Curious Aspects of Three-Dimensional N=1 SCFTs
AApril 19, 2018
Prepared for submission to JHEP
Curious Aspects of Three-Dimensional N = 1 SCFTs
Davide Gaiotto, Zohar Komargodski, , and Jingxiang Wu Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Israel Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY
Abstract:
We study the dynamics of certain 3d N = 1 time reversal invariant theories.Such theories often have exact moduli spaces of supersymmetric vacua. We propose severaldualities and we test these proposals by comparing the deformations and supersymmetricground states. First, we consider a theory where time reversal symmetry is only emergent inthe infrared and there exists (nonetheless) an exact moduli space of vacua. This theory has adual description with manifest time reversal symmetry. Second, we consider some surprisingfacts about N = 2 U (1) gauge theory coupled to two chiral superfields of charge 1. Thistheory is claimed to have emergent SU (3) global symmetry in the infrared. We propose adual Wess-Zumino description (i.e. a theory of scalars and fermions but no gauge fields)with manifest SU (3) symmetry but only N = 1 supersymmetry. We argue that this Wess-Zumino model must have enhanced supersymmetry in the infrared. Finally, we make somebrief comments about the dynamics of N = 1 SU ( N ) gauge theory coupled to N f quarks ina time reversal invariant fashion. We argue that for N f < N there is a moduli space of vacuato all orders in perturbation theory but it is non-perturbatively lifted. a r X i v : . [ h e p - t h ] A p r ontents ABC
Model 5 N = 1 Abelian Gauge Theory with a Charge 2 (Super)Field 6 N = 2 QED and Supersymmetry Enhancementin a Wess-Zumino Model 13 N = 2 Gauge Theory 16 N = 1 Duality Between SQED and a Wess-Zumino Model 186 N = 1 Supersymmetric SU ( N ) Gauge Theory with N f Quarks 21A Detailed Analysis of the Wess-Zumino Model (4.2)
24B Further Checks of the N = 2 Dualities 25
In theories with four supercharges (both in 2+1 and 3+1 dimensions) it is well known thatthere are powerful non-renormalization theorems [1–3]. This is due to the holomorphic na-ture of the superpotential. These non-renormalization theorems have many applications. Inparticular, they allow in many cases to determine the space of supersymmetric ground states.In such theories with four supercharges, these spaces are complex (in fact, K¨ahler) manifolds.Understanding the space of ground states is a crucial step before one can study the behaviornear various interesting singular points.Here we will study 2+1 dimensional theories with N = 1 supersymmetry, namely, tworeal supercharges. These theories have a real superpotential, W . Since one does not havecomplex analysis at one’s disposal, it is typically hard to find exact results. It would seem that W can receive corrections since it is easy to write real functions invariant under the globalsymmeties. One aspect of this problem was recently studied in [4], where it was shown that N = 1 theories often have walls in parameter space and it is possible to obtain exact resultsnear these walls using only the leading radiative correction to W . The models studied [4] do– 1 –ot have time reversal symmetry due to various Chern-Simons terms. Some related aspectsof Chern-Simons-matter theories with N = 1 supersymmetry were recently discussed in [5].Here we will study some N = 1 models with time reversal symmetry. Such modelsoften have exact real continuous manifolds (with singularities) of supersymmetric groundstates. A related fact is that the renormalization of W is severely restricted, and often W cannot be corrected at all. These non-renormalization theorems are due to the fact thatthe superpotential is a pseudo-scalar under time reversal symmetry. This property of thesuperpotential was already noted in [6]. Here we will review this fact and develop someapplications of it.As a simple example, consider the model of 3 real scalars and three Majorana fermions,embedded into the three real superfields A, B, C with superpotential W = ABC .
We argue that the full theory has a moduli space of vacua consisting of 3 real lines thatintersect at an N = 1 SuperConformal Field Theory (SCFT).Next, we will study the theory of a charge 2 superfield coupled to a U (1) gauge field in atime reversal invariant fashion. This theory has N = 1 and N = 2 versions and we find a dualdescription in both cases. The dual description consists of a pure U (1) TQFT tensored witha charge 1 superfield coupled to a U (1) gauge field with a Chern-Simons term at level 3 / U (1) + charge 2 ←→ U (1) ⊗ (cid:2) U (1) / + charge 1 (cid:3) Time reversal symmetry in the dual description on the right hand side is therefore emergentin the infrared. Interestingly, the theory on the right hand side has a moduli space of N = 1vacua even though it has no microscopic time reversal symmetry. We will explain the basicmechanism that allows such exact moduli spaces of vacua to exist without time reversalsymmetry.We then discuss a new duality between an N = 2 SQED theory and a Wess-Zuminomodel. This duality has two surprising aspects. First, the symmetry of the infrared fixedpoint is enhanced from U (2) to SU (3). Second, the dual theory is a Wess-Zumino like modelbut we do not have an N = 2 description of it. The N = 2 supersymmetry arises in theinfrared and only an N = 1 symmetry is manifest in the flow N = 2 U (1) + 2 charge 1 ←→ N = 1 W = TrΦ , where Φ is in the adjoint of SU (3) (i.e. 8 real scalar degrees of freedom). W is an N = 1superpotential. We see that on the right hand side there is emergent supersymmetry (and R -symmetry) in the infrared and on the left hand side there is emergent SU (3).This duality also has a purely N = 1 version (where the Wess-Zumino model has 7 realscalar fields rather than 8), but in that case there is no enhanced global symmetry and no– 2 –nhanced supersymmetry. However, there is a moduli space of vacua, which we match acrossthe duality.We close with brief remarks on N = 1 time reversal invariant non-Abelian gauge theories.More specifically, we consider SU ( N ) gauge theories minimally coupled to N f fundamentalmultiplets. We show that for N f < N there is a moduli space of vacua to all orders inperturbation theory but it is non-perturbatively lifted.The outline of this note is as follows. In section 2 we explain how time reversal symmetryacts in the context of N = 1 supersymmetry. We give some examples of applications of thefact that W is a pseudo-scalar. In section 3 we discuss the theory of a charge 2 particlecoupled to a U (1) gauge field. We discuss the N = 1 and N = 2 versions of the theory andfind dual descriptions in both cases. We outline the connection of these dualities to somenon-supersymmetric dualities. In section 4 we discuss QED with two charge 1 particles, andagain discuss N = 1 and N = 2 versions of the theory, finding dualities in both cases, andin particular, in the latter case, we find an enhanced global symmetry in the infrared. Onthe other side of the duality, we find enhanced supersymmetry (and R symmetry) in theinfrared. In section 5 we make some comments about N = 1 supersymmetric non-Abelian SU ( N ) gauge theories with time reversal symmetry. We show that for N f < N there isa moduli space of vacua to all orders in perturbation theory but supersymmetry is brokennon-perturbatively on this moduli space. We take the sigma matrices to be as usual σ = (cid:32) (cid:33) , σ = (cid:32) − ii (cid:33) , σ = (cid:32) − (cid:33) . (2.1)Below we will use the Lorentzian signature ( − , + , +). We will denote the corresponding γ matrices by γ , , γ = iσ , γ = σ , γ = σ . (2.2)A Majorana spinor is a real two dimensional vector λ α . We define ¯ λ ≡ λ T γ . As usual,¯ λλ is a Lorentz invariant and ¯ λγ µ ∂ µ λ is likewise a Lorentz invariant. In our conventions,when we add these terms to the action, they both have to be multiplied by a factor of i .Time reversal symmetry acts as follows (in addition to the obvious action on space-timecoordinates, where the sign of x is reversed): T : λ → ± γ λ . (2.3)One is free to choose the sign in this transformation rule. It is important to remember that T is anti-linear. This will be used throughout below. The Majorana mass term is odd under– 3 –ime reversal symmetry (whatever sign in (2.3) we use) T ( i ¯ λλ ) = iλ T γ γ γ λ = − i ¯ λλ . Thekinetic term is of course even under time reversal symmetry. In theories with N = 1 supersymmetry, the superspace consists of the usual coordinates x µ and the Majorana Grassmann coordinates θ α . Since the θ α are Majorana, time reversalsymmetry must act on them as in (2.3): T : θ → ± γ θ . The Lagrangian is determined by thereal superpotential W , L = i (cid:90) d θ W . The factor of i is important for the theory to have a Hermitian Hamiltonian. As with thefermion mass, (2.3) implies that id θ is odd under time reversal symmetry. Therefore to writetime reversal invariant theories we need W to be a pseudo-scalar T : W → − W . (2.4)In addition, W clearly needs to be invariant under all the global symmetries and gaugesymmetries. These conditions will turn out to be very restrictive, as we will see.Let R be a real superfield (therefore it contains a real boson and a Majorana fermion).The bottom component of R will be denoted by R (cid:12)(cid:12) . Suppose that R is a scalar, that is,under time reversal symmetry T : R → R . Clearly we cannot write any time reversalinvariant superpotential W ( R ) which does not contain superspace derivatives since there isno way to make it odd under time reversal symmetry as required in (2.4). Of course, withoutsupersymmetry we could easily write such a potential, but the fermionic terms accompanyingany nontrivial function W ( R ) are necessarily odd and hence we cannot write a potentialconsistent with N = 1 SUSY T : R → R , W ( R ) = 0 . (2.5)In particular, any time reversal invariant theory of a real scalar superfield must have an exactreal flat direction.If R (cid:12)(cid:12) is a pseudo-scalar, T : R → − R , then we can easily write analytic superpotentials(without superspace derivatives) such as W ( R ) = R + R + ... , containing only odd powersof R and thus preserving time reversal symmetry T : R → − R , W ( R ) = R + R + · · · . (2.6)The most general case can always be analyzed by simply decomposing the superfieldsinto real superfields and using the rules above. The action on the kinetic term is T ( i ¯ λγ µ ∂ µ λ ) = T ( i ¯ λγ ∂ λ ) + T ( i ¯ λγ i ∂ i λ ) = ( − i )( − λ T γ γ γ ∂ γ λ +( − i )( − λ T γ γ γ i ∂ i γ λ = i ¯ λγ ∂ λ + i ¯ λγ i ∂ i λ = i ¯ λγ µ ∂ µ λ . Therefore, the kinetic term is time reversal invariant. Below when we establish various non-renormalization theorems, except when mentioned explicitly, we arealways concerned with the terms in the superpotential which do not vanish for constant bottom components,i.e. potential terms. Of course, also the terms with superspace derivatives need to obey the same selectionrules under time reversal symmetry. – 4 –f we start with a theory with N = 2 supersymmetry in 2+1 dimensions, we can rewriteit as an N = 1 theory. The N = 1 superpotential would be a pseudo-scalar but the original N = 2 superpotential may or may not be a pseudo-scalar, depending on how the time reversalsymmetry is defined to act. ABC
Model
Let us consider a toy model which exhibits some of the ideas above. Take three real superfields
A, B, C and consider the superpotential W = gABC . (2.7)( g is the coupling constant). This is an interacting, super-renormalizable model; it is stronglycoupled in the infrared.First let us consider the classical supersymmetric ground states. The equations for thoseare AB = AC = BC = 0 , and hence there are solutions with A = B = 0 and arbitrary (real) C , A = C = 0 andarbitrary (real) B , B = C = 0 and arbitrary (real) A . These are three real lines whichintersect at the origin. Let us consider the fate of this moduli space quantum mechanically.The global (unitary) symmetries of the model are generated by the Z action A →− A, B → −
B, C → C and by the permutation symmetry S acting on A, B, C . In total,this group has 24 elements. This group turns out to be isomorphic to S . This is not yetsufficiently constraining as this by itself would allow as we integrate out high momentummode (near the ultraviolet, where the original degrees of freedom are still useful) to generatenew vertices such as A + B + C , which is indeed invariant under the S . Such a term wouldlift the moduli space.Now let us add time reversal symmetry into our considerations. Since the superpotentialhas to be odd under time reversal symmetry, we can by no loss of generality choose A tobe a pseudo-scalar superfield and B, C are scalar superfields. All the other choices can beobtained by composing this time reversal symmetry with S .This significantly restricts the vertices that can be generated as we integrate out highenergy mode in the ultraviolet (where, again, the original degrees of freedom are still useful).For example, A + B + C cannot be generated since it is even rather than odd under timereversal symmetry. Similarly, no vertex of the sort A n + B n + C n with integer n can begenerated. We may, however, generate various new irrelevant operators such as ∼ (cid:90) d θ ABC ( A + B + C ) , We thank M. Rocek for making this observation. – 5 –hich is both irrelevant and does not lift the moduli space. It is indeed easy to see that everyvertex that is generated as we perform the renormalization group transformations has to beproportional to
ABC and hence cannot lift the moduli space. It is easy to prove that the classical moduli space continues to exist non-perturbativelyby going on the moduli space and computing the Coleman-Weinberg [7] potential. Let usconsider the branch, where, without loss of generality B = 0 , C = 0, and A is arbitrary.Then, we can integrate out the heavy B, C fields and we obtain some effective superpotential W eff = W eff ( A ). We can choose A to be a scalar under time reversal symmetry and hence W eff = 0 necessarily follows. (Time reversal symmetry is not spontaneously broken for largeenough A since the theory is arbitrarily weakly coupled there. )The moduli space of 3 real lines meeting at a point therefore survives in the full quantumtheory. At the intersection there is an N = 1 SCFT. The model has no relevant deformationswhich preserve all the symmetries and N = 1 supersymmetry. We can deform the modelby the mass term (cid:82) d θ m (cid:0) A + B + C (cid:1) such that m is odd under time reversal but allthe global symmetries are maintained. For either positive or negative m we have 5 gappedtrivial vacua. These 5 gapped vacua split into one that preserves S and in the other 4 vacuathe unbroken symmetry is isomorphic to S and therefore these 4 vacua are related by thesymmetry breaking pattern S −→ S . N = 1 Abelian Gauge Theory with a Charge 2 (Super)Field
Here we consider a U (1) gauge field (with no Chern-Simons term) coupled minimally ( N = 1supersymmetrically) to a charge 2 multiplet Φ. The theory is time reversal invariant, whichmeans that the Chern-Simons level vanishes (hence the subscript 0 on the gauge group). Weare using the usual convention, where integrating out a charge 1 fermion shifts the Chern-Simons level by ± depending on the sign of the mass.The most general superpotential is again some function W = W ( | Φ | ). Hence, thereis no way to write a superpotential that preserves time reversal symmetry. This leads to anon-renormalization theorem: If we start from a superpotential that vanishes at tree level,then the superpotential vanishes in the full quantum theory as we integrate out high energydegrees of freedom near the ultraviolet. We see that in this case the implication of timereversal symmetry is even stronger than in the ABC model. To see that it is convenient to think about the generated vertices as some polynomials in
A, B, C . If anyof the terms does not contain all the three fields, we can choose the time reversal symmetry to be such that itacts on all the fields in that vertex as scalars and hence such a vertex must violate some combination of thetime reversal symmetry and the unitary symmetries. We thank M. Rocek for a discussion of this point. We thank N. Seiberg for a discussion of this point. A single charge 1 fermion cannot be coupled to a dynamical U (1) gauge field while preserving timereversal symmetry because of an (ABJ-like) anomaly [8–10]. This is why we study the model of a U (1) gaugefield coupled to a charge 2 multiplet, which suffers from no such anomaly and time reversal symmetry can bemaintained. The non-supersymmetric version of this model was considered in [11] and the non-supersymmetricmonopole-deformed version in [12]. – 6 –he exact vanishing of the superpotential shows that the time reversal invariant theoryhas a moduli space given by arbitrary expectation values of Φ (cid:12)(cid:12) divided by the gauge symmetry.This leads to a moduli space isomorphic to R + M vac (cid:39) R + . (3.1)The effective theory on this R + requires some attention. At the origin of R + there is a certainSCFT. Let us now consider what happens away from the origin. Due to the fact that Φ hascharge 2 under the gauge symmetry, there is an unbroken Z gauge theory on the modulispace. The most familiar version of Z gauge theory is described by the k matrix k = (cid:32) (cid:33) , however, here we have a Dijkgraaf-Witten modification [13] of this Z gauge theory. Theeasiest way to understand it is through the fact that our theory has a vanishing quantumChern-Simons level (which is why the theory is time reversal invariant), but the bare Chern-Simons level (in absolute value) is 2.This modification of Z gauge theory can be described by the k matrix k = (cid:32) (cid:33) . (3.2)It is easy to see that this modified Z gauge theory is isomorphic to U (1) × U (1) . Thereforewe conclude that on the moduli space (3.1) the effective theory consists classically of themodulus ρ parameterizing R + as well as the TQFT U (1) × U (1) . The origin is a singularpoint, where there is a SCFT. The TQFT on the moduli space is time reversal invariant (seefootnote 7), which is of course important for the consistency of our picture.Let us now break time reversal symmetry explicitly by adding a mass term in the super-potential W = m | Φ | . For m > U (1) TQFT and of course also for m < U (1) TQFT. It is important to remark that the Witten index at negative m is − m it is +2 and therefore the Witten index jumps. It jumps at m = 0 by theappearance of the exact moduli space. (Hence, the index at strictly m = 0 is ill defined.)It will prove useful to make a few comments about the model where we also add a quarticterm W = m | Φ | + λ | Φ | . (3.3)For fixed nonzero λ there is now no moduli space at m = 0. If λ > m one vacuum with U (1) TQFT and for negative m we have two ground states, one with U (1) TQFT and one with U (1) × U (1) TQFT. For λ < Here and below we will use the fact that U (1) (cid:39) U (1) − as spin TQFTs. – 7 –hese various phases as well as the moduli space of vacua (at λ = 0) with the particularTQFT on the moduli space (3.2) will be later compared to a dual description of this theory.To understand the duality in the next subsection it is useful to extend the theory (3.3)even further. We add another neutral real scalar superfield S and consider the theory as afunction of some superpotential terms for the superfield S . The tree-level superpotential isgiven by W = S | Φ | −
12 ˜
M S + ˜ ξS and we study the model as a function of ˜ ξ and ˜ M . Time reversal symmetry forces S tobe a pseudo-scalar (this we see from the first term in the superpotential) and as a resultthe parameter ˜ M is time reversal odd and the parameter ˜ ξ is time reversal even. Quantumcorrections to the superpotential are allowed. We will keep only the one-loop correction for S and later explain why this is sufficient for our purposes. The one-loop corrected superpotentialis given by W Tree+1 − Loop = S | Φ | −
12 ˜
M S − S | S | + ˜ ξS . (3.4)In the limit of large | ˜ M | the model clearly reduces to the minimal model we analyzed before,with a | Φ | correction in the superpotential (3.3). When we take | ˜ M | to be strictly infinite thenwe recover precisely the minimal model of a charge 2 multiplet with a vanishing superpotentialand moduli space of vacua isomorphic to R + .A very important fact to note is that the model has enhanced N = 2 supersymmetry(and R symmetry) for ˜ M = 0. The phases of the model as a function of ˜ M and ˜ ξ are • ˜ M >
1. For positive ˜ ξ we have a gapped SUSY vacuum with U (1) TQFT in the deepinfrared. For negative ˜ ξ we have two gapped SUSY vacua, one with the Z gauge theory(recall that because of the Dijkgraaf-Witten term it is isomorphic to U (1) × U (1) )gauge theory and one with U (1) TQFT. The Witten index is constant as a function of ξ . • ˜ M < −
1. For positive ˜ ξ we have a SUSY vacuum with U (1) TQFT. For negative ˜ ξ wehave two SUSY vacua, one with U (1) TQFT and one with U (1) × U (1) TQFT. TheWitten index is again constant as a function of ξ . These are the same phases that wesaw for ˜ M >
1. This is not surprising, since under time reversal symmetry ˜ M → − ˜ M (and ˜ ξ is fixed). • | ˜ M | <
1. For positive ˜ ξ we have two gapped SUSY vacua, each carrying a U (1) TQFT.For negative ˜ ξ we have a gapped SUSY vacuum with U (1) × U (1) TQFT. The Wittenindex is constant as a function of ˜ ξ .We draw all these phases in the following figure.– 8 – (1) ( A ) ( B )( C ) ( E ) I = 2 ( D )( F ) ˜ M ˜ M = 1 ˜ M = U (1) ⇥ U (1) + U (1) I = 4 U (1) ⇥ U (1) + U (1) I = 4 U (1) I = 2 I = 2 + 2 U (1) + U (1) I = 4 U (1) ⇥ U (1) There are 6 regions, labeled A,B,C,D,E,F clockwise and in each region we record thevacua and their Witten indices. The bold vertical line is at ˜ ξ = 0, positive ˜ ξ is to the rightand negative ˜ ξ is to the left.While as a function of ˜ ξ the Witten index does not jump, if we travel vertically on thediagram (i.e. change ˜ M ) the Witten index jumps and there are two walls at ˜ M = ±
1. Thewalls may move as we include higher loop corrections, but the number of such regions and the(gapped) vacua in each region would not change, which is why we are allowed to keep just theone loop correction in the superpotential. An important fact is that ˜ M = ∞ and ˜ M = −∞ should be really identified (with a twist in the horizontal direction), as they describe thetheory (3.3) with λ = 0. We also see that at large | ˜ M | we find precisely the phases we haveseen in (3.3) for positive and negative λ . We begin with a quick review of the N = 1 supersymmetric model of a charge 1 superfield ˜Φcoupled minimally to a U (1) / gauge field. We study the phases of the model as a functionof the mass in the superpotential W = m | ˜Φ | . In [4] it was shown that there is a wall at m = 0 (i.e. the Witten index jumps there due to the appearance of a new vacuum at infinity)and due to a radiative correction there is a conformal field theory at some m ∗ > U (1) TQFT.Here we will generalize this model a little bit and add an additional neutral field S aswell as a superpotential W = S | ˜Φ | . This model naturally has two deformation parameters, M and ξ W Tree = gS | ˜Φ | + 12 M S + ξS . – 9 –his model has no microscopic time reversal symmetry. If we take | M | to be very large wecan integrate out S and the model reduces to the minimal model without S and some quarticcoupling | ˜Φ | .To study the critical points of this superpotential, it is important to add the leadingnontrivial radiative correction for S : W One − Loop = gS | ˜Φ | − M g S + ξS − g S | S | . (3.5)For a special choice M = this model has enhanced N = 2 supersymmetry as well as a U (1) R symmetry.The various phases of the model are • M > /
2. Here for positive ξ we have a gapped SUSY vacuum with U (1) TQFT inthe deep infrared. For negative ξ we have two trivial gapped SUSY vacua. The Wittenindex is constant as a function of ξ . • M < − /
2. For positive ξ we have a trivial gapped SUSY vacuum. For negative ξ we have a trivial gapped SUSY vacuum as well as a vacuum with U (1) TQFT. TheWitten index is constant as a function of ξ • | M | < /
2. For positive ξ we have two gapped SUSY vacua, one trivial and one withTQFT U (1) . For negative ξ we have a trivial gapped SUSY vacuum. The Wittenindex is constant as a function of ξ We will discuss the the lines M = ± momentarily. These are the walls where theWitten index jumps. Of course, these lines may take a different shape in the full theory, butthe various phases and the topology of the phase diagram are robust. It is useful to againplot the various phases we have described above along with the Witten index of each of thegapped phases. – 10 – U (1) T + T T U (1) + T U (1) + T T M = 1 / M = / ( A ) ( B )( C ) ( E ) I = 1 + 1 I = 2 I = 2 I = I = 1 I = ( D )( F ) There are again 6 regions, labeled A,B,C,D,E,F clockwise and in each region we recordthe vacua and their Witten indices. The bold vertical line is at ξ = 0, positive ξ is to theright and negative ξ is to the left. The dashed lines at M = ± are walls where the indexmust jump. The bold line at ξ = 0 is the naive line of SCFTs. (These SCFTs do not have tobe all distinct as we travel on the M axis.)Let us now make some observations – except for M = ± /
2, as we move horizontallyon the diagram, the Witten index does not jump. However, a SCFT must nonetheless occursomewhere around ξ = 0 because the number of vacua and their low-energy properties change.In particular, at ξ = 0 , M = 3 / N = 2 supersymmetry and it can studiedin great detail. As we move vertically on the diagram (away from ξ = 0) the Witten indexclearly jumps at the two walls at M = ± . For example, when we move from region F to Ea trivial SUSY vacuum disappears to infinity and as we proceed into region D a new SUSYvacuum with U (1) TQFT appears from infinity.On the two walls at M = ± the superpotential is W = gS | ˜Φ | ∓ g S − g S | S | + ξS = 0 . (3.6)and the critical point equations are S ˜Φ = 0 , g | ˜Φ | ∓ g S − g | S | + ξ = 0 . (3.7)On the M = 1 / S (and vanishing Φ) is asymp-totically constant and similarly on the M = − / S (and vanishing Φ) theenergy density is constant. In both cases at ξ = 0 there is a flat direction isomorphic to R + .This moduli space of vacua is not an artefact of the leading order approximation. The walls– 11 –xist in the full theory and since there is a phase transition on the wall, the only way thatthe figure can be consistent is that a moduli space opens up at the SCFT on the wall. Thisis a general mechanism that can lead to moduli spaces of SUSY vacua even in the absence ofa microscopic time reversal symmetry.Note a remarkable fact: consider the point M = − / , ξ = 0, at the intersection ofthe regions B,C,D,E. Around that point there is an emergent symmetry of reflections. TheSCFT at M = − / , ξ = 0 is therefore conjectured to have emergent time reversal symmetry.(As we will see, this is not the only point that has emergent time reversal symmetry in theinfrared.)Now simply tensor the theory of S, ˜Φ coupled to a U (1) / gauge field by a U (1) pureTQFT. Then, all the phases we found here can be seen to exactly coincide after the appropriateidentification with the phases of the charge 2 particle coupled to a U (1) gauge field.In particular, the point M = − / , ξ = 0 (tensored with a U (1) TQFT) is dual toour U (1) gauge field coupled to a charge 2 superfield with vanishing superpotential (i.e. | ˜ M | = ∞ ). Therefore, there is emergent time reversal symmetry at M = − / , ξ = 0. Inaddition, we see that ˜ M = 1 (or equivalently ˜ M = −
1) maps to M = 1 /
2. The regionsABCDEF in the second figure therefore map to EFCDAB, respectively, in the first figure.Another interesting special case of this duality is the map between ˜ M = 0 and M = 3 / M = 3 / , ξ = 0. This latter case isa duality between two N = 2 supersymmetric theories (of which one has emergent timereversal symmetry) and it can be subjected to stringent tests using the S and S × S partition functions. We study this N = 2 duality in the appendix B.It would be interesting to understand when the parameters M, ˜ M are relevant. Clearlythe SCFTs at M = 3 / ←→ ˜ M = 0 and M = − / ←→ | ˜ M | = ∞ are not the same since theformer does not have a moduli space of vacua while the latter does. The former has N = 1supersymmetry while the latter N = 2 supersymmetry. Therefore, the duality we find herehas both N = 2 and N = 1 versions. In both cases in one of the duality frames there isemergent time reversal symmetry. In the N = 1 version of the duality, the infrared SCFThas a moduli space of vacua isomorphic to R + .Note that starting from the duality that we established above U (1) + charge 2 ←→ U (1) ⊗ (cid:2) U (1) / + charge 1 (cid:3) , we can imagine deforming the theories by giving a mass to the bosons on the left-handside and a corresponding mass to the fermion on the right-hand side. This leads to a non-supersymmetric duality between U (1) +charge 2 fermion and U (1) ⊗ [ U (1) +charge 1 boson].But since U (1) + charge 1 boson is dual to a Dirac fermion, we recover precisely the dualitybetween U (1) + charge 2 fermion and U (1) ⊗ Dirac fermion, in agreement with [11, 12].Both sides of the proposed duality have a Z Symmetry Enhancement in N = 2 QED and Supersymmetry Enhance-ment in a Wess-Zumino Model
Here we study a certain N = 2 duality that exhibits surprising features. On one side of theduality the global symmetry of the infrared theory is not manifest. On the other side of theduality, the supersymmetry of the infrared theory is not manifest.We want to claim a duality between the following two theories:1. A 3d N = 2 U (1) gauge theory with vanishing Chern-Simons level and two chiral super-fields of charge 1. This theory has an unexpected IR enhancement of flavor symmetryto SU (3).2. A 3d N = 1 Wess-Zumino model eight real chiral fields φ a transforming in the adjointof SU (3), and a real cubic superpotential W = 16 d abc φ a φ b φ c , (4.1)where d abc = 2Tr[ { T a , T b } T c ], with T a , a = 1 , ..., su (3), satisfyingTr[ T a T b ] = δ ab . This N = 1 theory is conjectured to have N = 2 supersymmetry (as wellas a U (1) R symmetry) in the infrared.To support this surprising duality proposal, we will match the massive deformations ofthese theories, including the contact terms for background gauge fields. Additional argumentsfor this symmetry enhancement have also been suggested in [18, 19]. There is a simple mass deformation, with super-potential W = 16 d abc φ a φ b φ c + m a φ a . (4.2)The masses m a transform in the adjoint of the SU (3) flavor symmetry. We can alwaysconjugate them to the Cartan sub-algebra and to a Weyl chamber therein. Generic masseswill preserve a U (1) symmetry, but there is a co-dimension 1 locus where the preserved flavorsymmetry enhances to U (1) × SU (2).To be concrete, we can collect the real scalars and masses into traceless Hermitian 3 × φ a T a and M = m a T a . The superpotential becomes W = 23 TrΦ + Tr M Φ (4.3)and the equation for classical vacua isΦ ∗ + M = c × . (4.4)Observe that for M = 0 (i.e. the undeformed model) there are no solutions other than thetrivial one and hence there is no moduli space of vacua at the SCFT point.– 13 –aking M diagonal and generic, we see that Φ ∗ is the square root of a diagonal matrixwith positive entries and is also diagonal. Only two of the possible square roots are traceless,giving us two semi-classical vacua with unbroken U (1) global symmetry. The two vacuaspontaneously break time-reversal symmetry, as Φ is a pseudo-scalar.On the other hand, if M is special and preserves U (1) × SU (2), say, diagonal with entries( m, m, − m ), then we have two possibilities:1. For m < c = − m and Φ ∗ has a traceless 2 × × .That gives an C P worth of vacua, spontaneously breaking SU (2) × U (1) → U (1) .2. For m > c = 2 m and there are again two vacua, which preserve SU (2) × U (1). Thesetwo vacua are related by time reversal symmetry.In the ( M , M ) plane, we thus have three half-lines with an C P worth of vacua, and2 vacua that are related by time reversal symmetry elsewhere. Time reversal symmetryunder which Φ is a pseudo-scalar acts by the antipodal map on the C P . The analysis ofthese massive deformations is also done in detail (in components) in appendix A. The phasediagram is shown in Fig. 1.As C P is a K¨ahler manifold, the low energy non-linear sigma model on these threelines where the global SU (3) symmetry is explicitly broken to SU (2) × U (1) gains an N = 2supersymmetry in the deep infrared. (In other words, the C P model with a round metricand at most two derivative interactions with N = 1 supersymmetry must in fact have N = 2supersymmetry as well as a U (1) R symmetry.)We would like to argue that this supersymmetry enhancement is also a property of the IRSCFT at the origin of parameter space (i.e. at M = 0). The new infrared supercharge startsits life as a cubic in the fermions (the spins are symmetrized so the flavor indices must beanti-symmetrized and hence we have to use the f abc symbol of su (3)) and thus the predictionis that the dimension renormalizes down from 3 in the ultraviolet to 2 . ∗ takes the form δW = TrΦ ∗ δ Φ . (4.5)As the vacuum vev Φ ∗ is diagonal, the coefficient for the off-diagonal | δ Φ ji | is simply (Φ ∗ ) ii +(Φ ∗ ) jj = − (Φ ∗ ) kk , where k is the index different from i and j . Since Φ ∗ is traceless, we eitherhave two positive masses and one negative, or two negative masses and one positive. As thetwo vacua have opposite Φ ∗ , the pattern of masses is opposite in the two vacua and so arethe background Chern-Simons couplings. This is compatible with the spontaneous breakingof time-reversal symmetry.If we denote the U (1) background gauge connection as A i , with A + A + A = 0,then depending on which of the three chambers we are in the vacua will have background CScouplings ± A dA , ± A dA or ± A dA . – 14 – igure 1 . Phase diagram of the Wess-Zumino model and gauge theory. For the Wess-Zumino model,this is ( M , M ) plane and three half lines are given by L : M − M = 0 ( M < L :2 M + M = 0 ( M < L : M + 2 M = 0 ( M > t , m f ) plane and three lines are given by L : m f = 0 ( t > L : t + m f = 0 ( m f > L : t − m f = 0 ( m f < SU (2) × U (1) preservingdeformation, and CP worth of vacua due to spontaneous symmetry breaking. Otherwise, we have 2isolated vacua due to time reversal symmetry breaking. If we sit on a line with unbroken U (1) × SU (2), we choose a parametrization of Φ asΦ = φ + R R + iR XR − iR φ − R ¯ Y ¯ X Y − φ ; (4.6)where R i and φ are real and X , Y , and Z are complex. Plugging the expectation valueΦ ∗ = diag { m, m, − m } with m >
0, the mass terms can be written as2 m ( R + R + R ) − m ( | X | + | Y | ) − φ (4.7)In other words, we have an SU (2) doublet of fields with U (1) charge 1 and mass − m , a tripletof SU (2) with U (1) charge 0 and mass 2 m , and a singlet φ .A background Chern-Simons term for SU (2) induced by integrating out matter field withmass m i in the R i representation. This is given by [20] k SU (2) ,eff = k SU (2) ,bare + 12 (cid:88) i T ( R i ) sign( m i ) (4.8)where T ( R i ) is the quadratic index of representation R i defined byTr[ T aR T bR ] = 12 T ( R ) δ ab (4.9)– 15 –ith the normalization T ( ) = 1, T ( ) = 2. Therefore, k SU (2) ,eff = 12 . (4.10)(We will soon compare this do the dual gauge theory.) N = 2 Gauge Theory
We denote the two chiral superfields with the gauge charge +1 as Q and ˜ Q and the bottomcomponent in the vector multiplet V is denoted as s . The theory has four natural (i.e.visible in the microscopic theory) N = 2-preserving real mass deformations, associated to the U (1) t × SU (2) f flavor symmetry. Without loss of generality, we can consider the two massdeformations associated to the Cartan subalgebra. We will denote the generator associatedto the Cartan of SU (2) by m f and the FI parameter for the topological U (1) T symmetry isdenoted by t . The supersymmetric vacua can be found by looking for the critical points ofthe one-loop corrected potential, as in [20]. The equations are thus | Q | + | ˜ Q | = t eff (4.11)( s + m f ) Q = 0 (4.12)( s − m f ) ˜ Q = 0 (4.13)where the one-loop effective FI parameter is t eff = t + 12 ( | s + m f | + | s − m f | ) . (4.14)For t = m f = 0 the theory has no moduli space of vacua, as our Wess-Zumino model.When t > m f = 0, the theory has a C P of vacua parameterized by vevs of the chiralfields (and s = 0). This corresponds to the half line L in Fig.1. Turning on m f reduces thevacua to the two poles of C P . These two vacua are related by time reversal symmetry since s = ± m f in these two vacua. Hence, time reversal symmetry is spontaneously broken in thisgapped phase.When t <
0, something special happens when m f = ± t , i.e. on the half lines L and L :a Coulomb branch parameterized by the real scalar in the vectormultiplet, with t ≤ s ≤ − t exists. In this phase Q = ˜ Q = 0. The two chirals have opposite masses and contribute ( s − t ) − ( s + t ) = − t to the effective FI parameter. The real scalar combines with the dualphoton to give a new C P .For intermediate values of m f in the region II in Fig.1, we have vacua where both chiralshave either positive or negative mass and no vacuum expectation values. Therefore theeffective FI parameter is t ± s , so that s is fixed either to s = t or to s = − t . We have againtwo vacua and time reversal symmetry is spontaneously broken.The S symmetry of the phase diagram, permuting the three C P half-lines, is clearlysuggestive of an enhancement of the flavour symmetry to SU (3).– 16 –his conjectural S Weyl symmetry can be realized as a group of self-dualities of thetheory. The simplest way is to use the N = 2 particle-vortex duality to convert a chiralmultiplet to a U (1) theory coupled to a dual chiral multiplet. The resulting theory has anew manifest Z symmetry exchanging at the same time the two gauge fields and the twochiral multiplets. Together with the Weyl symmetry of SU (2) f , this generates the required S . The massive phase that we have found with m f = 0 for t < SU (2) × U (1) symmetry. In this case the doublet ( Q, ˜ Q ) (in one of the two vacua which arerelated by time reversal symmetry) acquires a positive mass, generating a background SU (2)Chern-Simons term after it is integrated out k SU (2) ,eff = 12 (4.15)This matches the result (4.10) in the Wess-Zumino model.We can look at the background Chern-Simons coupling in the massive vacua. Withgeneric m f and t , U (1) t and U (1) f ⊂ SU (2) are preserved. Again, integrating out fermionswith real mass m i will induce background Chern Simons terms given by k ab, eff = k ab, bare + 12 (cid:88) i n a,i n b,i sign( M i ) (4.16)where n a,i is the charge of i -th fermion under U (1) a , M i = m i + n g,i s is the effective mass,with gauge charge n g,i . In the region I, the result for two vacua is (cid:32) k JJ k Jf k fJ k ff (cid:33) I = (cid:32) (cid:33) , (cid:32) − − − (cid:33) (4.17)In the region III, the situation is similar (cid:32) k JJ k Jf k fJ k ff (cid:33) III = (cid:32) − − (cid:33) , (cid:32) − (cid:33) (4.18)Region II is a bit different since the signs of masses of Q and ˜ Q are the same and s is equal to FI parameter ± t , which forces gauge field to identify with A J . We then havenonzero k JJ . (cid:32) k JJ k Jf k fJ k ff (cid:33) II = (cid:32) − (cid:33) , (cid:32) − (cid:33) (4.19) The Weyl symmetry can be made manifest in both the index and S b partition functions of the theory, bymanipulations analogue to that particle-vortex duality. See appendix B There is also a more physical way to look at it. In the positive mass vacuum, integrating out Q and ˜ Q generates level one Chern Simons term π ada for the gauge field a . Then π ( ada + 2 adA J ) can be written as π (( a − A J ) d ( a − A J ) − A J dA J ). The first term is an trivial theory U (1) [21] and we have k JJ = −
1. Likewisewe have k JJ = 1 for the other vacuum. – 17 –n the other hand, we can also compute the k matrix in the three regions on ( M , M )plane of Wess-Zumino model side. The charges of the different fields under the global sym-metry are given by Q ˜ Q X Y Z φ φ U (1) f U (1) J X Y ZA A A φ a T a in the Chevalley basis. In particular, X = ( φ − iφ ), Y = ( φ + iφ ), Z = ( φ + iφ ). The masses of the three complex fields X , Y and Z are givenby the Hessian matrix of the scalar potential at the vacua. A simple calculation gives usthe same k matrix (4.17),(4.19),(4.18), as the gauge theory in the three regions respectively.Upon identifying A f = A and A J = A + A in the previous section, we get ± A dA , ± A dA , and ± A dA in the region I, II, III respectively. This serves as another consistencycheck of our duality. N = 1 Duality Between SQED and a Wess-Zumino Model
Here we would like to make some comments about an N = 1 duality which can be derivedfrom our N = 2 duality above. We will see that this N = 1 duality is also closely related tothe SQED -XYZ duality [22].The dual pair is1. A 3d N = 1 U (1) gauge theory with two chirals of charge 1, total Chern-Simons level0. This theory has an SU (2) f × U (1) t global symmetry.2. A 3d N = 1 Wess-Zumino model with seven real chiral fields: a complex SU (2) f doublet u α of U (1) t charge 1 and a real SU (2) f triplet R αβ . We also add a real cubicsuperpotential W = R αβ u α ¯ u β . (5.1)We denote ¯ u β = ( u β ) ∗ (i.e. it is just the complex conjugate superfield) and indices areraised or lowered using the SU (2) invariant tensor (cid:15) αβ .We will first motivate the duality and then connect it to the duality we have presentedin the previous section. This N = 1 duality has appeared before in a different context, forinstance, [23, 24].First let us compare the moduli spaces of vacua of the two theories before we add anydeformations. We begin with the U (1) with two superfields Φ , Φ carrying charge 1 under– 18 –he U (1) gauge symmetry. We have a classical moduli space of vacua, where the gaugesymmetry is broken (everywhere except at the origin). We can parameterize the moduli spaceby the expectation values of Φ and Φ while removing one overall phase which is gauged.Therefore we can parameterize the moduli space by | Φ | , | Φ | , and arg (cid:16) Φ Φ (cid:17) . The model hasa global SU (2) symmetry which has a U (1) subgroup that acts by shifting arg (cid:16) Φ Φ (cid:17) whileleaving | Φ | , | Φ | intact. Therefore, considering now the full theory and not just the classicaltheory, we see that the superpotential cannot depend on arg (cid:16) Φ Φ (cid:17) . And since | Φ | , | Φ | aretime reversal even, it cannot depend on them either. Therefore the classical moduli space ofvacua is not lifted. M vac = R . (5.2)At a generic point on the moduli space the SU (2) global symmetry is broken to U (1) and hencewe can equivalently parameterize the moduli space by the expectation value of | Φ | + | Φ | and the S = SU (2) U (1) worth of Nambu-Goldstone vacua fibered over it.Now let us consider the moduli space of vacua of the Wess-Zumino model (5.1). Thereis classically a moduli space parameterized by R αβ . Going far on this moduli space, we canintegrate out F, ¯ F and write an effective superpotential W eff = W eff ( R αβ ). This effectivesuperpotential is constrained by SU (2) invariance and by time reversal symmetry, which actsby R → − R . This leaves arbitrary SU (2) invariant terms with an odd number of R fields. Itis easy to see that all such terms vanish identically by simply diagonalizing R . Therefore, themoduli space is spanned by R ab and hence we get M vac = R , as in the Wess-Zumino model.Now we compare some deformations of the model, starting from the non- SU (2) invariantmass deformation. Without loss of generality we consider W = m | Φ | − m | Φ | . This mass deformation breaks the global SU (2) symmetry explicitly to U (1). At nonzeropositive m we have an effective theory of a U (1) gauge field without a Chern-Simons term.The matter fields are massive. Therefore, it can be dualized to a real compact superfieldand hence we have a circle of supersymmetric vacua. Similarly, for negative m we have acircle of supersymmetric vacua. In both cases, these circles of supersymmetric vacua can beinterpreted as due to the spontaneous breaking of U (1) t .The dual description of this mass deformation corresponds to adding a linear term in R ,which we can take without loss of generality to be δW = mR (remember that R = − R ).The equations of motion now take the form | u | − | u | = m ,u ¯ u = 0 ,R αβ u α = 0 . The solution is clearly R = 0, and for positive m we have nonzero u = 0, | u | = m , and fornegative m we have R = 0, u = 0, | u | = − m . So in both cases we have a circle of vacuawith a spontaneously broken U (1) t symmetry.– 19 –inally, we can have an SU (2) invariant mass deformation. In the Abelian gauge theorythis corresponds to W = m | Φ | + m | Φ | . At positive m we have a trivial supersymmetric vacuum (here we use the fact that U (1) is atrivial TQFT) and for negative m we have likewise another trivial supersymmetric vacuum.(The Witten index at positive m is +1 and at negative m it is − δW = mη αβ u α ¯ u β + c R αβ R αβ , (5.3)where η αβ is as usual the identity matrix. The coefficient c is unknown. Classically, withthe deformation (5.3) the R equations of motion still set u = 0 and due to c R is pinned tothe origin. (Even if c = 0, while R is classically arbitrary, there will be an effective potentialon the moduli space parameterized by R since there is no more time reversal symmetry.) Aslong as R is pinned to the origin (either due to c or due to a radiatively generated potential)we get a massive trivial supersymmetric vacuum with unbroken U (1) t × SU (2), as requiredby the duality.Therefore, under the duality, R αβ maps to a triplet of mesons and the singlet meson mapsroughly to | u | (with a possible admixture of R ).This pair of N = 1 theories is related by simple “flip” operations both to the dual pair inthe previous section and to the SQED -XYZ mirror symmetry. It is also related to the basic3d N = 4 mirror pair [22, 25].In order to see the relation to the former, we can “flip” the real U (1) t moment mapoperator in the 3d N = 2 U (1) gauge theory. Flipping means adding a new real multiplet R with linear superpotential coupling to S ( S is the real N = 1 superfield containing in thebottom component the real scalar of the vector multiplet), i.e. promoting the N = 2 FIparameter to a dynamical real superfield. This clearly preserves N = 1 supersymmetry. Thisnew term allows to integrate out R and S . No extra N = 1 superpotential terms can beinduced, as no gauge-invariant operators odd under time reversal symmetry (and invariantunder the global symmetries ) are available. We thus have the N = 1 U (1) gauge theory withtwo chirals of charge 1 at low energies.On the Wess-Zumino side of the duality, the flip breaks SU (3) to SU (2) f × U (1) t . The8 real chiral multiplets decompose into a complex SU (2) f doublet u α of U (1) t charge 1,a real SU (2) f triplet R αβ and a singlet, to be identified with S . The flipping operationagain removes S and leaves a Wess-Zumino model of seven real chiral operators and cubicsuperpotential W = R αβ u α ¯ u β , (5.4)which is the model we have studied here. Again, no extra terms compatible with the symme-tries are available other than (5.4). Note that the other SU (2) singlet deformation, u α ¯ u α , is redundant (in the sense that it can be removedby a change of variables). – 20 –ow we will explain the relation of this N = 1 duality with the familiar N = 2 mirrorsymmetry. We again start from the gauge theory side ( N = 2 SQED ) and flip the real U (1) t moment map operator. The result is, again, the N = 1 U (1) gauge theory with twomultiplets of charge 1. No superpotential can be generated. Notice the enhancement of flavorsymmetry from U (1) to SU (2) f × U (1) t . This enhancement is due to the fact that in N = 1theories, there is no difference between charge 1 and charge -1 multiplets.On the XY Z side, the operator we are flipping is the real moment map | X | − | Y | . Thereal superpotential is deformed from Re( XY Z ) to W = Re Z ( XY + ¯ X ¯ Y ) + i Im Z ( XY − ¯ X ¯ Y ) + R ( | X | − | Y | ) (5.5)If we form a doublet u α = ( ¯ X, Y ), then the superpotential becomes W = Re Z ( u ¯ u + u ¯ u ) + i Im Z ( u ¯ u − u ¯ u ) + R ( u ¯ u − u ¯ u ) = R αβ u α ¯ u β (5.6)which has enhanced SU (2) f × U (1) t global symmetry, with a triplet R αβ = (Re Z, Im Z, R ).This is precisely our dual N = 1 Wess-Zumino model.What we have shown is that the N=1 duality described here can be derived either fromthe standard mirror symmetry or starting from our new duality in the previous section. Wecould have of course derived this duality direction from the 3d N = 4 mirror symmetrybetween SQED and a Free twisted hyper. We can do so by an N = 1 S operation, couplinga U (1) N = 1 gauge field to the free twisted hypermultiplet, in such a way that it “ungauges”the U (1) gauge field on the SQED side.The Wess-Zumino model real superpotential is indeed the N = 1 description of thecoupling between the three real scalars R αβ in the N = 4 gauge multiplet of SQED and thehypermultiplet flavors u α .Alternatively, we can “flip” all three real moment map operators for U (1) t in the N = 4mirror pair. The result is the same. To summarize, the dualities we have considered are allconnected by the ’flip’ operation as in the figure below. N = 1 Supersymmetric SU ( N ) Gauge Theory with N f Quarks
An interesting class of time-reversal invariant non-Abelian gauge theories is given by SU ( N ) gauge theory (the subscript indicates the quantum Chern-Simons level) coupled to N f mul-tiplets of quarks Q i in the fundamental representation ( i = 1 , ..., N f ). The number of funda-mental representations is constrained such that N = N f mod 2 . (6.1)This is a necessary and sufficient condition for the time reversal invariant theory to exist.(We will see below that if this condition is not obeyed one finds various nonsensical results.)– 21 –uppose the superpotential vanishes classically. Then the theory has time reversal sym-metry along with U ( N f ) global symmetry. In fact, if we require time reversal symmetryand U ( N f ) symmetry, it is not possible to write any superpotential which is a function of the Q i . Therefore, to all orders in perturbation theory, the renormalization group (around theultraviolet, where the Q i are good degrees of freedom) does not generate a superpotential ifwe take the superpotential to vanish at tree level.That means that the large moduli space of supersymmetric ground states that exists inthe classically massless theory persists to all orders in perturbation theory.Let us assume that N f < N c . In that case we can parameterize the moduli space by theexpectation values of the mesons M ij = Q † i Q j . (6.2)To see that, we can use the gauge symmetry and U ( N f ) symmetry to bring the N × N f matrix of Q i to the form Q = a ... a ... ... ... ... ... ... a N f ... ... ... ... ... ... (6.3) We ignore various discrete identifications. – 22 –ith a i ≥
0. For generic a i the global symmetry is broken as U ( N f ) → U (1) N f . (6.4)(And the gauge symmetry is broken as SU ( N ) → SU ( N − N f ).) Acting on (6.3) with thebroken U ( N f ) U (1) Nf generators we therefore obtain a N f − N f + N f = N f real-dimensional modulispace. This space is parameterized by the N f mesons (6.2). (The mesons can be thought ofas a Hermitian N f × N f matrix.)The degrees of freedom on the moduli space to all orders in perturbation theory aretherefore these N f massless mesons but, importantly, (at a generic point on moduli space)also the SU ( N − N f ) vector multiplet. The two sectors are not entirely decoupled. Thevector multiplet effective action to first approximation is the canonical one, independent ofthe M ij but there are some irrelevant operators tying the two sectors together. It is easy towrite the leading terms that couple the two sectors, but we would not need them here.Thus far the analysis was to all orders in perturbation theory. However, non-perturbatively,the pure SU ( N − N f ) vector multiplet theory breaks supersymmetry [26]. In the infrared onehas a Goldstino along with the (spin) time reversal invariant U ( N − N f ) N − Nf ,N − N f TQFT [12](this TQFT makes sense due to (6.1)). The vacuum energy is set by the gauge coupling.Therefore, non-perturbatively, our N f real-dimensional moduli space is lifted.Since the coupling between the mesons and the vector multiplet vanishes for | M | → ∞ ,we see that the vacuum energy is asymptotically constant. Unlike in the analogous theoriesin four dimensions (with four supercharges) [27], there is no supersymmetric ground state atinfinity.Note that the case N f = N − N f < N c as there is alwaysa nontrivial unbroken gauge group (which in turn leads to dynamical supersymmetry breakingand lifts the moduli space non-perturbatively).The cases N f ≥ N c are more interesting as a generic point on the moduli space breaksthe gauge symmetry completely. Therefore, there is no mechanism to lift the moduli spaceat a generic point and hence there would be a real moduli space of SUSY vacua in the fulltheory. We leave the analysis of N f ≥ N c for the future. Acknowledgements
We thank O. Aharony, J. Gomis, S. Razamat, M. Rocek, N. Seiberg, A. Sharon, and R. Ya-coby. The research of D.G and J.W. was supported by the Perimeter Institute for TheoreticalPhysics. Research at Perimeter Institute is supported by the Government of Canada throughIndustry Canada and by the Province of Ontario through the Ministry of Economic Devel-opment and Innovation. Z.K. is supported in part by an Israel Science Foundation center forexcellence grant and by the Simons Foundation grant 488657 (Simons Collaboration on theNon-Perturbative Bootstrap) – 23 –
Detailed Analysis of the Wess-Zumino Model (4.2)
We first write the superpotential explicitly, using the d -symbols of SU (3), in the basis ofGell-Mann matrices: W = 16 (cid:20) √ φ (cid:0) ( φ ) + ( φ ) + ( φ ) (cid:1) + 32 (cid:0) φ φ φ + 2 φ φ φ + 2 φ φ φ + φ ( φ ) + φ ( φ ) (cid:1) − (cid:0) φ φ φ + φ ( φ ) + φ ( φ ) (cid:1) − √ φ (cid:0) ( φ ) + ( φ ) + ( φ ) + ( φ ) (cid:1) − √ φ ) (cid:21) . The critical points in the absence of mass deformations can be taken without loss ofgenerality to be diagonal matrices and hence it is sufficient to look for critical points withnonzero φ , φ , and with all the other φ ’s vanishing. The relevant critical point equations arethus ( φ ) − ( φ ) = 0 ,φ φ = 0 . The only solution is thus the trivial solution φ a = 0 and in particular, there is no modulispace of vacua in the undeformed theory. Now we deform the superpotential by linear termsfor φ . Again, without loss of generality we can add a linear term for just φ and φ , so wenow consider the superpotential W = 16 d abc φ a φ b φ c + m φ + m φ . We again look for solutions where only φ , φ are activated and the equations for the criticalpoints are modified to ( φ ) − ( φ ) + 2 √ m = 0 ,φ φ + √ m = 0 . Clearly, there are two solutions (unless m = m = 0, in which case, as we explainedabove, there is only one solution). The two solutions are related by φ , φ → − φ , − φ , whichis nothing but time reversal symmetry (there is a choice of time reversal symmetry thatcommutes with SU (3), for which all the φ a are pseudo-scalars – this is consistent with (2.4)).For some special choices of m , m the assumption that only φ , φ are activated is notjustified. Indeed, there are three lines in the m , m plane where the mass perturbationpreserves SU (2) × U (1) symmetry, namely – 24 – m = 0. In this case we see that the solution for m > φ ∼ ±√ m and φ = 0. For m < φ is nonzero and φ = 0. Therefore the global SU (2) × U (1)symmetry is broken to U (1) × U (1) for m < m > • m = −√ m . Repeating the analysis, one find that the global SU (2) × U (1) symmetryis broken to U (1) × U (1) on the half-line with m > • m = √ m . One again finds that the global SU (2) × U (1) symmetry is broken to U (1) × U (1) on the half-line with m > SU (2) × U (1) is spontaneously broken to U (1) × U (1) there is a C P sigma model at low energies. The C P is parameterized by additional scalars that aremassless. Time reversal symmetry acts as an antipodal map on the target space. B Further Checks of the N = 2 Dualities
We want to analyze in detail the duality we stated in section 3 U (1) + charge 2 ←→ U (1) ⊗ (cid:2) U (1) / + charge 1 (cid:3) (B.1)In particular, we can try to compare the superconformal index, which for a 3d SCFT withflavor symmetry U (1) N , is defined by [28, 29] I T ( m ; q, ζ ) = Tr H (cid:32) ( − F e − βH q ( E + j ) / (cid:89) a ζ e a a (cid:33) (B.2)where E , j , e a are energy, the third component of the angular momentum rotating S , andflavor charges. ζ a is the fugacity corresponding to each global symmetry, with a runningfrom 1 to N . Trace is taken over the Hilbert space H on S at a certain magnetic flux m background. Since the generator for E + j commutes with the Hamiltonian given by H = { Q, Q † } = E − R − j (B.3)( R is the R charge of the state) thus we have a fugacity q corresponding to this symmetry.However, instead of working in the fugacity basis and manipulating hyper-geometric functions,it is much easier to work in the charge basis. We refer the reader to [30] for more details. Herewe just mention a few useful points related to our discussion. We can rewrite the index of thetheory T , I T ( m ; q, ζ ) = (cid:80) e ∈ Z N I T ( m, e ; q ) ζ e , and work with the index at a fixed backgroundmagnetic flux m and electric charge e . There is an Sp (2 N, Z ) action on the 3d SCFT withflavor symmetry U (1) N . Accordingly, index transforms as I g ◦T ( gγ ; q ) = I T ( γ ; q ) (B.4)for g ∈ Sp (2 N, Z ) and γ = ( m, e ) T is the symplectic charge vector.– 25 – theory T M for any 3 manifold M can be built from the basic block T ∆ , the theory for asingle ideal tetrahedron [30] consisting only of a single free chiral multiplet. There is a trialityenjoyed by the tetrahedron index I ∆ ( m, e ; q ) = ( − q / ) − e I ∆ ( e, − e − m ; q ) = ( − q / ) m I ∆ ( − e − m, m ; q ) (B.5)This simply comes from the particle vortex duality I ∆ ( m, e ; q ) = I σ e ST ◦ ∆ ( m, e ; q ) = I ( σ e ST ) ◦ ∆ ( m, e ; q ) (B.6)Before calculating the index on both sides of the duality, it is important to understand firsthow our theory on both sides are related to the tetrahedron theory T ∆ . We now show thatthe duality (B.1) follows simply from the particle vortex duality ∆ ↔ ( ST ) ∆, which wewrites as | D A φ | − AdA + . . . ↔ | D b ˜ φ | + 12 bdb + 2 bdc + cdc + 2 cdA . . . (B.7)To avoid clutter, we normalized the Chern-Simons term such that minimal allowed ones lookslike AdA and 2
AdB (i.e. the usual 1 / π factor is implicit). And . . . denotes the super-partnerparts. Also, we use the upper case and lower case letter for U (1) background and dynamicalgauge field respectively. We rename c → c − b , and integrate out c on the right hand side, wehave | D b ˜ φ | − bdb − bdA − AdA . . . (B.8)Now on both sides, we rescale A → A followed by ST operation. | D a φ | + 2 adA + . . . ←→ | D b ˜ φ | − bdb − bdc − cdc + 2 cdA + . . . (B.9)Now if we rename c → c − b on the right hand side, we are left with | D a φ | + 2 adA + . . . ←→ | D b ˜ φ | + 32 bdb − bdA − cdc + 2 cdA + . . . (B.10)Therefore, we end up with a single chiral coupled to U (1) / tensored with a decoupled U (1) − ∼ = U (1) on the right hand side, which is dual to U (1) with a single charge 2 chiralon the left hand side. Note that we can integrate out dynamical field c on the right handside, and the last two terms just yield AdA .Now it is pretty straightforward to calculate the index on both sides of Eq. (B.10), denotedas I U (1) / ⊗ U (1) ( m, e ) and I U (1) ( m, e ). Knowing the relation between the theory T and ∆, I T ( m, e ) = I ∆ ( m ∆ , e ∆ ) (B.11)we just need to find ( m ∆ , e ∆ ) in terms of ( m, e ). To this end, recall that the presence ofChern Simons terms modifies Gauss’ law. A state with nonzero gauge flux M is not gauge A toy example to see why this happens: consider the Lagrangian L = k π (cid:15) µνρ A µ ∂ ν A ρ − A µ J µ . Thevariation of an Chern-Simons term is k π (cid:82) S δA (cid:82) S F , where m = π (cid:82) F ∈ Z is the magnetic flux. Obviouslyeach magnetic flux carry gauge charge k . And the equation of motion is k π (cid:15) µνρ F νρ = J µ simply tells thatmagnetic field is proportional to the matter field charge density. So monopole operator has to be dressed withchiral matter operators in order to have gauge charge zero. – 26 –nvariant. Therefore, we require a state with gauge flux M and flavor flux m dressed withchirals of charge e ∆ to be gauge invariant, and has flavor charge e . The magnetic flux felt bythe chiral is denoted as m ∆ . Gauge invariance then gives us the relation between them.In particular, the theory on the left hand side of (B.10) has index I U (1) ( m, e ) = I ∆ (2 e, − e − m I U (1) / ⊗ U (1) ( m, e ) = I ∆ ( − e + m , e ) (B.13)which equals to I U (1) ( m, e ) up to affine shift of the charge due to the triality property of idealtetrahedron index (B.5)[30], an indication that the duality indeed follows from the particlevortex duality.The duality (B.1) should also be subject to the check of S partition function. For 3d N = 2 gauge theory on S , partition function from the localization [31, 32] is Z S [∆ i , ∆ t ] = 1 |W| (cid:90) Cartan dσ (cid:89) a (cid:2) e iπk a tr ( σ a ) − π ∆ t tr σ a det Ad (2 sinh( πσ a )) (cid:3) (cid:89) i det R i e (cid:96) (1 − ∆ i + iσ ) , (B.14)where the function (cid:96) ( z ) is given by (cid:96) ( z ) = − z log(1 − e πiz ) + i πz + 1 π Li ( e πiz )) − iπ σ a is the adjoint scalar in the vector multiplet of the corresponding U (1) gauge group.∆ i is the R charge for i -th chiral field. ∆ t is the topological charge and corresponds to pureimaginary FI parameter. And free energy is defined by F S = − log | Z S | . It flows to a SCFTat IR fixed point where free energy is maximized with respect to R charge ∆ i and topologicalcharge ∆ t .In particular, for the theory U (1) with charge 2 chiral on the left hand side of the duality,we have Z S [∆ , ∆ t ] = (cid:90) + ∞−∞ dσ (cid:2) e − π ∆ t σ (cid:3) e (cid:96) (1 − ∆+ i σ ) , (B.15)The only global symmetry is U (1) t , and only ∆ t maximization is needed. We obtainnumerically, F S = 0 . . . . , ∆ t → U (1) ⊗ [ U (1) / + charge 1] we have Z S [∆ , ∆ t ] = (cid:90) + ∞−∞ dσ (cid:2) e iπ σ − π ∆ t σ (cid:3) e (cid:96) (1 − ∆+ iσ ) × (cid:90) + ∞−∞ dσ (cid:2) e iπ σ − π ∆ t σ (cid:3) (B.17)After minimization, we get F S = 0 . . . . , ∆ → / , ∆ t → S partition function matches nicely.Finally, we examine the duality discussed in the section 4. It tells us N = 2 U (1) gaugetheory with two chirals of the charge +1 has a SU (3) enhancement of the global symmetryfrom SU (2) × U (1). Consequently, the Weyl symmetry group will be S instead of S . Wecan check this S by looking at the index. We denote the electric charge and magnetic chargeof the gauge invariant operator to be e i and m i , i = 1 , U (1) of SU (3). As before, we need to find e ∆ , e ∆ , m ∆ , m ∆ in terms of e i and m i , suchthat I T ( m , e ; m , e ; ) = I ∆ ( m ∆ , e ∆ ) I ∆ ( m ∆ , e ∆ ) (B.19)Again, by requiring the gauge invariance of the states, we obtain I T ( m , e ; m , e ; ) = I ∆ ( e − m ,
12 ( − e − e − m + m )) I ∆ ( e + m ,
12 ( − e + e − m − m ))(B.20)We now show explicitly I T ( m , e ; m , e ; ) is invariant under the Weyl group action. Oneexample of S action on the charge m i and e i is given by e (cid:48) = 12 ( − e − e ) e (cid:48) = 12 ( − e + e ) (B.21) m (cid:48) = 12 ( − m − m ) m (cid:48) = 12 ( − m + m ) (B.22)Note that we use 2 T and 2 T / √ U (1) . So there is no √ I T ( m (cid:48) , e (cid:48) ; m (cid:48) , e (cid:48) ; ) = I ∆ ( 12 ( − e − e + m − m ) , e + m ) I ∆ ( 12 ( − e − e − m + m ) ,
12 ( − e + e + m + m ))(B.23)which is equal to (B.20) due to the triality property of the ideal tetrahedron index. Otherelements of the Weyl group can be checked in the same way. References [1] M. T. Grisaru, W. Siegel and M. Rocek,
Improved Methods for Supergraphs , Nucl. Phys.
B159 (1979) 429.[2] N. Seiberg,
Naturalness versus supersymmetric nonrenormalization theorems , Phys. Lett.
B318 (1993) 469 [ hep-ph/9309335 ].[3] N. Seiberg,
Exact results on the space of vacua of four-dimensional SUSY gauge theories , Phys.Rev.
D49 (1994) 6857 [ hep-th/9402044 ].[4] V. Bashmakov, J. Gomis, Z. Komargodski and A. Sharon,
Phases of N = 1 Theories in 2+1Dimensions , .[5] F. Benini and S. Benvenuti, N =1 dualities in 2+1 dimensions , . – 28 –
6] I. Affleck, J. A. Harvey and E. Witten,
Instantons and (Super)Symmetry Breaking in(2+1)-Dimensions , Nucl. Phys.
B206 (1982) 413.[7] S. R. Coleman and E. J. Weinberg,
Radiative Corrections as the Origin of SpontaneousSymmetry Breaking , Phys. Rev. D7 (1973) 1888.[8] A. J. Niemi and G. W. Semenoff, Axial Anomaly Induced Fermion Fractionization and EffectiveGauge Theory Actions in Odd Dimensional Space-Times , Phys. Rev. Lett. (1983) 2077.[9] A. N. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions , Phys. Rev. Lett. (1984) 18.[10] A. N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Actionin Three-Dimensions , Phys. Rev.
D29 (1984) 2366.[11] C. Cordova, P.-S. Hsin and N. Seiberg,
Time-Reversal Symmetry, Anomalies, and Dualities in(2+1) d , .[12] J. Gomis, Z. Komargodski and N. Seiberg, Phases Of Adjoint QCD And Dualities , .[13] R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology , Commun.Math. Phys. (1990) 393.[14] A. Kapustin and N. Seiberg,
Coupling a QFT to a TQFT and Duality , JHEP (2014) 001[ ].[15] M. Dierigl and A. Pritzel, Topological Model for Domain Walls in (Super-)Yang-Mills Theories , Phys. Rev.
D90 (2014) 105008 [ ].[16] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,
Generalized Global Symmetries , JHEP (2015) 172 [ ].[17] D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, andTemperature , JHEP (2017) 091 [ ].[18] D. Gang, Y. Tachikawa and K. Yonekura, Smallest 3d hyperbolic manifolds via simple 3dtheories , Phys. Rev.
D96 (2017) 061701 [ ].[19] D. Gang and K. Yonekura,
Symmetry enhancement and closing of knots in 3d/3dcorrespondence , .[20] K. Intriligator and N. Seiberg, Aspects of 3d N=2 Chern-Simons-Matter Theories , JHEP (2013) 079 [ ].[21] N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2+1 Dimensions andCondensed Matter Physics , Annals Phys. (2016) 395 [ ].[22] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg and M. J. Strassler,
Aspects of N=2supersymmetric gauge theories in three-dimensions , Nucl. Phys.
B499 (1997) 67[ hep-th/9703110 ].[23] M. Gremm and E. Katz,
Mirror symmetry for N=1 QED in three-dimensions , JHEP (2000)008 [ hep-th/9906020 ].[24] S. Gukov and D. Tong, D-brane probes of special holonomy manifolds, and dynamics of N = 1three-dimensional gauge theories , JHEP (2002) 050 [ hep-th/0202126 ]. – 29 –
25] A. Kapustin and M. J. Strassler,
On mirror symmetry in three-dimensional Abelian gaugetheories , JHEP (1999) 021 [ hep-th/9902033 ].[26] E. Witten, Supersymmetric index of three-dimensional gauge theory , hep-th/9903005 .[27] I. Affleck, M. Dine and N. Seiberg, Dynamical Supersymmetry Breaking in SupersymmetricQCD , Nucl. Phys.
B241 (1984) 493.[28] Y. Imamura and S. Yokoyama,
Index for three dimensional superconformal field theories withgeneral R-charge assignments , JHEP (2011) 007 [ ].[29] A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional FieldTheories , .[30] T. Dimofte, D. Gaiotto and S. Gukov, , Adv. Theor. Math. Phys. (2013) 975 [ ].[31] S. S. Pufu, The F-Theorem and F-Maximization , J. Phys.
A50 (2017) 443008 [ ].[32] D. L. Jafferis,
The Exact Superconformal R-Symmetry Extremizes Z , JHEP (2012) 159[ ].].