Effective actions in \( \mathcal{N}=1 \) , D5 supersymmetric gauge theories: harmonic superspace approach
HJE
1, D5 supersymmetric gauge theories: harmonic superspace approach
Tomsk 1
Russia 1
Novosibirsk 1
Russia 1
I.L. Buchbinder 1 2 4
N.G. Pletnev 0 1 3
Novosibirsk, Russia
0 National Research Novosibirsk State University
1 Tomsk , 634061 Russia
2 National Research Tomsk State University
3 Department of Theoretical Physics, Sobolev Institute of Mathematics
4 Department of Theoretical Physics, Tomsk State Pedagogical University
We consider the o shell formulation of the 5D, N = 1 super YangMills and super ChernSimons theories in harmonic superspace. Using such a formulation we develop a manifestly supersymmetric and gauge invariant approach to constructing the oneloop e ective action both in super YangMills and super ChernSimons models. On the base of this approach we compute the leading lowenergy quantum contribution to the e ective action on the Abelian vector multiplet background. This contribution corresponds to the `F 4' invariant which is given in 5D super eld form.
Extended Supersymmetry; Superspaces; Supersymmetric E ective Theories

N
E
Dedicated to the memory of Boris Zupnik
1 Introduction
2 Review of the 5D, N = 1 harmonic superspace approach
2.1 5D SYM theory in harmonic superspace 2.2
The super ChernSimons model
2.3 5D N = 1 hypermultiplet in harmonic superspace
3 The background
eld formulation for quantum N
3.3 The leading contribution to e ective action of the 5D SCS multiplet
4 The leading contribution to the e ective action of a 5D SYM multiplet 19
5 The leading and nexttoleading contributions of the ghosts and matter
super elds
6 Summary
1
Introduction
The study of the quantum structure of the supersymmetric
vedimensional eld
theories attracts recently considerable attention, mainly due to attempts to
nd the e ective
worldvolume action for multiple M5branes [1]. It was conjectured in [2, 3] that the
sixdimensional (2,0) superconformal eld theories on a stack of M5branes are equivalent to
vedimensional super YangMills (SYM) theories. To establish this correspondence, the
KaluzaKlein reduction of the general 6D (
1,0
) pseudoaction with a nonAbelian gauge
group G was performed in the series of papers [4, 5] and the 5D e ective action for the
KaluzaKlein zeromodes was derived. This duality serves as an important constraint on
the models for multiple M5branes. Of course, not all consistent vedimensional theories
arise in such a circle compacti cation.
On the other hand, the 5D theory has a global U(
1
) symmetry and the current j =
?F ^ F is always conserved. The corresponding conserved charge is the instanton number.
Such a conserved current can be coupled to vector super eld what allows us to identify
the scalar component
of this vector super eld as the gauge coupling <
>
Using this observation the authors of papers [8{13] proposed that the maximally extended
1
gS2YM
[6, 7].
{ 1 {
5D supersymmetric gauge theory describes the 6D, (2; 0) superconformal eld compacti ed
on circle without introducing the KaluzaKlein reduction. This proposition was based on
the observation that the KaluzaKlein momentum along the circle can be identi ed with
instanton charge in the 5D theory. The latter is a topological charge carried by soliton
con gurations, which are analogous to monopole and dyon con gurations in 4D. Such an
attractive hypothesis means, in fact, that adding the 1/2BPS particle soliton states with
instanton number k and a mass formula M
full nonperturbative particle spectrum and detegrS2YmMines the nonperturbative completion of
R5
k to 5D SYM theory gives us the
4 k
the theory under consideration. This might be an argument in favor of UV
niteness of
5D SYM perturbative theory and thus be an argument for the consistent quantum theory.
In the strong coupling limit the 5D SYM should de ne the fully decompacti ed 6D, (2; 0)
theory, which, in its turn, is expected to describe the lowenergy dynamics of multiple
M5branes.
An additional important motivation to study 5D supersymmetric gauge theories comes
from the existence of the corresponding super ChernSimons (SCS) theory. This theory is
interesting since it has a conformal xed point in
ve dimensions and can admit a
holographic duality [6, 7]. There are several reasons why the 5D supersymmetric ChernSimons
theory can be interesting in quantum domain. First of all, the ChernSimons terms can be
generated by integrating out the massive hypermultiplets in the SYM theory when the
hypermultiplets transform in complex representations of the gauge group [2, 3]. If we consider
the masses of the hypermultiplets as the UV cuto , then this leads to the generation of the
ChernSimons term in the oneloop correction to the classical theory. Hence inclusion of the
SCS term into the action can be useful in some cases if we want to have a complete
description of the theory. SCS theory can also be important in the relationship between 5D SYM
and 6D, (2,0) theories. In particular, one can argue [14, 15] that the 5D ChernSimons term
can be generated by the anomaly terms in the sixdimensional theory. By focusing on a
certain class of anomalyfree sixdimensional theories the authors of [14, 15] formulated the
explicit constraints on the spectrum and supersymmetry content of the sixdimensional
theory in terms of the
vedimensional ChernSimons couplings. Therefore it would be
interesting to compute the perturbative quantum corrections in such 5D theories. In
particular, it was demonstrated that massive fermions running in the loop generate constant
corrections to the 5D ChernSimons terms of the form kABC AA^F B^F C + AAA^tr (R^R),
where AA denotes collectively the graviphoton and the vectors from the vector multiplet,
F A are the corresponding eld strengths, and R is the curvature twoform.
Though 5D and 3D ChernSimons theories share some interesting properties, such as
quantization of the level k, nevertheless there are also some di erences. The most important
di erence is the presence of local degrees of freedom in the higher dimensional case. This
peculiar fact makes it attractive and interesting to perform a more detailed analysis of 5D
SCS theories.
It is known that nontrivial observables exist in supersymmetric gauge theories which
are not very sensitive to details of the UV completion. Quantum e ects which nontrivially
contribute to such BPS observables are often highly constrained. With using the
procedure of localization of the path integral were studied of the various observables in 5D
{ 2 {
supersymmetric theories [16{21]. It was shown that the partition function for the
maximally extended SYM on S5 captures the physical aspects of the 6D, (2; 0) theory in a
surprisingly accurate and detailed manner. In particular, the N 3 behavior of this partition
function in 5D supersymmetric gauge theory is in agreement with the important results
obtained for 6D, (2; 0) theory from the supergravity duals and conformal anomaly [22, 23].
The study of integral invariants in halfmaximally and maximally extended
supersymmetric theories such as supergravity and SYM attracts an attention because they can be
viewed, on the one hand, as possible higher order corrections to the string or brane e ective
actions and, on the other hand, as quantum
eld counterterms. It is well known on the base
of the power counting arguments that the 5D SYM is perturbatively nonrenormalizable.
Therefore we should expect an in nite number of divergent structures at any loop what
leads to in nite number of counterterms. However, the superspace arguments and the
requirements of onshell supersymmetry rule out the rst divergences in Ddimensional SYM
theory. Actually the divergences can appear at L loops where D=4+6/L.1 Construction of
the various supersymmetric, gauge invariant functionals in quantum
eld theory is
conveniently formulated in the framework of the e ective action. The lowenergy e ective action
can be represented as a series in supersymmetric and gauge invariants with some coe
cients. In general, the supersymmetry together with conformal symmetry imposes rigid
constraints on these coe cients. In some cases, they can be determined exactly [33, 34].
For example, the leading term in the lowenergy e ective action on the Abelian vector
eld background is F 4 which is generated only at one loop and is not renormalized at
higher loops. A possible new nonrenormalization theorems for Abelian F n was
conjectured in [35]. Recently the authors of [36{38] systematically analyzed the e ective action
on the moduli space of (2,0) superconformal eld theories in six dimensions, as well as their
toroidal compacti cation to maximally SYM theories in
ve and four dimensions. They
presented an approach to nonrenormalization theorems that constrain this e ective action.
The rst several orders in its derivative expansion are determined by a oneloop calculation
in vedimensional SYM theory. In general, the functional form of the e ective action at
the rst several orders in the derivative expansion can be obtained by integrating out the
massive degrees of freedom in the path integral. However, it is di cult enough to perform
exactly such an analysis for supersymmetric models in the component formulation.
Construction of the background eld method in extended supersymmetric gauge
theories faces a fundamental problem. The most natural and proper description of such theories
should be formulated in terms of a suitable superspace and unconstrained super elds on
it. Some time ago a systematic background
eld method to study the e ective actions
in 4D, N = 2 supersymmetric eld theories was developed in a series of papers [39{44].
This method is based on formulation of N = 2 theories in harmonic superspace [45, 46]
and guarantees the manifest N = 2 supersymmetry and gauge invariance at all stages
of calculations. The method under consideration gives the possibility to calculate in a
straightforward manner not only the holomorphic and nonholomorphic contributions to
the lowenergy e ective action but also to study the full structure of the e ective action.
1For a discussion of this issue see [24{32] and references therein.
{ 3 {
HJEP1(205)3
Evaluation of the e ective action within the background eld method is often accompanied
by using the proper time or heat kernel techniques. These techniques allow us to sum up
e ciently an in nite set of Feynman diagrams with increasing number of insertions of the
background elds and to develop a background eld derivative expansion of the e ective
action in a manifestly gauge covariant way.
The 5D SCS theories are superconformally invariant and, hence, their e ective actions
must be independent of any scale. Unlike the 4D, N = 2; 4 supersymmetric theories where
holomorphy allows one to get the chiral contributions to e ective action [47], in the 5D case
the contributions to the e ective action can be written only either in full or in analytic
superspaces. Then, taking into account the mass dimensions of the harmonic potential
V ++ and super eld strength W as well as the dimensions of the superspace measures
d5xd4 + and d5xd8 , one obtains that the most general lowenergy e ective U(
1
)gauge
invariant action in the analytic superspace is the SCS action [48, 49]. The nexttoleading
e ective Abelian 5D action can be written only at full superspace in terms of the manifestly
gauge invariant functional
= R d5xd8
W H( W ); where
is some scale and H( W ) is
the dimensionless function of its argument. The requirement of scale invariance means
an equation
d
d R d5zd8
WH( W ) = 0, where the only solution is H = c ln W : Any
perturbative or nonperturbative quantum corrections should be included into a single
constant c. The component Lagrangian in the bosonic sector corresponding to the above
e ective action is 13 (F 4 + (@ )4 + : : :), where F is the Abelian strength of the component
vector eld from 5D, N = 1 vector multiplet and
is the corresponding scalar component.
In this paper we derive the leading contribution to the lowenergy e ective action in
the 5D SCS theory using the harmonic superspace description of the theory and
propertime techniques. The result precisely corresponds to the above analysis and has the form
R d5xd8
W ln W . Besides the e ective action in the 5D SCS theory, we calculate the
leading oneloop lowenergy contribution in the 5D SYM theory. Although this theory is
not superconformal and is characterized by the dimensional coupling constant, its leading
contribution to the e ective action has the same functional form as in the 5D SCS theory
and does not depend on the scale and coupling constant. Also, we consider the e ective
action in the 5D hypermultiplet theory coupled to a background 5D vector multiplet.
The leading lowenergy contribution to e ective action in this theory was calculated in
paper [49], where it was shown that this contribution is the 5D SCS action. In the given
paper we calculated the rst nexttoleading term in the lowenergy e ective action for the
theory under consideration and found that this term again has the same functional form
as the leading term in 5D SCS theory.
The paper is organized as follows. Section 2 is devoted to a brief review of harmonic
superspace formulation of the 5D, N = 1 supersymmetric eld models such as the SYM
theory, the hypermultiplet theory and the SCS theory. In section 3 we consider the Abelian
5D SCS theory and develop the manifestly supersymmetric and gauge invariant procedure
for calculating the e ective action. This procedure is based on the background eld method
and propertime technique. We
nd the exact expression for oneloop e ective action in
terms of functional determinants of di erential operators in analytic subspace of harmonic
superspace and calculate the leading lowenergy contribution to this e ective action. In
{ 4 {
section 4 we develop the analogous procedure for 5D SYM theory and calculate the leading
lowenergy contribution to oneloop e ective action. Section 5 is devoted to the study of
the rst nexttoleading contribution to e ective action in the 5D hypermultiplet theory
coupled to a 5D vector multiplet background. The last section is devoted to the summary
of the results.
2
Review of the 5D, N
= 1 harmonic superspace approach
Various supersymmetric theories with eight supercharges admit the o shell super eld
formulations in terms of formalism of the harmonic superspace. The harmonic superspace
approach for the 4D, N = 2 theories was originally developed in [45]. The formulation
for the 5D, N = 1 models has been given in [48, 50, 51]. The harmonic superspace
approach for the 6D, (1; 0) SYM theories was considered in [52{55] and for the 6D, (1; 1)
SYM in [56]. The construction the superde Rham complex in vedimensional, N = 1
superspace and its relationship to the complex of sixdimensional, N = (
1, 0
) superspace via
dimensional reduction was considered in [57]. All these harmonic superspace formulations
in spacetime dimensions 4, 5 and 6 look almost analogous modulo some details. In a
series of papers [58{62] an extensive program of constructing the manifestly supersymmetric
formulation for the 5D, N = 1 supergravitymatter models was realized and the universal
procedures to construct manifestly 5D supersymmetric action functionals for a di erent
supermultiples were developed. These super eld results are in agreement with the earlier
component considerations of the same models [63{65].
The study of the structure of the lowenergy e ective action in the 5D superconformal
theories looks useful from the point of view of the classi cation of theories consistent at
the quantum level. In ref [49] a manifestly 5D, N = 1 supersymmetric, gauge covariant
formalism for computing of the oneloop e ective action for a hypermultiplet coupled to a
background vector multiplet was developed. It was demonstrated, as a simple application,
that the SCS action is generated at the quantum level on the Coulomb branch. The above
paper was the only one where explicit oneloop harmonic superspace calculations were
done. We believe that until now the possibilities of the covariant 5D harmonic superspace
methods were nonsu ciently explored to study the e ective action in 5D supersymmetric
gauge theories. The aim of this paper is to develop the general manifestly supersymmetric
and gauge invariant methods for 5D quantum supersymmetric gauge theories and apply
these methods to calculate the lowenergy e ective action in 5D SCS and SYM theories.
In this section we brie y review the superspace description of a vector multiplet in 5D
supersymmetric gauge theories. Our aim here is to x our basic notation and conventions
(for details see [50]).
The 5D gammamatrices
a^ are de ned as follows f a^
; ^bg =
f 1; 1; : : : ; 1g: The matrices f1; a^
; a^^bg form a basis in the space of 4
charge conjugation matrix C = ("^ ^) and its inverse C 1 = ("^ ^) are used to raise and
lower the spinor indices. The matrices "^ ^ and ( a^)^ ^ are antisymmetric, while the matrices
( a^^b)^ ^ are symmetric. The anticommuting variables i^ are assumed to obey the
pseudo2 a^^b1; with
a^^b =
4 matrices. The { 5 {
^
i: They obey the anticommutation relations
by the coordinates zM = (x^ ^; i^) where i = 1; 2 and x^ ^ = ( a^)^ ^xa^.
Majorana reality condition ( i^)? = i^ = "^ ^"ij j^. 5D, N = 1 superspace is parameterized
The N
= 1 harmonic superspace R5j8
S2 extends the conventional 5D, N
Minkowski superspace R5j8 with the coordinates zM
SU(2)=U(
1
) parameterized by harmonics, i.e., group elements
= (xm^ ; i^); by the twosphere
D0; D
=
2D
D++; D
= D0:
The main conceptual advantage of harmonic superspace is that the N = 1 vector multiplet
as well as hypermultiplets can be described by unconstrained super elds on the analytic
subspace of R5j8
S2 parameterized by the coordinates
M = (xAm^ ; +^; ui ); where
xaA^ = xa^ + i +^ a^
^ ^
^
;
= ui
i
:
(xa^;
^; ui ) satisfying the constraints D+^
= 0 is an analytic super eld
( ; u): It
The important property of the coordinates
M is that they form a subspace closed un
der N = 1 supersymmetry transformations. In the coordinates
derivatives D+^ = ui+Di^ have a short form (see eqs. (2.5)) and therefore the super eld
M the spinor covariant
leads to reducing the number of the anticommuting coordinates and, hence, to reducing
the number of independent components in comparison with general super elds. However,
all component elds depend now on extra bosonic coordinates ui .
In harmonic superspace a full set of gauge covariant derivatives includes the harmonic
derivatives which form a basis in the space of leftinvariant vector elds of SU(2):
nDi^; Dj^o =
u
i 2 SU(2); u+i = ui ; u+iu
i = 1 :
D
(2.1)
= 1
(2.2)
(2.3)
(2.4)
^
(2.5)
(2.6)
The generator of the SU(2) algebra D0 is an operator of harmonic charge, D0 (q) = q (q).
In the analytic basis, parameterized by the coordinates
derivatives: D^ = ui Di^ and the harmonic derivatives take the form
M (2.2) the spinor covariant
D+^ =
In this basis eqs. (2.1), (2.4) leads to
+
+^
D
n
D+^; D +^o
D+^; D ^
o
D^ =
^
;
^
+
; D^ = 0; D
; D^ = D^ ;
{ 6 {
These relations are necessary integrability conditions for the existence of the analytic
super elds. Since the
eld in the harmonic superspace depend on the additional bosonic
variable ui, we must de ne rules of integration over harmonics (that is, over the group
manifold SU(2)=U(
1
)). The basic harmonic integrals have the form [46]
Z
du = 1;
Z
duu(+i1 : : : ui+n uj1 : : : ujm) = 0; n + m > 0 :
(2.7)
It means that the harmonic integrals are nonzero only if the integrand has the zero
U(1) charge.
2.1
5D SYM theory in harmonic superspace
To describe a YangMills supermultiplet in 5D conventional superspace we introduce the
gauge covariant derivatives DA = DA + iAA where DA = (@a^; Di^) are the
at covariant
derivatives and AA is the gauge connection taking values in the Lie algebra of the gauge
group. The operator DA satis es the gauge transformation law DA ! ei (z)
DAe i (z); y =
with a super eld gauge parameter (z). The gauge covariant derivatives are required to
obey some constraints [66]
HJEP1(205)3
iFa^^b =
1
a richer gauge freedom than the original group
eiv0(z;u) = ei (z;u)eive i (z); D+^
= 0:
The  and  transformations generate, respectively, the socalled  and groups. In the
frame the spinor covariant derivatives D+^ coincide with the at ones, while the harmonic
2The properties of these matrices are discussed in [50, 61].
n i
D^; Dj^o
=
i
D^; Da^ =
2i"ij (D^ ^ + "^ ^iW);
^ i
i( a^)^ D ^W; Da^; D^b =
1
4
^ ^ i
a^^b
D^D ^iW = iFa^^b;
with the matrices
a^ and
Wy = W and obeys the Bianchi identity
a^^b de ned above.2 Here the eld strength W is Hermitian,
(i j)
covariant derivatives acquire connections D
is an analytic super eld, D+^V ++ = 0 and its transformation law is
= D
+ iV
. The real connection V ++
V ++ 0 = ei V ++e i
iei D++e i :
(2.12)
The gauge freedom the form
can be used to impose the WessZumino gauge in
In this gauge, the super eld V ++ contains the real scalar eld , the Maxwell eld Am^ ,
the isodoublet of spinors i^ and the auxiliary isotriplet Y (ij). The analytic super eld V ++
turns out to be the single unconstrained potential of the pure 5D, N = 1 SYM theory and
all other quantities, associated with this theory, are expressed in its terms. In particular,
the other harmonic connection V
using the zerocurvature condition [D++; D
turns out to be uniquely determined in terms of V ++
] = D0: The result looks like [69]
V
(z; u) =
1
X( i)n+1 Z
n=1
du1 : : : dun u+u1+
V ++( ; uu11+) u::2+: V: +::+(un+;uun+) ;
where (u1+u2+) = u1+iu2+i. The details of the harmonic analysis on S2 = SU(2)=U(
1
)) were
designed in the pioneering works [45, 46, 67{69]. In the basis the connections A^ , Am^
and the eld strength can be expressed in terms of V
using the relations (2.10):
A^ =
D+^V
;
W
=
D+^D+^V
;
F^ ^ =
i
8
i
2 D(^D ^)W:
+
i[V ++; W ] = 0: In the Abelian case, the super eld W
The super eld strength W
satis es the following constraints D
++
W
= D++
W
= 8i R du(D )2V ++ does not
depend on harmonics and moreover, in this case there is no distinction between W
W. The Bianchi identity (2.9) takes the form
+ +
where we have used the notation (D+)2 = D
of [50] built an important covariantly analytic super eld G
+^ +
++:
D^ : Using these identities, the authors
This super eld can be transformed to the form
G
++ =
D+ 4 (V
W);
)
2 ( ) =
where (D )4 =
312 (D )2(D )2 and 12 (D+)4(D
Unlike the 4D, N = 2 case, the chiral superspaces are not Lorentzcovariant in the
case of 5D, so that to construct the super eld actions we can use only the full superspace
1 ^ ^D^ ^ ( ):
4 D
{ 8 {
(2.14)
(2.15)
+
and
(2.16)
(2.17)
(2.18)
or else the analytic superspace. In the full harmonic superspace the 5D SYM action has
the universal form [69]
= ' + i^ i^ + ^ i ^
i H^ ^ + : : : ;
subject to the constraint D(^iDj^)
= 0: The expansion of
is
The SYM equations of motions (D+)2W = 0; 2W = 0 have a vacuum Abelian solution
)2Z where Z is the linear combination of the Cartan generators of the gauge
group [48]. This vacuum solution spontaneously breaks the gauge symmetry, but it
conserves the 5D supersymmetry with the central charge and produces BPS masses of the
Zcharged elds. In addition, let us note that onshell, the degrees of freedom (1 + 3)B + 4F
in W match the degrees of freedom in a 5D tensor multiplet described by the super eld
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
SSYM V ++
=
1
X1 (
1
)n
2
gSYM n=1
n
tr
Z
d13zdu1 : : : dun
V ++(z; u1) : : : V ++(z; un) ;
u1+u2+ : : : un+u1+
where gS2YM is the coupling constant of dimension [gS2YM] = 1. The SYM action in terms
of the component elds de ned by (2.13) is
SSYM =
1
2
gY M
Z
d ( 4)i
+ 2 tr GS+Y+M:
Z
( a^^b)^ ^Ha^^b is dual to the 3form
eld strength Fa^^bc^ of the 2form gauge
eld Ba^^b. In this situation the vector massless representation is equivalent under duality
to a tensor representation.
The other universal procedure to construct 5D manifestly supersymmetric actions is
based on ideas developed in [70, 71]. Let us consider two vector multiplets, namely a U(
1
)
vector multiplet V ++ and a YangMills vector multiplet VS+Y+M. They can be coupled in a
gaugeinvariant way, using the interaction
where GSYM corresponds to a nonAbelian multiplet and is de ned in eg. (2.17). If we
assume that the physical scalar eld in V ++ possesses a nonvanishing expectation value
< V ++( ; u) >= i( +)2 gY21M , then one gets
The integration in (2.22), (2.23) is carried out over the analytic subspace of the harmonic
superspace
Let us consider the 5D harmonic superspace formulation of a supersymmetric ChernSimons
(SCS) model. The o shell nonAbelian SCS action in ve dimensions was rst constructed
in [65]. SCS model possesses remarkable properties, which makes this theory very
interesting for various applications. First of all, in
ve dimensions the pure U(N ) and SU(N )
ChernSimons theories have superconformal xed points [6, 7]. Another reason to study
the SCS models is that they can be generated by integrating out the massive
hypermultiplets in the SYM theory [2, 3, 14, 15] when the hypermultiplets transform in complex
representations of the gauge group. For example, if we consider the masses of the
hypermultiplets as the UV cuto , we obtain the ChernSimons term in the oneloop correction
to the classical theory. Hence inclusion of this term can be important in some cases to
get a complete description of the theory. In a manifestly supersymmetric setting, where
the entire vector supermultiplet is taken into account, the corresponding oneloop
calculation was given in [49], both in the Coulomb and nonAbelian phases. Using the covariant
harmonic supergraphs and the heat kernel techniques in harmonic superspace [39{44], it
was shown that the hypermultiplet e ective action contains the SCS term. Finally
ChernSimons theory can be important in the relation between the 5D SYM and 6D (2; 0) theories.
In particular, one can argue [14, 15] that the 5D ChernSimons term can be generated by
the anomaly terms in the sixdimensional theory.
In general, in spacetime with dimension (2n 1), the action of the ChernSimons theory
can be constructed using the ChernSimons form
2n 1, de ned as d (2n 1) = tr [F n] where
F = dA + iA ^ A is the gauge eld strength twoform and its powers F n are de ned by
the wedge product. In three spacetime dimensions this form gives rise to the famous 3D
ChernSimons action. In the 5D ChernSimons theory the action is given by
SCS =
k Z
12
d5x"a^^bc^d^e^tr
Aa^F^bc^Fd^e^
iAa^F^bc^Fd^e^
2
5 Aa^A^bAc^Ad^Ae^ ;
where k is the ChernSimons level which plays the role of the coupling constant. Here Aa^
is the gauge eld with the gauge group SU(N ) or U(N ). The eld Aa^ transforms under the
gauge transformation g as Aa^ ! g 1
Aa^g
ig 1@a^g: Under this transformation, SCS gets
an additional term
SCS given, modulo a total derivative, by SCS = 2 ik Q(g); where
Q(g) takes only integer values. Like in the case of threedimensional Chern Simons theory,
gauge invariance of the partition function leads to k 2 Z. However, despite the fact that the
action of the theory CS does not depend on the metric and thus the theory is topological,
the ve (and, generally, any higher odd) dimensional case admits local propagating degrees
of freedom [72, 73].
Unlike the component construction of [65], a closedform expression for the nonAbelian
SCS action has never been given in terms of the super elds. Here there exists only a unique
de nition of the variation of the SCS action with respect to an in nitesimal deformation of
the gauge potential V ++. However, as it was noted in [62], it is unknown how to integrate
this variation in a closed form to obtain the action as an integral over the superspace. The
component formulation of the nonAbelian SCS model can be constructed in the framework
of the superform approach [62], where a closedform expression for the nonAbelian SCS
action was given. The super eld analysis in the Abelian case is more transparent and the
SCS action was derived in the 5D N = 1 harmonic [48, 50] and projective superspaces [61].
The approach of constructing the manifestly supersymmetric actions developed in [48,
50] leads to Abelian SCS action in the form
The action (2.25) is invariant under the gauge transformations equations of motions for the model with such an action are
V ++ = D++ : The
The Abelian SCS theory with the super eld action (2.25) leads to the following
component action:
F a^^bFa^^b
i
2 Yij
1
2
Y ij Yij
(2.27)
i j
:
(2.25)
(2.26)
The action (2.27) clearly shows that the ve dimensional Abelian SCS theory is a nontrivial
interacting eld model (see e.g. [74]).
The theory (2.27) is superconformal at the classical level and the coupling constant
g2 is dimensionless. The latter can mean a renormalizability of the theory. However, the
action (2.27) does not involve a quadratic part and perturbative calculations, based on the
weakness of the interaction term with respect to the free part, are impossible. However, we
can use the presence of the vacuum moduli space <
>= m in the Lagrangian (2.27). This
allows one to decompose the Lagrangian into the free and interaction parts and construct
the S matrix in a conventional way. However in this case, conformal symmetry of the
original Lagrangian is broken spontaneously and the mass parameter in the action makes
the theory nonrenormalizable.
2.3
5D N = 1 hypermultiplet in harmonic superspace
Here we brie y discuss the hypermultiplet formulation in harmonic superspace.
On massshell, the 5D, N = 1 massless hypermultiplet contains two complex scalar
elds forming an isodoublet of the automorphism group SU(2)A of the supersymmetry
algebra (2.1) and an isosinglet Dirac spinor eld. The description of the o shell hypermultiplet
in terms of the analytical supspace of harmonic superspapce is completely analogous to the
description of a 4D, N = 2 hypermultiplet. Like in the fourdimensional 4D, N = 2
supersymmetric theory [45], the o shell hypermultiplet coupled to the vector supermultiplet
is described by a super eld q+( ) and its conjugate q+( ) with respect to the analyticity
preserving conjugation [46]. The classical action for a massless hypermultiplet coupled to
the background 5D, N = 1 vector multiplet is
(2.28)
Z
Shyper =
d ( 4)q+D++q+:
The hypermultiplet that transforms in a real representation of the gauge group,
onshell has scalars in the representations (1 + 3) of SU(2)A and an SU(2)A doublet of
pseudoMayorana fermions. It can be described by a real analytic super ed !( ). Such a super eld
is called the !hypermultiplet. The action describing an interaction of this hypermultiplet
with a vector supermultiplet is written in the form
In this section we will construct the background eld method for the super eld theory with
The harmonic super eld background eld method (see construction of this method for
4D, N = 2 SYM theory in [39{41]) is based on the socalled backgroundquantum splitting
of the initial gauge eld into two parts: the background eld V ++ and the quantum eld v+
To quantize the theory, one imposes the gauge xing conditions only on the quantum eld,
introduces the corresponding ghosts and considers the background eld as the functional
argument of the e ective action. Then, the original in nitesimal gauge transformations (2.12)
can be realized in two di erent ways: as the background transformations
and as the quantum transformations
V ++ = 0; v++ =
i v++; :
To construct an e ective action as a gaugeinvariant functional of the background super eld
V ++ we will use another form of writing the action (2.25)3
(3.1)
(3.2)
(3.3)
d ( 4)V ++ ++ =
G
d5xd8 du V ++V
W ;
(3.4)
and expand the action S[V ++ + v++] in powers of the quantum eld v++:
d13z1du1 : : : dzndun n! v++(
1
) : : : v++(n) v++=0
v++(
1
) : : : v++(n):
(3.5)
1
Here W , and v++ denote the  and frame forms of W, and v++ respectively
W
= eiv
W e iv; v++ = e ivv++eiv:
Each term in the action (3.5) is manifestly invariant with respect to the background gauge
transformations. The rst variation of the action is
where v++ = e iv V ++eiv. It depends on V ++ via the dependence of v++ on the bridge v.
The term linear in v++ determines the equations of motion. This term should be dropped
when one considers the e ective action.
To determine the second variation of the action it is necessary to express V
via
V ++.
A variation of V ++ =
ieiv(D++e iv) can be represented as e iv V ++eiv =
iD++(e iv eiv) = V ++: This equation is solved in the form
e iv eiv =
i
Z
u+u
1
du1 u+u1+
V ++(u1):
Z
u+u
1
du1 u+u1+
du1du2
1
u1+u2+ 2 g2(W)
1
g2(W)
= W :
V ++(u1) =
du1
Z
Vu+++u1+(u12) :
V ++(u1) V ++(u2) :
V
= iD
e iv eiv = D
2SSCS =
d5xd8
Calculating the second variation of the action yields the result
These two expressions illustrate the speci c feature of 5D, N = 1 SCS theory as a theory
with local coupling constant (since there is a nontrivial backgrounddependent factor in
the vector eld kinetic operator) [39{41].
In the framework of the background eld method, we should x only the quantum eld
gauge transformations. As to the super eld W it is invariant under these transformations.
Let us introduce the gauge xing function in the form
F (+4) = D++v++;
F (+4) = D
++
D
++
+ i v++;
;
Z
SFP = tr
d ( 4)du b
D
++ 2 c:
which changes by the law
under the quantum gauge transformations. Eq. (3.12) leads to the FaddeevPopov
determinant and to the corresponding ghost action
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
Next we average the (F (+4) f (+4)) with the weight
( 1 Z
2
1 =
V ++ exp
d13zdu1du2f (+4)(u1)W
ghosts and f (+4) = e ivf (+4)eiv is a tensor of the group. Note that we need the insertion
of the super eld W to balance the dimensions. The functional [V ++] is chosen from the
condition
1 V ++
=
Z
Df (+4) exp
The background eld dependent operator A^ has the form
and act on the space of analytic super elds. Thus
A^ =
nd DetA^ we represent it by a functional integral over analytic super elds of
the form
Det 1 A^ =
Z
and perform the following replacement of functional variables
itr
Z
D
(+4)
D
(+4) exp
d 1( 4)d 2( 4) (+4)(
1
)A^(1j2) (+4)(2) ; (3.16)
(+4) =
D
++ 2 ; Det
(+4) !
= Det D
++ 2
:
Further, repeating the construction of the e ective action from the work [67, 68] we obtain
Z
d13zdu1du2 1
(+4)
W
Z
Z
d13zdu (+4)
d ( 4) (+4) _2
1
W 2
W
D
;
2
where
is the deformed version of the covariant analytic d'Alembertian [50]
_
2
W =
_
2=
1
(D
)2:
Eventually we get the representation of [V ++] by the following functional integral
V ++
= Det 1=2
D
++ 2
_
Det(14=;20) 2
W
_
= Det(14=;20) 2
W
D'eiSNK[';V ++]
:
Z
(3.15)
(3.17)
(3.18)
(3.19)
(3.20)
Here Det(q;4 q)(H) is de ned as follows
Det(q;4 q)H^ = eTr (q;4 q) ln H^ ;
and the functional trace of operators acting on the space of covariantly analytic super elds
of U(
1
) charges q and 4
q is
Tr (q;4 q)H^ = tr
Z
H
d ( 4) (q;4 q)( ; );
where `tr ' is the matrix trace and H
(q;4 q)( 1; 2) is the kernel of the operator. Super eld
' in (3.20) is a bosonic real analytic super eld, the NielsenKallosh ghost, with the action
SCS[V ++] with the weight (3.14), one gets the
following path integral representation of the oneloop e ective action
Abelian 5D SCS theory
(S1C)S[V ++] for the
ei (S1C)S[V ++] = eiSSCS[V ++] Z
Dv++
DbDcD'eiSquant[v++;b;c;';V ++]
;
where
Squant =
SSCS v++; V ++ + Sgh v++; V ++ + SFP b; c; v++; V ++ + SNK '; V ++ :
Here Sgh[v++; V ++] is the gauge xing contribution to the action of quantum super elds
Sgh v++; V ++
=
=
The sum of the quadratic part
special values
=
1 has the form
SSCS (3.9) in quantum super eld v++ and Sgh (3.26) for
1 Z
2
d ( 4)duv++
D
d ( 4)duv++ _2
W v++:
(3.27)
eqs. (3.24){(3.25) completely determine the structure of the perturbation expansion to
calculate the oneloop e ective action
manifestly supersymmetric and gauge invariant form.4 The complete oneloop e ective
(S1C)S[V ++] of the pure N
= 1 SCS theory in a
4We restricted ourselves to the one loop approximation. However, the same construction will be valid
at any loop if we replace in (3.24) the quadratic part of the action for the quantum super elds with the
general nonquadratic action.
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
action is given by the sum of the contributions coming from the ghost super elds and from
the quantum super elds v++. It has the form
1
2
~(S1C)S = i
Here the rst term is the contribution from the NielsenKallosh ghosts, the second term
is the contribution from the FaddeevPopov ghost and the third term is the contribution
from the vector multiplet. In the next subsection we will consider the contribution to this
e ective action from the SCS multiplet, The contribution from the ghosts will be considered
The propertime representation of the contributions in (3.28) from the
The contribution of the ChernSimons vector multiplet to (3.28) can be written in the
propertime representation (see the analogous representation for 4D, N = 2 SYM theories
(S1C)S =
(2;2) is the vedimensional gauge covariant version of the projector [42{44, 67,
68] on the space of covariant analytic transverse super elds v++: The properties of the
(2;2)( 1; 2) were described in the paper [61].
T
For later, it is convenient to rewrite the projector
Here we follow the work [42{44]. We have5
(2;2)( 1; 2) in a di erent form.
T
(2;2)( 1; 2) =
T
_
=
A
(2;2)(1; 2)
A
(2;2)(1; 2)
L
The operator 2 is de ned by (3.19). As a result,
T
(2;2) can be expressed as
(2;2)( 1; 2) = _
T
1 hD1++; _21i _1
D1+ 4
D2+ 4 13(z1
z2)
21
21
1
_
21
D1+ 4
D2+ 4 13(z1
z2)
1
u1+u2+ 3
(3.29)
(3.30)
(3.31)
Note that the rst term does not vanish since [D++; _2] (q) = 14 (D+2
W)(1
q) (q).
5Here we are using a manifestly analytic form of the delta function:
(2;2)(1; 2) =
A
1
_
The next step is a representation of a twopoint function in the form
D1
where
expression for
T
(2;2):
= iD
D D
+ W(D )2 + 4(D
W)D
+ (D D
W):
(3.33)
This representation was obtained in the work [49]. As a result one gets the following
(2;2)(1j2) = u1 u2
T
+
D+2
W
(3.34)
1
4
1
_
21
D1
13(z1
z2):
1
2 D
The
d'Alembertian (3.19)
W
1
64
1
64
It is useful to compare this expression with a similar projector in 4D, N
= 2 SYM
theory [42{44].
3.2
The deformed covariantly analytic d'Alembertian _2
W
The operator in the quadratic part of the action for quantum elds plays a fundamental role in calculations of the e ective action in the framework of the background eld method.
In the case under consideration this operator is 2
In the analytic subspace the operator 2
_
W (3.18).
W (3.18) is rewritten in the form
+ 4
W(D
)2 =
W D
+ 4+4 D
+^
W
+
D^ D
+ 2
+
D
+ 2
W
D
+ 2 (D
)2:
rst term here is, up to the multiplier W, the standard covariant analytic
D
+ 2
+ 2
D
(D
)2 = W Da^Da^
1
4 D
+^ +
D^ W
D
1
4 D
+
D
+^
W
D^ +
^ +
D^ W
W
2
= W 2 :
_
The deformation of the operator de ned by (3.18) is stipulated by the super eld W as
follows
1
64
1
64
+^
D
W
+ 2
W
D
+
D^ D
+ 2
+ 2
(D
(D
)2 =
)2 =
1
64
1
64
D
D Da^
D+D
W
1
W G
D+^
W
This means that in this case the operator 2
Summing up all together we obtain
_
2
W = W
a^
D Da^
+
3
W
D^ +
" D+^
WD+^W +
W
1
2W
1
+^ +
D^ W
D+^
iD^ ^D
#
D
^
+
D+^
1
32
D+2
W
W
^ +
D^ W
(3.35)
W
This is a nal result for the operator 2
W acting on analytic super elds.
The deformed covariantly analytic d'Alemberian 2
possesses a useful property
[D+^; _2W ] = 0: It is important to note that the coe cient at the harmonic derivative
++ and that on the equations of motion (2.26) this term vanishes!
++ = 0 and therefore
D D+
W
=
D+D
W :
_
W takes the form
The leading contribution to e ective action of the 5D SCS multiplet
Our next aim is to calculate the leading lowenergy quantum contribution to the
oneloop e ective action. To do that it is su cient to evaluate the oneloop e ective action
(S1C)S[V ++] for an onshell background vector multiplet. The representation (3.29) provides
a simple and powerful scheme for computing the e ective action in the framework of 5D,
N = 1 super eld propertime technique.6
_
2
Hence, we get
If the background gauge multiplet is onshell, G
W does not involve any harmonic derivative D
_
++ = 0, the analytic d'Alembertian
, but 2 contains the factor D
+2
WD
.
1
_
21
u1+u2+
= u1+u2+
1
_
21
+
1
_ :
21
Taking into account this relation we obtain that the second term in the operator
absent. Therefore the projector is reduced only to (D+)4 13(z1
z2): Then
=
i Z ds Z
2
s
d ( 4)eis_2W D
Now we replace the deltafunction in (3.37) by its Fourier representation
5(x1
x2) D1
+ 4 8
( 1
2) =
eipa^ a^ 4
+
1
2+ ;
Z
d5p
(2 )5
6Note that Tr ln W does not contribute to (3.28).
where a^ = (x1
2i( 1+
+
2 ) a^
2 is the supersymmetric interval. Then we push
eipa^ a^ through all the operator factors in (3.37) to the left and then replace it by unity in the
coincidence limit. This leads to the following transformation of the covariant derivatives
Da^ ! Da^ + ipa^; D^ ! D^
2ip^ ^( 1
is no enough operators D+^ to annihilate ( 1
2) ^ in D^ vanishes in the coincidence limit since there
2 ). To get the expansion of the e ective
action in the background elds and their derivatives we should expand the exponent of the
operator and calculate the standard momentum integrals. Thus, we expand the exponent
eis_2W in powers of spinor derivatives near e isp2 up to the fourth order in spinor derivatives
(it corresponds to expanding up to the fourth order in the proper time) and use the identity
2 (D1 )4 4( 1+
1
+
2 )j 1= 2 = 1. It remains to perform a standard integration over the
momentum variables and over the propertime. The nal result has the form
is some scale.7 One can show, using the methods developed in [51], that the
action (3.40) is superconformal. A component form of (3.40) in the bosonic sector corresponds
to the Lagrangian F34 .
The e ective action (3.40) can be treated as the 5D analogue of the nonholomorphic
potential in 4D; N = 2 supersymmetric gauge theories. This e ective action is also similar
to the action of the the socalled 4D improved tensor multiplet [75{77].
4
The leading contribution to the e ective action of a 5D SYM multiplet
It is interesting and instructive to compare the oneloop leading lowenergy e ective action
in 5D SCS theory and in 5D SYM theory.
5D, N = 1 SYM theory is described by the action (see (2.14), (2.19)):
where the coupling constant has dimension [gS2YM] = 1:
Let us consider the construction of the background eld method for the theory (4.1).
It is obvious that the second variation of the action has the form completely analogous to
the 4D, N = 2 case
2
S =
1
7It is easy to see that the e ective action (3.40) does not depend on the scale, since
Z d13zW ln
= ln
Z
d ( 4) D+ 4
(3.39)
(3.40)
(4.1)
(4.2)
Therefore we can simply repeat step by step all the stages of the construction of the
e ective action developed in [39{44]. In particular, we can use the same choice of the
gauge conditions on the quantum super eld D++v++ = 0 and the same procedure to
x gauge as in [39{41]. It gives the sum of the quadratic part of action (4.2) and the
corresponding gauge xed action in the form SGF
S2 + Sgh =
1
2
v++ 1 (D
2
1
gSYM
v++(
1
)v++(2)
)2v++
Z
Further we put gauge parameter
=
The derivation of the general expression for the oneloop e ective action in 5D SYM
theory is completely analogous to derivation of the expression (3.28). The nal result is
i
2
i
2
~(
1
)
SYM =
Tr (2;2) ln 2
Tr (0;4) ln 2 :
_
_
The rst term is the contribution of the ghosts and the second term is the contribution of
a SYM multiplet. Here the covariantly analytic d'Alembertian _2 is given by (3.19) and
has the form [49]
_2 =
+
1
64 D
1
)2j = Da^Da^ + D
2
W :
Now we consider a calculation of the contribution to the e ective action (4.3) from a
SYM multiplet. Contribution of the rst term in (4.3) will be calculated in section 5. In
the propertime representation, the contribution of SYM multiplet has the form analogous
to (3.29)
(
1
)
SYM =
The only di erence with (3.29) is the operator 2 instead of the operator 2 . To nd
the leading lowenergy contribution to e ective action it is su cient to consider the
onshell background super eld. Classical onshell equation for the 5D SYM theory look like
= 0. In this case the covariantly analytic d'Alembertian _2 and the projection
T
(2;2) are simpli ed and we have the e ective action in the following form
_
W
(
1
)
SYM =
Evaluation of this expression is realized completely the same way as it was done in the
subsection 3.3 to obtain the (3.40). The nal result has the form
(4.3)
(4.4)
(4.5)
(4.6)
We see that the functional form of the e ective action (4.6) generated by a SYM
multiplet coincides with the one generated by a SCS multiplet (3.40). Although the onshell
conditions for 5D SCS theory and for 5D SYM theory are di erent and, moreover, 5D
SYM theory is characterized by a dimensional coupling constant, the leading lowenergy
contributions in these two theories to the oneloop e ective action turn out to be the same
up to a numerical coe cient.
5
The leading and nexttoleading contributions of the ghosts and matter
super elds
The ghost contribution to the oneloop e ective action in both SCS and SYM theories is
de ned by the expression
i
2
(
1
)
ghost =
hyper = ( )iTr ln D
iTr ln G(1;1) ;
obtained variation.
(
1
)
hypermultiplets or from ghosts
The contribution from matter hypermultiplet super elds di ers only by a sign and by the
choice of the representation of the gauge group. Hence, to
nd the leading contribution
to (5.1) we can use the results from the previous analysis [49] of the e ective action for a
qhypermultiplet coupled to a background vector multiplet.8 It was shown in that paper
that the leading quantum correction is the SCS action. Therefore, further we will focus
only on the rst nexttoleading correction.
For calculation of the rst nexttoleading contribution to the e ective action (5.1) we
will follow the procedure proposed in the papers [42{44, 49]. This procedure is based on
calculating the variation of the e ective action with its subsequent restoration given the
Let
hyper be the oneloop contribution to the e ective action from either matter
(5.1)
(5.2)
(5.3)
HJEP1(205)3
where the upper sign corresponds to the contribution from matter and the lower sign to that
of ghosts. Further, for simplicity, we will consider for only the plus sign. Here G(1;1)( 1; 2)
is the hypermultiplet Green function satisfying the equation
D1
++G(1;1)( 1; 2) = (3;1)( 1; 2) ;
A
where A(3;1)( 1; 2) is the appropriate covariantly analytic deltafunction
(3;1)( 1; 2) = (D1+)4 13(z1
A
z2) ( 1;1)(u1; u2)1:
This equation is similar to the equation for the hypermultiplet Green function in 4D, N = 2
theories and we can use methods developed in [39{41]. The Green function G(1;1)( 1; 2)
8From a formal point of view, the e ective action (5.1) corresponds to a so called !hypermultiplet [46].
It was pointed out some time ago [39{41] that this e ective action is equal, up to the coe cient 2; to the
e ective action for a q hypermultiplet. Therefore, further we will take into account just the q hypermultiplet
To evaluate this expression one can use the propertime technique. The leading lowenergy
contribution goes from the terms without derivatives of W. It was calculated in [49] and
[1]
hyper =
1
(4 )2 sign(W)
Z
The variation of (5.6) corresponds precisely to the classical action (2.25) of the SCS gauge
has the form
theory.
hyper =
Z
V ++G(1;1)
(5.5)
can be written in the form
Here 2 is the covariantly analytic d'Alembertian (4.4) and (u1+u2+) 3 is a special harmonic
distribution [67, 68].
The variation of the e ective action (5.2) under the background super eld V ++ is
written as follows (see [42{44, 49] for details)
(5.4)
(5.6)
(5.7)
(5.8)
Our main purpose in this section is to nd the rst nexttoleading correction to the
SCS action. First of all one considers the Dyson type equation relating the free and full
hypermultiplet propagators [39{41]
Z
G(1;1)(1j2) = G(1;1)(1j2)
0
d 3( 4)G(1;1)(1j3)iV ++(3)G(1;1)(3j2) :
0
Substituting eq. (5.7) into the variation (5.5), one nds
Z
(
1
)
hyper =
d 1( 4)d 2( 4) V ++(
1
)iV ++(2)G(01;1)(1j2)G(1;1)(2j1) :
Taking into account the explicit form of the propagator (5.4), we rewrite this expression
as follows
(
1
)
hyper =
Z
d 1( 4)d 2( 4) V ++(
1
)
1
21
D1+ 4 D2+ 4 13(z1
z2)
Now we use the spinor derivatives from the rst deltafunction to restore the full N = 1
superspace measure according to the rule R d ( 4)(D+)4 = R d13z. So far, we did not consider
any restrictions for the background super eld, therefore (5.8) is the exact representation
for the oneloop hypermultiplet e ective action. In principle it can be a starting point for
calculations of di erent contributions to the oneloop e ective action.
To compute the rst nexttoleading quantum corrections to (5.6) it is su cient to
expand in (5.8) the covariant analytic d'Alembertian in powers of the derivatives of the
background super elds. Let us remember that the Green function of the hypermultiplet is
antisymmetric with respect to the permutation of its arguments. This allows us to rewrite
the expression (5.8) in the form
The next step is the expansion of the operator 2 in the denominator in power series of the
derivatives of W. In this expansion we keep only the leading terms with two derivatives.
It leads to
(
1
)
Now the usual steps lead to
Z d13zdu1du2
V ++(
1
)
u1+u2+ 2 D2
2 iV ++(2)
+
2 (2
W2)3
D2
4
D2
+ 4 13(z2 z1):
W
2
+ 2 ln W =
1
As a result we obtain the rst nexttoleading contribution to the oneloop hypermultiplet
e ective action in the form
Z
[1]
hyper = chyper
Here chyper is a numerical coe cient depending on details of the hypermultiplet action
(such as the number of components, the representation, whether hypermultiplet super elds
are commuting or anticommuting). We see that the e ective action (5.10) has the same
functional structure as the e ective actions generated by the SCS multiplet (3.40) and by
the SYM multiplet (4.6).
6
Summary
We have considered the vedimensional N = 1 supersymmetric eld models such as the
Abelian ChernSimons theory, the YangMills theory and the hypermultiplet theory coupled
to a background vector multiplet, formulated in the harmonic superspace approach. In
all these models we calculated the universal four derivative contributions to the oneloop
In the 5D SCS theory we have developed the background eld method in harmonic
superspace and represented the oneloop e ective action in terms of functional determinants
of the operators acting in the analytic subspace of harmonic superspace (eq. (3.28)). The
above expression contains the contributions from the ghost super elds and from a SCS
vector multiplet super eld.
We studied the structure of the latter contribution to the
e ective action and evaluated the leading lowenergy e ective action (eq. (3.40)).
The same consideration has been realized in 5D SYM theory as well. We developed
the background eld method for this theory, found the oneloop e ective action in terms
of the functional determinants of the di erential operators acting in the analytic subspace
of harmonic superspace (eq. (4.3)). We studied the structure of the oneloop contributions
to the e ective action from a SYM vector multiplet and calculated the leading lowenergy
contribution of this multiplet. Although 5D SYM theory is not superconformal, its coupling
constant is dimensional, and the expressions (3.28) and (4.3) are di erent, the leading
contribution to the e ective action in 5D SYM theory (eq. (4.6)) has the same functional
form as in 5D SCS theory.
In 5D hypermultiplet theory in a vector multiplet background eld we have calculated
the rst nextto leading contribution to the e ective action (eq. (5.10)). The corresponding
leading contribution is the SCS action and was found in the paper [49]. We have shown
that this rst nextto leading contribution (5.10) again has the same functional form as
the leading contribution in SCS theory.
We have found the manifestly 5D, N = 1 superconformal form of the term
W
the e ective actions of the SCS theory, SYM theory and hypermultiplet theory. The next
step of studying of the oneloop e ective action in the theories under consideration is a
construction of the full lowenergy oneloop e ective actions where all the powers of the
Abelian strength are summed up. In other words, the next purpose is to construct the 5D
super eld HeisenbergEuler type of the e ective action. Like in 4D, 3D superconformal
2 = 12 D4 ln W: We hope to study this issue in the forthcoming work.
gauge theories [78, 79] it is reasonable to expect that this e ective action will be expressed
in the terms of so called superconformal invariants which transform as scalars under the
5D, N = 1 superconformal group. For example, such a scalar invariant can be a super eld
F 4 in
Acknowledgments
The authors are very grateful to E.A. Ivanov and S.M. Kuzenko for useful comments. The
authors are thankful to the RFBR grant, project No 150206670a and LRSS grant, project
No 88.2014.2 for partial support. Work of I.L.B was supported by Ministry of Education
and Science of Russian Federation, project No 2014/387/122. Also I.L.B is grateful to the
DFG grant, project LE 838/122.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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