JHEP11(2016)162 Springer April 11, 2016 : November 21, 2016 November 28, 2016 : : 0) theory compactified Received , , 10.1007/JHEP11(2016)162 Accepted Published doi: a Published for SISSA by [email protected] and Savdeep Sethi , b . 3 Daniel Robbins 1512.03999 a The Authors. Supersymmetric Effective Theories, D-, Flux compactifications, c

Studying a quantum field theory involves a choice of space-time manifold and , [email protected] George P. and CynthiaTexas W. A&M Mitchell University, Institute forCollege Fundamental Station, Physics TX and 77843-4242, Astronomy, U.S.A. E-mail: [email protected] Enrico Fermi Institute, University ofChicago, IL Chicago, 60637, U.S.A. b a Open Access Article funded by SCOAP Keywords: M-Theory ArXiv ePrint: space-time metric, global symmetry backgroundfor and this a landscape, choice we of study tensorto the fluxes. two supersymmetric dimensions. six-dimensional As Different (2 evidence choicestheories, of metric which and can flux preserve give differing rise to amounts distinct of two-dimensional . Abstract: a choice of backgroundmore for choices are any possible globalstring when symmetries theory, the specifying of additional the theWe propose data background. theory. the corresponds existence In We to of the argue a a landscape context that choice of of many of field branes theory in backgrounds, tensor characterized fluxes. by the Travis Maxfield, A landscape of field theories JHEP11(2016)162 9 space-time dimensions. Can D 7 11 – 1 – 19 15 21 4 12 6 4 22 1 ]. That spheres can preserve supersymmetry is surprising, and counter to the 1 There has been much progress on both these questions in recent years, starting with The second question involves a special class of backgrounds, distinguished because 3.2 Supersymmetric3.3 cycles World-volume supersymmetry 3.4 Closure and equations of motion 2.1 M-theory on2.2 a CY 4-fold Heterotic flux2.3 vacua Constructing the dual heterotic 3.1 Symmetries of the M5- action existence of a landscape ofon field theories backgrounds obtained that from involve a a parent choice theory of by metric compactification andsupersymmetric a sphere choice compactifications of and flux. functions the [ use of localization to compute partition they give rise toa the Kaluza-Klein degrees reduction. ofbroader freedom and Included of can in a includeprovides lower-dimensional this non-compact a quantum spaces class natural field like are association a theoryderived compact Taub-NUT between by in manifold. geometry compactifying spaces, the and This but parent the reduction the theory. lower-dimensional class field We will is theory see that suggests the ing” implies backgrounds thatstring preserve theory some suggests degree a ofground” more supersymmetry. usually general refers We to class will a ofglobal see choice symmetries interesting of that of backgrounds. space-time the metric field Thegeneral and theory. term notion a Again choice of “back- we of background. will background see for that any string theory suggests a more 1 Introduction This project is concernedclassic: with imagine two a supersymmetric closelyone quantum related classify field questions. the theory in interesting The rigid first backgrounds for question this is theory? quite Usually the term “interest- B Theta expansions A Spinors and Gamma matrices 3 Coupling an M5-brane to 11-dimensional supergravity Contents 1 Introduction 2 Flux backgrounds JHEP11(2016)162 symmetry. It is . We would like R µν T ]. From this perspec- 2 = 2 theories obtained by 0) theory. The theory has , D , we answer the question of how 3 = 2 has a long history, including = 6 (2 ]. 4 D , D 3 , we describe a class of M-theory flux 2 ]. – 2 – 10 – 7 0) theory to , Lorentz symmetry and the Spin(5) ], and more recently a prominent role in the program L 6 , 5 1) , 0) theory by studying an M5-brane coupled to 11-dimensional Lorentz and R-symmetry group. Kaluza-Klein reduction gives , = 2 field theory. The preserved supersymmetry on the string, R D 0) superconformal symmetry. In section Spin(5) , × The metric couples to the field theory stress-energy tensor L = 2 sigma model geometry that can emerge from wrapped M5-branes in the . 1) , µν D g From the perspective of field theory, we understand how to introduce a background The paper is organized as follows: in section Since this example is derived from M-theory, it falls into the on-shell category. Despite Both closely related questions are framed entirely within quantum field theory. How- Backgrounds that do not admit killing spinors, but still preserve some supersymmetry, the field theory? Unlikeare the tensors under background both metric the case,natural Spin(5 to the expect flux the couples operatorsusing to to the operators full be which (2 part ofthe the flux stress-tensor couples supermultiplet, to constructed the (2 basic evidence for the fieldkind theory of landscape. Itpresence will of also flux. help set our expectations for the metric a similar understanding for background fluxes. Namely, which operators are activated in applications to entropy [ of associating geometry to field theory [ vacua. In certain cases,duality between there certain exist M-theory flux heterotic backgrounds dual and theories, heterotic string which vacua we provides the also describe. This this restriction, we willincluding still the see flux evidence degreesand of for off-shell freedom. a backgrounds landscape We for will of introduction. theories comment on The with the reduction maximal relation of supersymmetry between the at on-shell (2 the close of this a Spin(5 a string supporting a along with the spectrumical of type excitations, of will thesupergravity depend fluxes. Calabi-Yau on space, the the following choice data: of metric the for topolog- the space, and the choice of enlarged the space of supersymmetric string vacua [ ever, it is goingexamples to be that useful follow for fromof us M-theory. a to Calabi-Yau support We 4-fold. will our The claim consider M5-brane by an studying supports M5-brane a the wrapping particular a class of 4-cycle can be understood asiliary off-shell supergravity fields configurations are for not whichbeautiful required the structures to supergravity that aux- satisfy have theirof been equations supersymmetric uncovered of and so motion. far non-supersymmetricmind are backgrounds. We is will part The of argue quite generalization a that analogous we much the have to larger in landscape the discovery of flux vacua in string theory, which greatly explanation for thefield preserved theory supersymmetry to comes ative, from non-dynamical backgrounds coupling admitting background killing the supergravity spinorsexamples theory supersymmetric are [ on-shell of supergravity this configurations.wrapping type Many submanifolds can of be a ten found or in eleven-dimensional metric. string theory and M-theory by studying branes usual intuition that a supersymmetric background requires a killing spinor. The nicest JHEP11(2016)162 0) , = 4 2) or , D ]. The 15 = 6 to D = 6 is the (2 ]. See also the recent D 13 = 2 theories with (0 D ]. 17 = 2, we expect the landscape en- D ]. 18 = 2, and so on. For example, potentially ]. This formalism has been used to describe = 6 to 16 D – 3 – D = 4 to = 4 work of [ ] and for Euclidean M5-branes in [ D D 12 , ]. We would like to be able to compute such partition functions 11 1 = 5 supersymmetric Yang-Mills [ = 6 to D D 0) theory perspective is an outstanding question. , = 6 theory provides a tensor generalization of the SYM vacuum structure. In D ] which also discusses some of the aspects for supersymmetric brane configurations = 6 (2 14 classes of vacuum equations; forsolution example, spaces Hitchin of equations the in vacuum variousfor equations contexts. this can work The be was to very find interesting.the vacuum equations A for prime M5-branes motivation in fluxM-theory backgrounds examples because that possess heterotic dual descriptions, we can build the flux solutions toof explore. compactifications with It differing is amountsor not of hard lower supersymmetry to dimensions, from imagine from generalizing flux the generalizations Supersymmetric YM theories studied on rigid backgrounds give rise to interesting quantity. Whether it, or aD higher-dimensional generalization, is computable from the While we focus onlargement reductions to be from much broader. There are already many more on-shell examples from The computation of supersymmetricvance in partition recent years functions [ hason been more a general backgrounds significantpossible. that ad- For involve example, flux. it ismore There world-sheet not are supersymmetry. hard hints For to such that construct models, many this the elliptic might genus be is a computable Our initial comparison ofa on-shell subset fluxes of with configurations off-shellappear that auxiliary to might be fields be related suggests to calledfluxes. that on-shell off-shell The flux supergravity backgrounds extent with configurations to auxiliarybe which fields understood this identified is more with precisely. true is unclear, but it is something that needs to supersymmetry. This is truethe both case for of fieldclosest half-maximal theory to supersymmetry and supergravity. an nicelyconformal studied This appropriate supergravity is in formulated off-shell in unlike recent supergravity [ backgrounds work formalism for [ in We study the on-shellexpect case an with off-shell fluxes extensionwhat that to degree. emerges exist By from to thiscurrently M-theory. some we do mean degree. not However, the admit we following: Interestingly, fully it theories off-shell is with unclear maximal formalisms, supersymmetry to with manifest maximal There are a couple of observations and future directions worth mentioning in closing: • • • • volume field theory, theaspects preserved of world-volume , the and resultingfluxes we equations to branes examine of appears some motion. inwork [ [ Some of thein early flux work backgrounds. on the coupling of supergravity. We describe the fixing of kappa symmetry and the emergence of a world- JHEP11(2016)162 ]. 23 (2.1) (2.2) – 19 . At a basic ] int 4 27 G and spinor indices by . ) . ) M,N,... y ( Z with metric, 8 , . 8 8 2 M denote a Calabi-Yau 4-fold. The 2 M ds 8 ( ) 4 ηdx y × M ( M 1 H w , 2 − ∈ e R ) 8 + . Let 2 – 4 – 4 M ( ]. 1 ηdx p ) y is denoted by 25 ( , − 1 w , A, B, . . . 2  24 2 e R π int 4 ]. What we are proposing is a field theory version of the = ]. There are many other cases one might consider but we 2 G 3 2 26  ds ) is the warp factor which depends on the internal coordinates collectively denoted y ( . We use the term “bosonic coordinates” here because we will later meet superspace perspective, this construction suggests webackgrounds should to enlarge include our these notion special of classes “interesting” of non-supersymmetric backgrounds. ever, one canbut also does consider not a preserve supersymmetry.namely choice If that that a higher truncation solves momentum toof the couplings supergravity flux can is SUGRA gives be permissible, a equations ignored, backgroundsupersymmetry of then which at is motion turning a special on scale from this the below kind bulk the perspective, Kaluza-Klein since scale. it breaks From a purely field theory supergravity classification. There areexample, fewer many constraints of the on key the tadpolerelaxed. constraints field found We theory in might therefore string side. reasonably theory expect and For in M-theory interesting field are new theory. classes of backgrounds Most of our discussion involves choices of flux that preserve supersymmetry. How- capture both heteroticThis torsion would as be wellnon-geometric an as backgrounds heterotic alternative [ to non-geometric physics doubled [ fieldThe theory classification of as supersymmetric anstudy solutions of approach of supergravity, G-structures to including [ defining flux, is the string from the wrapped M5-brane and its target space geometry is expected to w . The Minkowski metric for In addition to the manifold, we choose an internal space 4-form flux Let us label 10 or 11-dimensional bosonic coordinates by • • y level, the cohomology class of the flux satisfies the quantization condition [ However, we stress that supergravity doesof not supergravity, we see can this smoothly quantization condition. turn on At and the off level fluxes. As we turn off any flux, the warp Here by flux degrees of freedom. a, b, . . . coordinates which willbackground denote of by interest is a warped product of 2.1 M-theory onOur a motivation CY for 4-fold consideringcomes more from general string field theory. theory Webased will backgrounds begin on that by Calabi-Yau involve reviewing 4-folds fluxes thewill [ structure focus of on M-theory flux this vacua case because it provides a great deal of intuition about how to include 2 Flux backgrounds JHEP11(2016)162 ], 3 (2.9) (2.8) (2.3) (2.4) (2.5) (2.6) (2.7) ) and the y ( f ) and consider . ]. This leads to ) 2.7 28 is the Hodge dual ) i 8 y ˆ ? − ) relates y ( 2.4 8 . δ ) 8 n . =1 i X M  2 ( π χ int 4 , 1 G 24 = 0. If this is not the case, we have 4 − explicit M2-branes [ ∧ G , 8 , ) momentarily via equations of motion. = ) , int 3 y y ∧ X ( int ( to be non-compact. This is the usual no- M2 G 2 int 4 4 M2 w f G 8 π N 3 G G ) vol N 2 1 e 8 1 2 y ˆ ? M − ( − + 4 + − f  = – 5 – int π 8 int = = ) = 2 ˆ ? ) to go beyond the restriction imposed by super- int 4 G G y 4 3 ( G ∧ C ∧ f 2.7 ) = . We will fix int becomes Ricci flat. The last ingredient we need is a 3- 1 w , π d ? G int 3 2 G 2 8 defines the potential. The purely space-time components − G R 1 2 ) is again defined with respect to the unwarped Calabi-Yau ) vanishes only if e 3 8 M . This is not a terrible restriction from the perspective of − ˆ 8 2.7 , which requires M ∆( 2.7 dC ( 8 Z 8 M = 1 2 ˆ ? M 4 G ) = w 3 − , the integrability condition for this equation requires vanishing of the e 8 ˆ ∆( M below, and the second is to include is the volume form of 8 ) tends to zero and 3 y X ( w There are two ways to modify ( The right hand side of ( Supersymmetry, along with the supergravity equation of motion, For compact total membrane charge: denoted the modified warp factor equation, only non-compact spaces wrapped branes. The fieldthe theory on submanifold the on brane which reallythan it only the is sees entire wrapped. the compact local space. It normal bundle effectively to only seesgravity. a The local first geometry, is rather to include 8 derivative couplings in the M-theory effective action [ metric. net membrane charge on go result prohibiting flux onwork a strictly compact space within in the 11-dimensional confines supergravity. of If supergravity we then want we to should impose ( and also fixes the warp factor in terms of the flux: The Laplacian appearing in ( where hatted quantities refer torestricted the unwarped to Calabi-Yau the metric and Calabi-Yauwarp directions. factor In addition, equation ( requires that form potential, where of the potential are given by, where vol factor JHEP11(2016)162 = 2 = 2 (2.11) (2.10) N N 3, which ) requires K × 2.2 with a holon- 3 8 K . In the compact M 8 = = 4. To summarize 8 M N M , 2) , 2) flux, which is also primitive (2 . This flux can also be viewed , 8 G M + ]: . 4  4) , . (0 1 or 2 supersymmetries in 3 dimensions. ⊕ ) and quantization condition ( = 0 Re(Ω) ] , 3 2 2) 2.5 int , 30 , + = 0 G – 6 – (2 J 29 ∧ ]. A nice case of this type is = 1. ⊕ N ∧ 4 J from the case of a non-compact 0) J N , 8  of the Calabi-Yau [ (4 1 c M J = = 2 to int N = 2 supersymmetry requires a (2 G = 1 supersymmetry by a choice of flux that is neither primitive N N is, typically, an integral class with Hodge type: . In the non-supersymmetric case, there are equally good reasons to int p ` G a constant and Ω the holomorphic top form of 2) but rather of the special form [ 1 , c Aside from these generic possibilities, we could also consider a space Preserving the full However, for supersymmetric backgrounds, there is good reason to expect a solution The Calabi-Yau background preserves 4 real supersymmetries, which we call There are many generalizationswrapped of M5-branes, we the only fluxcompactification need of story the to but heterotic consider or forwe one type will our other I use string purpose class to the of oferotic 3 language flux flux dimensions understanding of vacua with vacua. has the flux. not Consider heterotic For yet simplicity, a string. been described, The though general there form has of been 3-dimensional considerable het- progress this brief review: 11-dimensionalYau supergravity flux admits vacua a whichFurther, large there preserve set is either of evidence non-compactwhich for Calabi- preserve a some large degree of set supersymmetry. of compact2.2 Calabi-Yau flux vacua Heterotic in flux M-theory, vacua omy group properly containedsupersymmetry. in The SU(4) amount of sodepend supersymmetry on the preserved the space particular by choice the itself ofwill background flux preserves play [ would a more role then in than our subsequent discussion. This example preserves with as spontaneously breaking with respect to the K¨ahlerform One can also preserve nor (2 Planck length expect that a solution doessupersymmetric not fluxes, exist. we really This should point restrictto will to two be non-compact derivative elaborated spaces, supergravity where on is a elsewhere. self-consistent. truncation For non- as spontaneous breaking fordifferentiate sufficiently the large case Calabi-Yau of volume. compact case, Here neither we supersymmetric again fluxessupergravity need equations nor to of motion. non-supersymmetric One fluxes must take give into account solutions higher derivative to corrections. to the the M-theory equations of motion in a perturbative expansion in the 11-dimensional in 3 dimensions. Thethat self-duality condition a ( generic A generic flux breaks all the background supersymmetries in a manner that can be viewed JHEP11(2016)162 (2.12) (2.13) as well that ad- y -flux and 3 surface. 3 8 H K and a 2-form M take the form I 7 φ , which take the 7 M M . ) y ( 4 . 8 ) is trivial. In these cases, 2 M M ds 2.12 . + 3 to which we turn momentarily. 2 From the heterotic perspective, the 3 heterotic torsional backgrounds is a ) i . ∗H T 3 surface and the heterotic string on 3 0 , A = K 16 + 3 Γ 3 surface has been studied quite exten- i AdS H ⊕ K dθ 3 ( , ) 3 3-fibered spaces with metrics: y – 7 – ( i , is associated to the ten-dimensional gauge-fields. 4 K u 2 0 , of 11-dimensional supergravity on the lattice of e , M 3 dB 16 3 C =1 i X = 3 + 2 H , which admits an elliptic fibration, along with a choice of 8 while Γ , including flux. isometry extending to a symmetry of both the 8 3 ηdx 1 M T 2) supersymmetry. If an M-theory dual exists for this case, it S M = 3 surface. This Kaluza-Klein reduction gives rise to 22 gauge- , ). There is also, typically, a that varies with 3 K 2 H , θ 2 ds = 1 space-time supersymmetry is preserved then the two-dimensional determine the structure of the 3-torus fibration, where the torus itself , θ i 1 N θ ]. This duality follows from the more basic 7-dimensional duality relating 3 surface to the heterotic string on with the A 1 K with self-dual 3-form field strength: 33 S , 2 ]. Later in our discussion, we will extend those analyses to the case of more 4 × B is associated to )[ 35 3-fibration to the heterotic string on a torsional space with a metric of the 6 3 , ]. Fortunately, we can get by with an understanding of certain special cases that , 3 K M 34 32 2.12 -flux. We are being fairly schematic about the form of these metrics because the , 3 = For the moment, we want to gain some intuition about wrapped M5-branes and flux To construct the dual heterotic string, we wrap an M5-brane on the First there is a special case that deserves mention. Some spaces What is important for us is a duality relating M-theory on a space Finding compact examples of Minkowski or 31 H 7 . The generic space-time gauge fields visible on the heterotic side arise in M-theory , the bosonic light modes supported on an M5-brane consist of 5 scalars 3 p so a more qualitative discussion` will suffice. Including onlygauge-field the lowest order interactions in lattice Γ The reduction ofsively the [ M5-brane actiongeneral on 4-submanifolds of a Let us revisit theT duality between M-theory onfrom a reducing the 3-formintegral potential, 2-forms of the fields associated to the cohomology lattice Γ world-sheet theory has (0 corresponds to a geometry, flux compatible with taking an F-theory limit. 2.3 Constructing the dual heterotic string M the dilaton. Essentially,we one can of decompactify the thestring. circle circle If to fibrations minimal find in a ( 4-dimensional compactification of the heterotic detailed structure is not important for us. mits a form ( M-theory on a The connections has coordinates ( as include dual descriptions of M-theory on difficult task. The knownform Minkowski of examples a correspond 3-torus to fibration spaces, over a base in [ JHEP11(2016)162 ] 36 (2.15) (2.16) (2.14) couples i ; see [ s have mass φ g I combination φ 2 B . If we fermionize i − A 2 found on M5-branes F − 3 i 0) supersymmetry in 6 8) extended world-sheet on the lattice of 2-forms C 3 depends on two basic , , dφ 2 K B and 2 . B 2 / 0) theory, the scalars 3 , 0) degrees of freedom in terms of  , 4 4 p . . 3, the resulting chiral boson ` 3 V I 3) chiral bosons corresponding to the 3 C K  , ∂φ − I ∼ 2 2 s 0) theory, reduce , dB x ∂φ – 8 – 6 = d 3 surface. From these two parameters, one can and the 7-dimensional string coupling 2-cycle of 3 s , g K Z ` . th 4 6 p H i 3 ` V T ∼ 2 s of the ` 4 V 3 would not naturally give the usual string non-linear sigma model without does not appear in the terms that involve 2 derivatives in the equations of K p and the volume ` p 3 surface. The result is 22 chiral bosons. These bosons assemble themselves into 0) chiral bosons, the result is a collection of chiral fermions coupled to a space-time ` ) on , K The second comment concerns parameters. M-theory on We want to make two observations about this well-known duality. The first con- The heterotic world-sheet can be presented in many ways. To study interacting world- 0) chiral bosons of definite chirality, and (3 2.16 , 1) supersymmetry in order to couple the theory to world-sheet supergravity. For the , The scale motion, but onlyof appears ( in terms involving higher derivative interactions. Reduction In the conventional field theory normalizationdimension for 2 the (2 with kinetic terms proportional to, direct motivation for considering flux backgrounds. scales: determine the heterotic stringfor length a review: to the corresponding space-time gauge-fieldthe via (16 the combination gauge-bundle. The world-sheetif theory we is do an not interactingheavily include non-linear studied flux sigma class degrees model. of of heterotic Indeed freedom, world-sheets we from would the be wrapped unable M5-brane! to construct This the is most one found on D-branes, thetakes the gauge-invariant form, combination of After Kaluza-Klein reduction on the (0 case of toroidal compactification, thesupersymmetry. world-sheet theory enjoys (0 cerns the coupling to space-time gauge fields. In analogy to the sheets, it isleft-moving usually world-sheet convenient fermions to couplednon-chiral describe scalars to and the the their space-timeget (16 right-moving gauge fermion space superpartners, bundle, geometry which along andfreedom couple with flux. with to these 8 the Supersymmetry conventions. tar- acts The fundamental only heterotic on string the only right requires moving minimal degrees of dimensions. For a singlethe M5-brane, dual the heterotic theory string emerge isof from free. the the (2 To see(16 how the massless modesnon-chiral of scalars parametrizing a There are also 16 chiral fermions. Together these fields enjoy (2 JHEP11(2016)162 , 32 | (3.1) 11 which (2.17) 8 M has mass appearing M 2 i , for reasons B A . 3.2 I φ 3 p ]. as π` Z 37 , √ 19 = 2 I . Throughout the following 2, respectively. ˆ φ / 2 1 , −  flux vanishes when pulled-back to . This is a caveat in comparing the ) , I σ I 4 , we can write φ ( ˆ φ 4 a G ∂ 1 and M V Θ I − , ˆ √ φ ) 3 can preserve differing amounts of super- σ , into these fields. We will not perform this = x∂ K 3. With different choices of metric and flux, ( p 2 ` I M K d × ˜ φ – 9 – 3 at all. In a standard field theory reduction, one X × Z 4 K 3 0 V K 1 πα ) = 4 = σ ( 8 = 2 string with a torsional target space geometry. For a ). Viewed as coordinates on the supermanifold A σ Z M D ( ⇒ 2 B 0) theory with expectations from a dual heterotic description. , I 3 discussed earlier, with the choice of connections K ∂φ . Choosing coordinates on I does not involve 32 | I ∂φ 11 φ over 3 gives a physical heterotic string for this background. The two choices of 3 K 8 T Z via → M x p M 2 1. This differs from the standard field theory dimensions, which would result 0 ` | d 6 − are coordinates on Σ. In addition to the embedding, the world-volume supports Z 3 of :Σ i ) determined by the M-theory flux. The fundamental heterotic string is described σ K Z Before proceeding, we should make a few remarks regarding our assumptions and We can therefore conclude that in this special case of M-theory/heterotic dual pairs, Let us consider the case of 2.12 and Θ are naturally assigned mass dimension our reliance on thediscussion, example we scenarios will described onlyKilling assume in spinor that section and the thatdescribed 11-dimensional the in M5-brane background is that admits Lorentzian. section, at we least Beginning will in one section assume that the where a two-form gauge potential X dimension from absorbing powers of therenormalization, Planck however, length, for the discussion of this section. The world-volume of the M5-braneifold, is described by its embedding into a target superman- this motivation, we now turnin to a general the background question of of 11-dimensional describing supergravity. the couplings on3 an M5-brane Coupling an M5-brane to 11-dimensional supergravity symmetry. Different choices ofmodes. metric We therefore also already give findactivating different a the large numbers flux class of of degree space-time interactingcase; of two-dimensional massless we theories freedom. can from wrap There ansolves is M5-brane the no or supergravity reason multiple equations M5-branes at of on all motion, any to 4-submanifold subject restrict of to to mild this conditions special on the flux. With by a world-sheet theoryeither with this geometry aswrapping a submanifold target correspond space. to Wrapping dual an heterotic M5-brane descriptions on [ the wrapped M5-brane gives a given flux, different choices of metric on natural reduction of the (2 we find a dualthe heterotic fibration description of consistingin of ( a torsional geometry. The geometry is The redefinition of would have arrived at dimensionless scalars introducing JHEP11(2016)162 . (3.2) (3.3) (3.4) (3.5) before . One 2 a, b, . . . i σ B indices which ! ] — involves the ij H 39 , ij A, B, . . . e 38 . H g 2 and spinor indices det 4 , − N √ M , where we describe some specific η . + , . N , etc. For more details regarding our  3 a A M,N,..., , B 3.2 k . ij C 3 E ∗ e ,M H ijk a∂ i C Z , ) j M ∗ a + ∂ A i 5 denote the tangent directions. Z − ijk ∂ ∗H ij jk E . 2 ( ∧ g g 6 ) whose equation of motion is responsible for H B k k −  3 σ ,... − ) Z – 10 – v v dB ( 1 p j H a , p ∂ 1 2 = = = det π` : A 3 ij (2 = ij ij − + = 0 Z e π i i e H H 6 H H v µ r ∂ 1 C is 2

∗ = 5 σ Z 6 ij M d g T Z 5 ) where Z I M 5 T M ,X T + µ = is not logically identified with the world-volume coordinate X 5 µ = ( M is the induced metric on the world-volume: X S M ij . The 11-dimensional superspace coordinates will have ) the supervielbeine of the 11-dimensional background. The auxiliary scalar g X Z ( A A i, j, . . . E . In a trivial Minkowski supergravity background, truncating the PST action to the As an effective field theory describing the interactions of the brane degrees of freedom, The action which describes the self-interaction of the M5-brane is complicated by the Note that Let us list our index conventions. M5-brane world-volume coordinates will be de- p The tension of the M5-brane, ` 1 , using the field theory normalization in which the two derivative terms are independent p the PST action includes an` infinite set of irrelevantof deformations suppressed by powers of only couples via its normalized derivative, which shows up in the definition of The metric with presence of a self-dualstrategy three-form, to overcome nonlinearly this relatedintroduction complication of to — an the auxiliary the fieldensuring scalar PST self-duality. strength field formulation The of action [ is M5-brane: choosing static gauge. Thisdiscussion. notational distinction At will times, be weby necessary will underlining for also the clarity need respective in to coordinatespinor our denote indices: and later orthonormal gamma frame matrix indices, conventions, which see we appendix do examples, will we return to the backgrounds described innoted section decompose into 11-dimensional bosonicWe coordinates further split the 11-dimensional coordinates into those tangent and transverse to the the classical world-volume. Finally, only in section JHEP11(2016)162 (3.7) (3.6) 0) tensor , -symmetry R holds. flux vanishes on 4 4 G G is zero. This is consistent 3 , these symmetries are H a ]. These are described further , 41  Θ ) and Θ G ( i . M ∇  i X γ Θˆ .  A g the quadratic action for the world-volume K 2 = det ], schematically: A − – 11 – 13 Z p K σ δ 6 , though we will later make some comments on what d Z Θ=0

a ) and acts on the 11-dimensional coordinates as = K Z of the 11-dimensional background, each of which is generated ( only if the previously mentioned assumption about 3 A quad 3 S K H ]. includes both the aforementioned couplings to the bulk connection as well as . ) 40 B G ( ∇ Note that in terms of the world-volume fields Our primary interest will be in the supersymmetries enjoyed by theories like the one In the presence of a non-trivial supergravity background, the action for the tensor mul- Also assumed is that the background value of the world-volume three-form We actually will be concerned with the supergauge transformations [ 2 3 happens when this is not true. with the Bianchi identity for in appendix apply to a subsetconfigurations of is the equivalent local to symmetries; findingrealized backgrounds thus, symmetries for finding are which actually manifestly a linearly supersymmetricfreedom. subset realized brane of when We restricted these will to nonlinearly always thewe assume physical define degrees that as of the having a background nonzero admits odd superisometries, which The superisometries includefermionic the symmetries generated bosonic by the isometries Killing spinors of thegenerally background nonlinearly realized, as i.e. they are well spontaneously broken as symmetries. the This will also The M5-brane action enjoys bothtries global are the and superisometries local symmetries.by a Among super-Killing the global vector symme- superisometries of the backgroundwill and describe are, next. in general, a subset of them. This is3.1 what we Symmetries of the M5-brane action where couplings to the bulk flux. Its precise form willabove, be when given later. supplemented with the bosons. These supersymmetries descend from the For example, under the simplifying assumptionpull-back that to the 11-dimensional the classicalfermions brane takes configuration, an especially simple form [ multiplet [ tiplet includes couplings to the backgroundcouplings fields, like to the the case of connection topologicalgauge twisting on fields where in the the bulk field tangent theory. Supergravity bundle fluxes act induce as yet more background field theory couplings. two derivative terms yields the superconformal action for a single, abelian (2 JHEP11(2016)162 is a (3.8) (3.9)  (3.10) (3.11) (3.12) . 4 ]: , = 0 42 A B , a are the inverse supervielbeine. The

. E A K a = 1 A ) + = 0 2 κ κ , E ) Γ κ , B κ

3  = 0 C B K ,

]. There is a large literature on these topics, B ) ∆ , κ

ι κ A – 12 – ∗ ψ Γ 44  , Z ) = 0 δ − κ 43 1 + Γ = ∆ = = ((1 + Γ  (1 2 A A B tr (Γ = Z ∆ κ κ δ B δ

) about the supersymmetric embeddings of M5-branes? It Θ δ 3.12 in the theory. On the brane, the only fermions are Θ. Furthermore, ψ . i , is supersymmetric if a fermionic symmetry transformation preserves its contains information on the embedding and the background fields, so this B ) acts as a projector, leaving only 16 independent components in the kappa κ κ ). These symmetries only include the superisometries and the kappa symmetry. satisfies ) is an arbitrary, Majorana spinor and κ σ 3.10 ( κ Now we must ask, when does a brane configuration possess manifest supersymmetries The local symmetries include world-volume diffeomorphisms and the fermionic kappa- We have assumed that the M5-brane world-volume is Lorentzian. If it is Euclidean, then this equation 4 What can we learnturns from out ( thatgeneralized submanifolds calibrated satisfying — this geometries condition [ are described byis calibrated modified by — factors of or puts a constraint onconfiguration. which Or, superisometries put differently, are theembeddings respected supersymmetric in cycle by a condition given the tells background brane us preserve which plus some brane supersymmetry. background 3.2 Supersymmetric cycles Projecting onto the subspacecycle condition, unaffected also by called kappa the symmetry calibration condition, yields of the [ supersymmetric The matrix Γ for all fermionic fields only those fermionic symmetriesviolate ( that act onTherefore, Θ we by require: a Θ-independent shift could possibly bosonic nature. In othersupersymmetry words, if a the variation symmetry of generated each by fermion the vanishes: fermionic parameter nents of Θ unphysical. Thismultiplet agrees with contains the 16 field and theory not multiplet structure 32 since fermionic the degrees tensor ofon freedom. its world-volume? As inically the denoted case of supergravity, a bosonic field configuration, schemat- where matrix Γ and so (1 + Γ transformation. These surviving kappa transformations act to render half of the compo- symmetry. The latter act on the embedding and world-volume fields as follows, JHEP11(2016)162 . . 4 ]. 8 8 3 [ CY M . On 4 takes 4 (3.15) (3.16) (3.13) (3.14) ⊂ M κ e Σ , but in CY 4 4 e Σ ) do exist and to 3.12 are simply related to , which is conformally e J 8 metric, the volume form , which is that : M 4 J 2 ]. 42 CY . 4 e Σ

J , , . ∧ 8 e Γ 4 . This is the same form that Γ Σ J

= M : A J ∝ 5 a generalized K¨ahlerfour-cycle on W 4 ∧ Γ 4 e × Σ to zero, which we will do. Under this last

J e 1 Σ , ··· e 3 J 2 . Analogously, one can show that calibrated 0 4 H R ∧ – 13 – is conformally related. Let us motivate this with ) = e Σ are conformally related to those of the 4 e 8 J = = Γ 8 (and called a generalized K¨ahlerform), then keeping B M ) by the dual spinor, now in the presence of fluxes, 11

M ) = J κ 4 Vol(Σ M Γ e Σ 3.12 to a Killing spinor bilinear [ is a K¨ahlercycle, i.e. a submanifold whose volume form is 4 J of the square of the form K¨ahler Vol( 4 to which CY 4 run over the directions of the world-volume in the orthonormal ⊂ 4 CY 5 , which is conformally related to the ) after multiplying that equation by the dual spinor and integrating 8 ... M 3.12 in the presence of flux. Similar statements should hold in flux backgrounds takes a very simple form: 8 κ M conformally a Calabi-Yau 4-fold. Also, the M5-brane wraps a four-cycle while further relating 8 4 M When we include flux, we can take the same underlying topological cycle as Σ We claim that in this case, the supersymmetric cycles of For now, let us restrict to the scenario described in section has positive world-volume chirality. we see thatsupersymmetry this on generalized the calibrationK¨ahlercycles brane condition of wrapping is different exactly dimensionsalso or what lead calibrated we to special need appropriateCalabi-Yau Lagrangian to supersymmetric cycles preserve generalized of the calibrated cycles on the conformal track of the conformal factors we can show that When we haveMoreover, this if condition, we we again can multiply ( call the new metric on of this new cycle hasthe changed other by hand, a if conformal werelated factor; use to we’ll the the denote Killing previous the spinor K¨ahlerform new to cycle construct by a bilinear given by the restriction to Σ This follows from ( over Σ the world-volume, and the supersymmetricK cycle condition reduces to the requirement that those of the underlying an example. Suppose Σ where the indices 0 frame. Our conventions are describedin the in absence appendix of flux. It is the 11-dimensional gamma matrix that measures chirality along In this case, theFurthermore, we Killing will spinors make of flux the to simplifying the assumption braneallows that world-volume, us the evaluated to pull-back in setassumption, of the the Γ the classical background 4-form configuration, value of vanishes. This discuss some of the properties of a few examples. with and it will be enough for our purposes to point out that solutions to ( JHEP11(2016)162 , 5 j 1). , (3.18) (3.17) = (1 ]. In this are a basis 4 N α ] for the case l 48 ]. In this case, the 49 β ) l e e ) ω ∧ 3. The forms e ω ∧ α + c.c. K ∧ ∗ l , we can characterize the su- j 4 ∗ αβ bω ∧ ], and especially [ f bω CY e ω 47 + ∗ – h .  yields a two-dimensional theory with 4 and 44 + 4 ) + Re ( ]: 3, the resulting two-dimensional theory ˜  8 f e ) + Re ( ω vol + c.c. K 30 CY e ∧ ω ∧ M Re(Ω) j + ' ∧ 3 2 4 ω ∧ hω 4 ω + e ω but are otherwise unconstrained. The forms – 14 – vol + ∗ Re ( J  h β C l Re ( e ) a ∧ c + ∈ a ∧ 2 J + ˜  α 3 and Σ ˜  + l − ∧ 3 appears in [ ˜ f  b, h K a ∧ f αβ K ∧ f hω × aj × , the possible supersymmetric fluxes can preserve either all 3 cj 1). + + (4 + and 3 , 2 K = 2 = R K 4 4 ∈ = (0 G G = 0, i.e. ]. N αβ c 2 45 − comprise a triplet of complex structures on -structures are only modified by a conformal factor. This is a special case is actually a a, c, f ω G 4 4) supersymmetry, without accounting for flux. As before, it is possible for , CY 2) supersymmetry. Alternatively, a special Lagrangian cycle yields , = (0 , and Im If the As described in section Starting with a generic Calabi-Yau 4-fold, which we take to have holonomy SU(4) With this relation between the cycles of N See, for example, [ ω = (0 2) flux. 5 , we must have 4 The constants Re of anti-self-dual two-forms. To satisfy our assumptions regarding the pull-back of the flux, fluxes to partially breakcase, they the can supersymmetry preserve all, ofsupersymmetric half, the flux or on pure one a quarter metric of background the supersymmetries. [ The most general, For either a K¨ahleror specialthe Lagrangian cycle, world-volume. this Therefore, flux(2 has under a our non-vanishing assumptions, pull-back to we can only includehas the primitive or half ofpreserves the two supersymmetries supersymmetries of doesthe not the world-volume. satisfy metric-only our As background. assumption a reminder, regarding this its However, component pull-back the of to flux the flux that is M5-brane wrapping a K¨ahlercycleN inside of a There is a thirdresulting possibility, theory which has is to wrap a Cayley submanifold [ on the classical world-volume. Tothat give some can examples result of from theassumptions two-dimensional our below. field construction, theories we describe the cycles andand not fluxes a that proper satisfy subgroup,We our the will supersymmetric focus 4-cycles have on been two studied and types: classified. K¨ahlerand special Lagrangian. In the absence of flux, an of the broader conceptof of M-theory generalized on calibrations eight-manifolds. [ persymmetric M5-brane embeddings withflux, flux but by with the the corresponding added embeddings requirement without that the flux satisfy our assumption that it vanish whenever the JHEP11(2016)162 . } 1). e ω , (3.22) (3.20) (3.21) (3.19) 3 then Im 0) com- = (0 , K e ω, × N 3 Re ˜ , K { 2) or , and flux with a (4 = (0 } 4 ω N G and the fermions Θ are a Im 11 1) supersymmetric even if the ω, , M . .  , Re ) b in j, a = 0 N 6 { .  (  e Γ K  ⊗ S 1 + Θ = 0  κ + Σ) . Our gauge choice is ) 1 2 P 6 T κ 3 submanifold of, for example, ( , – 15 – = = ] : − K b S N + a = 0, the flux preserves only half of the background  ) (1 + Γ Θ Γ[ Γ  h  = 2 theory is actually (0 P ∈ ∈ + ( = 0. = D 1) is a very weak condition on the resulting sigma model. P φ Θ h a , = b ^ SU(2) rotations acting on = × a : Z are a section of the normal bundle to Σ 1) supersymmetry regardless of whether the background possesses any space-time I , φ To make this more precise, we fix the kappa-symmetry gauge. Define the projection In summary, wrapping an M5-brane on a four-cycle inside of a Calabi-Yau 4-fold in the = 2 string to be completely non-supersymmetric since there are no ambient superisome- Generically, kappa-transformations and superisometriesThe move subset away from which preserves this the gauge gauge slice. choice satisfies which project 11-dimensional spinors onto thedefined subspaces with with respect positive to or negative the chirality, world-volume Σ matrices physical. We will fix thediffeomorphisms gauge so allow as us to to make removebosons this 6 manifest. of Additionally, the the world-volume 11(chiral) scalar section degrees of of freedom. theembedding Ultimately, spin the bundle associated to the pulled-back tangent bundle of the 3.3 World-volume supersymmetry Next, we come tophysical the question fields of on determiningvolume? the the world-volume. supersymmetry We transformations saw of First, before the what that are the the kappa-symmetry renders physical only fields half on of the the world- Θ variables needs (0 supersymmetry. In addition, (0 It is, therefore, possible thatflux the breaks all the bulk supersymmetries. presence of flux canThe lead remaining to possibility two-dimensional is field theponent theories strangest; so with that namely, to all the turnD ambient on supersymmetries a are broken. Wetries. might expect the However, resulting ifwe we should were have constructed wrapping the a physical heterotic string. The world-sheet heterotic string supersymmetries. Off this locus,ground it supersymmetries, preserves it only a issubgroup quarter. of necessary the To for preserve SU(2) theThis all is flux of only the to possible back- be if invariant under a nontrivial At generic points on the locus JHEP11(2016)162 to ) in K (3.28) (3.23) (3.30) (3.27) (3.29) (3.24) (3.25) (3.26) 3.27 -dependent in the chiral K κ + A 0) theory in a trivial 1 , − ) give the supersymmetry B to leading order in a Θ- . − − I ) will be linearly realized. κ ] φ 1 ), which combines with . , (1 + = Θ . : a − 50 . − µ K  − 3.12  , ( A Γ K  Θ ) κ = − , we obtain the following expansion K ! − are nonlinearly realized. In other + + B 1 a Θ P B + ) − + + − − . κ A  ) ) A (1 ! σ, φ 2 κ B 1 AK ( 1 1 − B I − − ) − − µ ) + , K e − B 1 0 0 , i 2 B K M (1 + Γ

(1 − 1 – 16 – Γ   + − − µ ) − A (1 + B ) is solved by + Γ (1 + Θ P e  B Γ = must take the form [ −

i − I 2 − κ = Γ = = Θ 3.22 dropped out. We should also include a ) = = − − κ − (1 + − Θ − Γ Θ M Θ ) − κ σ, φ δ Θ δ ( = 1, Γ K δ = I 2 κ are then given by, σ, φ µ ( ), the leading-in-Θ transformation is + e I − κ i 2 I ie 3.28 + : − = I δφ ) = 0 and Γ κ commuting. Then, ( B solves the 11-dimensional gravitino variation condition, so it is a Killing spinor of and  A Returning to the bosons, the transformations of the To get a more concrete handle on these transformations, we decompose Γ expansion are We see thatwords, the the supersymmetries linear supersymmetries associatedbackground. are to chiral, as they are for the (2 of the superisometry: where the spacetime. Using ( Note that the dependenceworld-volume diffeomorphism, on whose rolescalar is degrees to of fixdiffeomorphism, freedom; the the above gauge however, expression associated it is exact.Θ to will to We the obtain will play the be choice leading no interested of transformations. in role expanding In ( appendix in our analysis. Ignoring this The transformations of Θ with basis To satisfy tr (Γ preserve the gauge choice.transformations Together, of their Θ action on Only the subset of these transformations also satisfying ( This constraint determines the kappa-transformation, JHEP11(2016)162 . We = 0. (3.33) (3.32) (3.34) (3.35) (3.31) B φ = Θ = 0 . Let us φ B ]: . 39 and  A ij γ i v 5 i given in appendix ]. In this expansion, we v e Γ. Only when 4 A i , 51 3 ) i K  e ( H 2 and, hence, κ i ) 1 κ i κ are the off- and on-diagonal com- e H j B 5 . . expansion, only the first term would 6 ...i M , (1 + Γ φ 1 Γ ...i i ij and 1  g i M 1 8 is isomorphic to the submanifold γ MN A A to emphasize that we have not truncated to 6 Γ 6 − = 2 E e − ...i A 1 } i Θ j ijk  Z – 17 – , we will use a world-volume momentum expan- i γ N as the pull-back of the bulk vielbeine using the 1 , γ κ j 6! ∂ i jk i e γ ) reduce to the world-volume vielbeine. e = = { H M i i M i e γ M v e γ i ie ∗ 1 2 e − X + = e γ = 0, = ( ij  φ ) M δB ˜ i H e . At still has a complicated form. contributes only bosonic terms at leading order, as can be seen from 1 X κ κ -dependent bulk vielbeine φ are the pull-backs of the bulk gamma matrices. Specifically, det(1 + i q γ . However, we have used the gauge freedom of world-volume diffeomorphisms = 0 which is block diagonal. So, in a φ = φ κ Γ are chosen so that the world-volume Σ To get a more concrete handle on Γ Without employing some expansion, there is little more we can glean about the world- After all of this discussion, we have yet to specify Γ At leading order in Θ, the 2-form has the transformation 11 cause the bosonic Γ sion. This is theassign degree same 1 expansion to used derivatives withworld-volume in respect bosons, a to as world-volume similar coordinates well context and as in degree to 0 [ derivatives to of the bulk objects with respect to the normal do they agree. volume supersymmetry from these expressions. ponents, respectively, and theysupersymmetries, Γ have a complicatedthe structure. discussion above. In This a doesn’t Θ necessarily expansion make the of expressions the simpler, though, be- and the associated chirality matrix is Note that this generally differs from the pull-back of the matrix These matrices furnish a representation of the six-dimensional Clifford algebra, The matrices where we have defined bosonic embedding remedy this. The kappa symmetry projector for the M5-brane appears in [ framing at survive at leading order. have retained the any order in and bulk frame rotationsM to simplify the expressionsThe bulk a metric little. restricted to Specifically, this coordinates submanifold on is block diagonal, and we can also choose a where we have used the Θ-expansions of the supervielbeine and JHEP11(2016)162 ), are − 3.27 (3.36) ) terms K 2 Θ  ( 2, while the / O .  − on the world-volume Θ 3 . C 4 ) which parts of JK G Γ B.6 .  2 + ) as long as the coefficient has /  2 I Θ Γ  , ( − p ] = 1 − O Θ − 5 that carries a product metric. To be [Θ . 11 . This is because, as we will see, there is IJK ijk × M − M ω p H i K 8 , 2. Thus, 6+ / , – 18 – 1 ) + M M ] = 1 − i − 3 e = Θ H 11 µν . In general bulk objects, like connections, that depend p (Γ M ] = [  6+ I , it is simple to deduce from ( + and derivatives with respect to those coordinates can be φ . For example, the formulae for relating Christoffel symbols +  i M p  has degree I ∂ p decomposes as [ 2 will come from Γ  − 6+ / 5 − 11 M Θ M . Since we are interested in the linearly realized supersymmetries, M A Iµν ω i 8 and so must be given degree 1. On the other hand, the 4-form seems to . This has been our working assumption for several reasons, as we discussed ). 4 2 G 3.6 ], it was argued that it is consistent to assign degree 1 to all geometric objects dB is a submanifold of on the world-volume. This is because the components of 6 51 4 , there could in principle be terms of the form − G To obtain the supersymmetries to leading order in this expansion, we need the degree With this prescription for the degree expansion, it is simple to include the Similar care must be taken to define the degree of the 4-form flux, when there is a non- In [ Generally, this degree expansion is inconsistent with our truncation of the supersym- It is a drastic simplification that these terms doexpansions not of depend the on following quantities: the only terms ofno degree degree 1 0 piecewhich to are parameterized by important. They are is no 4-form along thedegree world-volume, 1 there to are no inconsistenciesabove associated eq. with ( assigning that may contribute to the Θ transformation. Returning to the exact expression ( zero mix with appear along with connectionsflux in degree various 1. formulae, These and two we considerations would are also in like tension. to However, assign if the we assume that there where Σ on the coordinatesassigned of degree 0. Thisor derivatives assignment involving cannot beto made spin consistently connections for can objects mix with derivative type, indices and therefore degree. unless they are associatedmore with specific, a suppose subspace of order transformations raise the degree. metry transformations to leadingon order Θ in Θ. Specificallydegree in zero. the The action coefficients of aredegree typically supersymmetry assignment bulk requires objects some like care. fluxes or connections, and their supersymmetry parameter This expansion is often employed in theThe expansion leading of supersymmetric supersymmetry effective transformations field theories. preserve this notion of degree, while higher- coordinates. Additionally, the world-volume fermions are assigned degree 1 JHEP11(2016)162 - 1) φ , are (3.40) 4 (3.38) (3.39) (3.37) G .  − Θ in degree is the , JK µ n µ Γ  e  + .  ABCD I  M I Γ ] Γ − jk Θ SO(5) subgroup of SO(10 C I ) × ABCD φ φ . G ( i 1) [ I , 1 I ∂ 288 Iijk e , and this is only necessary when the G − contributions have higher degree since I I I n IJK φ ) + 3  e φ = Θ = 0. Also, in such a case, the inverse + ω φ  ( + i 8 φ ABC ijk , when acting on any field, yields a bosonic Γ ijk ˆ γ Iijk 2 ) ) +  G – 19 – − ijk I dB Θ φ H MABC and G µν 1 + = ( 24 1 1 36 (Γ  ijk − ijk  -dependence of ) + 6  + φ H n  +  dB  I , I M Γ  Γ ∇ − i = ( has the structure of a product as mentioned before. Note that − ˆ γ = Θ Θ I 11 matrices, evaluated at n ijk I , because higher-order in φ )  i ) Γ γ φ H φ M G ( + ∇ , ( M I  − ∇ I in ) e ) Θ is the covariant derivative on the combined normal and tangent bundles. φ 6 must satisfy the bulk Killing spinor equation, φ shifts by a two-form to remove this piece, leaving ( ij are the pull-backs of Γ to the classical world-volume. Specifically, they are 0 = ( ), should be replaced with their “hatted” versions, where “hatted” means I 2 Iµν ∇ ˆ = 0. γ 3. 2 I I  φ γ ω I , has no degree 0 part. Its first non-vanishing component is related to the + e ( B φ K I  i I 8 1 2 i ie ) = 0, which leaves manifest only the SO(5 µ I and × e e − − − + 1 3  σ, φ = = = ( K µ I − ij I = The three-form expansion is more straightforward. All of the components of Furthermore, expanding this component of the vielbeine as well as e Θ δφ Both 8 6 δB δ world-volume fields. Specifically, wetransformations, demand that with the generators commutator of two supersymmetry and are chiral when pulled back to the world-volume, but we drop the + subscript to avoid clutter. vielbeine, evaluated at 3.4 Closure andWe equations can of use the motion closure of the supersymmetry algebra to derive equations of motion for the The derivative The matrices ˆ the previously defined Assembling the above results, we find the leading supersymmetries on the world-volume: The last term is anunder exact which form on the world-volume. We can fix the bulk gauge symmetry M of degree 1, sodependence. in To first the order degree in degree: expansion of the three-form, we must retain all of the same as expanding in they are related to curvaturesare on only the forced world-volume. to So,complement maintain in of the a Σ leading degreethese expansion, details we are important for us since they directly affect one of our primary examples, also yet to (partially)ing fix therotations local on Lorentz the symmetry. frame.vielbeine, This This choice is ensures accomplished thatconnection the on by other the choos- normal off-diagonal bundle, component which of is the of degree 1. The degree counting of the frame is subtle for the aforementioned reasons. We have JHEP11(2016)162 -field (3.44) (3.48) (3.41) (3.42) (3.43) (3.45) (3.46) (3.47) and Θ. B ), as well φ ρσJ ] 3.39 ν G 1 .  − J  | -field, we can com- ρ I Θ ] | I B ν Γ Γ B  ρσ ) ρ ijk ] µ 1 [ . ν  ˆ γ Γ ρ .  B 2 Γ , 1  2  ijkI ¯  νρ ρ ]. = 0 i − ρστI G Γ , + 3¯ B − 2 11 G 1 1 ( ¯  24 − ijk 2) I  i ρ µ φ ] is quite interesting. In particular, [ + Γ − SUSY transformations ( ↔ ∂ ρIJK 2 − 16 , 2 , − I ¯  is defined to be (this is consistent with 1 − G (1 i φ 2  1 Θ − 1 − ρ  ⇒ H − ρστ  = 0 − IJ Γ − I . µν H 2 1 Γ JK I − Γ ¯   i  ] B ) Γ φ i 2 ˆ ν γ Θ ρ 1 1 ρ G )  ∂ ρστI Γ  ( I − Iijk − G 2 – 20 – I ) µν ( i G ¯  ∇ G 1 Γ Γ i 1 ρστ = I  ∇ )) ν 2  -field and Θ i ρ µν φ −  ρ µν Γ ˆ γ  KLMIJ B Γ Γ 3¯ ξ 2 − Γ  2 2 ρστ ¯  µ 2 ¯   [ 3.40 i − ¯  i = ∂ µν i − 6 ). In this case, we find the result: − Γ ( 2 − ijk 0 = ¯ iKLM  = ¯  + 2 + JKL ) ] G ν 3.46 ν = − Λ 1 dB G 48 ( -field gauge transformation with parameter 1 satisfy the bulk Killing equation. In particular, we needed µν  B 2 − B µνρI  ] ), we see that the first line is a diffeomorphism with parameter − G 2 1 JKL | Θ  , δ i I ρ | 1 and 3.41 , given in ( δ Γ ∇ Γ [ 1 i 2 µ  [  ˆ γ − ijk Γ H 2  + 6¯ = 6 conformal supergravity formalism of [ 0 = ¯ D The similarity of our supersymmetry variations and equations of motion to those listed The constraints of closure yield no new information when acting on the scalar fields. Taking the commutator of SUSY transformations acting on the This equation of motion agrees with the result foundin in the [ where the fluxed covariant derivativethe acting on bulk Θ definition given above in ( Acting on the fermioneasier fields, to find we the can fermion equation findof of the motion motion by fermion for acting with equation supersymmetry of on the motion. equation However, it is and the second line is a and the third lineequation will of need motion is to vanish on-shell, from which we conclude that the Then considering ( which implies that To reach this point we hadas to the use fact the that will limit our discussion to the supersymmetry transformations topute linear that order in symmetry action on that field, if we restrict to on-shell configurations. For simplicity, we JHEP11(2016)162 7 (A.1) (A.2) (A.3) coordinate, { -dimensional field is a combination of our D ij µ V to denote the } I,I and their { ijkI ) are very closely related to their . G and } , , 3.39 MN η , . is our 5 µ, µ = 0 = 0 { ... 0 = 2 ij i abc o iIJK T e Γ

e Γ G , , are diagonal in this basis. N I – 21 – e Γ is diagonal. Because of the (anti-)commutation µ Γ I Γ e Γ Γ = Γ , h n M Γ  -dimensional supersymmetric background and the on-shell coupling of D . These matrices will be field-independent and satisfy the 11-dimensional are off-diagonal, while Γ b a µ directions along and transverse to the brane, respectively. Along the M5-  } M being the world-volume directions. Γ 5 ... We will often work in a basis in which This work was performed, in part, at the Aspen Center for Physics, which is supported Specifically, the auxiliary tensor they refer to as 7 the matrices Γ normal bundle connection and the flux component with 0 relations In the presencetransverse of to the branes, brane. we We willorthonormal use will the want indices tobrane, group we can these define matrices a chirality into matrix, those along and A Spinors and GammaOur matrices 11-dimensional spinors will carrythe upper form indices, so theClifford Γ algebra: matrices will naturally be of by National Science Foundation grant PHY-1066293.by T.M. NSF and Grant S.S. No. are PHY-1316960. supported,Focused D.R. in Research part, is Grant supported DMS-1159404, byInstitute the and for grants PHY-1214344 by Fundamental Physics and the and NSF George Astronomy. P. and Cynthia W. Mitchell ically, the “Strings, BranesTexas and A&M, the Holography” Great workshop“F-theory Lakes held at Strings at the Conference Interface the of held Mitchell ParticlesKIAS-YITP at and Institute Workshop the Mathematics” on at workshop University “Geometry in of Aspen,like in Michigan, and to Gauge the the thank Theories joint the and respective String organizers Theory.” of S.S. these would workshops for their hospitality. Acknowledgments It is our pleasureDave to Morrison, thank Ergin Sezgin Dongsu andwere Bak, Dimitrios presented Katrin Tsimpis at Becker, for a Jerome helpful number Gauntlett, discussions. of Greg These conferences Moore, results earlier this year, prior to publication. Specif- equations after an identification ofThe their equations auxiliary of fields with motionand our also fascinating flux relation match degrees between after of thetheory freedom. this off-shell to formalism identification. a for rigid coupling Thisa a suggests brane to a higher-dimensional surprising supergravity. We hope to explore this relation further. our supersymmetry variations appearing in eq. ( JHEP11(2016)162 e Γ is (B.1) (B.2) (A.4) (A.5) (A.6) (A.7) , leads to the a  C , , ]. For reference: 4 5 LM , , , 52 3 2 , , ΘΓ a NP QR , . = 0 = 1  4 5 G k k , , Θ 3 2 if if , , . M NP QR Γ  , , ] = 0 = 1 , , Θ Θ LM  k k k k QR a ]  R Γ if if NP QR P ...M ...M Θ . 1 1 P QR M , , M , M M T T M Γ k k 4 1 δ a , NP QR Γ Γ a Γ Θ N N )   [ [ + i G 2 − ...M ...M L + − 1 1 M a = 0 δ δ  = ( M M o 1 Γ Γ as an intertwiner. This means that for a spinor Θ + e NP QR Γ 1 , the charge conjugate spinor has the opposite = C − + 8 NP QR , – 22 –  + 24  C M D C C −C + M ( k  NP QR ω n M T , ( M M Γ , = defined as Θ ...M T = C a 1 NP QR  a NP QR MN  Θ ΘΓ a M NP QR T e Γ M − Γ ) Θ i  2 Γ Γ G a a LM k C  i M   6 Γ − T MN Γ  M M ...M + 1 M 1 M ω NP QR 1 i 288 a 2 ω 1 M ΘΓ M Θ = Θ a D 1 4 144 − e − δ Γ  C k = = = = = = = = . This matrix acts as an intertwiner between the equivalent repre- a a a ab ...M M a M M a C 1 ) , and ( M a ω E M M E E E NP QR NP QR are taken to be anti-commuting. The above is a particular result of the ΘΓ M  NP QR NP QR 1 LM T is off-diagonal, which follows from D M R ( T C , M ω Furthermore, in 11-dimensions, there is an antisymmetric charge conjugation matrix and its inverse will be used to raise and lower spinor indices. The antisymmetry of ab with chirality. B Theta expansions The expansion of the bulk supervielbeine in the Θ coordinates is found in [ diagonal, which itself follows from theof definition of given six-dimensional chirality Furthermore, it will be important to note that with respect to the basis in which where Θ and (anti-)symmetry of the gamma matrices themselves C sentations: C following symmetry properties for bilinears in Majorana fermions: JHEP11(2016)162 , in A K M (B.7) (B.5) (B.6) (B.3) (B.4) G . as  . a Θ) = 0 NP QR  , LM is: a G   a 2 (Γ  M = 0  Θ P QR  G B Γ NP QR Θ A . Taking a fermionic . N M P QR E A a ], which refers to this com- a T ) M  Γ  δ K 41 K M  ( LM  LMNP QR + 8 B N ΘΓ A ΘΓ ΘΓ L i 3 ΘΓ a . +  + NP QR B + 8 a super-two-form. Using the known Θ A a A M a  B 2 − δ  Γ , , E Θ a ]. a M NP QR =  B  ω 53 Θ B K M Θ LM ) = 0 A A – 23 – G QR R ∂ E NP QR a K a Γ Θ (  MNP QR  A Θ G 2 + Γ , A Θ M  Λ A a  + 1 E d   A 288 a  NP ) to find a Θ expansion of ΘΓ  Θ + E NP Γ − M 3 NP QR B M i M 2  B.5 ∂  C , ω Γ G ΘΓ  ΘΓ B − K Θ M i M 24 MN i a 2 K L M 24 a Γ + e − δ −  = = ΘΓ 3 = = = + = A MN C a a A , M M M a K E  M a NP QR ω δ  E M MNP 1 4 K E E G , we can use ( is required to satisfy δ . We will also need the inverse supervielbeine, which satisfy ω E M A  i 8 − M A i  ΘΓ ω E + 1728 M a i ∂ 2 a local Lorentz transformation and Λ  + B = = A a M = 1 supergravity in 4-dimensions, this is described in [ L K The spinor We are interested in the preserved supergauge transformations — those that leave The 11-dimensional supergravity in superspace possesses superdiffeomorphism invari- K N with expansion of i.e. one whose Θ = 0 component is fermionic, the result up to order Θ invariant the bulk superfields.ponent These action will that correspond are to preserved the by a supersymmetries given of background. the In com- superspace, these satisfy form potential by exactare superforms. a The subset supersymmetries ofof of all the three component-field transformationsbination action rather of than transformations any as onedescribed supergauge in individually. transformations. 11-dimensional At For supergravity low the in order [ case in Θ, they were ance as well as invariance under the local Lorentz rotations and shifts of the super-three- analogy with This implies the Θ expansion It will at times be more convenient to refer the combination JHEP11(2016)162 ] , ]. 05 12 (B.8) (B.9) (B.10) (B.11) , ) must BPS SPIRE (1999) , JHEP 8 IN , /  B.6 ][ 08  1 JHEP JHEP , , -models σ (1996) 155 P QRS JHEP 4) arXiv:1310.0818 . 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