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JHEP08(2018)004 f N Springer July 21, 2018 June 13, 2018 August 2, 2018 : : : SCFTs there is a moduli Accepted Received < N = 1 Published f N N , a ) gauge theory coupled to N = 1 time reversal invariant theories. Published for SISSA by https://doi.org/10.1007/JHEP08(2018)004 N = 1 supersymmetry. We argue that this = 1 SU( N N and Jingxiang Wu b,c [email protected] , = 2 U(1) gauge theory coupled to two chiral superﬁelds of charge . N 3 Zohar Komargodski 1804.02018 a The Authors. c Duality in Gauge Field Theories, Nonperturbative Eﬀects, Supersymmetric

We study the dynamics of certain 3d , [email protected] Simons Center for GeometryStony and Brook, Physics, NY, Stony U.S.A. Brook University, E-mail: [email protected] Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo,Department ON of N2L Particle 2Y5, Physics Canada Rehovot and 76100, Astrophysics, Israel Weizmann Institute of Science, b c a Open Access Article funded by SCOAP Gauge Theory, Supersymmetry and Duality ArXiv ePrint: some brief comments aboutquarks the in dynamics a of timespace reversal of invariant vacua fashion. to We all argue orders that in for perturbation theoryKeywords: but it is non-perturbatively lifted. surprising facts about 1. This theorypropose is a claimed dual Wess-Zumino to description have (i.e.fields) a emergent with theory manifest SU(3) of SU(3) global scalars symmetry andWess-Zumino symmetry but model fermions in only but must have no the enhanced gauge infrared. supersymmetry in We the infrared. Finally, we make Such theories often have exact modulidualities spaces and of we supersymmetric test vacua. theseground We propose proposals states. several by First, comparing we thein consider deformations the a infrared and theory and supersymmetric where there timehas exists reversal a (nonetheless) symmetry dual an is description exact only moduli emergent with space manifest of time vacua. reversal This symmetry. theory Second, we consider some Abstract: Davide Gaiotto, Curious aspects of three-dimensional JHEP08(2018)004 22 . The models ], where it was 21 W 17 4 6 quarks 9 f . Since one does not have N W ) 12 15 = 1 supersymmetry, namely, two 4.2 ( ]. This is due to the holomorphic N 3 – 1 3 24 – 1 – gauge theory with = 2 gauge theory = 2 QED and supersymmetry enhance- 0 12 ) N N N = 2 dualities N 5 model 1 = 1 theories often have walls in parameter space and it is possible to obtain ABC N = 1 duality between SQED and a Wess-Zumino model ] do not have time reversal symmetry due to various Chern-Simons terms. Some 4 can receive corrections since it is easy to write real functions invariant under the N = 1 supersymmetric SU( = 1 Abelian gauge theory with a charge 2 (super)field W Here we will study 2+1 dimensional theories with N ment in a Wess-Zumino4.1 model Massive deformations4.2 of the Wess-Zumino Massive model deformations of the 2.1 The N 3.1 A dual description with emergent time reversal symmetry that global symmeties. Oneshown aspect that of thisexact problem results was near these recently walls studied usingstudied only [ in the [ leading radiative correction to states. In such theoriesmanifolds. with four Understanding supercharges, the these space spaces ofthe ground are behavior states complex near is (in various a fact, interesting crucial singular Kähler) step points. before one canreal study supercharges. These theoriescomplex have analysis a at real one’s superpotential, disposal, it is typically hard to find exact results. It would seem In theories with four superchargesthere (both in are 2+1 powerful and non-renormalization 3+1nature dimensions) of theorems it the [ superpotential. is well TheseIn known non-renormalization that particular, theorems have they many allow applications. in many cases to determine the space of supersymmetric ground 1 Introduction 6 A Detailed analysis of the Wess-ZuminoB model Further checks of the 5 An 3 4 Symmetry enhancement in Contents 1 Introduction 2 The action of time reversal symmetry JHEP08(2018)004 = 1 = 1 N TQFT N 2 cannot be is an W W , = 2 versions and we 3 N = 2 supersymmetry arises = TrΦ N W + charge 1 2 = 1 and = 1 supersymmetry were recently / 3 N N = 1 = 2 SQED theory and a Wess-Zumino U(1) with superpotential N ⊗ 2 is severely restricted, and often ←→ N ABC . W – 2 – = U(1) A, B, C W = 1 version (where the Wess-Zumino model has 7 ←→ N = 2 description of it. The = 1 models with time reversal symmetry. Such models often = 1 symmetry is manifest in the flow N N N + charge 2 0 ]. Here we will review this fact and develop some applications of it. 6 = 2 U(1) + 2 charge 1 U(1) N = 1 SuperConformal Field Theory (SCFT). 2. Therefore, loosely speaking (the details will be presented in the main N / ]. 5 This duality also has a purely We then discuss a new duality between an Next, we will study the theory of a charge 2 superfield coupled to a U(1) gauge field As a simple example, consider the model of 3 real scalars and three Majorana fermions, Here we will study some -symmetry) in the infrared and on the left hand side there is emergent SU(3). R real scalar fields rather thanno 8), enhanced but in supersymmetry. that However, caseacross there there the is is duality. a no enhanced moduli global space symmetry of and vacua, which we match in the infrared and only an where Φ is insuperpotential. the adjoint We see of that SU(3) on (i.e. the 8 right real hand scalar side degrees there of is emergent freedom). supersymmetry (and symmetry. model. This duality haspoint two is surprising aspects. enhanced from First,model the but U(2) symmetry we to of do SU(3). the not infrared have Second, fixed an the dual theory is a Wess-Zumino like Time reversal symmetry in the dualin description the on infrared. the right Interestingly, the hand theory sidevacua on even is the though therefore right emergent it hand has side nomechanism has microscopic a that time moduli allows reversal space symmetry. of such We exact will moduli explain the spaces basic of vacua to exist without time reversal find a dual description intensored both with cases. a The dual chargeterm description 1 at consists superfield level of 3 a coupledbody pure to of U(1) a the paper), U(1) the gauge duality field is with a Chern-Simons We argue that theintersect full at theory an has a moduli space of vacuain consisting a of time 3 reversal real invariant fashion. lines that This theory has tential is a pseudo-scalar under timewas reversal symmetry. already This noted property in of [ the superpotential embedded into the three real superfields discussed in [ have exact real continuous manifoldsA (with related singularities) fact of is supersymmetric that the groundcorrected renormalization states. at of all. These non-renormalization theorems are due to the fact that the superpo- related aspects of Chern-Simons-matter theories with JHEP08(2018)004 f γ + . λ i N 0 γ (2.1) (2.2) (2.3) . The 0 ∂ 0 ¯ λλ γ i there is a 0 γ . As usual, − 0 0 γ γ = T T < N λ λ 2 λ 0 f symmetry) in the 1) γ ≡ N 0 − R γ ¯ λ )( 0 . i γ − T ! 1 iλ − ) = ( λ . 1 0 0 i 1 ∂ ) = 3 = 2 versions of the theory and i

σ . We deﬁne ¯ λλ ¯ λγ = N i = 1 supersymmetric non-Abelian i α = ( ( 3 λ 2 T T N )+ there is a moduli space of vacua to all λ , σ 0 , γ . Therefore, the kinetic term is time reversal +). We will denote the corresponding λ . ∂ , 0 λ 1 = 1 and 0 ! µ γ i σ + ) gauge theories minimally coupled to ∂ < N 0 ¯ , λγ µ − N i ) we use) = N ( f − i 0 ¯ λγ 1 T → i N – 3 – 2.3

= 1 time reversal invariant non-Abelian gauge λ = ) = : = λ is likewise a Lorentz invariant. In our conventions, λ N 2 µ i T ∂ ∂ λ , γ i µ µ 2 ∂ ¯ λγ is reversed): ¯ λγ i , σ µ iσ i = 2 versions of the theory, ﬁnding dualities in both cases, ( 0 + T ! x = ¯ λγ N λ 0 0 = 1 supersymmetry. We give some examples of applications of γ ∂ 0 1 1 0 0

N ¯ λγ i = 1 = = 1 and σ λ 0 N γ i 2 is a pseudo-scalar. In section 3 we discuss the theory of a charge 2 particle , ∂ i 1 , γ 0 W 0 γ γ 0 γ T λ ) gauge theories with time reversal symmetry. We show that for 1) − A Majorana spinor is a real two dimensional vector Time reversal symmetry acts as follows (in addition to the obvious action on space-time The outline of this note is as follows. In section 2 we explain how time reversal symme- We close with brief remarks on N is a Lorentz invariant and The action on the kinetic term is )( 1 i is anti-linear. This will be used throughout below. The Majorana mass term is odd under − ¯ time reversal symmetry (whatever sign inkinetic ( term is of course even under time reversal symmetry. ( invariant. coordinates, where the sign of One is free to chooseT the sign in this transformation rule. It is important to remember that λλ when we add these terms to the action, they both have to be multiplied by a factor of Below we will use thematrices Lorentzian by signature ( 2 The action ofWe time take reversal the symmetry sigma matrices to be as usual On the other side of theinfrared. duality, we In find section enhanced 5 supersymmetrySU( (and we make some comments about moduli space of vacuanon-perturbatively to on all this orders moduli in space. perturbation theory but supersymmetry is broken the fact that coupled to a U(1) gaugefind field. dual We descriptions discuss the innon-supersymmetric both cases. dualities. We In outlineand section the again 4 connection discuss of we these discussand dualities QED in to with particular, some in two the charge latter 1 case, particles, we find an enhanced global symmetry in the infrared. theories. More specifically,fundamental we multiplets. consider We show SU( thatorders for in perturbation theory but it is non-perturbatively lifted. try acts in the context of JHEP08(2018)004 (2.5) (2.6) (2.4) since there 2 ). Of course, 2.4 is a scalar, that is, R . are Majorana, time reversal ··· , containing only odd powers α + θ ... . 3 pseudo-scalar + R 3 . The Lagrangian is determined by + R θ ) = 0 0 . Suppose that + R

γ R ( R R to be a W. . Since the ) = θ W . α → 2 W R ) = θ d ) are necessarily odd and hence we cannot ( θ → − . Clearly we cannot write any time reversal R : R ( Z ( – 4 – R i T W W , then we can easily write analytic superpotentials = 2 supersymmetry in 2+1 dimensions, we can = 1 superpotential would be a pseudo-scalar but W R,W : = → R ): is odd under time reversal symmetry. Therefore to T N N → L θ R = 1 SUSY 2.3 2 R,W R : → − : id N T R T : will be denoted by → − ) which does not contain superspace derivatives T R R R ( : , W T W = 1 supersymmetry, the superspace consists of the usual coordinates N = 1 theory. The ) implies that N clearly needs to be invariant under all the global symmetries and gauge 2.3 is important for the theory to have a Hermitian Hamiltonian. As with the = 2 superpotential may or may not be a pseudo-scalar, depending on how i W N be a real superﬁeld (therefore it contains a real boson and a Majorana fermion). is a pseudo-scalar, R

R and thus preserving time reversal symmetry The most general case can always be analyzed byIf simply decomposing we the start superfields with a theory with If Let In theories with Below when we establish various non-renormalization theorems, except when mentioned explicitly, and the Majorana Grassmann coordinates 2 R µ the time reversal symmetry is defined to act. we are always concernedcomponents, with i.e. the potential terms terms.same in selection Of the rules course, under superpotential also time which the reversal do terms symmetry. not with superspace vanish for derivatives need constant to bottom obey the into real superfields and using the rules above. rewrite it as an the original (without superspace derivatives) such as of In particular, any timeexact reversal real invariant flat theory direction. of a real scalar superfield must have an invariant superpotential is no way towithout make supersymmetry it we odd could underaccompanying easily time any reversal write nontrivial symmetry suchwrite function as a a required potential, potential in consistent but with ( the fermionic terms In addition, symmetries. These conditions will turn out to be veryThe restrictive, bottom as component we of willunder see. time reversal symmetry The factor of fermion mass, ( write time reversal invariant theories we need x symmetry must act on themthe as real in superpotential ( JHEP08(2018)004 A → (2.7) A . If any = 0 and . In total, C with integer action A, B, C = n 2 . Such a term This is not yet 4 Z C A S 3 . A, B, C + , 4 4 ] potential. Let us 7 n S C B + n . A 4 acting on S 3 , S ) 2 C , . These are three real lines which + A 2 = 0 B + BC 2 A = = 0 and arbitrary (real) gABC . ( – 5 – are scalar superfields. All the other choices can B = AC = W = cannot be generated since it is even rather than odd B,C θ ABC A 2 , which is indeed invariant under the 4 d 4 AB C C and hence cannot lift the moduli space. Z + + 4 ∼ 4 = 0 and arbitrary (real) B B ABC + C + 4 and by the permutation symmetry 4 = A A C B and consider the superpotential model , → B B,C ABC A, B, C → − It is easy to prove that the classical moduli space continues to exist non-perturbatively This significantly restricts the vertices that can be generated as we integrate out high Now let us add time reversal symmetry into our considerations. Since the superpoten- The global (unitary) symmetries of the model are generated by the First let us consider the classical supersymmetric ground states. The equations for We thank M. RocekTo for see making that this it is observation. convenient to think about the generated vertices as some polynomials in is the coupling constant). This is an interacting, super-renormalizable model; it is 3 4 can be generated. We may, however, generate various new irrelevant operators such as A, B g by going on the moduli space and computing the Coleman-Weinberg [ of the terms does notit contain acts all on the all threethe the fields, time fields we reversal can in symmetry choose that and the vertex the time as unitary reversal scalars symmetries. symmetry and to We hence be thank such such M. that a Rocek vertex for must a violate discussion some of this combination point. of which is both irrelevantevery and vertex does that not is lifthas generated the to moduli as be space. we proportional perform to It the is renormalization indeed group easy transformations to see that useful). For example, under time reversal symmetry. Similarly,n no vertex of the sort tial has to be oddto under be time a reversal pseudo-scalar symmetry,be superfield we obtained and can by by composing no this loss time of generality reversal choose symmetry with energy mode in the ultraviolet (where, again, the original degrees of freedom are still this group has 24 elements.sufficiently constraining This as group turns this outmode by (near to itself the be would ultraviolet, where isomorphic allow the to new as original degrees vertices we of integrate such freedom out as are stillwould high useful) lift momentum to generate the moduli space. and hence there arearbitrary solutions (real) with intersect at the origin. Let us consider the fate of− this moduli space quantum mechanically. strongly coupled in the infrared. those are Let us considersuperfields a toy model which exhibits some of the ideas above.( Take three real 2.1 The JHEP08(2018)004 = N (3.1) we have there is a and in the m is arbitrary. + 4 ] and the non- S R A 11 ). Hence, there 2 | ) 5 Φ | gauge theory on the ( 2 divided by the gauge Z W

= 0, and is odd under time reversal = m ,C W and therefore these 4 vacua are = 0 3 + S B R = 1 supersymmetry. We can deform model. such that The theory is time reversal invariant, N = 1 SCFT. The model has no relevant ]. This is why we study the model of a U(1) . 2 6 . 10 + N 3 C – ABC R 8 S + ' 2 fields and we obtain some effective superpoten- −→ – 6 – ]. B vac 4 12 + to be a scalar under time reversal symmetry and S M 2 B,C A A requires some attention. At the origin of θ m depending on the sign of the mass. + 2 d R 2 1 R since the theory is arbitrarily weakly coupled there. ). We can choose A A ( eff W = 0 necessarily follows. (Time reversal symmetry is not spontaneously broken = eff = 1 Abelian gauge theory with a charge 2 (super)field W eff The exact vanishing of the superpotential shows that the time reversal invariant the- The most general superpotential is again some function The moduli space of 3 real lines meeting at a point therefore survives in the full N W We thank N. SeibergA for a single discussion charge of 1 this fermion point. cannot be coupled to a dynamical U(1) gauge field while preserving time 5 6 reversal symmetry because ofgauge field an coupled (ABJ-like) to a anomalycan charge [ 2 be multiplet, which maintained. sufferssupersymmetric from The no monopole-deformed such non-supersymmetric version anomaly in and version [ time of reversal symmetry this model was considered in [ The effective theory oncertain this SCFT. Let usthat now Φ consider has charge what 2 under happens the away gauge from symmetry, there the is an origin. unbroken Due to the fact degrees of freedom nearreversal the symmetry ultraviolet. is even We stronger see than that in in the thisory case has the a implication modulisymmetry. of space This time given leads by to arbitrary a expectation moduli values space of isomorphic Φ to the Chern-Simons level by is no way to writenon-renormalization a theorem: superpotential if that we preserves startthen time the from reversal superpotential a symmetry. vanishes superpotential This in that the leads full vanishes to quantum at a theory tree as level, we integrate out high energy Here we consider a U(1)1 gauge field supersymmetrically) (with to no Chern-Simons awhich term) charge coupled means 2 minimally that ( multiplet the Φ. group). Chern-Simons We are level using vanishes the (hence usual convention, the where subscript integrating out 0 a on charge 1 the fermion gauge shifts related by the symmetry breaking pattern 3 quantum theory. At the intersectiondeformations which there preserve is all an the symmetriesthe and model by the mass term but all the global5 symmetries gapped are trivial maintained. vacua.other For These 4 either 5 vacua positive gapped the or unbroken vacua negative symmetry split is into isomorphic one to that preserves Then, we can integrate outtial the heavy hence for large enough consider the branch, where, without loss of generality JHEP08(2018)004 ) 7 . 2 3.3 (3.3) (3.2) to be 2 while U(1) S matrix − k × is = 0 by the 2 0 it flows to m m . The origin is a m < 2 gauge theory. The = 0 is ill defined.) 2 matrix 0 we have at positive m Z U(1) k = 0) with the particular × λ λ > 2 and consider the theory as S . The tree-level superpotential ] of this S is time reversal even. Quantum 13 0 the situation is simply reversed. = 0. If ˜ ˜ ξ ξS . we have two ground states, one with m 4 + | λ < as spin TQFTs. 2 ) the effective theory consists classically 2 m Φ , . . | − . Time reversal symmetry forces 2 λ | ˜ 3.1 ! ! ˜ MS gauge theory is described by the M Φ + | 1 2 U(1) 2 2 0 2 2 0 2 2 2 0 | m Z ' − TQFT and of course also for

Φ and – 7 – 2 | 2 = 2 gauge theory is isomorphic to U(1) | ˜ ξ TQFT. For = = m Φ 2 2 | W k k Z = S as well as the TQFT U(1) + = W U(1) R × W 2 ) will be later compared to a dual description of this theory. gauge theory can be described by the 3.2 TQFT and for negative 2 Z 2 there is now no moduli space at λ parameterizing ρ is time reversal odd and the parameter it is +2 and therefore the Witten index jumps. It jumps at ˜ m M TQFT. It is important to remark that the Witten index at negative 0 it flows in the infrared to U(1) 2 TQFT and one with U(1) 2 These various phases as well as the moduli space of vacua (at To understand the duality in the next subsection it is useful to extend the theory ( It will prove useful to make a few comments about the model where we also add a Let us now break time reversal symmetry explicitly by adding a mass term in the This modification of m > Here and below we will use the fact that U(1) one vacuum with U(1) 7 parameter is given by and we study the modela as pseudo-scalar a (this function we of see from the first term in the superpotential) and as a result the TQFT on the moduli space ( even further. We adda another function neutral of real some superpotential scalar terms superfield for the superfield quartic term For fixed nonzero m U(1) For the U(1) at positive appearance of the exact moduli space. (Hence, the index at strictly singular point, where thereinvariant (see is footnote a 7), SCFT. which The is TQFT of course on important thesuperpotential for moduli the space consistency is of our time picture. reversal It is easyTherefore to we conclude see that on that theof moduli this space the ( modified modulus tum Chern-Simons level (which isChern-Simons why level the (in theory absolute is value) time is reversal 2. invariant), but the bare however, here we have aeasiest Dijkgraaf-Witten way modification to [ understand it is through the fact that our theory has a vanishing quan- moduli space. The most familiar version of JHEP08(2018)004 ˜ 2 2 ξ Z (3.4) TQFT. TQFT. are 2 2 ˜ ξ to be strictly TQFT in the U(1) U(1) and | 2 × ˜ × ˜ M 2 M | 2 = 2 supersymmetry TQFT. For negative N 2 2 ˜ ξS . (1) U + | . These are the same phases + 2 2 . ξ S ) ) ) | 2 + A B C S (1) (1) R ( ( ( 1 2 (1) U U ). When we take U − TQFT. The Witten index is constant . ˜ 3.3 ξ 2 2 =2 =2 I I ˜ =2+2 MS I TQFT and one with U(1) 1 2 2 − 1 ˜ M 2 – 8 – | =1 = Φ ˜ ˜ | 2 2 M M S we have two gapped SUSY vacua, one with the 2 (1) (1) = ˜ ξ U U (1) + + U we have two gapped SUSY vacua, each carrying a U(1) Loop 2 2 we have a SUSY vacuum with U(1) we have a gapped SUSY vacuum with U(1) ⇥ ˜ ξ − we have a gapped SUSY vacuum with U(1) ) ) ˜ ξ ) 2 ˜ 1. This is not surprising, since under time reversal symmetry ξ (1) (1) ˜ E F ξ the model clearly reduces to the minimal model we analyzed D U U ( ( = 0. The phases of the model as a function of ( (1) | is fixed). Tree+1 ⇥ ⇥ ˜ U ˜ . ˜ M M ˜ 2 2 M > ξ ξ | W 2 2 (1) (1) correction in the superpotential ( ) gauge theory and one with U(1) 2 =4 U U 4 =4 =4 I | (and I I Φ | ˜ M U(1) 1. For positive 1. For positive 1. For positive × − < 2 | → − symmetry) for ˜ ˜ ˜ ˜ M and later explain why this is sufficient for our purposes. The one-loop corrected TQFT. For negative The Witten index is constant as a function of M < we have two SUSY vacua, oneThe with Witten U(1) index isthat again we constant saw as for aM function of M > deep infrared. Forgauge negative theory (recall thatU(1) because of theas Dijkgraaf-Witten a term function it of is isomorphic to R | We draw all these phases in the following figure. A very important fact to note is that the model has enhanced S • • • before, with a infinite then we recover precisely the minimalsuperpotential model and of moduli a charge space 2 of multiplet with vacua a isomorphic vanishing to (and for superpotential is given by In the limit of large corrections to the superpotential are allowed. We will keep only the one-loop correction JHEP08(2018)004 ξ 0 1. as ∞ > = 2 S (3.5) = ∗ = TQFT ˜ N m M ˜ 2 M and some S TQFT. The 2 is to the right ˜ ξ TQFT. to be very large 2 | M we find precisely the | . | | S ˜ | M | S 2 g = 0, positive 1 4 ] it was shown that there is a ˜ 4 ξ − ξS . . ξS + λ . In [ 2 2 + | ˜ 2 Φ | S MS 2 . m 1 2 ξ ξ = + Mg 1 2 2 we have two trivial gapped SUSY vacua. The | W – 9 – ˜ Φ ξ − | 2 = 1 supersymmetric model of a charge 1 superfield | gS ˜ Φ we have a gapped SUSY vacuum with U(1) | N gauge field. We study the phases of the model as a : = symmetry. ξ . This model naturally has two deformation parameters, S gS 2 2 we have a trivial gapped SUSY vacuum. For negative = 0. We also see that at large | / R 3 ˜ = Tree ξ Φ λ | the Witten index does not jump, if we travel vertically on (the Chern Simons level) this model has enhanced W S ˜ ξ 3 4 ) the Witten index jumps and there are two walls at ) for positive and negative Loop = ˜ − M = 3.3 ) with W and the model reduces to the minimal model without One S M 3.3 . W 4 | ˜ Φ | 2. For positive is to the left. should be really identified (with a twist in the horizontal direction), as they / 2. Here for positive ˜ ξ 1 / 1 − −∞ = 0 (i.e. the Witten index jumps there due to the appearance of a new vacuum at ξ = m in the deep infrared.Witten index For is negative constant as a function of M < we have a trivial gappedWitten SUSY index vacuum is as constant well as as a a function vacuum with of U(1) M > ˜ M The various phases of the model are To study the critical points of this superpotential, it is important to add the leading Here we will generalize this model a little bit and add an additional neutral field While as a function of There are 6 regions, labeled A,B,C,D,E,F clockwise and in each region we record the • • and ˜ For a special choice supersymmetry as well as a U(1) nontrivial radiative correction for This model has nowe microscopic can time integrate out reversalquartic symmetry. coupling If we take wall at infinity) and due to awhere radiative two correction trivial there vacua is merge a into conformal a field theory supersymmetric at vacuum some well with as U(1) a superpotential M 3.1 A dual descriptionWe begin with with emergent a time quickΦ reversal review symmetry of coupled the minimally tofunction of a the U(1) mass in the superpotential and the (gapped) vacuakeep in just each the region one would loopand not correction change, in which the is superpotential. whydescribe An we the important are theory fact allowed ( is to phases that we have seen in ( vacua and their Witten indices.and The negative bold vertical line is at the diagram (i.e. change The walls may move as we include higher loop corrections, but the number of such regions JHEP08(2018)004 (3.6) is to the = 0) the ξ ξ = 0, positive . are walls where the index ξ 2, as we move horizontally = 2 supersymmetry and it 1 2 / T = 0 1 2 + N ) ) ) ξS 2 = (1) T A B C = ( ( ( + U (1) . For example, when we move from | M 1 2 M U S | 1 1 S =2 = 2 I = I g =2 I 1 4 M − 2 2 / 2 / 1 M S – 10 – =1 4 that SCFT has 2 = axis.) / g ξ M M 1 4 momentarily. These are the walls where the Witten M 1 2 = 3 ∓ we have a trivial gapped SUSY vacuum. The Witten TQFT appears from infinity. 2 the superpotential is 2 | ξ ˜ Φ 1 2 = ,M we have two gapped SUSY vacua, one trivial and one with | T = 0 is the naive line of SCFTs. (These SCFTs do not have ξ + T ξ M gS ) ) ) = 0 = 2 T + F E D = ξ ( ( ( (1) T M W U = 0 because the number of vacua and their low-energy properties is to the left. The dashed lines at 2+1 =1 ξ I ξ =1+1 . For negative = I 2 I 2. For positive / 1 < | M TQFT U(1) index is constant as a function of | On the two walls at Let us now make some observations — except for There are again 6 regions, labeled A,B,C,D,E,F clockwise and in each region we record We will discuss the lines • region F to E aD trivial a SUSY new vacuum SUSY disappears vacuum to with infinity U(1) and as we proceed into region on the diagram, the Witten indexsomewhere does not around jump. However, achange. SCFT must In nonetheless occur particular,can at studied in greatWitten detail. index clearly As jumps we at move the vertically two on walls at the diagram (away from the vacua and their Wittenright and indices. negative The boldmust vertical jump. line The is bold at to line be at all distinct as we travel on the the various phases wegapped have phases. described above along with the Witten index of each of the index jumps. Of course,various these phases and lines the may topology take of a the different phase shape diagram in are robust. the full It theory, is but useful to the again plot JHEP08(2018)004 1 + 4. S / 1 pure × (3.7) 2 2 = 3 = 0. In S [U(1) M , ξ gauge field. ⊗ 2 and / 0 2 1 3 2. The regions S − / TQFT) is dual to = (and vanishing Φ) = 0 and , = 1 2 are relevant. Clearly S ˜ M . M are not the same since ˜ M M . B = 0. This latter case is = 0, at the intersection ∞ M, gauge field by a U(1) , ξ , ξ = = 0 (and vanishing Φ) is asymp- 2 4 2 | / ξ / / 3 S ˜ 1 + charge 1 M + 2 − | = 3 / S 3 1) maps to | = 2 . − ←→ | M g = 1 version of the duality, the infrared + 1 2 2 = 0 there is a flat direction isomorphic U(1) = M R 2 wall for positive / N ξ / 1 − ˜ 1 + charge 2 fermion and U(1) ⊗ M − S − 2 0 2 = g = = 0 (tensored with a U(1) = 1 supersymmetry. Therefore, the duality we 1 2 – 11 – M U(1) ∓ , ξ N M ˜ Φ coupled to a U(1) 2 2 | = 2 duality in the appendix / S, ˜ 1 Φ ←→ | N − = 1 versions. In both cases in one of the duality frames = 0 and = , g N ˜ M = 0 is therefore conjectured to have emergent time reversal M = 2 supersymmetric theories (of which one has emergent time , ξ = 1 (or equivalently ˜ Φ = 0 2 + charge 2 ←→ / S N 0 ˜ 1 4 M = 2 and / − 2 wall the energy density for negative N / U(1) = = 3 = 1 M M M gauge field coupled to a charge 2 superfield with vanishing superpotential (i.e. ). Therefore, there is emergent time reversal symmetry at 0 ∞ . This moduli space of vacua is not an artefact of the leading order approximation. + = Note that starting from the duality that we established above It would be interesting to understand when the parameters Now simply tensor the theory of In particular, the point Note a remarkable fact: consider the point On the | = 1 supersymmetry while the latter R ˜ M we can imagine deformingside the and theories a bya giving corresponding non-supersymmetric a mass mass duality to to between the the U(1) bosons fermion on on the the left-hand right-hand side. This leads to N find here has both there is emergent time reversalSCFT symmetry. has In the a moduli space of vacua isomorphic to reversal symmetry) and itpartition can functions. be We study subjected this to stringent teststhe using SCFTs the at the former does not have a moduli space of vacua while the latter does. The former has | addition, we see that ABCDEF in the second figureAnother therefore interesting map special to case EFCDAB,This of respectively, this leads in duality the to is first emergenta the figure. time duality map reversal between between two symmetry at TQFT. Then, all the phasespriate we identification found with here the can phases be of seen the to charge exactly 2 coincide particleour after coupled the to U(1) appro- a U(1) of the regions B,C,D,E.The Around SCFT that at point theresymmetry. is an (As emergent we symmetrysymmetry will in of the see, reflections. infrared.) this is not the only point that has emergent time reversal to The walls exist in theway full that theory the and figure since can therewall. be is This consistent a is is phase that transition athe on a general absence the moduli mechanism of wall, space that the a opens can only microscopic up lead time at to reversal the moduli symmetry. SCFT spaces on of the SUSY vacua even in totically constant and similarlythe on energy the density is constant. In both cases at and the critical point equations are JHEP08(2018)004 (4.1) (4.2) ]. (3), satisfying 19 , su 18 Dirac fermion, in agree- ⊗ 1-form symmetry with a ’t transforming in the adjoint 2 2 = 2 supersymmetry (as well a ] (and references therein) for Z φ N 17 . – , a c 14 φ Φ (4.3) a φ b 8 the generators of m φ M a + φ c ,..., φ + Tr abc b d 3 φ 1 6 a = 1 φ – 12 – = 2 QED and supersymmetry enhance- = TrΦ . The superpotential becomes a a 2 3 abc , T d W a N a = 1 6 T SU(2). m +charge 2 fermion and U(1) × ] for details and to [ = W 0 + charge 1 boson is dual to a Dirac fermion, we recover = 1 12 W , M ], with 11 = 2 duality that exhibits surprising features. On one side of c T = 1 theory is conjectured to have } and N b a N ,T T symmetry, but there is a co-dimension 1 locus where the preserved a a transform in the adjoint of the SU(3) flavor symmetry. We can always 2 T φ { a ]. Both sides of the proposed duality have a m . This = 2 U(1) gauge theory with vanishing Chern-Simons level and two chiral ab 12 = 1 Wess-Zumino model eight real chiral fields , δ 2 1 = 2Tr[ N symmetry) in the infrared. 11 N R ] = abc b d symmetry to SU(3). of SU(3), and a real cubic superpotential superfields of charge 1. This theory has an unexpected IR enhancement of flavor T a A 3d A 3d To be concrete, we can collect the real scalars and masses into traceless Hermitian The masses To support this surprising duality proposal, we will match the massive deformations We want to claim a duality between the following two theories: ment in a Wess-Zumino model 3 matrices Φ = T 2. 1. × flavor symmetry enhances to U(1) 3 conjugate them to the Cartanwill sub-algebra preserve and to a a U(1) Weyl chamber therein. Generic masses of these theories,arguments including for the this symmetry contact enhancement terms have also for been4.1 background suggested gauge in [ fields. Massive deformations Additional There of is the a Wess-Zumino simple model mass deformation, with super-potential where Tr[ as a U(1) of the duality, the supersymmetry of the infrared theory is not manifest. 4 Symmetry enhancement in Here we study athe certain duality the global symmetry of the infrared theory is not manifest. On the other side precisely the duality between U(1) ment with [ Hooft anomaly. This anomalyof is the matched duality. exactly as Wesome in refer more the to background non-supersymmetric [ on analogue anomalies of one-form symmetries. charge 1 boson]. But since U(1) JHEP08(2018)004 . . 2 2 × = 2 2 (4.5) (4.4) U(1). N U(1) × is simply → 2 is traceless, | j i ∗ takes the form Φ . The analysis U(1) δ 1 ∗ | 5 in the infrared. × P . C worth of vacua, and U(1) gains an 1 . Since Φ × j P (3)) and thus the predic- C SU(2), say, diagonal with su model with a round metric and 1 × i P C global symmetry. The two vacua 2 = 0). The new infrared supercharge symbol of . 2 block which is a square root of 1 . is the square root of a diagonal matrix 3 2 M × abc = 1 supersymmetry must in fact have × ∗ Φ 3 f δ 1 ∗ c N = , the pattern of masses is opposite in the two ∗ symmetry.) – 13 – M = TrΦ R + 2 ∗ δW Φ . has a traceless 2 is the index different from 1 ∗ k is special and preserves U(1) M and there are again two vacua, which preserve SU(2) and Φ worth of vacua, spontaneously breaking SU(2) is diagonal, the coefficient for the off-diagonal m m 1 , where 2 ∗ k k P ) − ∗ = 2 = 0 (i.e. the undeformed model) there are no solutions other than the C ), then we have two possibilities: = c (Φ m c M ) plane, we thus have three half-lines with an − 2 0, 22 diagonal and generic, we see that Φ 0, − = is a Kählermanifold, the low energy non-linear sigma model on these three j j ,M M ) 1 ∗ 11 m > m < P m, m, M C That gives an These two vacua are related by time reversal symmetry. + (Φ For For i i It will prove useful to analyze the background Chern-Simons couplings in the massive We would like to argue that this supersymmetry enhancement is also a property of the As On the other hand, if Taking ) = 2 supersymmetry as well as a U(1) 1. 2. ∗ (Φ we either have twopositive. positive As the masses two andvacua vacua have and one opposite so negative, Φ arespontaneous or the breaking two of background negative time-reversal Chern-Simons symmetry. masses couplings. and This one is compatible with the vacua. The superpotential expanded to quadratic order around the vev Φ As the vacuum vev Φ IR SCFT at thestarts origin its of life parameter as a space cubicbe (i.e. in anti-symmetrized at the and fermions hence (the we spins have totion are use is symmetrized the that so the the dimension flavor renormalizes indices down must from 3 in the ultraviolet to 2 lines where the global SU(3)supersymmetry symmetry is in explicitly the broken deep to infrared.and SU(2) (In at other most words, the twoN derivative interactions with In the ( 2 vacua that areunder related which by Φ time isof reversal a these symmetry pseudo-scalar massive elsewhere. deformations actsphase Time is by diagram also reversal the is done symmetry antipodal shown in in map detail figure on (in the components) in appendix A. The with positive entries and is also diagonal.giving us Only two two of the semi-classical possiblespontaneously vacua square break roots with are time-reversal unbroken traceless, symmetry, U(1) as Φ is a pseudo-scalar. entries ( Observe that for trivial one and hence there is no moduli space of vacua at the SCFT point. and the equation for classical vacua is JHEP08(2018)004 0), 0), , a = 0, (4.7) m > < 3 − f 11 A m M + 2 = 0 ( A = 0 ( U(1) preserving f + × 22 m 1 M A + t − : 11 . 2 ) (4.8) 0). For the gauge theory i 8 2 8 L M φ φ , with > m : i 6 ; (4.6) 0), 1 A 11 − L     ] ) M 8 2 t > | 20 ) sign( φ i ¯ 2 Y Y X | R − = 0 ( ( = 0 ( + T 2 3 f 2 22 | R i iR m M , and a singlet X X : | − + m ( 2 1 1 8 + 2 1 SU(2), we choose a parametrization of Φ as L m φ are complex. Plugging the expectation value + R . 11 3 × − 2 Z – 14 – 3 M ) 2 3 bare : R dA iR , 1 R ¯ 3 XY + − A L + 8 , and 1 SU(2) φ 2 2 R Y 0, the mass terms can be written as k R , or     = 3 + X 0), and 2 1 m > eff , background gauge connection as dA R Φ = 2 < ( 2 A m 11 representation. This is given by [ with 0). On the three solid half lines, we have an SU(2) SU(2) 2 i k ) plane and three half lines are given by M , } < 2 R worth of vacua due to spontaneous symmetry breaking. Otherwise, we have 22 m f 1 2 M dA , are real and m 1 − = 0 ( CP 11 8 A φ 22 in the M i M = 0 ( ) plane and three lines are given by m, m, m f f + and { . Phase diagram of the Wess-Zumino model and gauge theory. For the Wess-Zumino m i m , 11 t R − M t A background Chern-Simons term for SU(2) induced by integrating out matter field If we denote the U(1) If we sit on a line with unbroken U(1) = diag : : 2 ∗ 2 3 triplet of SU(2) with U(1) charge 0 and masswith 2 mass Φ In other words, we have an SU(2) doublet of fields with U(1) charge 1 and mass where then depending on which ofCS the couplings three chambers we are in the vacua will have background L this is ( L deformation, and 2 isolated vacua due to time reversal symmetry breaking. Figure 1 model, this is ( JHEP08(2018)004 : t T 3 L (4.9) ≤ − (4.11) (4.14) (4.10) s reduces and 2 ≤ f . We have L t t m − = and the bottom s ˜ Q and or to t . Turning on Q = 1 , we have vacua where both s . 1 ) | ]. The equations are thus , i.e. on the half lines f t defined by 20 m i in figure . The theory has four natural (i.e. ab − R 1 = s δ ) s L f | . eff t R m 1 2 ( + = = 0= 0 (4.12) (4.13) | T = f 2 1 2 ˜ | Q Q of vacua parameterized by vevs of the chiral m ) ) is fixed either to ˜ eff Q , f f 1 | s + = 2 gauge theory ] = and the FI parameter for the topological U(1) – 15 – m m P b + R s | f C T 2 ( N + − SU(2) | is denoted as a = 2-preserving real mass deformations, associated R m 1 2 k ) = 2. Therefore, s s Q T ( ( 3 | V ( + N in the region II in figure t T Tr[ , so that . These two vacua are related by time reversal symmetry f s 1 = . m P 1 = 0. The two chirals have opposite masses and contribute C eff t P t ˜ ) = 1, Q C . The supersymmetric vacua can be found by looking for the 2 t ( to the effective FI parameter. The real scalar combines with the = T t flavor symmetry. Without loss of generality, we can consider the Q − f ) = = 0, the theory has a t = 0 the theory has no moduli space of vacua, as our Wess-Zumino model. f SU(2) in these two vacua. Hence, time reversal symmetry is spontaneously broken 0, something special happens when + f m f × s = 0). This corresponds to the half line m ( t ) is the quadratic index of representation m s 1 2 i 0, t < = R − t ( = ) t > T t s For intermediate values of When For − s ( 2 1 dual photon to give a new chirals have either positive orthe negative effective mass FI and parameter no vacuum is again expectation two values. vacua Therefore and time reversal symmetry is spontaneously broken. since in this gapped phase. a Coulomb branch parameterized by theexists. real scalar In in this the vectormultiplet, phase with When fields (and the vacua to the two poles of where the one-loop effective FI parameter is symmetry is denoted by critical points of the one-loop corrected potential, as in [ component in the vectorvisible multiplet in the microscopicto theory) the U(1) two mass deformations associated toassociated the to Cartan the subalgebra. Cartan We of will SU(2) denote the by generator (We will soon compare this do the dual4.2 gauge theory.) Massive deformations ofWe denote the the two chiral superfields with the gauge charge +1 as with the normalization where JHEP08(2018)004 s ˜ 8 1. Q . − 3 S = and (4.18) (4.15) (4.17) (4.19) Q JJ k ) can be written We then have J 9 . J adA half-lines, is clearly A are the same and is the effective mass, 1 ] and we have + 2 ˜ Q s 0 preserves the global ! P 21 [ ! 2 ! C g,i ada 1 ( 1 2 − n ) (4.16) π i 1 and t < . 1 0 4 − − 0 1 1 partition functions of the theory, + 0 1 M B − i 3 1 b Q

0 S − m

) (in one of the two vacua which , this generates the required = sign( . Then f ˜ i a Q = 0 for , b,i , M f ! n Q, , , ! 1 m a 1 2 a,i 1 − ! n − = 1 2 i 0 1 0 0 0 1 1 2 − X eﬀ ,

2 1

– 16 – + = 2 particle-vortex duality to convert a chiral mul- SU(2) for the gauge field k N SU(2) are preserved. Again, integrating out fermions bare = = = ada ⊂ I ab, II π 1 4 k III f ! ! ! = ff Jf ff Jf ). The first term is an trivial theory U(1) ) in the Wess-Zumino model. k k k ff k Jf J eff , which forces gauge field to identify with k k -th fermion under U(1) t i dA ab, fJ JJ fJ JJ 4.10 J k k k Weyl symmetry can be realized as a group of self-dualities of the k fJ k JJ and U(1) A k k 3 t

. In the region I, the result for two vacua is

− S

) g,i J n will induce background Chern Simons terms given by = 1 for the other vacuum. A i theory coupled to a dual chiral multiplet. The resulting theory has a new , U(1) JJ − t 1 2 m k a ( d symmetry exchanging at the same time the two gauge fields and the two chiral . ) symmetry of the phase diagram, permuting the three and J is the charge of 2 3 JJ f A Z S U(1) symmetry. In this case the doublet ( k a,i m − n × a (( Region II is a bit different since the signs of masses of We can look at the background Chern-Simons coupling in the massive vacua. With The massive phase that we have found with This conjectural The The Weyl symmetry can be made manifest in bothThere the is index also and a more physical way to look at it. In the positive mass vacuum, integrating out π 1 4 8 9 by manipulations analogue to that particle-vortex duality. Seegenerates appendix level one Chern Simonsas term Likewise we have is equal to FInonzero parameter In the region III, the situation is similar with gauge charge with real mass where This matches the result ( generic multiplets. Together with the Weyl symmetry of SU(2) SU(2) are related by time reversalSU(2) symmetry) Chern-Simons acquires term a after positive it mass, is generating integrated a out background suggestive of an enhancement of the flavour symmetry totheory. SU(3). The simplest way is totiplet use to the a U(1) manifest JHEP08(2018)004 ) f ), Z 5 22 iφ (5.1) (4.20) ,M and − 11 4 Y M φ , ( 1 2 X = X carrying charge 1 under 2 . We also add a real cubic . Φ in the previous section, we 8 , αβ 1 2 φ αβ R A 1 2 3 = 1 duality which can be derived = 1 duality is also closely related φ + . 1 N N β triplet A ¯ u f α = u matrix in the three regions on ( J k ), as the gauge theory in the three regions αβ A R global symmetry. 4.18 XYZ = 0 1 1 1 1 0 1 0 1 t XYZ and – 17 – ), ( 2 W ˜ with two superfields Φ Q 3 2 1 A in the region I, II, III respectively. This serves as U(1) 0 1 2 A A A 2 4.19 × 0 0 1 -1 0 0 0 1 -1 1 1 -2 0 0 Q = ). The masses of the three complex fields f dA 2 ), ( f 1 = 1 duality has appeared before in a different context, for ]. iφ A f J in the Chevalley basis. In particular, A (i.e. it is just the complex conjugate superfield) and indices N 4.17 22 + a ∗ 1 T ) U(1) U(1) a φ charge 1 and a real SU(2) β ( φ t u 1 2 , and 1 = = ( matrix ( β dA Z k 3 u of U(1) ]. ), A α = 1 Wess-Zumino model with seven real chiral fields: a complex SU(2) 7 -XYZ duality [ = 1 U(1) gauge theory with two chirals of charge 1, total Chern-Simons level u 24 1 iφ , = 2 duality above. We will see that this , = 1 duality between SQED and a Wess-Zumino model N 3 N + 23 N 6 N dA 2 φ ( A superpotential We denote ¯ are raised or lowered using the SU(2) invariant tensor 0. This theory has an SU(2) doublet 1 2 A 3d A 3d We will first motivate the duality and then connect it to the duality we have presented First let us compare the moduli spaces of vacua of the two theories before we add any The dual pair is On the other hand, we can also compute the 1. 2. = in the previous section.instance, This [ deformations. We begin with the U(1) 5 An Here we would like tofrom make our some comments about an to the SQED gives us the same respectively. Upon identifying get another consistency check of our duality. Y are given by the Hessian matrix of the scalar potential at the vacua. A simple calculation where we rewrite Φ = plane of Wess-Zumino modelsymmetry are side. given by The charges of the different fields under the global JHEP08(2018)004 . . . R R 2 1 2 1 Φ Φ Φ Φ (5.2) , and m ). There = we have a , as in the 2 5.1 3 | m 1 R u | , and arg = (remember that ). This effective | 2 αβ 11 Φ vac | = 0, R , ( | 2 M 1 mR u eff Φ | W = = δW . eff t W . 2 | 2 . , . So in both cases we have a circle of Φ | m , . m . Going far on this moduli space, we can we have nonzero 3 and hence we get while removing one overall phase which = 0 m = 0 − = αβ R m 2 − 2 2 ab α R = | ¯ = u 2 u 2 R | 2 1 | u 1 u αβ 2 symmetry. – 18 – vac Φ u | R t and Φ | − | M m 2 1 | 1 = u | = 0, intact. Therefore, considering now the full theory and 1 W worth of Nambu-Goldstone vacua fibered over it. | u 2 Φ | , U(1) | SU(2) = 0, 1 = 0, and for positive Φ = R | R 2 S are time reversal even, it cannot depend on them either. Therefore the and write an effective superpotential | 2 . This leaves arbitrary SU(2) invariant terms with an odd number of ¯ F Φ we have R | , F, | and the 1 m ). The equations of motion now take the form 2 we have an effective theory of a U(1) gauge field without a Chern-Simons term. Φ | → − | 22 2 while leaving R m Φ R | − 2 1 + Φ Φ , which we can take without loss of generality to be The dual description of this mass deformation corresponds to adding a linear term Now we compare some deformations of the model, starting from the non-SU(2) invari- Now let us consider the moduli space of vacua of the Wess-Zumino model ( = 2 | R 1 11 Φ The solution is clearly for negative vacua with a spontaneously broken U(1) in R positive The matter fields areand massive. hence Therefore, we it have a cancircle circle be of of dualized supersymmetric supersymmetric to vacua.be vacua. a In interpreted real Similarly, both as for compact cases, due negative superfield to these the circles spontaneous of breaking supersymmetric of vacua U(1) can Wess-Zumino model. ant mass deformation. Without loss of generality we consider This mass deformation breaks the global SU(2) symmetry explicitly to U(1). At nonzero integrate out superpotential is constrained byacts SU(2) by invariance and byfields. time It reversal is symmetry, which Therefore, easy the to see moduli that space all is such spanned terms by vanish identically by simply diagonalizing At a genericand point hence on we the can equivalently| moduli parameterize space the moduli the space SU(2) by the global expectation symmetry valueis is of classically a broken moduli to space parameterized U(1) by not just the classicalAnd theory, since we see thatclassical the moduli superpotential space cannot of depend vacua is on not arg lifted. symmetry is broken (everywherespace except by at the the origin). expectationis gauged. values We can of Therefore parameterize Φ weThe the can model moduli has parameterize a the globalarg moduli SU(2) space symmetry by which has a U(1) subgroup that acts by shifting the U(1) gauge symmetry. We have a classical moduli space of vacua, where the gauge JHEP08(2018)004 1 is = 2 R (5.4) (5.3) . The 2 t N c charge 1, or due to = 2 mirror t U(1) 2 c × moment map N f t = 1 U(1) gauge 1.) In the Wess − N ) and flip the real of U(1) 1 = 1 supersymmetry. α u it is N = 0 and due to m is unknown. Classically, . The flipping operation = 1 superpotential terms u S 2 c since there is no more time = 1 superfield containing in N , = 2 SQED doublet R N f αβ N . R 2 ). | 2 αβ 2 , R R Φ | β 2 is classically arbitrary, there will be an ¯ c u . No extra m α , is redundant (in the sense that it can be removed is the real S + = 1 duality with the familiar R u α The coefficient + ¯ u β S 2 αβ α ( ¯ | N u u 1 10 is +1 and at negative α and R -XYZ mirror symmetry. It is also related to the – 19 – S Φ u 1 | we have likewise another trivial supersymmetric ). = m R αβ maps to a triplet of mesons and the singlet meson m m 5.4 W is pinned to the origin (either due to = mη αβ = 0, while equations of motion still set ]. R R = W 2 c R 25 , δW 22 and a singlet, to be identified with ) the αβ 5.3 = 2 U(1) gauge theory. Flipping means adding a new real multiplet R (with a possible admixture of 2 = 1 theories is related by simple “flip” operations both to the dual pair | N SU(2), as required by the duality. u | and leaves a Wess-Zumino model of seven real chiral operators and cubic N × S we have a trivial supersymmetric vacuum (here we use the fact that U(1) triplet t f m = 4 mirror pair [ is as usual the identity matrix. N αβ η Now we will explain the relation of this On the Wess-Zumino side of the duality, the flip breaks SU(3) to SU(2) Therefore, under the duality, This pair of In order to see the relation to the former, we can “flip” the real U(1) Finally, we can have an SU(2) invariant mass deformation. In the Abelian gauge theory Note that the other SU(2) singlet deformation, with linear superpotential coupling to 10 symmetries are available other than ( symmetry. We again start from the gauge theoryby side a change ( of variables). a real SU(2) again removes superpotential which is the model we have studied here. Again, no extra terms compatible with the can be induced, asinvariant under no the gauge-invariant global operators symmetries odd )theory are under with available. two time We chirals reversal thus of symmetry have charge the (and 1 at low8 energies. real chiral multiplets decompose into a complex SU(2) operator in the 3d R the bottom component theFI real parameter scalar to of a dynamical theThis real vector superfield. new multiplet), term This i.e. clearly allows promoting preserves to the integrate out maps roughly to in the previous section andbasic to 3d the SQED pinned to the origin.effective potential (Even on if thereversal moduli symmetry.) space parameterized Asa by long radiatively generated as potential)unbroken we U(1) get a massive trivial supersymmetric vacuum with Zumino model, this deformation is mapped to a deformation of the superpotential by where with the deformation ( this corresponds to At positive is a trivial TQFT)vacuum. and (The Witten for index negative at positive JHEP08(2018)004 . ). 2 | = 4 and Y (5.6) 1 Z,R N − | 2 Im | operation, β Z, X ¯ u | in the S α t u ) (5.5) αβ 2 = (Re | = 1 R Y αβ = 4 mirror symmetry N R − | ) = = 1 description of the 2 2 N | ¯ u 2 N X | u ( − R 1 ¯ u = 1 U(1) gauge theory with two 1 ) + u ¯ N Y ( = 4 gauge multiplet of SQED ¯ R X N ) to side. − ) + 1 2 ¯ . This enhancement is due to the fact that u t XY 1 XYZ ( in the u Z − – 20 – αβ U(1) 1 Im R i ¯ u × 2 f u ( ) + global symmetry, with a triplet t ¯ Z Y ¯ ), then the superpotential becomes X Im i U(1) + . = 1 Wess-Zumino model. to SU(2) ¯ α × X,Y ) + 2 u 2 f N XY ¯ u ( = ( 1 Z u α u + = 1 gauge field to the free twisted hypermultiplet, in such a way that 1 side, the operator we are flipping is the real moment map = Re ¯ u and a Free twisted hyper. We can do so by an 2 N 1 u W ( Z XYZ = Re moment map operator. The result is, again, the = 1 theories, there is no difference between charge 1 and charge -1 multiplets. t W Alternatively, we can “flip” all three real moment map operators for U(1) The Wess-Zumino model real superpotential is indeed the What we have shown is that the N=1 duality described here can be derived either from On the N the hypermultiplet flavors mirror pair. The resultall is connected by the the same. ‘flip’ operation To summarize, as the in the dualities figure we below. have considered are between SQED coupling a U(1) it “ungauges” the U(1) gauge field on the SQED coupling between the three real scalars which has enhanced SU(2) This is precisely our dual the standard mirror symmetryWe could or have starting of course from derived our this duality new direction duality from in the 3d the previous section. If we form a doublet multiplets of charge 1.flavor symmetry No from superpotential U(1) canin be generated. Notice the enhancement of The real superpotential is deformed from Re( U(1) JHEP08(2018)004 f N (6.2) (6.3) (6.4) (6.1) × ) with N 6.3 ). The number f real-dimensional quarks 2 f f N N = ,...,N ). (The mesons can be f ).) Acting on ( 6.2 = 1 N f i N + f − N N ) symmetry to bring the − f mesons (                2 f . f N 2 f SU( . 0 0 0 0 N f N N . N j → Q ...... a ...... ) † i U(1) mod 2 In fact, if we require time reversal symmetry 2 N Q 0 gauge theory with a f → = 11 0 N – 21 – 1 ) ) 0 0 0 0 0 0 0 ij a f ...... = N N M                N matrix.) U( = f . In that case we can parameterize the moduli space by Q N c × are good degrees of freedom) does not generate a superpo- f i < N the global symmetry is broken as in the fundamental representation ( N Q f i i a N Q ) global symmetry. f N generators we therefore obtain a ) f f N N to the form U( i U(1) 0. For generic Q ) symmetry, it is not possible to write any superpotential which is a function of f gauge theory (the subscript indicates the quantum Chern-Simons level) coupled to ≥ = 1 supersymmetric SU( N 0 i . Therefore, to all orders in perturbation theory, the renormalization group (around ) i a Let us assume that That means that the large moduli space of supersymmetric ground states that exists Suppose the superpotential vanishes classically. Then the theory has time reversal sym- N Q N multiplets of quarks We ignore various discrete identifications. 11 f (And the gauge symmetrythe is broken broken as SU( moduli space. This spacethought of is as parameterized a by Hermitian the with To see that, wematrix can of use the gauge symmetry and U( in the classically massless theory persists to all ordersthe in expectation perturbation values theory. of the mesons metry along with U( and U( the the ultraviolet, where the tential if we take the superpotential to vanish at tree level. This is a necessary(We and will sufficient see condition below that for if the this time condition is reversal not invariant obeyed theory one to finds exist. various nonsensical results.) An interesting classSU( of time-reversal invariantN non-Abelian gaugeof theories fundamental is representations is given constrained by such that 6 JHEP08(2018)004 . , 3 2 ) ) 8 5 φ φ ( | → ∞ ( 3 3 1 φ M | √ as there is + c − 2 ) 4 2 ) φ real-dimensional 7 ( < N 3 φ 2 f f φ )). The vacuum en- N +( N + 2 6 ) 6.1 φ 6 5 ) φ φ 2 +( φ 4.2 2 ( ) 5 +2 φ 7 φ -symbols of SU(3), in the basis of 5 +( d φ 2 ) 1 4 φ φ ( +2 8 ], there is no supersymmetric ground state 6 φ φ 27 4 3 2 φ vector multiplet theory breaks supersymme- √ 1 for the future. – 22 – φ − c 0 2 ) N 2 f ) 2 3 N 7 ≥ + φ f ( − 1 does not exist in the sense that it cannot preserve time 3 2 N ) φ ] (this TQFT makes sense due to ( − 3 N + φ 12 N 2 ) +( 6 = 2 φ ) f ( 2 are more interesting as a generic point on the moduli space breaks 3 ). Hence our analysis is in fact valid for all vector multiplet. The two sectors are not entirely decoupled. The N φ c φ TQFT [ 0 N + ) 6.1 +( f 7 f massless mesons but, importantly, (at a generic point on moduli space) 2 ≥ N φ ) N 2 f 4 1 − f φ φ N − ( 2 ,N N f φ 8 N 2 N 2 φ − 3 N 3 2 but there are some irrelevant operators tying the two sectors together. It is easy ) √ − f ]. In the infrared one has a Goldstino along with the (spin) time reversal invariant ij N 1 6 2 − 26 The cases Since the coupling between the mesons and the vector multiplet vanishes for Note that the case Thus far the analysis was to all orders in perturbation theory. However, non- The degrees of freedom on the moduli space to all orders in perturbation theory are M = N Gell-Mann matrices: W Foundation center for excellence grant and byCollaboration the on Simons Foundation the grant Non-Perturbative 488657 (Simons Bootstrap) A Detailed analysis ofWe the first Wess-Zumino write model the superpotential explicitly, using the We thank O. Aharony,R. J. Yacoby. The Gomis, research S.Theoretical of Physics. Razamat, D.G Research and M. atCanada J.W. Rocek, Perimeter was through N. Institute supported Industry is Seiberg, by Canadaof supported the and A. Economic by Perimeter by Development Sharon, the Institute and the Government and for Innovation. Province of Z.K. of is Ontario supported through in the part Ministry by an Israel Science at a generic point andtheory. hence We there leave would the be analysis a of real moduli space ofAcknowledgments SUSY vacua in the full reversal symmetry ( always a nontrivial unbroken gauge group (whichbreaking in turn and leads lifts to the dynamical supersymmetry moduli space non-perturbatively). the gauge symmetry completely. Therefore, there is no mechanism to lift the moduli space moduli space is lifted. we see that the vacuum energyin is four asymptotically dimensions constant. (with Unlike four inat the supercharges) infinity. analogous [ theories perturbatively, the puretry [ SU( U( ergy is set by the gauge coupling. Therefore, non-perturbatively, our therefore these also the SU( vector multiplet effective action to firstthe approximation is the canonicalto one, independent write of the leading terms that couple the two sectors, but we would not need them here. JHEP08(2018)004 , , 8 3 8 φ φ m U(1) − √ , × 8 and φ are acti- 3 ∼ 3 U(1) sym- φ 8 φ , φ 0. × → − 8 U(1) there is a φ 3 > 8 × , φ 8 m φ plane where the mass 8 0. , m 0 has nonzero > 3 . U(1) symmetry is broken to 8 8 > m × φ m . , 8 8 m m = 0 = 0 = 0, in which case, as we explained + , . 3 8 3 3 0 and unbroken for = 0. Therefore the global SU(2) φ m m m are pseudo-scalars — this is consistent 3 8 3 3 = 0 = 0 < the assumption that only = 0 and in particular, there is no moduli φ a m = 2 3 √ √ 8 φ a 8 ) is parameterized by additional scalars that 8 φ 0. + 8 φ m + 8 are activated and the equations for the critical 1 c φ m + 2 , m φ 3 > ( 3 φ 3 P 2 φ b – 23 – 8 ) 8 − C m φ , φ 8 ’s vanishing. The relevant critical point equations φ m 8 a 2 φ ) φ ( φ φ 3 U(1) for φ − abc ( 2 d U(1) is spontaneously broken to U(1) × ) is nonzero and 6 1 3 U(1) on the half-line with × 3 U(1) symmetry, namely φ φ = ( × × 0, W < 8 m . Repeating the analysis, one find that the global SU(2) 8 . One again finds that the global SU(2) 8 m 3 m , and with all the other 3 √ 3 U(1) on the half-line with = 0. For √ − 3 . Again, without loss of generality we can add a linear term for just , φ × φ 8 φ = = 0. In this case we see that the solution for = )). For some special choices of φ 3 3 3 2.4 U(1) symmetry is broken to U(1) m metry is broken to U(1) m m and sigma model at low energies. The When the symmetry SU(2) Clearly, there are two solutions (unless The critical points in the absence of mass deformations can be taken without loss of 1 • • • P C are massless. Time reversal symmetry acts as an antipodal map on the target space. vated is not justified.perturbation preserves Indeed, SU(2) there are three lines in the above, there is onlywhich one is solution). nothing but The timethat two reversal commutes solutions symmetry with (there are SU(3), iswith related a for ( by choice which of all time the reversal symmetry points are modified to so we now consider the superpotential We again look for solutions where only The only solution isspace thus the of trivial vacua solution interms for the undeformed theory. Now we deform the superpotential by linear generality to be diagonal matricesnonzero and hence it isare sufficient thus to look for critical points with JHEP08(2018)004 m , and (B.7) (B.1) (B.2) (B.3) 2 S running ) action Z a ] for more ) (B.5) denotes the , the theory 30 N, q ∆, which we ; ∆ 2 ) T ... ST m, m ( . We now show that cdA . . . ) (B.6) − ∆ ↔ q e ! ; T a + 2 − e a ( ζ ∆ m, e cdc I ( a at a certain magnetic flux Y , and work with the index at ∆ m + e ◦ ) 2 2 2 / ζ 2 . There is an Sp(2 + charge 1 ) ) ) S / corresponding to this symmetry. 3 e 3 1 2 q j j bdc / q q ST ; + 3 e ) (B.4) on − − σ E q ( ( + 2 ; I R q H m, e γ ( ( U(1) − bdb T ) = ( βH T factor is implicit). And ) = I q I 2 1 − E ⊗ q ; e N π ; 2 . Accordingly, index transforms as Z + 4 F m = ] ∈ / commutes with the Hamiltonian given by ) = N 2 e | } 1) q − 3 † ˜ m, e – 24 – ; φ 29 j − e P ( U(1) b , ( ∆ − gγ + D 28

◦ ( Q, Q and electric charge e, ) = E H { ( ←→ ST ◦T ↔ | = 2 dualities e g is the symplectic charge vector. ∆ m σ I = I q, ζ I e T ; ] consisting only of a single free chiral multiplet. There is ... ) N − H m ) ) = Tr 30 + ( 2 ) = / T q m, e 1 (i.e. the usual 1 I ; q, ζ q ; , − + charge 2 AdA = ( m T 0 1 2 , is defined by [ ( m, e γ ( AdB T N − ∆ I ) = ( I 2 q U(1) | is the fugacity corresponding to each global symmetry, with for any 3 manifold M can be built from the basic block ; φ a ) and and 2 ) follows simply from the particle vortex duality ∆ A ζ are energy, the third component of the angular momentum rotating M Z D T m, e a | ( B.1 e . Trace is taken over the Hilbert space N, ∆ , AdA I 3 N j , Sp(2 E ∈ A theory g is the R charge of the state) thus we have a fugacity R the duality ( writes as To avoid clutter, welooks normalized the like Chern-Simons term such that minimal allowed ones This simply comes from the particle vortex duality Before calculating the index onhow both our sides theory of on the duality, both it sides is are important related to to understand the first tetrahedron theory for a single ideal tetrahedrona [ triality enjoyed by the tetrahedron index a fixed background magneticon flux the 3d SCFT with flavor symmetry U(1) for ( However, instead of working intions, the it fugacity basis is and muchdetails. manipulating easier hyper-geometric Here to func- we work just in mentionthe a the index few charge of useful basis. the points related theory We to refer our the discussion. reader We can to rewrite [ flavor charges. from 1 to background. Since the generator for flavor symmetry U(1) where We want to analyze in detail the duality we stated in section 3 In particular, we can try to compare the superconformal index, which for a 3d SCFT with B Further checks of the JHEP08(2018)004 ), . The (B.8) (B.9) simply B.10 µ (B.12) (B.13) (B.10) on the µ J c µ J A = is not gauge − dressed with νρ ρ ... F A M m ν + ... µνρ ∂ on the right hand µ is the magnetic flux. π + k A c 4 Z cdA µνρ ∈ cdA + 2 . The magnetic flux felt π k 4 e F + 2 tensored with a decoupled R = cdc with a single charge 2 chiral e , and integrate out π 2 1 2 b 2 2 L 0 / , cdc 3 and flavor flux ) has index − − 2 = 2 m 2 c m operation. − M ) (B.11) bdc m + 2 − B.10 4 → ∆ e AdA . . . e c , e − bdA − ST − − 2 ∆ e, − ∆ m bdb 2 , where ). To this end, recall that the presence of ( I 1 2 bdA F ). Knowing the relation between the theory ∆ 2 2 I bdb . ∆ − S I m, e A state with nonzero gauge flux ) = 3 2 − R . And the equation of motion is 2 | m, e – 25 – k ( ˜ ) = φ + 12 δA 0 AdA b ) = bdb followed by m, e 2 1 1 2 | ( 1 2 S D ˜ 2 R φ A m, e U(1) b 2 2 ( − m, e I π ( D k T 2 4 U(1) 0 | I ], an indication that the duality indeed follows from the → ←→ | ˜ ⊗ φ 2 b 30 / A . Gauge invariance then gives us the relation between them. U(1) 3 ) and D ... I ) in terms of ( ←→ | | on the right hand side, we are left with ∆ )[ ∆ b + ) up to affine shift of the charge due to the triality property of m U(1) ... , e I m, e B.5 − ( ∆ 2 + c m, e adA ( m 0 to be gauge invariant, and has flavor charge → U(1) + 2 adA c ⊗ ∆ 2 2 U(1) e / | on the right hand side, which is dual to U(1) I 3 φ + 2 2 a 2 2 | U(1) φ D I | a U(1) 2 D ∼ = | 2 − In particular, the theory on the left hand side of ( Now it is pretty straightforward to calculate the index on both sides of eq. ( A toy example to see why this happens: consider the Lagrangian and ∆, 12 particle vortex duality. variation of anObviously Chern-Simons each magnetic term flux is carrytells gauge that charge magnetic field isdressed proportional with to chiral the matter matter operators field in charge order density. to So have monopole gauge operator charge has zero. to be which equals to ideal tetrahedron index ( The theory on the right hand side has index we just need toChern find Simons ( terms modifies Gauss’ law. invariant. Therefore, we require achirals state of with charge gauge flux by the chiral is denoted as side, and the last two terms just yield denoted as T Therefore, we end upU(1) with a singleon chiral the coupled left to hand U(1) side. Note that we can integrate out dynamical field Now on both sides, we rescale Now if we rename and dynamical gauge fieldright respectively. hand We side, rename we have super-partner parts. Also, we use the upper case and lower case letter for U(1) background JHEP08(2018)004 , i ) in iσ 2 + i ∆ (B.20) (B.17) (B.14) (B.15) ∆ instead m − , 3 . It flows = 2 U(1) (1 | 1 S ` 3 ∆ e S i N m Z R | σ , t ] is 2 2 corresponding ) ∆ det , ∆ 2 log π e 2 ) 32 i , m 2 − − Y , iπ 12 2 1 = 1 m 31 − σ ) (B.19) = ∆ i 2 − )) 2 0 (B.18) 1 e 3 a + , , iπ ∆ ) i S m 1 e ) σ partition function. For 3d → . It tells us F , e πσ 2 m m 2 i t − 3 4 πiz ∆ dσ 2 S 2 ∆ − e ∆+ e 0 (B.16) m 2 ∞ ( and − ( e 2 + + i (1 → ∆ −∞ (2sinh( 1 ` e Li I − t Z e , e ) 1 1 π Ad maximization is needed. We obtain 3 1 ∆ − e × σ / t ( ∆ t 1 ) + − 1 2 det ∆ ( , e 2 a iσ , π 1 is the topological charge and corresponds 2 1 σ 2 2 → ∆ , t πz − ∆+ m 2 tr e ∆ t m − ( ..., m ∆ + i (1 2 π ` ∆ 1 dσ 2 − I e e – 26 – − 1 ∞ with charge 2 chiral on the left hand side of the 2 σ ) + e ) t , and only ∆ + t a 0 ∆ −∞ ∆ σ 989539 ..., πiz ; ) = I π . Z 2 2 ∆ 2 e I × tr ( − , e a 2 = 0 2 − ] = σ t + charge 1] we have 3 3 2 m iπk 2 S ∆ 989539 ; ; ) = -th chiral field. ∆ e / iπ . by looking at the index. We denote the electric charge and , i 1 2 F 3 of SU(3). As before, we need to find e 3 log(1 , e , e a [∆ 2 , partition function from the localization [ S 1 2 Y z = 0 3 . 3 dσ t S U(1). Consequently, the Weyl symmetry group will be 3 − m m S [U(1) dσ ( S Z ; ∞ ) is given by × 1 F T + ⊗ z ) should also be subject to the check of I −∞ ( ) = , e 2 partition function matches nicely. ` z Z 1 , such that Cartan ( 3 i B.1 ` Z m S ( m ] = 1 t T I |W| ∆ , and is the R charge for ]= is the adjoint scalar in the vector multiplet of the corresponding U(1) gauge i t [∆ i e 3 a ∆ S , σ i Z . We can check this Finally, we examine the duality discussed in the section The only global symmetry is U(1) In particular, for the theory U(1) The duality ( 2 [∆ = 2 gauge theory on 3 S S Z Again, by requiring the gauge invariance of the states, we obtain magnetic charge of theto gauge the invariant maximal operator torus toterms U(1) be of We see that the gauge theory with twosymmetry chirals from SU(2) of theof charge +1 has a SU(3) enhancement of the global After minimization, we get numerically, For the theory U(1) to a SCFT atand IR topological fixed charge point ∆ where free energy is maximizedduality, with we respect have to R charge ∆ where group. ∆ to pure imaginary FI parameter. And free energy is defined by N where the function JHEP08(2018)004 B , B Phys. 2 + 1 , (B.23) 3 in the √ ) 2 Phys. Lett. m , theories in Nucl. Phys. + ) (B.22) , (1983) 2077 ) (B.21) 2 1 2 = 1 e m m 51 N + + + ]. 1 1 2 e arXiv:1803.01784 e . So there is no 3 m , ]. 2 + − − ( ( 1 SPIRE 2 1 2 1 2 e ]. IN Phases of m − SPIRE ]. ]. = = ( ][ ]. + IN 1 2 0 2 0 2 1 e , ][ SPIRE Phys. Rev. Lett. ) m dimensions 2 , e , IN SPIRE ) SPIRE is given by SPIRE [ m 2 IN IN i IN e m [ + 2 + 1 ][ [ 1 − ; ) is invariant under the Weyl group action. m 1 ]. 2 and i m − , e – 27 – Improved methods for supergraphs 2 (1982) 413 2 Time-reversal symmetry, anomalies and dualities in m + Instantons and (super)symmetry breaking in e arXiv:1712.08639 m 2 SPIRE ; [ e (1973) 1888 − (1984) 2366 dualities in 1 IN arXiv:1802.10130 ) Radiative corrections as the origin of spontaneous 1 − 2 ][ [ , e e ]. Axial anomaly induced fermion fractionization and effective B 206 1 1 m ) D 7 − = 1 e 3 to generate the Cartan U(1) hep-th/9402044 2 ( 3 m ), which permits any use, distribution and reproduction in D 29 [ e ( − 1 2 √ N ( − / T − 1 2 SPIRE 8 I 1 (2018) 006 1 T ]. IN ∆ 5 m e I [ (2018) 123 ∆ − − I ( ( × Nucl. Phys. ) due to the triality property of the ideal tetrahedron index. Other Phys. Rev. 07 1 2 1 2 and 2 , , SPIRE Phys. Rev. CC-BY 4.0 (1994) 6857 3 , hep-ph/9309335 IN = = action on the charge ; ) = T [ [ B.20 0 1 0 1 0 2 This article is distributed under the terms of the Creative Commons 3 Parity violation and gauge noninvariance of the effective gauge field action in Gauge noninvariance and parity violation of three-dimensional fermions (1984) 18 e S m JHEP , e Exact results on the space of vacua of four-dimensional SUSY gauge theories Naturalness versus supersymmetric nonrenormalization theorems D 49 0 2 , 52 SciPost Phys. m ]. ]. ; , 0 1 d -dimensions , e (1979) 429 (1993) 469 0 1 SPIRE SPIRE m IN IN ( (2 + 1) Rev. Lett. three-dimensions (2 + 1) symmetry breaking gauge theory actions in[ odd dimensional space-times dimensions [ 159 318 Phys. Rev. We now show explicitly T A.N. Redlich, S.R. Coleman and E.J. Weinberg, A.J. Niemi and G.W. Semenoff, V. Bashmakov, J. Gomis, Z. Komargodski and A. Sharon, F. Benini and S. Benvenuti, I. Affleck, J.A. Harvey and E. 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