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32
UDC 530.1; 539.1
EXCEPTIONAL FIELD THEORY FOR E6(6) SUPERGRAVITY
E. Musaev
National Research University Higher School of Economics, Faculty of Mathematics ill. Vavilova, 7, 117312, Moscow, Russia.
E-mail: emusaev@hse.ru
A brief description of the supersymmetric and duality covariant approach to supergravity is presented. The formalism is based on exceptional geometric structures and turns the hidden U-duality group into a manifest gauge symmetry. Tensor hierachy of gauged supergravity appears naturally here as a consequence of covariance of the construction. Finally, the full supersymmetric Lagrangian is explicitly constructed. This work was presented on the International Conference "Quantum Field Theory and Gravity (QFTG'14)" in Tomsk.
Keywords: supergravity, extended geometry, dualities, exceptional field theory.
1 Introduction
1.1 Dualities in supergravity
Since the seminal work of Cremmer and Julia fl] it is well known that 11-dimensional supergravity compactified on a torus Td enjoys a hidden symmetry Ed(d). From the point of view of the underlying M-theory these are the so-called U-duality transformations that unify the perturbative T-duality, that relates Type IIA and Type IIB theories, and S-duality of Type IIB string theory.
To get the basic idea of the construction it is the most instructive to start with D = 11, N = 1 supergravity, whose field content is very simple. This introduction mainly follows the paper [2] by Cremmer and Julia that contains very clear and detailed review of their results presented in the letter [1]. The field content of eleven-dimensional supergravity is very-simple: graviton, 11-dimensional gravitino and the 3-form gauge field. Upon reduction on a d-dimensional torus Td, parametrised % the coordinates {xn}, the
D=
11 — d dimensions. Decomposing the 11-dimensional fields under the split ll=D+d one gets the following field content in 4 dimensions. From the vielbein we get one D-dimensional vielbein e^, d vector fields A^ and d(d + 1)/2 scalar fields gmn. The 3-form field reduces into a 3-form C^p, d number of 2-forms BMvm, d(d—1)/2 vectors AMmn and q = d(d— 1)(d—2)/6 scalar fields Cmnk-
Such constructed effective theory has in general SL(d) k Rq global (rigid) symmetry group, where the SL(d) part comes from the diflteomorphisms of the internal space of the form 5xm = Amxn. The abelian group Rq, that is the remnant of the gauge symmetry, acts on the axions Cmnk as constant shifts
In addition, in dimensions D = 3,4,5 one can dualise 1,2 and 3-forms respectively to obtain addition scalars when the p-forms enter the Lagrangian only by their derivatives. There are certain subtleties when this procedure is applied to the 11-dimensional supergravity because of the Chern-Simons-like terms F[C] A F[C] A C, which will not be described here. Very detailed inspection of the global rigid symmetries that survive this construction is presented in [3]. To be mentioned is that such dualisations are necessary in D < 5 to obtain the full U-duality group Ed(d) in the scalar sector.
Hence, the scalar fields can be nicely packaged into a matrix V that is an element of the coset Ed(d)/K(Ed(d)). By choosing a correct parametrisation of the coset the scalar potential can be written in the following form that is globally invariant under Ed(d) [3]
L scalar = 4 eTr[dM-1dM],
(2)
^Cmnk cmnk( COIlSt).
(i)
where M = V*V is the metric on the coset space. The involution * here denotes the usual transposition for D > 6 and is replaced by Hermitian conjugate and contraction with a certain symplectic matrix Q for D<5
It is possible to repeat the same story for the p-forms sector, taking into account that to have the global symmetry on the level of Lagrangian (not the EOM), in even dimensions D = 2n one has to add extra "magnetic" duals to n-forms. This is necessary since on the level of equations of motion the symmetry-is realised on the field strengths rather than the gauge n
with its Hodge dual forms a representation of the duality group.
It is important that the hidden symmetries in the
D
dimensional effective theory. Following analogy with
General Relativity one may ask what is the geometric origin of the duality symmetries and to what extent do they present in the initial 11-dimensional supergravity. The formalism of Exceptional Field Theory that is an attempt to make sense of these questions and to find a way to answer them is briefly described in this letter. For calculational details and more involved discussion the reader may refer to [4].
1.2 Basic conventions
In what follows we focus on the E6 exceptional field theory and hence it is useful to list few basic conventions and definitions that will be used [5]. A coset representative is denoted as usual by
vM € E6(6)
M " USp(8)' where the index convention is the following
M, N,O,P,... — 1, ..., 27, E6(6) indices
A, B,C,D,... — 1, ..., 27, local USp(8)
i,j,k,... — ^ ..., 8, local USp(8)
V-.v-.p-.v-,... — 1, ..., 5, GL(5) indices
a,b,c, d, .. . — 1, ..., 5, local SO(1,4)
VijVN - SN y M vij — s
' M V ij — sM,
VMM ^kl — 0, QklQlm — -sm,
vm vmm — skj - g^ij ^kl,
VMij — (vM )
VM^ki^lj ,
A[i1,...in] = 1 (Ai1...in + permutations).
C— üij, ^ C — üij^,
ClaC-1 — yT , CT — -C, Cf — C-1
Yabcde — abcde.
'i,iTvj — -üiküjl Tpi(C -1rT C)^k
for any expression of gamma matrices T.
1.3 Extended geometry
Following the constructon of Cremmer and Julia the hidden exceptional symmetries of lower dimensional maximal supergravities most straightforwardly can be reproduced in toroidal reductions of 11-dimensional supergravity. The formalism of extended geometry provides more geometric background to the exceptional groups in terms of extended geometric structures on an extended space (for review see [6-8]).
The extended space is constructed by adding extra directions to the would-be internal manifold that correspond to winding modes of M-branes [9,10]
X ,ymn,zmnklp,
(10)
(3)
(4)
The scalar matrix V] and the symplectic USp(8) matrix Qij satisfy a set of constraint
(5)
where the star denotes complex conjugation and the Kronecker symbol for pairs of antisymmetric indices is defined as Sj = 1/2(SjSj -S'Sj). In addition we use the convention that all (anti)symmetrisations of n indices are performed with a prefactor of 1/n!, i.e.
Infinitesimal coordinate transformations on this space consistent with the exceptional groups are defined as a generalisation of the well-known Hitchin's construction. Hence, one defines generalised tensors that live on the extended space as objects with the following transformation rule [11,12]
(LaT)M =AndNTM - 6PMLNKONAkTl
+XT (OK AK )TM = [A,T]M. (11)
The first and the last terms play the roles of translation and a weight term respectively. The second term reflects the exceptional group symmetry and involves the projection of the matrix dNAK on the U-duality algebra, since in general it does not belong to the structure group E6(6) [13]. This is very similar to General Relativity where, however, the group is GL(n) and any non-degenerate matrix belongs to its algebra.
GL
will be just trivial.
In addition one introduces a differential constraint on all fields in the theory that restricts dependence on the extended coordinates XM
(6)
dPMNdM ® dN — 0.
(12)
For the spinor sector we use symplectic Majorana spinors subject to the reality constraint
(7)
where the charge conjugation matrix C is defined by the following relations
This extra condition in particular implies existence of a trivial transformation given by AM = dMNK dNSK which itself transforms clS cl generalised vector. The Jacobi identity and closure of the algebra hold up to a trivial transformation as well. The latter leads to the notion of E-bracket that is an antisymmetrisation of the Dorfman bracket
(8)
[LAi , LA2] — L[Ai,A2]e , [Ai, A2]e = [A[i, À2]]d .
(13)
This implies the following relation for fermionic bilinears with spinor fields ^^d <p'
(9)
It is important to mention that in contrast to the E-bracket, the Dorfman bracket [, ]D is not antisymmetric nor symmetric. This will play a crucial role in construction of tensor hierarchy starting from the covariant derivative to be defined in the next section.
*
2 E6(6) covariant exceptional field theory
2.1 Covariant derivative for D-bracket and tensor hierarchy
In the formalism of Extended Geometry generalised tensors and the corresponding transformations are considered to be independendent of space-time coordinates xthat decouples the would be scalar sector.
In order to naturally incorporate the tensor and fermionic sector into the formalism the fields and all the gauge parameters are now allowed to depend on the external space-time coordinates. In the spirit of the ordinary Yang-Mills construction this implies that one has to introduce a long space-time derivative, that is covariant with respect to D-bracket [13]
SaD»TM = LaD»TM ,
D»
d» — L
AM
d» — [A», ]
D,
SaAM = d» AM — [A», A] M
D»A
M
[D», Dv] = —Lf,
m
-v]
»
FMv = 2d»AM — [A», Av ]M + l0dMNK dN Bk»v .
SAM = D»AM — 10dMNK dN ^k» ,
AB
M »v
» 2V i
\»^Mv]
+ dMNK ANFuvK.
D» Vm
SO(1,4) ab W» ^Mab
USp(8) Q»1 j QMij
USp(8) connection QM is defined according to the group properties of the matrix Vf as usual (see [5])
VklM D»VMij = 2Sk[i Q»ijj + P»ijmnQmk Qui, Q» € usp(8), P» e ee © usp(8).
(17)
Explicit form of the SO(1,4) connecti on wMab can be found following the same story but for the space-time
vielbein e», i.e.
ea» dM e» = ^Mab + nm
ab
(18)
where nMab = nMba. Finally, the internal USp(8) connection QMij is derived from an analogue of the vanishing torsion condition for the extended space vielbein Vj. The generalised torsion is given by
(14)
t m _ p m ™M P p l
INK = r NK — 6P K Li PN
I m Q p p + 2P K Nr PQ ,
(19)
where the gauge field A^f is identified with the vector field of the corresponding maximal supergravity (with all necessary dualisations).
Since the E-bracket does not satisfy the Jacobi identity, one has to deform the usual field strenth by a trivial transformation
that follows from the usual definition T(V, W)M = cvw M — lv WM, wher e £v is the covariant generalised Lie derivative. Hence one may write for the vanishing torsion condition
Vm M Dn Vkm — 6pmkplVm lDp Vn1 + 3 pmkqn tpqp = 0.
(20)
(15)
In the spirit of tensor hierarchy, gauge transformations of the 1- and 2-forms naturally have the following form
(16)
p
D = 5 maximal supergravity and naturally leads to tensor hierarchy as a consequence of generalised covariance.
2.2 Geometry and connections
The structure of EFT explicitly distinguishes beetween the two sets of coordinates: space-time {x»} and the extended space XM. Respectively, one has two local groups SO(1,4) Mid USp(8). Hence, there are four types of connections listed in the following table
This equation has the form of the familiar expression e'a] = 0 however deformed in accordance to the algebraic structure of the duality group Ee.
3 Supersymmetry transformations
Supersymmetry transformations of the fields of Ee(e) covariant supergravity are taken to be of the following form
See» = 2 ëiYar»,
S£4» = D»ei — zV2VMij (Vm(l»ek)
— 3Y»Vme ) Qjk,
SeXijk = ^Pjl»Qimem — ^ (VM[ijQk]
— 3VMm[iQjk^ QmrVmer,
%/2 V
SeVj = ÏVm SeA»M = V2
ki
4Qp[k Xlmn]eP + 3Q[klXmn]pe
QmiQnj,
The SO(1,4) connection is defined by the usual vanishing torsion condition V^ea^ = 0. The
e^»
1
V5
+ djNPA[»NSe, Ap],
iQik ek + ekY»X
ijk
i
M
Se B »v M = ^VMlj
2^i[»lv]ek Qjk — iXijk Y»v ek
m
P
where we define the full covariant derivative as
Exceptional Field Theory takes the following form
V
m e
— v m s- -t,
gFm Ypa e .
(22)
1
— + Sso(i,4)(Oab) + S„Sp(8)(Aij )
+ Sgauge(A ) + Sgauge(sM M
+ Sgauge(S mv ) + Ssusy (es)+ S(Om^v ),
that is the same as for the five-dimensional theory. Parameters of the transformation on the RHS are given by the following expressions made of the spinors e1,2 and the scalar matrix VMij
e = 2^1, AM = (e2ie1 VMij j) ,
Qab — -
iV/2 , ab^ k - '
(^eiiY VMe2 - VmeiiY
ab k
V Mij Q
jk
ab
ab , - AM 77
- A Wm + 2 A FM
SMM — -VMklQlm(e2k YMel™)
Mv
3i 10
(ta) N VMsi V Nki( e2k YM v e1 )
dMNK dN Okmv — 0.
(24)
L — R - 4 Mmn Fmv
1
F M F MvN _ 1 |p ij kl 12
Closure of supersymmetry transformations on the fields of EFT has the following structure
- ^MiYMvpDvrp - 2%/2iVMijQik^MkY[M<V'M(Yv]^vj)
- 3 XijkYMDßxijk - 4V2iVMmnQnpXpklVMXmkl 4i
+ g- VßijklXijk Yv
mCl
Ql
k
(23) + 4%/2V Mij Xijk YMV m+ Ltop - V(M, g).
Om^v — 2fc Y^v 9m ek — 9m e 2k Y^v ek
2 \
— (e2kesi)efMdMev]a + 3Vn^mVNkt(e2kY^vel)J .
Here and AM are the diflteomorphism parameters, Qab parametrizes the Lorentz rotations, SM^d SM are the gauge transformation parameters of the 2-form B^vK and the extra 3-form Ca^vp- Finally, as a consequence of the section condition the tensor Om^v is constrained by
The operator S(OM^v) leaves invariant the field that is the only way of how the 2-form field enters the Largangian. The same is true for the gauge transformation generated by Hence, the
superalgebra is closed up to the section condition.
4 Invariant Lagrangian
Given the définitions of the covariant derivatives that respect the E6(6) structure of the extended space the full supersymmetric Lagrangian for the covariant
Here VM [F] = VM [-F] encodes switch of the sign of the gauge field flux, Ltop is the topological term that includes the covariant version of the Chern-Simions Lagrangian and V is the scalar potential of [13,14]. Due to the lack of space we do not check explicitly supersymmetry invariance of the above Lagrangian. For more detailed consideration the reader is referred to the paper [4]. However, the E6(6) invariance is manifest since all the objects in the Lagrangian are covariant.
5 Outlook
In this note the U-duality covariant approach to supergravity is briefly described. The essential feature of the Exceptional Field Theory approach is the notion of extended space and the structure of extended geometry defined on it. We describe the construction of E6(6) covariant derivatives in both the space-time and the extended space directions. The corresponding vanishing torsion and algebraic conditions give necessary expressions for the SO(1,4) and USp(8) connections.
The final result is the supersymmetric manifestly E6(6)-covariant Lagrangian that includes all the fields of the maximal D = 5 supergravity. The 11-dimensional diflfeomorphism symmetry is not manifest in this construction, however upon solution of the section constraint one is able to restore the full 11-or 10-dimensional Lagrnagian.
As it was shown in [13], decomposition of the 27 extended space coordinates under the GL(6) subgroup of E6(6) and leaving only the coordinates in the 6, provides a consistent solution of the section constraint. This corresponds to the Kaluza-Klein decomposition of the full 11-dimensional supergravity.
An alternative solution is given by decomposition of the 27 under the GL(5) x SL(2) subgroup. This leads to Type I IB supergravity with manifest SL(2) duality-symmetry.
Relation between the described formalism and the embedding tensor approach to gauged supergravities is given by generalised Scherk-Schwarz reductions [15]. As it was shown in [14, 16] the reduction naturally provides all the gaugings in terms of generalised twist
2
a
matrices and their derivatives with respect to the full set of extended coordinates.
Acknowledgement
The author expresses his gratitude to theoretical dpt of CERN for warm hospitality during complection
of this letter. In addition I would like to thank ENS de Lyon and personally Henning Samtleben for genereous financial support and productive collaboration. Finally, I thank Tomsk State Pedagogical University and personally Vladimir Epp and Joseph Buchbinder for creating a wonderful atmosphere during the QFTG'2014 conference.
References
[1] Cremmer E. and Julia B. Phys. Lett. B80 (1978) 48.
[2] Cremmer E. and Julia B. Nucl. Phys. B159 (1979) 141.
[3] Cremmer E„ Julia B., Lu H., and Pope C. Nucl. Phys. B523 (1998) 73-144, [arXiv:hep-th/9710119].
[4] Musaev E. and Samtleben H. [to appear],
[5] Wit de B., Samtleben H., and Trigiante M. Nucl. Phys. B716 (2005) 215-247, [arXiv:hep-th/0412173],
[6] Aldazabal G., Marques D., and Nunez C. [arXiv:1305.1907],
[7] Berman D. S. and Thompson D. C. [arXiv:1306.2643],
[8] Hohm 0., Lust D„ and Zwiebach B. Fortsch. Phys. 61 (2013) 926-966, [arXiv:1309.2977],
[9] Hull C. JHEP 0707 (2007) 079, [arXiv:hep-th/0701203],
[10] Hohm 0., Hull C„ and Zwiebach B. JHEP 1008 (2010) 008, [arXiv:1006.4823],
[11] Coimbra A., Strickland-Constable C., and Waldram D. [arXiv:1112.3989],
[12] Berman D. S., Cederwall M., Kleinschmidt A., and Thompson D. C. [arXiv:1208.5884],
[13] Hohm O. and Samtleben H. Phys. Rev. D89 (2014) 066016, [arXiv:1312.0614],
[14] Musaev E. T. JHEP 1305 (2013) 161, [arXiv:1301.0467],
[15] Grana M. and Marques D. JHEP 1204 (2012) 020, [arXiv:1201.2924],
[16] Berman D. S., Musaev E. T., and Thompson D. C. JHEP 1210 (2012) 174, [arXiv:1208.0020],
Received, 12.11.20Ц
Э. Mycaee
ИСКЛЮЧИТЕЛЬНАЯ ТЕОРИЯ ПОЛЯ ДЛЯ E6(6) СУПЕРГРАВИТАЦИИ
Представлено краткое описание суперсимметричного дуальность-ковариантного подхода к супергравитации. В основе описываемого формализма лежит структура обобщенной геометрии, при этом скрытая симметрия П-дуальпости рассматривается в качестве локальной калибровочной симметрии. Тензорная иерархия полей деформированной супергравитации появляется естественным образом как следствие ковариантности подхода. Полностью суперсимметричный дуальность-ковариантный Лагранжиан выписан в явном виде. Эта работа была доложена на международной конференции «Quantum Field Theory and Gravity (QFTG'14')» в Томске.
Ключевые слова: супергравитаци, расширенная геометрия, дуальности, исключительная теория поля.
Мусаев Э. Т., доктор.
Национальный исследовательский университет «Высшая школа экономики».
Ул. Мясницкая, 20, 101000 Москва, Россия. E-mail: emusaev@hse.ru
