turn on external gauge fields which couple to the flavor symmetry
current. The gauge field B (x ) on will generally differ from B (x ) on . Since
1µ 1 1 2ν 2 2
M M
the external gauge fields B and B are connections on principal G bundles over and
1 2 1
M
, then the analogue of the requirement that and are diffeomorphic is that these
2 1 2
M M M
bundles are isomorphic. Equivalently, the worldvolume M too has a principal G bundlewhich
is isomorphic to those over and .
1 2
M M
In this paper we consider theories with U(1) flavor symmetries, in which case the bundle
isomorphism is locally represented by maps c (σ) and c (σ). The maps are “bifundamental”
1 2
under target space gauge transformations Λ and Λ , as well as under worldvolume gauge
1 2
transformations Λ, in the sense that
c c Λ +Λ, c c Λ +Λ. (2.16)
1 1 1 2 2 2
→ − → −
Theworldvolumetransformationlaw isaconsequenceoftheisomorphismsbetweenthetarget
space and worldvolume bundles. The target space gauge fields pull back to
µ µ
B (σ) = ∂ x B (x (σ))+∂ c , B (σ) = ∂ x B (x (σ))+∂ c . (2.17)
1i i 1 1µ 1 i 1 2i i 2 2µ 2 i 2
Both B ’s are inert under target space diffeomorphisms and gauge transformations (B
i 1µ
→
B +∂ Λ and B B +∂ Λ ) and transform as connections under worldvolume gauge
1µ µ 1 2µ 2µ µ 2
→
transformations, B B + ∂ Λ and B B + ∂ Λ. From the two B ’s we define
1i 1i i 2i 2i i i
→ →
average and difference gauge fields B and B , such that the partition function can be
ri aj
formally written as
Z[B (x ),B (x )] = Z[B (σ),B (σ);c (σ),c (σ)]. (2.18)
1µ 1 2ν 2 ri aj 1 2
As before, the variation of Z with respect to B and B define “average” and “difference”
ai rj
symmetry currents Ji and Jj respectively.
r a
Equations (2.15) and (2.18) immediately imply that the partition function does not de-
pend on the embeddings,
δZ δZ δZ δZ
= 0, = 0, = 0, = 0. (2.19)
δxµ(σ) δxν(σ) δc (σ) δc (σ)
1 2 1 2
It is instructive (and will become useful later) to see how this independence manifests itself
in terms of the Ward identities for target space operators.
For simplicity, supposethattheonlysources weturnonareexternal metricsandexternal
U(1) fields. The variation of the Schwinger-Keldysh generating functional W = ilnZ may
−
be written as
1
δW = ddx √ g Tµνδg +JµδB
1 − 1 2 1 1µν 1 1µ
Z (cid:18) (cid:19) (2.20)
1
ddx √ g Tµνδg +JµδB ,
− 2 − 2 2 2 2µν 2 2µ
Z (cid:18) (cid:19)
– 8 –
µν µν µ µ
where T and T are the target space stress tensors, and J and J are the target space
1 2 1 2
U(1) currents.
The reparameterization and U(1) symmetries imply that W is invariant under the com-
µ µ
bination of infinitesimal target space reparameterizations ξ and ξ , as well as infinitesimal
1 2
target space U(1) transformations Λ and Λ . Notating the combined variation as δ , the
1 2 χ
variation of the external fields under these infinitesimal transformations is
δ g = £ g = D ξ +D ξ ,
χ 1µν ξ1 1µν 1µ 1ν 1ν 1µ
(2.21)
δ B = £ B +∂ Λ = ξνG +∂ (ξνB +Λ ),
χ 1µ ξ1 1µ µ 1 − 1 1µν µ 1 1ν 1
and similarly for the 2 fields. Here £ is the Lie derivative along Xµ, D is the covariant
X 1µ
derivative using the Levi-Civita connection constructed from the metric g , and G =
1µν 1µν
∂ B ∂ B is the field strength of B . Plugging these variations into δW (2.20), we see
µ 1ν ν 1µ 1µ
−
that the invariance of W is equivalent to the Ward identities,
D Tµν = Gµ Jν BµD Jν, D Jµ = 0,
δ W = 0 1ν 1 1ν 1 − 1 1ν 1 1µ 1 (2.22)
χ ⇔ (D2νT2µν = Gµ2νJ2ν −B2µD2νJ2ν, D2µJ2µ = 0,
µν µν
where in the first line we used g to raise indices and in the second line we used g to do
1 2
the same.
Now consider expressing the generating functional as a functional of sources pulled back
to the worldvolume M. As the partition function does not explicitly depend on the maps xµ,
1
µ
x , etc., the variation of W may be expressed as
2
1 1
δW = ddσ √ g Tijδg +JiδB √ g Tijδg +JiδB , (2.23)
− 1 2 1 1ij 1 1i − − 2 2 2 2ij 2 2i
Z (cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
where √ g and √ g are understood to be the measure factors associated with g and
1 2 1ij
− −
g respectively. We decompose the variations of the worldvolume sources into variations of
2ij
the target space sources and maps, e.g.
δg (σ) = ∂ xµ∂ xν(δg +£ g ) ,
1ij i 1 j 1 1µν δx1 1µν
(2.24)
δB (σ) = ∂ xµ(δB +£ B +∂ δc ) ,
1i i 1 1µ δx1 1µ µ 1
µ
where the Lie derivative is taken with respect to the vector field δx (σ(x )) on .
1 1 M1
Given (2.24) we can match the variation of worldvolume quantities to target space ones.
µ µ
Whenδx = 0, δx = 0etc. comparing (2.23)with(2.20),we seethattheworldvolumestress
1 2
tensors and currents are related to the target space ones by pushforward, e.g.
Tµν = Tij∂ xµ∂ xν, Jµ = Ji∂ xµ. (2.25)
1 1 i 1 j 1 1 1 i 1
Allowing for nonzeroδx and following thesameprocedure,wefind,after integration by parts,
1
δW = ddσ √ g Tµνδg +JµδB E δxµ E δc
− 1 2 1 1µν 1 1µ− 1µ 1 − 1 1
Z (cid:26) (cid:18) (cid:19)
√ g 1Tµνδg +JµδB E δxµ E δc (2.26)
− − 2 2 2 2µν 2 2µ− 2µ 2 − 2 2
(cid:18) (cid:19)(cid:27)
+(boundary term),
– 9 –