Jhep12(2017)031

Home , Seiberg duality

JHEP12(2017)031 Springer October 2, 2017 : December 1, 2017 December 6, 2017 : : with ﬁxed Chern- N Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP12(2017)031 b [email protected] , . 3 1709.07872 and Andreas Karch The Authors. a . c AdS-CFT Correspondence, Duality in Gauge Field Theories, Chern-Simons

k We give simple string theory embeddings of several recently introduced duali- , [email protected] E-mail: Department of Physics andSan Astronomy, Francisco, San CA Francisco 94132, State U.S.A. University, Department of Physics, UniversitySeattle, of WA Washington, 98195-1560, U.S.A. b a Open Access Article funded by SCOAP ArXiv ePrint: Simons level Keywords: Theories Abstract: ties between 2+1-dimensional Chern-Simons matter theoriesOur using probe construction brane is holography. reliable in the limit of a large number of colors Kristan Jensen into string theory Embedding three-dimensional bosonization dualities JHEP12(2017)031 ] fol- 4 ]. These limit had 3 – N 1 17 scalars . (1.1) f N with ] for a proposal to extend the N 7 ) k U( ) gauge groups in the large . (See [ . Bosonization dualities map monopole k N F ↔ ≤ f ). In the former case, the net flavor charge, ) theory. The duality we focus on in this note N – 1 – N N is conserved identically (that is, irrespective of the ) fermions F ]. Here one has to be careful to distinguish whether 14 π f 6 2 ) or SU( tr( N 14 ∗ N = 15 j with 2 11 / 4 f N + 9 14 k 7 − ) 1 these dualities turn out to be much richer, as first described in [ ) gauge factor itself gives a conserved global charge, “monopole number.” N N N SU( U( ] and elaborated on in [ ⊂ 5 At finite 4.3 Future applications 3.1 Bulk theory 3.2 Boundary theory 4.1 Orientifolds 4.2 Baryons and monopoles resentation of the gauge group. The subscript on the gauge groups indicates the CS level. is the one that equates This duality is believedduality to somewhat be beyond this valid “flavor for bound.”) all Here all matter is in the fundamental rep- a global conserved charge.the What U(1) is special inThe the corresponding case current of 2+1equations dimensions of is motion) however by that thenumber to Bianchi baryon identity number for in the dual SU( with bosonic matter. lowing [ the gauge group is“baryon number” actually in U( the particle physics language, is gauged and so does not correspond to Much progress along these linesdimensions has involving been Chern-Simons made (CS) recently intween gauge non-Abelian relativistic theories CS field coupled gauge theories theories to in basedbeen 2+1 matter. on identified U( Dualities by be- equating thedescription putative in dual terms theories of todualities were one a termed and classical 3d the bosonization higher same since spin “holographic” they related theory a in CS theory one with dimension fermionic to up one [ 1 Introduction Dualities have long beensystems. one of our By most rewriting(and important the tools sometimes to quantum weakly understand dynamics coupled) strongly in low coupled the energy right degrees set of of freedom can variables, the often be correct identified. A Numerical analysis of a D5 probe brane on the cigar geometry 4 Applications and discussion Contents 1 Introduction 2 Level/rank duality 3 Adding flavors JHEP12(2017)031 = 4 (1.3) ]. Last ]. This N ] match. 6 . , 18 19 4 , ], deformed 3 – ] deformed 3d 21 1 , 20 23 , ]. Second, one can 4 22 ] mirror symmetry is real scalarsreal scalars (1.2) ] (with closely related f f 25 9 , N N ]. More recently general- 8 15 , 6 with with ]. These embeddings often help N . Similarly [ 25 ) k N N ) ] by gauging the R-symmetry, leading k 27 SO( USp(2 ↔ ↔ – 2 – ) to symplectic and orthogonal groups have been 1.1 , and in this limit agreement can be verified by the = 4 mirror pairs [ ]. N/k real fermions fermions 24 N = 1. (This latter deformation comes with some caveats, as f f f N N limit one can even match their dimensions [ the global symmetries and their ‘t Hooft anomalies [ N ] exchanges the Coulomb branch and Higgs branch of k N = 26 [ with with k ]: = 1 case of this duality has been shown in [ with fixed and 2 f / = 2 supersymmetry. Recently it has even been demonstrated that, at k ], as the renormalization group flow is not under complete control.) We = f 17 2 , N N / N ], as well as several new Abelian dualities [ N f 23 N + 16 , = k N 14 − + of – 22 N ) k 1 ]) to act as a seed duality from which one can derive a wide web of previously − 11 N = ) 10 and large one can easily check that the global symmetry charges of these operators match, k N N N Mirror symmetry In this work we present a simple string theory realizations of these 3d bosonization None of these dualities have been proven, but they are backed up by a substantial SO( USp(2 For this special case, there exists also an exact derivation on the lattice, exhibiting a duality between 1 can be systematically added to to mirrors with two lattice gauge theoriesrespectively in that the are continuum constructed limit [ to flow to the bosonic and fermionic side of the duality constructions has a long history startingto with the clarify work patterns of [ of thetheories dualities in and three generate dimensions, new two examples. famous dualities For can supersymmetric be gauge supersymmetric realized Yang-Mills via theories. brane In embeddings: implemented the via S-duality brane in type realizations IIB of string theory. [ Supersymmetry breaking deformations special case explained in [ will discuss these supersymmetric dualities in more detail below. dualities. Embedding field theory dualities into full fledged string theory using brane resulting topological field theoriesbut (TFTs) not agree least, on one bothto can trigger sides start a of with flow a the to3d supersymmetric duality a Seiberg non-supersymmetric [ parent duality daughter duality duality. to andmirror In flow deform symmetry this to it to way reduce in 3d [ to order bosonization the at non-supersymmetric bosonization large duality, at least in the operators on one sidefinite of the duality mapbut in into the monopole ’t operators Hooftshow large on that the at any otherThird, side. when At adding relevant deformations that drive the theory into massive phases, the amount of evidence.large First, the fieldexplicit theories computation are of tractable correlationincludes in functions a the and detailed the ’t mapping thermal Hooft between free operators limit, energy in that [ the is two theories. In particular, baryon izations of the bosonizationconstructed duality in ( [ whose coefficient is tunedAbelian to a critical point,work in making [ them “Wilson-Fisher”known scalars. dualities, The suchduality as of ordinary [ particle/vortex duality, the fermionic particle/vortex The scalar theory is accompanied with a quartic potential for the fundamental scalars, JHEP12(2017)031 ]. But 30 – 28 , one recovers 3d N = 2 supersymmetry . While originally N ] following the famous 31 ] identified a particular set “flavor” D-branes. The field 38 f N Seiberg duality ]. In terms of brane setups, Seiberg 37 ]. Scalar matter on the other hand is 32 ] of a 2+1 dimensional gauge theory, the ]. Using this strategy one can attempt to ]. Mirror pairs with ] that, at least at large 25 37 [ 23 , 21 2 , – 3 – 22 20 ] realize the gauge theory on the worldvolume of 25 ], Seiberg duality can be compactified to 2+1 dimen- 33 ] dealt with Seiberg duality in 3+1 dimensions, the same EGK 38 ] where, when lifted to M-theory, a brane setup realizing a mirror pair can be seen 39 ]. It has been argued in [ 37 = 2 an a priori different equivalence is ]. 3d Seiberg duality becomes particularly rich and interesting when CS terms 36 N – 34 For The embeddings in the style of [ Note that the realizations in terms of branes makes it appear as if Seiberg duality and mirror symmetry “color” D-branes, with matter being introduced by 2 moves still apply in 2+1 dimensions as well are two genuinely unrelatedduality statements. of dualities One [ canto however be show equivalent that to at a brane least setup in realizing some a cases Seiberg there dual exist pair. a with boundaries and defects2+1 contains dimensional a lot low of energyD5 extra description: branes data the that along relative should theof be positions brane compact irrelevant of moves for direction. NS5 that the branes branes takes in The and the this authors extra flavor original direction, of gaugethe presumably theory leaving [ original the into low work its energy of theory Seiberg unchanged. [ dual While by rearranging duality is also easilysimply implemented, using even though S-duality. In its the derivationcolor is construction D3 of not branes [ as are straightforwardvolume suspended as gauge between theory NS5 livesdimensional branes on gauge so an theory. interval that The and the full reduces 3+1 3+1 at dimensional dimensional low description world- energies in terms to of the a desired field 2+1 theory discovered in 3+1 dimensionssions [ [ are included [ bosonization by deforming the Seiberg dual pairs of [ localized at an intersectionEven ignoring with the instability 2 of ND the(D5’s 2 turn directions, ND into albeit intersections, NS5’s, S-duality with D1’schange changes the into the a type fundamental number of negative strings), of brane but massS-duality ND it is squared. directions is met via hard with to S-duality. serious see obstacles. So how implementing one 3d could bosonization via number of bosons andonly fermions. be The realized latterfermionic on introduces supermultiplets). 0+1 purely To get or fermionic aone matter, 1+1 2+1 can but dimensional dimensional introduce can gauge intersections theory branesSakai-Sugimoto (both with construction with fermionic in of matter 6 3+1 which ND dimensions allow directions [ purely as studied in [ N theory matter is localized atcolor the to brane flavor intersection branes. andon arises The from the type strings number of stretching of matter from of ND localized one directions, on brane such that but a isND brane not the directions. intersection the directions depends In other. that the are former Supersymmetric part case intersections of one correspond the introduces to worldvolume a 4 hypermultiplet ND of or matter 8 with equal starting with the supersymmetric mirrors [ have been engineered via S-dualityit for brane is embeddings difficult with to rotatedout imagine branes of [ how string one theory could this way. get the non-supersymmetric bosonization duality least in the Abelian case, one can flow all the way to the non-supersymmetric seed pair by JHEP12(2017)031 ) degrees 2 N ( O ], but the results look ] we embed level/rank 40 42 ]. Level/rank duality was first . In particular, both sides of 44 , N/k gauge theory with 43 ) CS matter theory. The full gauge N N ] between such 1+1 dimensional CFTs and 45 with fixed k – 4 – and ]. To have a higher spin dual, one needs to take a ’t we introduce our basic construction, embedding the 3 N – 3 1 , which arises as a gauge theory living on probe branes in N . In particular, we recover the SO/USp versions of the duality by 4 . In this case, the bosonic gauge theory has a small rank, is weakly k limit, that is large ) theory at level k N ) into string theory via a brane embedding. We describe applications of our 1.1 with fixed ]. While we will also invoke holography, this realization is quite distinct from the N 41 The rest of this note is organized as follows. In the next section we will review the con- We will see that the only low energy modes in the bulk geometry correspond to the Given these difficulties, in this work we will give an embedding of 3d bosonization into 2 Level/rank duality 3d bosonization is closely related tointroduced level/rank as duality [ an equivalencebased on between WZW 1+1 models. dimensional2+1 Using conformal the dimensional mapping CS field [ gauge theories theories, (CFTs) level/rank duality can also be read as an equivalence duality ( construction in section including orientifolds, and describe howof baryons the and string monopoles duality. are realized on both sides make some basic assumptionsto of argue the for strong the coupling desired dynamics low energy on theory the onnection boundary the between in 3d fermionic bosonization order side. and level/rank duality.duality Following into [ AdS/CFT. In section bosonized U( the bulk spacetime. Inlow energy this limit, way with AdS/CFT thetheory reduces fermionic whereas to CS the the theory being 3d bosonicidentification the bosonization theory low of duality is energy the in limit the the of low low the energy energy boundary limit theory of in the the bulk bulk theory. is While fairly the straightforward, we have to most of these degreesbulk of freedom and will field be theorymetric gapped. give spectrum rise After has to all, toany gapped for fluctuating fluctuations, agree. holography degrees so to of that be Most freedom at true in importantly, low the the all energy bulk. gravity fluctuations does of not the contribute of freedom. We engineerlow this energies theory we in can suchtheory argue a has that way a it that nice reduces mostdual holographic to excitations are dual. are a realized gapped. SU( The verythose At physical of differently: degrees probe they D-branes of embedded are freedom into fluctuations in the this of bulk holographic supergravity geometry. But fields of as course well also as in the bulk higher spin holographic dualsHooft of [ large the duality have alarge large number of colors.coupled, and surely In has our nofermionic classical case, supergravity side we dual. of will In the our appeal string story to theory will embedding, holography be the at a standard large nothing like 3d bosonization.ND Once flavor again, brane it to is a hard 2 to ND see flavor how brane thestring could required theory be change via accomplished of branes by aphy using the 6 [ a EGK completely moves. different approach based on flavored hologra- derive non-supersymmetric dual pairs as was for example done in [ JHEP12(2017)031 . π (2.1) kinetic . Their 2 = 4 SYM F /L this theory N angle gives a N ]. In order to θ 46 the only degrees [ 2 YM g = 0 versions of the 5 = 0 theories are an S /L f = f N × λ N 5 ] as we now review. The 42 = 0 are in fact TFTs and not f N , ) = 4 SYM, we introduce a spatially F , leaving behind a topological theory ) at 2 3 ∧ N 1.3 kg F tr( )–( term at low energies. In the presence of CS θ CS theory. 2 1.1 and becomes irrelevant at low energies, leaving k F – 5 – Z with anti-periodic boundary conditions for the − = 4 vector multiplet. These boundary conditions 2 2 ) L − 1 π YM and large ’t Hooft coupling N 8 N ], one can think of the 3d bosonization dualities as . This linear profile is consistent with the periodicity Lg . As an upshot, this compactification of = 4 SYM, but at energies below the compactification N = k 16 /L ∼ , θ − 3 N 6 2 S since the theta angle is only well defined up to shifts of 2 − 3 g k πkx = 0 or by following RG flows in phases where all matter fields f N ) = 2 3 x from loop corrections. At energy scales below 1 ( = 4 super Yang-Mills theory (SYM) in 3+1 dimensions with gauge θ ). Since a Chern-Simons gauge field contributes no dynamical degrees /L N 1.3 )–( ). In the limit of large N 1.1 reduces to a topological SU( direction for integer 3 /L x In 2+1 dimensions we have another option for a gauge field kinetic term: CS terms. We implement this idea of adding flavors to level/rank duality using holography. A direction on a circle of periodicity 3 reduces to a 3don CS term a at circle level withtheory anti-periodic that boundary in conditions thescale and UV 1 a is linearly simply varying varying theta angle of the Upon integrating by parts, the 4d theta term They are marginal and dominateterms over the the gauge bosonsat pick low up energies a governed massdimensional by of theory the order we CS obtained action. by compactifying In order to introduce a CS term in the 2+1 of freedom that remainterm has in a the coefficient of theorythe order are gauge bosons the without masslessstrongly a gauge coupled). kinetic bosons. term Thisdynamics in theory Their in the is the infrared infrared. believed (or, to formally, they be are completely infinitely gapped with no interesting x adjoint representation fermions in the break supersymmetry completely andscalar superpartners, give all whose mass themasses is fermions of no order masses longer 1 of protected order by 1 supersymmetry, will pick up holographic realization ofstarting level/rank point duality is wasgroup given SU( in [ has a dual descriptionreduce in the terms theory of to type a IIB 2+1 dimensional supergravity gauge on theory AdS at low energies, we compactify the are massive. As“adding emphasized flavors” in to [ level-ranksince duality. we are augmenting Of the course theory onThe this both non-trivial sides, is part but not it is motivates a thematter to conjectured on derivation argue dualities. the of that other fermionic bosonization side. matter on one side corresponds to bosonic dualities ( of freedom, the field theoriesthree-dimensional appearing CFTs. in ( Theseimportant level-rank check dualities of between the the former 3d either bosonization by setting dualities, since the latter can be reduced to the between the latter. The level/rank dualities of interest here are the JHEP12(2017)031 . /L N and 3 (2.2) (2.3) 5 S πkx k . ]. It is given ) = 2 3 3 47 x , . The dashed line ( 5 θ 3 . The 2d cigar ge- S 4 g ∼ 2 . It has the topology /r χ R 1 4 0 r + − 2 3 in the “probe limit” dx 3 ) x ) = 1 3 r r 퐃ퟓ ( ( 퐃ퟓ f f + 2 2 and the field theory compactification has to approach χ dx without a conical singularity we require R /L , 0 is displayed in figure + 3 r 3 2 1 flavor D5 and anti-D5 branes extending from the x = dx f πkx r – 6 – N + and flux so that the backreaction of an axion field strength of order 2 = 2 = 4 SYM compactified on a circle with anti-periodic axion r 2 ). Of course the full type IIB supergravity equations dt χ − N − N 2.2 ∼ 2 2 2 r s R g + 2 r ) is dual to the massless bulk axion field, so in order to turn on a ) is the “blackening function” 2 ) r r r F ( . dr ( f ∧ f . D7 /L 2 2 F ) 2 R and in the field theory the bulk axion πR = k/N λ θ g = = 0 2 r 0 Holographic bulk dual of = 4 SYM on a circle with anti-periodic fermions is well known [ . Note that /α as N r 4 is fixed in terms of the curvature radius L R 0 r The operator tr( Let us see how the compactification on a circle with antiperiodic boundary conditions In the standard normalization of the axion field, in which the asymptotic value of the axion is in fact 3 is suppressed by ( k to the extra sourceof of energy the density axion from on the the axion background field geometry strength. canjust be But the this neglected Yang-Mills backreaction thetawhereas Newton’s angle, constant we scales as have an axion kinetic term with no powers of the string coupling, at large corresponds to a constant axion field-strengthof and motions is in an the exact fixed solution geometry toof of the motion axion ( equation also include Einstein’s equations, which require us to change the geometry due of a disc. Forthat it to smoothly terminateradius at linearly varying The metric can be written as where ometry spanned by the two coordinates for fermions and a linearlydual growing for axion plays out inby the the dual doubly-Wick geometry. rotated The AdS-Schwarzschild holographic geometry, also known as the AdS-soliton. Figure 1. boundary conditions for fermionsthe and 3d a Minkowski part linear of thetaboundary the to angle. spacetime. the Not Hall The D7 displayedcorresponds branes are to at the an the internal alternate tip D5/anti-D5 of embedding the that cigar is are also introduced in discussed section in section JHEP12(2017)031 ] f ]. as n has 42 47 2 1 (2.4) , x 2 probe . This R . These H N 1 and 1 x k , t -direction and the r ]. Flavor branes change , the theory is dominated 41 ] are localized in the radial embedding is quite distinct /L 42 5 for the properly normalized energy. S . This source is provided by the 0 space is compact, the low energy CS theory. In the × r KK CS theory in the infrared. To un- 5 1 , m N S = 2 ) theory. Correspondingly, the probe The D7 worldvolume theory includes k ) , , as also displayed in figure k k − r R 0 4 1 ) L r N units of background 5-form flux ∼ . But note that the Minkowski metric on = 5 2 S r CS theory, realized as a low-energy limit of 0 N r R k . Their − – 7 – ∼ ) N ) is that the constant axion field strength requires N to the KK 2.3 ) gauge theory. worldvolume of the D7 branes, the mass of these extra modes is m H k CS gauge theory. To conclude, the construction of [ k plane, that is at 5 S ∧ 3 N × x ) F 1 k , − 2 ∧ for the worldvolume U( R r again corresponds to an energy A N ]. The D5s intersect the color D3 branes along a 2+1 dimensional /R 49 ] for a review). Having realized level/rank duality in holography, we probe D7 branes located at 48 , the compactification radius of the k red-shift factor. Rescaling time, and hence energy, so that if the Minkowski metric is prop- /R 2 /R ] (see [ 2 0 r 41 The field theory was engineered to give SU( From the point of view of the = 4 on a circle, is in fact its level/rank dual U( 4 flavor D5 branes as in [ simply of order 1 an overall erly normalized the energy 1 Adding flavors to holographybranes can [ be accomplishedcan use straightforwardly this using strategy flavor tohere probe explicitly is add that flavors this to yields level/rank 3d duality. bosonization What as we aim anticipated. to We show add matter by introducing N limit, the infrared limit of the holographic duality is level/rank duality. 3 Adding flavors description is a 2+1 dimensionalWZ U( couplings of theinduces form a CS term ofbrane level theory is alsolimit gapped, is as simply it a has topologicalshows to U( that be the to holographic agree dual with of the SU( field theory. Its low energy reproduces field theory expectations.by So the at Hall energiesprobe probe below branes branes. 1 receive asame Due mass is as to expected they for the the need cigar fermions. to geometry, Since fluctuate the scalar “up” internal into fluctuations the of the Hall derstand how this isbulk. encoded in First the note bulk,The that we corresponding all need mass to IIB gap identify supergravity the modes low in energy the limit cigar in geometry the are gapped [ from the flavor probethe branes matter that content arise of inHall the flavored probe holography field branes [ theory; that theydirection. support reach the They to are axion asymptotic simply in a infinity.the the source In that spatially solution can contrast, varying of support field the [ the theory axion field theta strength angle. which encodes a source at the originintroduction in of the “Hall” probe branes spanwell as the the three entire non-compactprobe internal field sphere. branes theory As can directions for be the neglected axion for field strength, the backreaction of the A second problem with the solution ( JHEP12(2017)031 ) ) 2 1 ∗ k r r f S − ( n r = 4 χ × (3.1) 5 5 2 4 S S S N color D3 direction r x 3 N geometry. x 5 o o o S anti-D5 branes ; turning them × R f n The resulting gauge ) will develop a non- r 6 ( χ gives the corresponding 0 1 2 3 x xx x x x 2 flavor D5 branes intersect − f r , n Embedding of the probe branes 2 S D7 D5 , that is (b) in the AdS-soliton = 4 SYM. 2 χ g S 2 N 5 R-symmetry corresponding to rotations + sin 2 R . The D5 worldvolume gauge field is dual S 2 symmetry manifest, defect hypers so as to preserve 3d S g f R χ g geometry. Writing the metric on the internal direction. n 2 U(1). In the bulk geometry the triplet mass – 8 – 5 SU(2) 3 S x × × ) = 0. For generic embeddings, the fall-off of SU(2) × r L L + cos ( 5 o o o × 2 χ L . We will also add a second stack of dχ with metric = 2 1a 5 S S g o o o o o o x x x ) SYM coupled to ]. In fact this theory is a prototypical example of a defect CFT. N o 51 , flavor D5 and anti-D5 branes. in b) the same 49 f n -invariant embedding = 4 SU( R ) flavor symmetry current. f Brane realization of the gauge theory. N 0 1 2 3 4 5 6 7 8 9 Brane setups realizing the field theory as well as its holographic dual. In a) x xx x x x x n ], where D6 branes (along 0123567) gave rise to 3+1 dimensional defect hypermultiplets in a 4+1 di- (a) 50 The supersymmetry-preserving mass terms are triplets under SU(2) Let us briefly review the physics of the D5 probes before we compactify the D3 D5 This construction is a lower-dimensional version of oneThe anti-D5s of alone the would give earliest rise to holographic the realizations same matter of content, but preserving the opposite half of the in a form that makes the SU(2) by contracting at the pole of the internal sphere. The undeformed theory corresponds 5 6 5 ∗ fermion bilinear condensate. The requirement that the braneQCD terminates [ smoothly at mensional gauge theory (living on D4 branes along 01234) compactifiedsupersymmetry. on a supersymmetry breaking circle. trivial profile. This “massive”r D5 brane smoothly terminatesto at the a SU(2) finite radialnear coordinate the boundary encodesterm the gives dual the field triplet theory mass, data. the coefficient The coefficient of of the the subleading leading the D5 branes wrapto the the first U( on breaks thedeformation R-symmetry causes the to D5 SU(2) brane to slip off the internal worldvolume in the backgroundS AdS theory is supersymmetry [ The field theory has ain manifest the SU(2) 456 and 789flavor planes symmetry. respectively. In In the addition, holographic the dual flavors can bulk, be the rotated probe by D5 a branes U( span an AdS separated from the D5 branes along the and turn on the linearadds theta a angle. whole The hypermultiplet’s D5theory, branes worth preserving are of half 4 fundamental ND of representation flavor the branes matter 16 and to so supersymmetries the each of one gauge Table 1. branes intersect Hall D7 branes. defect, as can be seen in table JHEP12(2017)031 - ), R and 2.2 ξ . Here R-symmetry, i doublet label, i ) worldvolume R ψ f R . A n , the breaking is ]. These SU(2) f R i n 50 e m/r SU(2) ,i † × = ξ geometry these massive L h χ 5 S is an SU(2) i breaking condensates, which × baryon number symmetries on R 5 with the cigar geometry of ( B 5 as has been explicitly demonstrated D5 branes with zero triplet mass and 5 S , all the SU(2) triplet condensates van- f R n . By charge conservation, the probe D5s 0 r = – 9 – r ]; they are given by sin U(1). The resulting Goldstone bosons correspond 41 × L SU(2) even for vanishing triplet mass, leading to a spontaneous smooth coincident branes with a single U( 0 → r f n R direction and turn on the linear theta angle, and correspondingly ≥ by slipping off in the internal space, we now can have embeddings 3 ∗ x ). This breaking is geometrically encoded by the D5s and anti-D5s 0 . This type of embedding is very familiar from the Sakai-Sugimoto r f 1 SU(2) n > r × ∗ U( singlet mass. This operator is not dual to a light supergravity or brane L r R → with the cigar geometry and turn on the constant axion field strength. The ) f 5 n ]. Since this embedding preserves SU(2) U( 32 × ) Here is where the introduction of the stack of anti-D5 branes helps. Instead of termi- As we will discuss in more detail below, the field theory also allowsThe task a at supersymmetry hand is to understand what happens to the extra flavor degrees of freedom f n denote fermions on the D5 and anti-D5 stacks respectively, the D5 and anti-D5U( stacks to theirconnecting diagonal into subgroup. a stackgauge of For field general but twowere asymptotic encoded regions. in the Unlike asymptotic the fall-off SU(2) of a bulk field (the slipping mode), the non-local model [ ish and the only non-vanishingψ condensate is the non-local singlet and a Wilson lineand inserted ensures that between the theunbroken, non-local two this bilinear fermions is condensate runs gauge-invariant. breaks from While the leaving one independent the defect U(1) R-symmetry to the other nating at a finite with no slipping mode turned on,where thereby the preserving D5 the and full anti-D5 SU(2) as smoothly connect displayed in in the figure bulk into a single “U-shaped” D5 brane only new light degreesunder of the freedom CS are gaugethe the fields TFT living Goldstone living on bosons, on the thetip which Hall Hall of do D7s. D7 the not We branes. cigar, need carryD5s with a and They charge new flavor Hall completely brane light D7s. decouple that degrees from can of reach freedom all arising the way from to the the intersection of flavor must terminate at an breaking of SU(2) to fluctuations of the D5in embedding the in the closely internal relatedbreaking embeddings case do of not D6 serve our defects purpose in of adding the matter D4/D6 to level/rank intersection duality. [ The 3.1 Bulk theory The correct low energystart physics with of this the side bulk ofit theory the is is duality. no easier Once longer we to ancondensate. replace understand, option AdS and to The have so geometry a we itself single will ends stack of at once we compactify the replace AdS supersymmetry breaking circle compactification will giveof mass to freedom. many of the flavor Weholographic degrees need bulk as to well identify as in the the light boundary degrees field theory. of freedom left behind, both in the embeddings can be found analytically [ breaking SU(2) fluctuation, but rather to a true stringy mode. It will play an important role below. fixes the condensate in terms of the mass. In the full AdS JHEP12(2017)031 ) f n we U( 1b . In this × ) , we get the 1 0 k r intersection. is the radius = 2 R t S r strictly above the × 1 0 , 2 ) breaking associated f R . So at energies below , where once again do not add n > r 0 t /R /R U( corresponds to field theory r 1 , → > r 2 ) CS gauge field. However, there R t f ), in terms of which the area functional . r r n N 8 t ( . ) r 3 ) 2 k x 0 U( t circle determines how far down the r x ( 2 × 3 f f ) fundamental scalars (in addition to the 4 x 2 between the stacks, the D5s reach all 4 t f − r r f ) 8 n t L/ r n r ) 1 + ( r . f ( f p f = 2 as displayed by the solid U in figure 2 p N / complex scalars in the hypermultiplet this gives – 10 – 7 p f 1 2 − f r N f ) n 2 r r ( f CS gauge theory lives on the Hall D7 branes localized we can integrate = L ∼ . On the sphere we only have conformal Killing spinors 0 3 ]. The area of the worldsheet computes the condensate. L 2 N to get the brane separation at infinity. Choosing x ) S t 52 k r ), albeit for even CS theory with : the turning point appears at a value 1.1 charged scalars coupled to the U( N 1 ) as discussed in the previous section), the field theory we are left with f k = 1 for this calculation) ) gauge field as in the Sakai-Sugimoto model. But once again they only n f R /L n = 0 which corresponds to the maximally separated case. 0 3 = 2 x f TFT and a decoupled set of Goldstone bosons, not a CS-matter theory. N N , we only want to keep the zero modes of fields on the internal space. For scalars ) 2 does not appear explicitly in k S 3 x (which, when accounting for the redshift factor of the In the special case of maximal separation What is the extra light matter? From the embedding as displayed in table The separation of the two stacks along the In formulae, the D5 embedding is characterized by a function 7 /R Since This can be integratedsimple for solution generic decoupled Goldstone bosons). This3d bosonization is pair exactly ( the field content of the bosonicis given side by of (setting the are no fermion zerowhich modes give on rise to1 2+1 dimensional fermions ofenergies mass below of 1 orderin 1 the bulk is a U( bifundamental hypermultiplet. ThisSince we hypermultiplet are lives only on interestedof in the the matter that hasthis mass is less the than singlet 1 s-wave,us so exactly from the 2 the way down to thecase bottom new of massless the matter is cigar, localized at the D5/D7can intersection! see that the D5/D7 intersection is 4 ND, so the localized matter is a U( to the smooth connection ofworldvolume the U( two brane stacks.give They us are neutral expected to light be fieldsadding encoded decoupled fundamental in flavors from to the the level/rank CSis duality. theory a Said and U( another do way, the not low-energy serve theory our purpose of embedding in figure tip of the cigar.at Since the the tip U( any of light the charged matter cigar, toworldvolume these the modes CS on generic the gauge embeddings U-shaped theory. D5exception Just with brane are like are the the gapped Goldstone gravity by bosons the modes, cigar from most geometry. the of The U( the only line, it is dualwith to the a string string endpoints worldsheet.the lying The Sakai-Sugimoto on model relevant worldsheet in the has [ U-shaped the D5, topology as of discussedU-shaped a for disc embedding the reaches. related case of The generic case is depicted by the dashed “alternate” operator that condenses here is not encoded in a local bulk field. Since it involves a Wilson JHEP12(2017)031 ξ ) in f (3.2) (3.3) . We U(1) n I × ψ symmetries. B symmetry as . In their pro- k B ≤ f N ) deformed by a potential ) is also accompanied by a 1.1 1.1 symmetry. Here we focus on ] that the basic bosonization ]. ξ 7 7 U(1) , invariance demands that the triplet , × ! J ψ R 2 3 ) global symmetry is broken down to Ψ ) flavor symmetry is manifest. This indi- f J m m f I N into a single 4-component fermion Ψ 1 ∗ 2 U(1) n i M m m ξ × † I U(

R – 11 – 2 identity matrix. An unbroken U(1) × and so it does not play any role in the discussion. ) = = Ψ and × f i J L m n ), or the manifest symmetry is enhanced to U(2 I ψ = 4 SYM vector multiplet, the hypermultiplet scalars SU(2) f L n M × N L U( × ) f ). The low-energy theory in this phase is a sigma model of n ), and in the latter, we would find the duality on the nose. κ takes the form = 0. Additionally, if our theory was time-reversal invariant, that U( − 1.1 2 2 M / → m flavors, only a U( f = 0 since these terms are odd under time-reversal. ) f 3 N f ) global symmetry of our model, however we would like to emphasize n f n m )-invariant potential, tuned to criticality. In our brane construction, while U( n f = × = 2 n U( is a Hermitian 4 by 4 matrix. SU(2) 1 ) ) may be extended slightly above the flavor bound f × κ J m N I ) 1.1 f = 1 case for illustration and the two Abelian factors are the two U(1) M n 2 + / ) = U(2 f f f Naively one might expect fermion masses to be protected by symmetry. In the ultravi- Komargodski and Seiberg have recently conjectured [ Our work is not yet done. The bosonic side of the duality ( n N N where each entry issymmetry multiplied would by set awould 2 set where masses vanish and so the The fermions are neutral underIf SU(2) this whole symmetry werestrained. unbroken, fermion Let mass us terms groupcan would the write indeed fermions the be most strongly general con- mass term invariant under the diagonal U(1) Once again, we arematter. not In interested both in the bulkis the and whether latter boundary light they since fermions yield they survive a in do decoupled the sector. not field So correspond theory. olet the to real our question charged field theory has a SU(2) To understand thestrong low coupling energy dynamics limit ofAs the on with gauge the the theory, adjoint and field scalars soare of theory we the expected are side, to on we somewhatsymmetries. all must shaky The become footing. understand only massive candidates the via for light loop matter corrections are the since fermions they and Goldstone are bosons. unprotected by the U( that this global symmetryunrelated to is the present symmetry both breaking above pattern suggested and in3.2 below [ the flavor Boundary bound, theory and so is of the original duality ( duality ( posal a newU( phase appearsthe in modes which of the the symmetry U( breaking. This broken global symmetry somewhat resembles U( there are cates that our construction eitherwhich realizes breaks the U(2 bosonic side ofthe ( infrared. Both cases are interesting. In the former, we would end up with a deformation JHEP12(2017)031 ) f n /N (3.4) (1 and an angle. = 2 O ) gauge θ f PT N N even in the , so we know N positive mass ) B k f n level/rank dual. ) with generic real U(1) ) come in pairs and N ) × 3.3 k 3.3 R . CS TFT at low energies. − 2 | f k massless fermions together 2 − N f ) m | n N + 4 light fermions all of the effective light fermions we must tune one 2 f f ) 3 n n m massless fermions. So this has to be the − . This symmetry is only approximate to 1 f 1 n m m ( massless fermions we see that the two massive = p = 2 – 12 – 3 f f m n 3 N . m f n + 1 + symmetry at maximal separation. It is the combination of m k 2 is spontaneously broken by the chiral condensate in the U- − breaking triplet masses in the UV. While we can not follow = . The eigenvalues of the mass matrix ( Z ξ 2 R to m , whereas in order to have 4 CS gauge theory in the infrared. Here we show that it is indeed m k 3 is even. 2 massless fermions. To have 2 U(1) f − m k N f 1 × + n k ψ m − ) light fermions. The extra light matter only arises for maximal separation of = CS gauge theory with massless fermions with positive masses for the remaining fermions, ending 2 when N f 2 2 topological field theory, or more precisely its U( f | n / L/ 2 For generic separation the bulk low energy physics remains U( n and complex f f = m N N 8 3 or even 4 | positive mass fermions gives us exactly the theory that 3d bosonization demands, + + x = 2 m k k f f f − − n n ) ) N If our field theory identification is correct, we should also be able to deform our CFT Here we give some additional evidence that this is the correct identification by studying At this point we have to appeal to the holographic dual to determine which of these The U shaped embeddings we are interested in preserve SU(2) But time reversal is not a symmetry once we turn on the linearly varying N N and There is also an approximate 8 1 = 0 and SU( exchange of the D5sand and is anti-D5s, broken and by itx the sets linear theta angle, but it is exact for certain values of D5 and anti-D5 locations, like up with a SU( possible to describe this phase fromwe the probe can brane turn side as on well.explicitly the In the order SU(2) to RG reach this flow phase masses, in we the can field analyze theory the to dual see probe how this brane affects embedding the to effective confirm IR that fermion it describes an the stacks. presence of the flavorlight D5 branes. fermions a This phase negativefermions appears mass, we to already cancelling accounted correspond the for giving above CS our and contribution leaving of a SU( thewith 2 massive deformations of the CFT.and anti-D5 First stacks, of the all U-shapedHall note, D7 that D5 branes for brane at all. generic embeddingorder separations This in to is of the get consistent the 2 bulk with D5 the does fact not that we intersect have the to tune one parameter in options is the correct numberreassuring of to light note fermions that pickedwith among by 2 the the dynamics three of optionsSU( the we theory. have, It 2 is right option. coupling, mass terms to vanish.fermions For have the both case positive oftheory mass 2 is and shifted so, from in this phase, the CS level of the SU( m are given by So for generic masseshave we 2 have no massless fermions, but for specially tuned cases we can Furthermore, U(1) shaped embedding. So thebelow real question the is scale how of many chiral massless symmetry fermions breaking. remain at energies that the effective mass matrix in the infrared has to take the form ( JHEP12(2017)031 f n at a 0 r fermions f N charged scalars, f U-shaped D5s that n f n , or they reach = 2 0 f N > r ∗ r ) generators. This color-flavor massless fermions requires four nf f , at least assuming the Higgsing is n N ) f negative mass. But given that we only N scalars. The hypermultiplet is still living f − n . Nevertheless, this does not change the low f k 2 N S ], which, regrettably, does not apply here as supersym- – 13 – as the right number of massless fermions in the 25 f with n N massless fermions. ) f k n . The transition to the Minkowski embeddings is first order. ) flavor and the broken U(2 ∗ f ) flavor symmetry in this phase is given by a linear combination m n f a positive mass and n ). There are two qualitatively different embeddings that one could U( . If we give both D5 and anti-D5 the same triplet mass, the probes r f 0 ( but a slightly smaller n r × χ radius is reduced compared to the massless embedding, but this does 2 ) 2 S f S n . The theory can be Higgsed down to U( The global U( 2 S 9 N ) × k 1 , 2 For generic small triplet masses, the D5s and anti-D5s have to end on the Hall D7 Turning on the triplet mass corresponds to turning on a non-trivial boundary condition The maximal Higgsing recalls the S-rule of [ R 9 tunings (since all fourconjecture real that mass we indeed parameters have need 2 to vanish), it is much more naturalmetry is to completely broken.preferred. Nevertheless we assume that this symmetry breaking pattern is energetically confidence that our identificationfield of theory 2 description is correct.having Certainly no the light existence fermions. of In thisfermions principle phase and we is giving could inconsistent with also 3 tuned obtain this one phase parameter by (the starting stack with separation), 4 whereas 4 of the original U( locking leads tobackground a gauge flavor fields CS associatedCS term with this term in global is the flavor exactlypositive symmetry. low-energy what masses. The theory, one The resulting that fact would flavor that is get we can in a find the contact this dual phase term in description the for from holographic dual giving gives us great split on the D7s.phase. Splitting More all precisely, of this theD5/D7 exactly flavor intersection corresponds D5s a to across vacuum giving thethe expectation the Hall U( value. hypermultiplet D7s As living takes at there usmaximal. the are to a massive not change the physicsfinely of tuned the choice scalar of zero-mode. equalthe D5 low So energy and physics. anti-D5 at masses the is face irrelevant of from it the point itbranes of since appears they view that hit of this it at different locations. We can think of them the tip of the cigarcan at still reconnect into awrap U-shape, an even equatorial though now atenergy the physics, bottom which of remains theon U U( they no longer the Minkowski embedding has largerabove action a finite and triplet only mass becomesWe give the the preferred details configuration ofmeans the corresponding that probe as brane farpossible analysis as in deformations the the of appendix. embeddings ourThe with For CFT, us only small we embeddings this mass of can interest are completely are concerned, the ignore ones which the where describe Minkowski the embeddings. D5 the and anti-D5 probe branes reach for the slipping mode imagine: the D5 branesfinite either angle slip and off either the endin internal on sphere the the at probe D7s brane orconfirm meet literature from up a we with numerical call the analysis the anti-D5s. of former In the parlance “Minkowski worldvolume common action embeddings.” that for It small is triplet easy masses to JHEP12(2017)031 ] , ) k + − 53 ) 1.2 ) we N 1.3 and O5 . − ) ) into string F π 2 ) and ( fermions arises tr( 1.1 ∗ f on the color D3s 1.2 gauge theory with N = N ). Fortunately this j N ) ) N k ) k k and SO( k − ) N gauge theory with 2 / ) flavor symmetry the gauge groups are f f scalars, baryon number is gauged and so N fermions, we have a U(1) global symmetry n f + f k N − N ) – 14 – N ) gauge group on the D5 brane respectively. These f n CS theory with CS theory with ) or USp(2 . The D5 matter branes again add purely bosonic matter in the f 2 N N / ) . n ) f f k k N n + k ) flavor symmetry we obtain SO( − = 2 ) f f n N N and USp(2 k − ) ) with N 1.3 Lo and behold, we see that holographic duality at low energies in this particular geom- bosons describes the only light degrees of freedom of the dual string and probe brane f gave rise to SU( that is simply baryon number.the It bulk acts low on energy theit matter U( is fields by not an a overallhave physical phase a global rotation. U(1) symmetry. In monopole All number physical whose states identically conserved are current neutral is under it. Instead we and ( 4.2 Baryons andOne monopoles of the more interesting aspectsside of 3d of bosonization the is that duality it to maps monopole baryon number number on on one the other side. On the field theory side, which is, for an USp(2 and Hall D7s respectively,USp(2 whereas for the SO( bulk and fermionic matterhas to the the same boundary pattern fieldwhich as gives theory. in the extra the Assuming shift unitary that in case, the the Chern-Simons we mass level again to matrix exactly give reproduce masses the to dualities ( half of the fermions, what is the resultingbefore gauge group orientifolding) on and theanalysis the D3 is branes D7 simple, (which since branes started bothwith out (which the the with D3 realized and D5s. an the the SU( D7 Inthat U( have we this a obtain case relative the number it opposite of has 4 projection been ND for directions well the known D3s e.g and from D7s than the we D1-D5-D9 do system for of the [ D5s. That theory as a holographiccan duality. augment To the similarly construction with obtainorientifold O5 orientifolds. the plane on The real top most dualities ofwhich the natural ( give U-shaped idea rise D5-branes. is to We an to have two SO( symmetry superpose choices, O5 groups an appear as a global flavor symmetry in our duality. The real question is construction to re-derive generalizationsthe of details 3d of the bosonization operator to mapping. SO/Sp We groups,4.1 also comment as on well potential Orientifolds as future applications. In the previous section we embedded the unitary 3d bosonization duality ( 4 Applications and discussion Having successfully embedded 3d bosonization dualitiesuse into our string construction theory, to we gain wouldory new like construction to insights. should The make intuitive geometric itof nature easy the of to duality. the understand string In new the- this deformations section and we generalizations present two applications where we use the string theory etry reduces to 3d bosonization.as The the SU( low energy limitN of the boundary field theory,description. whereas the U( JHEP12(2017)031 strings flavors, f N ] a baryon N 54 scalars, so there strings attached. f n carry charge under N ψ = 2 f N ) gauge so that it sources one k fundamental fermions with color scalars provided by the N N sitting at the bottom of the cigar and 5 S Chern-Simons theory are a single unit of N ) k , built from – 15 – fundamental charges attached to it in order to N fundamental fields, so that the whole operator is gauge- ψψ . . . ψ ] in order to study defects in 2+1 dimensional CS theories ∼ N 55 such baryons. N ) f different monopoles that one can get this way. As for the dual baryon, N . By charge conservation the baryon vertex requires N 2 ) f R . The monopole needs N } 0 , each fermion can be in one of the two spin states and be one of the ..., has recently been used [ across which the CStransform level under jumps. 3d bosonization It once flavors would have be been interesting added. to seefactor how on these both defects sides. By gauging background fields, 3d bosonization dualities in- anomaly matching. Sooperates it in the is presencerealization a of of very boundaries pure natural CS and question theory defects. via to compactified Exactly ask D3 the how branes same 3d we holographic also bosonization employed in here 0 The simple bosonization dualities discussed in this work involve a single gauge group CS gauge theories in 2+1 dimensions necessarily have non-trivial edge states by , This matching of monopoles and baryons can be further solidified by looking at all The baryon vertex is exactly what we need to realize a monopole in the bosonic dual. How can we see that operators charged under baryon number turn into monopoles 0 2. 1. , 1 further questions for which we believe the stringy embedding will serve as a useful tool: ψψ . . . ψ also leading to (2 4.3 Future applications We have demonstrated two useful applications of our construction above. There are many { make a gauge-invariant operator; in theending bulk on these the are vertex, runningharmonic between in the a vertex monopole andare background the is really flavor D5s. spin-1/2, (2 and But there the are basic scalar is a monopole operator in the D7-brane gauge theory. the quantum numbers. Let’sOnce take it a is single dissolved D5unit into vertex of since the magnetic this Hall flux is D7s, on the we a simplest can single monopole. pick D7; a more U( precisely, its GNO charges are equivalent to The simplest monopole operatorsmagnetic flux in accompanied a by U( invariant. Now, note thatD7s. the The baryon baryon-vertex D5 D5 andother is Hall and D7s completely the form embedded D5 a 2 dissolves inside ND itself the system. inside Hall the Correspondingly they D7s attract as each a unit magnetic flux. So the baryon D5 at a point in Without flavor branes thesebaryon strings an would infinite have mass. todual run But operator out for is to us indeed theanti-D5 these a boundary branes. strings baryon and can built give end from the on the the defect probe fermions D5’s, coming from so the that D5 the and under the duality?the In U(1), the but fieldthe the theory, name only the suggests, gauge fundamental the fermions indices invariant baryon antisymmetrized. operator It that is carriesvertex, well that U(1) known is, that baryon a the charge D5 bulk is, brane dual as wrapping of the a entire baryon is [ JHEP12(2017)031 ]. 56 D7 branes as f N gauge theory with a known N massless Dirac fermions localized f N angle hence means we are turning on an 3+1-dimensional hypermultiplets. Then can be neglected, but D6 brane sources θ f 12 N F – 16 – , dual to 3 S , where 12 denotes the two circle directions. In the probe × 5 have to be introduced at the tip of the cigar. The low 12 F 4 fibered over a 3d geometry that has the topology of a cigar S 1 , 2 M ]. The string theory embeddings presented in here may provide D4 branes. We can compactify two directions on circles, one with 57 N and an we expect that at strong coupling there is a “chiral symmetry breaking” 4 ] we can use these D7 branes to make a “fractional topological insulator.” S 3.2 58 ], extended along AdS 41 negative on one halfhalf, of and therefore the vanishes field onof antipodal theory the points modes circle, of of the and this circle. theory largeto are At gapped domain and weak except coupling, walls positive for most at onsection antipodal the points other of the circle. Using the same logic we used in which can realize anstudied odd in this number paper, but of toanti-D5 flavors. add flavors brane in One probes, a different option dual way.in is Rather to [ than to defect adding D5 take hypermultiplets, and the weas can same in add setup [ That is, we turn on a position-dependent hypermultiplet mass which is large and realize more dualities ofother “flavored than topological 2+1. field theories,” maybe in dimensions number of flavors, and it would be useful to identify a different stringy embedding RR 2-form field strength limit, the backreaction ofwrapping the the constant internal energy physics of this typewe IIA discussed here. construction appears It identical would to be the interesting to IIB see description if with similar constructions we can periodic and one with anti-periodicinternal boundary conditions. Thetimes dual a geometry circle. has an Aftera the 3+1 first dimensional circle gauge compactification theoryRR we whose 1-form. theta can Turning angle view on is the a given spatially field by varying theory the as Wilson line of the theory as the lowholographic energy dual. limit Clearly of this a UVstringy completion full-fledged descriptions is large one not unique. couldthese pursue, So there dualities. potentially are giving many yet One other worldvolume another obvious of perspective alternative on starting point is the theory living on the dualities or make contactsymmetric with case known ones [ asguiding has principles been on demonstrated how in to do the this super- when supersymmetry is broken. String theory may be ableperhaps to help reproduce these organize more patternstheory complicated among dualities on as them. a well stack and matter Relatedly, of theory, given M2-branes perhaps that at these the anM-theory. quiver low-energy orbifold bosonization singularity dualities is can a also quiver be Chern-Simons embedded in volving product gauge groups with bifundamental matter can be constructed [ Relatedly, our construction only realizes the 3d bosonization dualities with an even Our stringy embedding of bosonization relied on realizing a Chern-Simons matter Compactifying 3d bosonization to lower dimensions should allow one to derive new 5. 4. 3. JHEP12(2017)031 . ∗ m appears in our ) Chern-Simons only Minkowski k N 5 S ≤ × f ) we want to map out 5 N r ( χ ) symmetry breaking, while the f n U( , black hole embeddings become possible → ∗ ) m f n U( – 17 – × ) f n D7 branes located at the tip of the cigar, and indeed, D5s ) which gaps half the fermions, leaving SU( k f fermions in the deep infrared. However we cannot be sure if N f N U( is not a Euclidean horizon but rather the bottom of the cigar. A D5 → 0 ) r by slipping off the internal sphere, as well as “black hole” embeddings f 0 N = r U( ). We consider both “Minkowski embeddings” where the D5 smoothly ends > r × ∗ ) 2.2 r f , the dominant solution is always a black hole embedding. Correspondingly the ∗ N m theory coupled to this is indeed whatdual happens D7 without brane numerically probes solving in for the cigar the geometry. embeddings of the construction. U( Finally, we would like to understand how the flavor bound To find the embeddings we need to solve the equations of motion for the slipping mode. By solving the equations of motion for the slipping mode 6. while Minkowski embeddings also still exist.below Somewhat surprisingly we find that, fortransition masses from the topological phase to gapless phase isSince first we order do and not happens turn on at any worldvolume gauge fields, the action for the slipping mode is embeddings exist. Iffind more out than which one onethe embedding regulated, corresponds is on-shell, possible to Euclidean foronly a action Minkowski a lower of embeddings given free the exist. mass energy. D5embeddings we This probe. exist need is To and We do to to findnegligible be the so, at that expected; large effects we for mass. in need large of Below AdS masses a to the critical evaluate blackening mass, function defining the cigar become “black hole” embeddings correspond to5-7 strings. a gapless We refer phase to withboth these light as phases modes the have “topological” coming a and from decoupled “gapless” the sigma phases. model (Strictly speaking, and sothe are phase gapless.) diagram of the theory. That is, determine the range of masses for which the brane cannot simply endthe there main unless text, there our iscan setup an end has object on D7s. for So itcorrespond “black to to hole” a end embeddings phase are on. where consistent.for the As The the low “Minkowski discussed energy embeddings” Goldstone in theory bosons is a of TFT the and a U( decoupled sigma model geometry ( at a finite where the D5 branecase, reaches but the follows tip theof of standard the naming the cigar conventions cigar. at in the The probe latter brane is literature. really The a tip misnomer in our grant numbers DE-SC0013682 and DE-SC0011637. A Numerical analysis ofIn a D5 this probe appendix brane we on solve the the cigar geometry equations of motion for a single D5 brane in the cigar Acknowledgments We would like to thank Francescoof Benini KJ and Nati and Seiberg AK for was useful supported discussions. in The part work by the US Department of Energy respectively under JHEP12(2017)031 T (A.1) 0.8 by reaching 0.6 ∗ and (minus ) is unity. r 1 . For Minkowski − 2.2 r r m 0.4 ) goes to infinity at this r ( 0 ) at large χ r . ( 0.2 2 appearing in ( 0 χ 0 χ r 2 Free energy as a function of mass. r , but to be precise one also needs an extra (b) ) + 0.0 gives the condensate as a function of r 1.4 1.2 1.0 0.8 0.6 0.4 /L ( F T 1 and so fixes the condensate as a function 2a − 0.5 only Minkowski embeddings exist, but ∗ f r 10 p χ – 18 – 2 . Figure 2 cos 2 of about ∗ 0.8 T r m − shows that as soon as the black hole embedding exists, its = L in the asymptotic expansions of 2b 0.6 2 − ). This will not be important for us here so we ignore these conversion factors r π triplet condensate and free energy as a function of triplet mass in the topo- m (2 R λ/ 0.4 ]. Figure √ 59 ). This gives the flavor mass in units of 1 . The mass and condensate are given by the coefficient of r 2 ( S χ 0.2 The SU(2) 2. The absence of a conical singularity requires that to the gapless phase. π/ ∗ Condensate as a function of mass. m To determine which configuration is the preferred one we want to determine the free Our results are displayed in figure fall-off in Strictly speaking, when we are referring to mass here, we simply display the coefficient of the leading ) = (a) 0.0 = is the effective brane tension; it is equal to the standard D5 tension times the volume of r 1 10 0.0 ( -0.2 -0.4 -0.6 -0.8 − of the system ism given by the black hole embedding, andr the system hasconversion factor a of transitionin at what follows. renormalized by integrating thesuitable action local up to counterterms. aembeddings The “cut-off in slice” [ counterterms at havecorresponding large been free radius worked energy and out is adding range for always of the below masses D5 that we can brane of always the ignore Minkowski the embedding. Minkowski embeddings. So The in true this ground state mass. We see that abovefor a masses mass below this both Minkowski and black holeenergy embeddings corresponding are to the possible. two possibleEuclidean solutions. on-shell action. For this As we simply usual, need the to on-shell calculate action the is infinite but can be regulated and χ point. This picks a uniqueof solution mass. for a For given theon black the hole D7 brane. embeddings we This require also picks that a the unique D5 solution brane for ends a orthogonally given mass. T the internal times) the coefficient of embeddings we require that the embedding smoothly truncates at a finite logical (black) and gaplessone (red) whenever phases it respectively. exists. The We gapless work phase in dominates units the where topological the parameter simply the area of the D5 brane. The corresponding Lagrangian density is Figure 2. c JHEP12(2017)031 09 01 ] 02 ]. 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