JHEP03(2018)154 Springer b March 18, 2018 March 26, 2018 January 5, 2018 : : : Accepted Received Published = 2 conformal supergravity , N and Bindusar Sahoo c,d Published for SISSA by https://doi.org/10.1007/JHEP03(2018)154 Ivano Lodato b [email protected] , [email protected] , = 2 conformal supergravity. We also show how this dilaton Weyl . N 3 1712.05365 Subramanya Hegde, The Authors. a c Extended Supersymmetry, Supergravity Models, Superspaces

We construct the dilaton Weyl multiplet for dilaton Weyl multiplet in 4D supergravity ] in superspace. , 1 [email protected] = 2 E-mail: [email protected] Indian Institute of ScienceVithura, Education Thiruvananthapuram and 695551, Research, India Indian Institute of ScienceHomi Education Bhabha and Road, Research, Pashan, PuneDepartment 411 of 008, Physics India andFudan Center University, for 220 Field Handan Theory Road, and 200433 Particle Shanghai, Physics, China George P. and CynthiaTexas Woods Mitchell A&M University, Institute College for Station, Fundamental Physics TX and 77843, Astronomy, U.S.A. b c d a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: Weyl multiplet, the equations of motionfields, can be following used to a eliminatecomponent the similar supergravity multiplet auxiliary pattern includes asvariant two in formulation gauge of five vectors and andmultiplet six is a dimensions. contained gauge inby two-form the M¨uller[ The and minimal resulting provides 32+32 24+24 a Poincar´esupergravity multiplet introduced Abstract: in four dimensions. Beginning from an on-shell vector multiplet coupled to the standard Daniel Butter, N JHEP03(2018)154 ]. This 6. This 11 3 – ≤ 6 ], and later 9 d 12 ≤ dimensions must correspond 10 11 d 8 . This is not the only exchange which D – 1 – 14 ], which are relevant to our discussion here. 8 – 2 ]. As the name suggests, these variant Weyl multiplets contain a 13 1 1 dimensions. One important example of such correspondence, central to − = 2 dilaton Weyl multiplet in components = 2 dilaton Weyl multiplet in superspace d N N The dilaton Weyl multiplet was discovered first in six dimensions in [ Off-shell superconformal multiplets with 8 supercharges exist for 3 3.1 Analysis of the constraint on the scalar fields scalar field of non-vanishing Weylthe weight. corresponding standard Because Weyl the multiplet, aWeyl off-shell bosonic multiplet degrees degree must of of freedom be freedomscalar from match removed the of in standard Weyl exchange weightoccurs. for two, the Supersymmetry typically dilaton. implies denoted a This dilatino field partner is to an the auxiliary dilaton, and this is exchanged entails that certain fieldto representations those and in structures in this paper, is the existence ofof a off-shell variant version degrees of of the Weyl freedom, multiplet, known with as the same the number dilatonon Weyl in multiplet. five dimensions [ higher derivative ones, see e.g. [ includes the Weyl multiplet of conformalin supergravity this itself, paper. which will These be Weyl multipletsand, our in chief differing in concern spacetime dimensions fact, are the deeply connected various cases can be connected via dimensional reduction [ as well as quantumbe description, gauge fixed superconformal to invariantsator obtain couplings multiplets, short the of matter physical supergravity multipletsknowledge Poincar´etheory. which can of This are general used procedure irreducible to requires representationsutmost fix of compen- conformal importance the symmetries. to superconformal The algebra the is construction hence of of invariant couplings in supergravity, especially 1 Introduction The superconformal multiplet calculusof is general a matter crucialconformality tool couplings implies for to the the gauge absence construction of theories, and a both analysis physical in energy scale, flat necessary and for curved both space. a classical Though 5 The 6 Conclusion and future directions 3 The vector multiplet and the multiplet of its4 equations of The motion Contents 1 Introduction 2 The standard Weyl multiplet and its (un)conventional constraints JHEP03(2018)154 ]. 13 = 2 gauge N ). This is two i χ ]. He considered a 1 is traded for a gauge ab T = 4 dilaton Weyl multiplet d is traded for a dynamical gauge − abc T = 4 was possible via a string-inspired d . In the dilaton Weyl multiplet, only one = 6, the short matter multiplets involving . D µ replaced by a gauge two-form and 0) worldsheet (with the left-moving sector of d , A – 2 – − ab -forms is precisely what one would expect from p T = 4 dilaton Weyl multiplet should be possible, but, = (2 d N = 5 and d = 6 two-form. In fact, this . Finally, an auxiliary tensor field is exchanged for a set of i d χ This suggests that constructing general higher derivative actions, = 5, this set of 1 d ] for its superspace form), consistent with its reduction from six dimensions. 16 = 4 dilaton Weyl multiplet differs from its higher dimensional analogues , as well as a gauge one-form d 15 , while in five dimensions, an antisymmetric tensor ] that a variant Weyl multiplet in 1) worldsheet and an constructed already in superspace even earlier by M¨uller[ µν µν , 17 B B ] (see [ 14 = (1 almost In this paper, we present a modern analysis, inspired by the five dimensional case [ It has long been suspected that a This exchange is helpful for a few reasons. First, in two-derivative actions, the auxiliary Actually, in five dimensions, the off-shell two-form multiplet can be defined in the presence of a central N plays the role of a Lagrange multiplier and so there must be another scalar degree of 1 -forms. In six dimensions, an anti-self-dual tensor not equivalent in fouractual dimensions. dilaton Weyl multiplet, Coupling while to the24+24 current an component analysis on-shell leads matter instead multiplet. vector to multiplet Here a large we leads irreducible focus to on the the dilaton Weylcharge multiplet, [ leaving coupled to a linear multiplet compensator. There the dilaton Weyl multipletconstruction was of found the current in multiplet twothe for different, standard a yet Weyl non-conformal equivalent multiplet vector ways: to multiplet via and an the by on-shell coupling vector multiplet. The two methods are in fact one-forms. As with dimensionally reducing the was larger 32 + 32 Poincar´esupergravitysupergravity multiplet, multiplet, which which it he turns called out can the be new interpreted minimal as the dilaton Weyl multiplet and right-moving modes of a stringcontent compactified of to the four standard dimensions, Siegel Weylan derived multiplet the and field the dilatontype Weyl multiplet II). from, The respectively, with the auxiliary anti-self-dual tensor the dilaton and dilatino as an added bonus) whileto remaining the off-shell. best ofby our Siegel knowledge, [ it hasargument. not By been formally explicitly tensoring constructed. together two It copies was of first (super) noticed Yang-Mills from the left built in. In the standardmatter Weyl multiplet, multiplets, any but two-form must intwo-forms come are both from on-shell. a superconformal such as those descending from stringnaturally theory, provides would a be means difficult. to The introduce dilaton a Weyl multiplet single two-form into the gravity multiplet (with freedom for it tothe remove reason to that constructdegree theories a of physical with action freedom eight (and becomesaction, supercharges similarly a while require with Weyl another two compensator mustcompensating compensators: whose be multiplet kinetic one is paired term needed scalar with the to yields 5d provide and the the 6d Einstein dilaton Einstein Weyl action. multiplets is Another that advantage they of both come with an off-shell two-form p two-form two-form D with a fermionic auxiliary JHEP03(2018)154 1 and (2.1) D 2) con- conven- | , 2 ij , ab T , connections i µ = 6). The field ] to remove the -supersymmetry ] several decades ψ 1 d S . 19 . These latter three a µ = 0 , the f , -form gauge fields that ab p µa µ ω = 0 De i , and a dilatation connection , a gravitino χ R 2 3 a ν µ γ e − transformations, and the ordinary 3 2 U(1) bij R µ + × T i R µν ) abij Q T ( 8 1 R + µ – 3 – ]. In section 2, we review the standard Weyl = 5 and a tensor multiplet for µa 18 ) d A ( , γ ˜ R . In section 3, we analyze the on-shell vector multiplet ) are the supercovariant curvatures associated with local − ab Q + i = 0 ( T b a R µν appearing in the table indicate the (anti)self-dual components µνa of the Weyl multiplet for this reason. These auxiliary fields -symmetry group SU(2) ) ) R , and ∓ P ab i ( M ) and T = 2 standard Weyl multiplet was worked out in [ χ ( R A , R ( N b D R µ e ), , and the special conformal (or K-boost) connection M i ( gauging the µ R φ auxiliary fields µ ), A P ( . Complex conjugation raises/lowers SU(2) indices, reverses the chiral weight and R and In five and six dimensions, the standard Weyl multiplet with eight supercharges shares ab j appearing both explicitly in these constraints and implicitly in the definitions of the T i . The remaining gauge fields are the Lorentz spin connection i µ µ these auxiliary fields are(the removed. dilaton), In a their place, dynamicallead one to fermion introduces a (the new a dilatino),Weyl multiplet dynamical multiplet of and scalar 24 can field a + 24 be setshell off-shell of viewed multiplet degrees (a as of vector the freedom. multiplet standard In for both Weyl multiplet cases, the coupled dilaton to a certain on- properties. The tensor of chirality (for fermions), and switches the duality property ofa a very tensor. similar structure,to including introduce a an similar alternative auxiliary Weyl multiplet, field known content. as There the it dilaton is Weyl multiplet, possible where called the ensure that the off-shelldegrees bosonic of and freedom fermionic degrees toWeyl of yield multiplet. freedom the match, The total providing field 24along 7 + content + 24 with 8 of off-shell the the degrees chiral standard of weights Weyl freedom multiplet and of is Weyl the tabulated weights standard in of table the fields, as well as their symmetry translations, local Lorentz transformations, chiral U(1) supersymmetry (also referred toχ as Q-supersymmetry). Thecurvatures covariant fields themselves are necessaryderivative for actions, completing these the covariant fields field appear representation. without In kinetic all terms two- and are generally Here V b connection connections turn out totional be constraints composite and determined in terms of the so-called 2 The standard WeylThe multiplet structure and of its the (un)conventionalago. constraints It is ansisting irreducible of field the representation following of field the content. superconformal There algebra is SU(2 a vierbein multiplet, with some minorthe modifications auxiliary to fields itsand conventional constraints solve to its separate multipletsection out of 4. field In equations.linear section multiplet The compensator, 5, giving dilaton we a superspace Weyl show description multiplet how of the is to dilaton summarized Weyl modify multiplet. in M¨uller’sconstruction [ the latter discussion to another work [ JHEP03(2018)154 (2.2) (2.3) (2.4) . . = 0 ] b — — — — — — +1 +1 | ] µa ν | ) Chirality e for fermions A ij ( c any dependence on the | ˜ R a i [ a . µ T − f 2 2 ij ij b b 1 / / c | 0 0 0 0 0 a 1 1 − µ T [ − − µνa Chiral and T ij ) weight (c) 1 4 µb µi M T ( φ + R b 1 ] 16 ν ν b e 2 − , a , 1 / i µ 0 0 0 1 2 i 1 [ − µa χ 3/2 χ Weyl − µ µν De , e De γ weight (w) γ – 4 – 1 4 1 2 1 2 − i = 0 + − ab + η µ = 2 vector multiplet to the standard Weyl multiplet γ µνi ) (old) µν (old) (old) µi µa N (old) µνi ) 1 3 1 1 1 1 2 2 − ) φ f Q ( j Irreps SU(2) Q M  . That is, one wishes to shift the auxiliary fields as = = ( ( R µ µ R R γ D ij = = T (new) (new) µi µa i · j φ f ab − ab µ , and γ T , γ i V . Weights and chiralities of standard Weyl multiplet fields. 1 8 (new) µνi ij χ − , ) , ε + h.c. i (new) µν , j a 1 2 ij i µ − i ) Q µ ), these (un)conventional constraints correspond to curvature shifts µ = 0 = ]: we couple an µ ij D ( µi µ χ b i ab 2 A e ψ ∗ =  V Field T ψ ab M ) 13 R a µν µ ( j a T ij ) i γ D Table 1 R µ ab i P V  ( T ( R = ¯ = 2 i a µ µ δe δψ To construct the 4d dilaton Weyl multiplet, we will follow the same procedure used in The transformations under Q- and S-supersymmetry will also change accordingly, Comparing with ( These redefinitions actually correspondstraints to making a different choice of conventional con- and use the equationsis of helpful motion to to eliminate extractauxiliary the from fields auxiliary the fields. composite To gauge do fields this efficiently, it equations for the on-shellbe multiplet reinterpreted as are constraint linear equationsWeyl for multiplet in the auxiliaries the for auxiliaries Weyl these themselves.off-shell other multiplet This physical degrees auxiliaries exchanges fields, of the and while freedom keeping can unchanged. the total number of five dimensions [ JHEP03(2018)154 have (2.8) (2.6) (2.7) (2.5) S j δ .   j j j η i η nor ) . ij ij V T Q ( T δ · · R γ , · γ µ , γ γ ) 1 relative to the stan- 12 µ η 1 i γ ( 24 and the non-covariant i η + S 1 3 + h.c. a ab δ i − ) µ i − f jk  Q D and µ ( a ) + ac γ γ R + j K T K (h.c. ; traceless)) i µ i kj ) jk  γ (Λ − i ) ab V bc  j K ( A T + h.c.) T  δ ( i i + ¯ R i 1 kj η ), the supersymmetry algebra is al-  · R µa  j µ i 2¯ b · e µb ) + η γ ψ γ T (¯ − 2.3 ε γ a K  i 2 ij ( ( 1 8 itself, the shifts must appear as explicit i D  3 1 4 V δ ab 2¯ . M δ depend on the auxiliary fields as − Q δ − + T i δ i i − j j χ ¯   K   a  1 2 j D ) +  – 5 – ) γ i i ξ j ] ) 1 (h.c.; traceless) ij , ( A b −  1] ) + h.c. + Λ ( V µ i a  + h.c. − ( e + h.c.) + a γ R χ µi DT + h.c.) + cov . They can be easily worked out, but are not relevant ] i µ a i i 2 R · , and Λ µi γ ( [ K · 6 ] i  i µ ψ µ ¯  ψ · δ η j i  ψ j [2 γ γ j φ + h.c. i i ¯ ab ¯  η , χ , γ µ γ ) have induced shifts in Λ µ − η ¯ ¯ η η ε γ 1 2 γ − i γ 1 6 ab i ab i ab − 2 ( ba 1 3 1 3 η ij T γ )] = 2.2 γ j A i (¯ − [ j 2 + [ + 2 ¯ T i + δ 2 χ  i b  − + i ¯ j   M ( i 4¯ j 2] χ i  δ χ 1 4 µ D + µ  χ 1 Q µ j 2 µ γ γ ¯  µ − i [1 γ + h.c. + 2Λ  ) γ , δ − i γ ij ] abij  ij i 1 j ) i ¯  j ), without redefining A ε   µi 1 ¯  T )] = ( ξ µi 1 2 i   ab · ψ 2 ¯ ( ( 6 φ ( ) = = = 6¯ DT ) R + h.c. while − + ab Q · i Q i − ab Q ). The supersymmetry algebra becomes ab δ a K 1 γ ab η ( cov 6 Dγ [ γ i γ i µ  µi µi ( , δ γ ε i Λ ¯ ) η  R µ δ ¯   φ φ φ i 1 i i η i j [ 2 1 2 γ 12 1 2 1 8 ¯ ¯   (   i 2 i S  1 2 2 − − − + − δ [ 2) must be modified, introducing new field-dependent structure constants (or = 2 ¯ = 8 ¯ = = = = = | = 2¯ i i 2 j µ µ ij i ab µ , µ µ µ ξ δχ φ δb ab δA δV In the presence of the modified constraints ( δω δT cov δ new terms in thebeen above redefined, commutator. their We commutator emphasize remains that unchanged, because neither The shifts in thedard connections constraints. ( This isdiffeomorphism simply because when one shifts connections within the covariant where tered. One findsof that, SU(2 as withstructure the functions conventional constraints, the superconformal algebra Here we have refrained frompieces giving in the the transformation transformation law rulefor of for our computation.obey It the is new a constraints. simple exercise to verify that all the fields appearing above JHEP03(2018)154 ], is (3.1) (3.2) 13 , one finds . µ , the algebra  µ W . X W j ] ν µνij ] on ψ Q . + XT , δ j ij Q i ε Ω δ ] = 0 1 4 ν  γ Xη −  ij µi ij ab + 2 [ ¯ j ψ µν ¯  XT ij ij ε ¯ ij XT Y ε + h.c. − ij 1 4 ). + = 0 (3.3) ε j . Computing [ µ  ) + h.c. (3.4) j l 1 4 λ
ψ − ¯  i 2.3 X Γ − ¯  ˆ ab X j k − F ] j ( ¯ ν · 1 ¯  X abij  µν ψ jl i ij γ − F 2 ε ε F – 6 – ij ¯  + XT +  ik ab ij j ij = ε 1 2 F ε + 2 ε Ω ) will in general contain also a local gauge transfor- ] j 1 4 + µν ν b + 2 ˆ Ω i γ = 4 2.6 F ) + D µ  j i λ γ − i Γ ab i µ i [ ¯  ˆ is the superconformally covariant field strength associated ( 6 F Ω ¯ DX ψ  ij i ,  ε ij − µν j ε F = ¯ = 2 = = 2¯ + ab − i ˆ µ Xχ F ij ] and is given as Ω 2  ν δX δ b δY µ δW − W D µ W j [ ∂ Ω 6 D , ) with field-dependent parameter = = 2 ˆ F λ j ( µν Γ F gauge δ ] there are two ways of achieving this: one is to use the superconformally invariant At this point we want to obtain the constraints which will allow us to trade the auxiliary We add here that because the vector multiplet contains a gauge field In the above equation, 13 in [ action of the vectorauxiliary multiplet fields coupled via to the themore equations suitable standard of for Weyl motion. getting multiplet TheIn the and second equations this eliminate of procedure, procedure, the motion as directly we outlined without obtain in referring [ the to multiplet the action. of equations of motion by starting from the mation fields of the standard Weyl multiplet for the fields of the vector multiplet. As was explained This multiplet contains 8+8 degreestwo of from freedom the off-shell, complex and 4+4 scalar plus on-shell two (for from the the bosons, of on-shell supersymmetry gauge transformations field). ( or in terms of with the gauge field The supercovariant field strength satisfies the following Bianchi identity with We have written these transformationassociated rules with already the using unconventional constraints the ( new composite connections Now let us introduce ansuperconformal abelian algebra, vector transforming multiplet. under This Q is and an S-supersymmetries off-shell and representation K-boosts of as the 3 The vector multiplet and the multiplet of its equations of motion JHEP03(2018)154 . k . In (3.6) (3.7) (3.8) (3.5)  µ (3.11) (3.10) jk ˜ W 6 DY + k  b = 0. Successive γ  ij ab Y + . . XT ab = 0 , 1 4 −  , with the final constraints ab − } ab = 0 ¯ − . . a XT − ab − 1 2 J − ij ¯ T , XT ˆ ¯ − F ab = 0 XT C 1 2  , T 1 2 ˆ i a ab  F· ij − Γ + 1 8 ε − D ab , ab + ij jk = ab + + ε Y XT + − { 1 2 ab = 0 (3.9) XT ] XT + 2 1 4 XD − XT bc 1 2 j = 0  1 2 G − − ,T – 7 – ab i  a ) as the Bianchi identity − [ = 2. The only covariant combination of fields of F − ab − X D abij ab 2 − 3.8 T Xχ w = ab T ˆ F 2 D F ij F  ˆ  ε F· = −  + b ab i 1 8 a i − = cd D Ω Ω D i + G + 6 ab − D χ = − G T ) l b abcd ), but its real part gives a non-trivial equation, = 0 = = ¯ = ab Γ XD ¯ ε i J k a + C i ij 2 ( 3.3 Γ − ¯  G Y J + jl  b X ε i 2 J ik ε D = 0 is a complex equation. Its imaginary part is automatically satisfied + b + 2 k ) J j Ω k Γ i ¯ χ (   given by µν = 2¯ = 2 G j In our case, the starting point of the analysis is the algebraic equation of motion of ij which has the lowest Weyl weight Γ δ δY ij or equivalently particular, we can write the equation ( with This can be interpreted as the Bianchi identity of a different vector gauge field Here we have defined The equation by the Bianchi identity ( The multiplet of equationsreading: of motion is then applications of Q-supersymmetry give components of the equationQ-supersymmetry. of motion multiplet are obtained by successive applicationY of Weyl weight 2 that we can write down, which is also S-invariant, is equation of motion with the lowest Weyl weight, which is also S-invariant. The other JHEP03(2018)154 from ) and (3.20) (3.17) (3.18) (3.19) (3.16) (3.12) (3.14) 9 µ ) using ) when . ˜ ˜ 3 λ W ( 3.10 . , gauge δ    ab X ˜ j − G ] ν G i ψ + h.c. G· 1 8 + + + . j . ab − Ω ˜ ] bcd F − G·G ν F γ 1 4 H   , , the Bianchi identity ( 1 F· in terms of the field strength of + + h.c. (3.13) = 0 equation of motion, we can µi − 1 8 [ X µν h.c. = 0 abcd j µ + ¯ ¯  ab C ψ G ε X − as ψ T − i ij 1 3! j ¯  = j + iε χ ¯ + h.c. (3.15) = 2 X = F·F Ω  − ij k 1 4 ab k X 6 D j  Ω iε − G·G Ω 1 1 a + ¯ 2 a  1 4 X − γ i k γ j 2 k ] − k X ¯  + Ω ν ¯ Ω j ,T ¯ can be obtained from the definition ( – 8 – 1 2 Ω ij ψ 6 + D Ω 1 2  1 2 ab k µ i ε = + ab 4 γ G ¯ + Ω j j − i + ¯  1 2 − χ Ω ¯ F·F G ] ¯ X X ij i X X ν 1 4 = + . Then one can work out the transformation of = 0, to eliminate γ iε − ˜ λ j ab ¯  + connection for a two form gauge field. log X i log = F 2 ab k a µ a [ R µ + Ω ¯ ψ ˜ ). The result is F W 6 , D ij ¯ XD and ¯ XD δ  XD k  iε 2.6 µν 1  X ¯ ab Ω 2 X G − + T 1 2 − − | ] X ν ) can be easily re-written in the form +  X ˜ | a W ¯ 1 2 µ X = 2 [ D 2 3.18 ∂ = ab + , with as follows and find a constraint on the complex scalar D XD µ = 2 T ] commutator picks up a field dependent gauge transformation ˜ µ should be interpreted as the supercovariant field strength of a new vector gauge W Q D µν ˜ W . Using the above equation one can solve for G , δ µν µ Q G ˜ δ W In the next subsection we will analyze this last constraint in detail, and show that it Finally, using the real and imaginary parts of the Next, we use the equation Γ and µ Let us denote the term in the parentheses as can be used to trade the U(1) 3.1 Analysis ofThe the constraint constraint ( on the scalar fields eliminate The [ it acts on From the above transformationinvariant law, field one strength can work out the form of the superconformally the transformations of the knowledge of the supersymmetrysupersymmetry transformation algebra of ( field W The supersymmetry transformation of Here, JHEP03(2018)154 ¯ X ] (3.21) (3.22) (3.23) (3.24) b . just as we XD a [ bcd gauge field is . In addition, µν j D H µ 2 ), B R )Ω ˜ − symmetry gauge | W + abcd R X G 3.21 ε | i i 1 and 3! 2 − µ − , = as well as the zero-form + i ] by supercovariantizations W k F Ω ( νρ µν h.c. Ω · H a . B G abi − γ ] ) γ µ [ k j a cd Q ¯ ˜ γ Ω Ω ( W G i a ¯  ). ¯ 1 2 3 4 R ab γ ), one can find that the three form [ ij ) ¯ ε G i − X + 2.3 ; the “Higgsed” U(1) − ) gives ] 4 i 3 8 ) for a two-form field 1 G X 16 symmetry field strength in terms of the 3.20 i Ω νρ + + we require F ai R ] i 3.20 − ¯ 3.21 µ h.c. (3.25) ψ ) becomes a Bianchi identity for the three- [ ) + to differ from cd Ω − µν is a gauge field, this is only possible because ¯ F − X W i ab X i F H are the standard field strengths of the gauge ) i µ 3 4 ( ab δB 3.18 1 2 [ – 9 – ] · Q Ω A (Ω ν ( a F + b + γ ˜ ¯ γ ] 3 8 i R W i i D ¯  . µ ¯ η νρ [ Ω i ij X ) constraint in ( ab = i ¯ a ∂ B ε 4 X γ  ] ¯ i µ ψ 1 8 k M [ 3 2 ¯  ( − ∂ bcd ¯ Ω X = 2 ] − − + R b H . To obtain 1 2 γ a fcd = = µ [ k µν := 3 ) to obtain the U(1) ˜ − ) as a constraint that determines the U(1) H D Ω   W G d  ¯ µνρ X X D bcd bcd a 3.20 [ 3.22 should transform as the exterior derivative of a one-form as well H and H H and D as shown below. log abfc 2 ] µ µν ε ν a − abcd abcd | 2 H transforms under supersymmetry as W B respectively. We expect . For the non-supersymmetric case, the solution to the Bianchi identity ε ε W − µ | X ¯ 1 6 1 6 µ µ | X ∂ [ ˜ abc W i ˜ 2 X ∂   W | X S is inert under the one-form transformation of H Q i 4 δ − δ + = 2 H a = ) for and ab µν µ 3.9 ) ¯ F XA W A ( the Bianchi identity of the three form field strength as given in ( X The two-form Now we must solve the Bianchi identity ( We can further use ( R 1. 2i Of course, transformations of involving the gravitino which we find below. as under the zero-formit gauge should transformations transform corresponding under to field supersymmetry. strength From ( would be where fields The dual of this appears in the solved ( what is determined by the above combination. three-form field strength and other covariant objects as We can then interpretfield ( in terms of theit other combines fields. with Because the phase of the complex scalar form field strength Now expanding out the covariant derivative in ( With the above definition, the constraint ( JHEP03(2018)154 , ), a K ), it , the ,Λ 3.25 a (3.26) (3.27) i µ η i 3.20 e ] , ν i  ]. ψ µ Q [ and two U(1) , γ i , δ i ] ¯  S ρ µν δ ¯ ψ X B U(1) R-symmetry, ν , two abelian gauge X γ × µ X i [ i ¯ ψ µ + 2 + h.c. (3.28) ¯ . The latter two will be ψ i i X µ 1 D Ω  X ˜ µ W Λ 3 µν γ ] as well as [ 2 1 is S-invariant. From ( γ i 2 i − Q ¯  ¯  . The fermionic field content is i − . ¯ and µν , δ X i Ω X µν ] Q η B µ µν X δ µ B + ] picks up a field dependent vector νρ G γ W i γ Q ˜ 4 λ + 2 Ω − µ i , a complex scalar , δ [ , with j µ j − ¯ µν i Q ψ  ˜ δ γ µ µν 2) superconformal algebra, this multiplet W µ i | X µν V ¯  γ ¯ B i 2 X 3 2 F ¯
, µ j 2 X 4  λ − ψ µb − i 1 + – 10 – G e ¯  i ab i − ] γ ij ν Ω ] ab M ] ν + = 2 dilaton Weyl multiplet now. It possesses 24+24 ˜ ab M iε . W i νρ Λ − Λ Q γ µ + Λ − N [ 1 4 δ µ F i [ a ∂ µ µ [ ¯ µ ψ respectively: + · e ˜ ¯ and Ω W j + 2 X γ µ ¯ D µ XW 1 2 i 1 ] 2 3 j 2 Λ i ν ψ − ) when it acts on  respectively. j + ψ i 1 µ ¯ and vector-gauge transformation parameterized by − i − X ] ¯ µ  ). The commutator [ [ ˜ ν W ij and Λ ij ˜ (Λ γ are the gauge transformation parameters associated with the gauge W ε i ε , the SU(2)-gauge field µνρ W + Λ V and another SU(2) doublet of spinors whose left and right chiral ¯  ˜ λ µ ˜ λ δ 3.24 i 1 has zero Weyl and chiral weights. From the above transformation of Q H ¯ b , 16 = i X µ δ + h.c. λ µ µ µ ψ [ − µν X , ψ := µi and , and a two-form tensor gauge field , U(1) A . The above analysis also shows that Λ i B D µ W µ  ψ Λ λ and U(1) W a µ 1 2 ˜ ˜ i + 2 2 µνρ W W γ and D i = 2 dilaton Weyl multiplet in components W ), Λ H =  − A given by ( µi and = ¯ = 2 Λ and N µν ψ , i a j µ µ µ i µνρ µ δB W W as H δe δψ the closure of the supersymmetry algebra, both [ the supersymmetry transformation of the three form field strength as given in ( In what follows, we give how the independent fields transform under all the superconfor- Apart from the generators of the SU(2 , one can obtain the explicit form of the fully supercovariant three form field strength , (Λ 3. 2. µνρ µν M ab denoted U(1) mal transformations and the othertiplet: gauge transformations Q-susy, present S-susy, K-boosts, in the localdilatation, dilaton Lorentz U(1) Weyl transformation, mul- SU(2) Λ doublets components are denoted by Ω also has one vector gaugegauge symmetry symmetries corresponding to corresponding transformation to of the gauge fields Let us summarize our resultdegrees for of the freedom.dilatation The gauge independent field bosonic fieldsfields are as follows:as the follows: vierbein the gravitino, whose left and right chiral components are denoted by the SU(2) gauge transformation 4 The B H with The parameters fields also follows that After incorporating the above operations, we obtain JHEP03(2018)154 (4.2) (4.1) j Ω may be i ) (4.3) j µ µ Λ is as given b (Λ ˜ λ − V can similarly i δ ), i Ω  3.4 ) + . A ˜ λ ( j Λ µν  D ˜ i 2 W together with a phase j G i Λ δ 4 ˜ µ λ µ − X ∂ V D − ) + 1 2 i + Λ λ ( Ω µν + 3 2 field strength that appears in µa corresponds to transformation i F W  µν e  is as given in ( δ R 4 λ γ V ) a K i δ + λ µ ¯  − i ) + ] ), X iA η ν ( and Xη Λ + + 2.7 S µ i δ [ ˜ µ W ∂ b Ω + 2 δ ˜ ( λ j , + h.c. + Λ µ µν ) +  λ 1 2 ). The U(1) (h.c.; traceless) ∂ γ + 2 µ W − i K µi ] δ ∂ ¯  + − i ν ψ i ¯ i (Λ ·G – 11 – 3.22 i symmetry. The Weyl fermions Ω µ k ψ X  ¯ η i µ K ψ γ [ µ 1 2 ab R δ j + corresponding to local Lorentz transformation, K- ] commutator acts on this multiplet as γ ij V γ η ] i symmetry are composite gauge fields which are de-  Q j ν ¯  − µ + h.c. + i ab k 4 i ). Here ¯ ˜ ) + , δ R are as given in ( A µ + h.c. + X j W µ Λ + 2¯ ) and ( ε Ω Q ω − ( i Q ψ η j µ δ X i − δ 1 4 6 j D ψ Ω ¯  2.3 3.28 M  µ X i j [ and δ µ ¯ ¯  6 k ) − D X γ ˜ + 2 µ ¯ i i i µ W A ·F µ ij X i ¯  , and  V φ 1 2 ] γ ) + Λ 1 ij γ µ iε k j ν i K ξ , i ε − ∂ ¯  2 ). The [ ij ( + a ψ µ i 1 )  − ] µ f X ,Λ [ = − − 1 4 Ω ν + 2 ε 1 4 and vector-gauge transformation respectively. γ , + Λ D cov i j ¯ i 3.23 ( ab X j  W j + ¯  ab Ω δ γ ˜ i µ − W Ω i Q µ ¯ µ ) is given in terms of the three-form field strength and other covariant − X Λ δ µ D ab M γ ω is as given in ( µi + (Λ µ i µ i µ γ [ Λ X i ¯  i φ 2.3 ∂ φ )] = µ ¯  i 1 4 j 2 ij 2 6 ¯ W  Ω DX  ij i = 2 dilaton Weyl multiplet in superspace  , U(1) 1 2 1 2  ε iε ( + 2 − + = 2 dilaton Weyl multiplet was already constructed in superspace by M¨uller W Q = = ¯ = 2 = = = = 2¯ N , δ i j µ µ µ N ) i ) and Λ µν Ω 1 ˜ µ δX δb  W δ V ( δ δW ], but with the addition of a compensating tensor multiplet that accomplishes the δB A few comments about this multiplet are in order. As with the dilaton Weyl multi- δ Q 1 3.15 δ [ eliminated by a choice of K-gauge. 5 The The 4d in [ plet in five dimensions,symmetry. it However, possesses here a that scalarthat field field appears can that in be can a usedbe be complex to used scalar used gauge-fix to to the gauge-fix gauge-fix U(1) S-supersymmetry. the Weyl As with the standard Weyl multiplet, where the parameters in ( under U(1) The gauge fields boosts, S-supersymmetry and U(1) termined from the constraintsthe ( constraint ( objects as shown in ( where JHEP03(2018)154 . , W =0 G i θ | (5.3) (5.4) (5.5) (5.1) (5.2) − + = d ab F W F , , . = 4 ˜ . W = 2 conformal . − = 0 = 0 ab ¯ ¯ X b T N X ˙ βj − ˙ ab αi − ab ) ˙ should be a reduced αi ) G W iW , i X G i + , X + ]. ), with = 0, one can introduce a + 8 , X αj X − X F . ( + + F , ab X ab ∇ X αj ( 3.12 ¯ ij ij ij W X ∇ X ε ε iW ∇ ˙ ij kl ij β ] and [ + ˙ ε ˙ ε αα α + ¯ ˙ ∇ for a new one-form ) β ¯ 21 . ˙ ˙ b X i ¯ jl α αα ˙ X γ 4 β )  ε ˜ ˙ ˙ ( β b W − α 4 ab ˙ i − ik α . γ ¯ ε − ∇ W ¯ ˙ ∇ = = β − ab = d ˙ ¯ , ˙ β ]), whose component formulation can be = α = ( = X b ˙ α ) 4 ˙ W G βj X b ˙ ) αi ˙ i X ¯ βj ab ˙ ˙ αi X αi G ab = 2 γ ij ˙ αi γ ( G F ( − ∇ ab 1 8 F – 12 – + 2 i 8 follow by applying superspace spinor derivatives. G i , X + ˙ D β + ¯ , , X − ˙ X , α X = 0 imposes that the superfield ¯ − ¯ ab , ¯ ˙ X ∇ αβ X αj ˙ αβ ˙ F replaced by and ¯ β α , F ˙ ¯ αj X -symmetry group, along with the additional component ∇ ∇ ˙ = 0 α i ¯ ∇ ¯ ∇ , ) ¯ ij ij R ∇ X X χ . The result is a minimal 32 + 32 supergravity multiplet. ˙ αβ ε α ε X ¯ ab αβ ) ij ij ˙ R X α α ) ˙ γ ε ε αi ) αβ α ab ij ˙ ( b can now be combined into a complex super two-form α  ) ab = 2 conformal supergravity. Note that the 5d dilaton Weyl ¯ γ ε ∇ γ 4 1 αβ α b 8 with γ ( )  G γ ( b 1 8 αβ − − 4 ( N i G 8 i γ i − − = = = 2( − − and to SO(2) = = = ( ab = = = 4 ) αi b F R ab ) αi βj G implies that (at least locally) ab αi b ) i ], G F αi b αi βj G i 1 G F G ] (we follow the conventions in [ αi βj G i + F G + 20 + F ( , F ( F − ab ( W Following [ Recall in superspace that a vector multiplet is described by a closed two-form tensor This is the superspaceThe component version constraints of on the component equation ( Encoded in the anti-self-dual part of the last equation is a constraint on the super-Weyl The closure of chiral, meaning that it obeys the two constraints If the superfield obeyssecond in closed addition two-form the on-shell constraint The superspace Bianchi identity d exactly identified with multiplet has also been constructed in superspace, see [ whose tangent space components are given by The further breakdown offields, the seems to havesuperspace prevented construction, its with appreciation. thedilaton additional Weyl Here tensor multiplet. we multiplet Aside givesuperspace removed, from a that [ notational modified leads changes, version to we of the also employ his breaking of SU(2) JHEP03(2018)154 = θ (5.9) (5.6) (5.7) (5.8c) (5.10) (5.11) (5.8a) (5.8b) to H . . ¯ X ]. -supersymmetry . , are the zero-form 22 X Q ) ˙  β ˜ λ in superspace, that ¯ ¯ γ X X G ∧ G ) .) The dimension-3/2 a . B 1 4 X ˜ ˙ λ αk γ ( . ( and ¯ ˙ ∇ + H ). The j βi k ∧ G ξ B δ λ ı ¯ X ξ ∇ ˜ λ ). The supercovariant field ı 2 ˙ α k β 1 4 + ) accompanied by the gauge 5.8c ˙ . α − − ∇ as ) ˙ αi ˙ ˜ α G F ∧ F 3.24 W = ab G α ξ ,  , 1 4 ξ ) ı γ a ı Λ = d αi cov ξ G ˙ βj γ ∧ F − δ = ˜ = λ ,  ( = ( λ ˜ γk 1 4 1 2 W ∧ H ˜ ˙ 1 4 αi W H d and similarly for , 1 4 ˜ ). are given by − W ∧ ab = (0 − connection has been exchanged for a two- ˜ ) 1 4 cov ξ W F − H . (By redefining the Λ transformation, one H δ ξ R A X 3.26 ı λ ˜ ∧ W F − d ξ , , W ξ ) reduces to ( 1 4 − λ ı + dΛ transforms as W − and = 0 projection of ( – 13 – ¯ − = , X ∇ 5.6 B = 0 B θ ) and cov ˜ ξ F d λ a ¯ ˜ δ ξ W ∧ ξ X ı ı W ∧ G ˙ ¯ 1 4 βj W X − 1 4 X = dΛ + d = ( ˙ γk , + W ∧ βi B B H ξ δ W 1 4 H − ı ∇ X ∇ ξ W ( ı , so that ( β = − ξ is identified by projecting the super-three-form cov ξ ı α  δ ) =0 = = d = − θ | W ∧ F ab B µνρ B abcd γ = 1 4 ε γk βj,a H µνρ λ cov ξ cov ξ + H δ = ( = descends from a covariant Lie derivative of δ H = 0 exactly reproduces ( B θ ) corresponds to the B = abc abαi gauge transformation as d H = d , H = d λ 3.27 µνρ ( H θ 2 H = 0 abc H γβα H The connection between superspace and component results is straightforward. The At the component level, the Higgsed U(1) In this section, we employ superspace conventions for differential forms, see e.g. [ = 0, i.e. is gauge invariant which implies that 2 θ Projecting to This can be rewritten using transformations This leads to component three-form d strength transformation of is, the usual Lie derivative with parameter could write the and dimension-2 tangent space components of The remaining pieces are given by where Λ is thegauge one-form transformation gauge parameters transformation of parameter and form. This two-form can befield identified strength, directly in superspace by constructing its three-form H JHEP03(2018)154 ] 2 ]. 2 4 R (1987) 329 B 282 ], that all curvature squared 8 – 6 Nucl. Phys. , ] using superspace. ]. We leave this for future work. 17 , 24 1 ]. In this paper we have filled the gap by = 2 dilaton Weyl formulation. These higher N 13 – 14 – , 12 , 9 ] or directly in the four dimensional context. The former ), which permits any use, distribution and reproduction in 7 ]. This gives all three possible curvature squared invariants supergravity in superspace 5 = 2 N CC-BY 4.0 This article is distributed under the terms of the Creative Commons ]. Furthermore, it would be interesting to see the relevance of the dilaton ] by dimensionally reducing it on a circle using the procedure outlined Minimal 23 ]. 13 , , 9 10 = 2 supergravity using the standard Weyl multiplet. It would be interesting to SPIRE N ]. This is currently a work in progress. IN [ Using the standard Weyl multiplet in four dimensions, a higher derivative invariant In principle, we can also obtain this multiplet from the five dimensional dilaton Weyl M. Muller, 10 [1] References 1620742 and the Mitchell Institute forUniversity. Fundamental Physics and Astronomy at Texas A&M Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. entropy of five dimensional black holes given in [ Acknowledgments The work of D.B. is partially supported by NSF under grants PHY-1521099 and PHY- procedure would havebetween the the important curvature squared advantage invariantsshown of in in five establishing [ and four useful dimensions,Weyl explicit multiplet similar and relations to what the was newnon-BPS and invariants small in black the holes, context and possibly of use black it hole to entropy, understand including the for puzzles involving the in 4d show, in complete analogyinvariants to can also the be fivederivative obtained dimensional invariants in case could the [ in 4d principlethe be 5d dilaton obtained Weyl either invariants by [ dimensional reduction from based on the logarithmA of particular a linear conformal combinationgives primary the of supersymmetrization chiral this of superfield with theinvariant was was Gauss-Bonnet the constructed constructed term. well-known in in A [ Weyl separate [ squared supersymmetric invariant [ explicitly constructing the dilaton Weylits multiplet equivalence in to four the dimensions. multiplets We constructed have in also [ shown multiplet [ in [ In this paper, wefrom constructed the the standard dilaton Weyl Weyl multipletU(1) multiplet in symmetries in its and four auxiliary dimensions. field oneWeyl content. multiplet. vector-gauge This It transformation The differs contains which existence twoknown of are additional two in absent different in five versions the of and the standard six Weyl multiplet dimensions was [ already 6 Conclusion and future directions JHEP03(2018)154 ] , = 2 ]. ] , B D 53 N (2012) ] SPIRE 03 (2001) 667] IN Phys. 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