aa r X i v : . [ h e p - t h ] M a y
2d (0,2) appetizer
Mykola Dedushenko and Sergei Gukov
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
Searching for the simplest non-abelian 2d gauge theory with N = (0 ,
2) supersymmetry and non-trivial IR physics, we propose a new duality for SU (2) SQCD with N f = 4 chiral flavors. The chiralalgebra of this theory is found to be so (8) − , the same as in 4d N = 2 SU (2) gauge theory withfour hypermultiplets. INTRODUCTION
Two-dimensional theories with N = (0 ,
2) supersym-metry play an important role in string theory, quantumfield theory, and connections with pure mathematics.They describe world-sheet physics of heterotic strings.In quantum field theory, they can be found on two-dimensional objects (dynamical and non-dynamical) ina 4d theory with
N ≥ N = (0 ,
2) theory is a chiral CFT(a.k.a. chiral algebra). Yet, until recently, not muchwas known about strongly coupled gauge dynamics of 2d N = (0 ,
2) theories.The first exactly solvable example of a 2d N = (0 , N c colors, N f chiral flavors and e N f Fermiflavors also contains 2 N c − N f + e N f anti-fundamental fla-vors and N f (2 N c − N f + e N f ) singlet “mesons” that can-cel anomalies. This quite elaborate 2d SQCD exhibits arich phase diagram, with dynamical SUSY breaking anda peculiar triality symmetry that mimics symmetries ofsmooth 4-manifolds (under ‘handle slides’) and resemblesSeiberg duality in four dimensions. In particular, the tri-ality symmetry permutes N f , e N f , and 2 N c − N f + e N f ,and signals SUSY breaking when these numbers violatetriangle inequalities.In this paper we propose the simplest 2d (0 ,
2) SQCDwhich, compared to the above class of theories, is onthe edge of dynamical SUSY breaking and, nevertheless,has non-trivial IR physics described by a (0 ,
2) Landau-Ginzburg (LG) model with a cubic superpotential. Tobuild this theory we take the simplest non-abelian gaugegroup SU (2). Since 2d (0 ,
2) vector multiplet containsleft-moving fermions, we need to cancel gauge anomaly,which in a theory with SU ( N c ) gauge group and N f fun-damental flavors looks like − N c + 12 N f . (1)In particular, we see that in a theory with N c = 2 thesimplest way to cancel gauge anomaly is to add N f = 4chiral multiplets in the fundamental representation of thegauge group.In the rest of this note we describe the IR dynamicsof this theory and propose a new duality with a (0 ,
2) Landau-Ginzburg model of one Fermi superfield Ψ cou-pled to six chiral multiplets Φ i , i = 1 , . . . ,
6, via a (0 , J = ΨPf(Φ) , (2)where we find it convenient to think of Φ as ∼ = ∧ under SO (6) ∼ = SU (4) global symmetry group.Given a possibility of a dynamical SUSY breaking,one might feel suspicious or, perhaps, even skepticalabout the IR duality between SU (2) SQCD and theLG model (2). Such concerns are further supportedby c -extremization [10], which leads to negative centralcharges on both sides of the duality and usually is a signalfor either dynamical SUSY breaking or lack of a normaliz-able vacuum. Despite all such indications, we argue thatboth sides of the proposed duality do not break SUSYand, furthermore, have equivalent IR physics. VACUA
Let us start by analyzing the space of classical vacua.On the gauge theory side, the classical space of vacua isa “complex cone” on the GrassmannianGr(2 ,
4) = V ⊗ //U (2) , (3)where V ∼ = C denotes the fundamental representationof U (2). One way to see this is to imagine (formally)that our gauge group is U (2) rather than SU (2). Then,the space of vacua in such fictitious theory would be theK¨ahler quotient (3). Going back to our original theory,that is replacing the gauge group U (2) by SU (2), in (3)has the effect of removing the D-term constraint and aquotient by U (1). Hence, the resulting moduli space ofvacua is a complex cone on Gr(2 , C P :Gr(2 ,
4) = { Φ Φ − Φ Φ + Φ Φ = 0 } . (4)Since C P = C //U (1) is a K¨ahler quotient, just as onthe gauge theory side discussed earlier, we can “ungaugethe U (1)” by removing the U (1) quotient and the mo-ment map constraint associated with it. The result, ofcourse, is simply the hypersurface in C :Φ Φ − Φ Φ + Φ Φ = 0 . (5)This is the same five-complex-dimensional space of vacuawe found in SU (2) SQCD with N f = 4. According to atheorem of Tian and Yau [11], this quadric admits a coho-mogeneity one Ricci-flat metric that comes in a family ofnon-compact Calabi-Yau 5-folds obtained by deformingthe complex structure in (5):Φ Φ − Φ Φ + Φ Φ = ǫ. (6)The conical singularity at ǫ = 0 is at a finite distance inmoduli space [12].In quantum theory, there are several possible scenariosfor the physics associated with this singularity. For in-stance, one option is that the conical singularity at Φ = 0is simply resolved in quantum theory, cf. [13]. Anotheroption is that the singularity is hiding extra degrees offreedom and / or another branch of vacua emanatingfrom it.Note, the Pfaffian Calabi-Yau (6) is a non-compactanalogue of a 2d N = (2 ,
2) model considered in [14].
COMPACTIFICATION FROM 4D
An insight into quantum physics of the proposed dualpair of 2d (0 ,
2) theories comes from connecting themto another dual pair, namely 4d N = 1 gauge theoriesrelated by Seiberg duality.Seiberg duality [15] relates 4d N = 1 SQCD with gaugegroup SU ( N c ) and N f fundamental flavors q i to a similartheory with SU ( N f − N c ) gauge group, N f fundamentalflavors, singlet “mesons” M ij ∼ q i e q j , and a certain su-perpotential. As we explain in the rest of this section, aspecial instance of this 4d duality, when compactified ona 2-sphere with a partial topological twist, gives preciselythe proposed dual pair of 2d N = (0 ,
2) SQCD and theLandau-Ginzburg model with the Pfaffian superpotential(2).To describe topological reduction a la [7, 16–18] of a4d N = 1 theory on S — or, more generally, on a genus g Riemann surface F g — we need to describe what hap-pens to three basic ingredients: N = 1 vector multiplets,chiral multiplets, and the superpotential interaction. Westart with a vector multiplet, whose topological reduc-tion on a genus g Riemann surface is relatively simple: itgives one 2d (0 ,
2) vector multiplet (with the same gaugegroup) and g adjoint (0 ,
2) chiral multiplets, see e.g. [19]:4d N = 1 vector on F g −−−−−→ ( N = (0 ,
2) vector , + g N = (0 ,
2) chirals . Topological reduction of 4d N = 1 chiral multipletsis more subtle and depends on the choice of the R -symmetry, under which all chiral fields must carry in-teger R -charges. Of course, the R -symmetry should also be non-anomalous, e.g. for SU ( N c ) theory with N f fun-damental flavors the R -charges of the chiral multipletsmust satisfy N f X i =1 R i = 2( N f − N c ) . (7)Assuming these conditions are met, the topological re-duction of a 4d N = 1 chiral multiplet with R -charge R yields 2d spectrum of (0 ,
2) chiral and Fermi multipletscontrolled by the following cohomology groups:2d N = (0 , ( H ( K R/ ⊗ L ( m )) chirals ,H ( K − R/ ⊗ L ( − m )) Fermi , (8)where K is the canonical bundle of F g and we also alloweda coupling to general background flux m . In particular,for a genus-0 compactification on S , each 4d chiral mul-tiplet contributes to the 2d N = (0 ,
2) field content either1 − R chirals if R < R − R > cf. [7].Finally, the superpotential terms in four dimensions,upon the topological reduction on F g , yield superpoten-tial E or J terms of the effective N = (0 ,
2) theory intwo dimensions.Now, let us apply these simple rules to 4d N = 1SQCD with N c = 2 and N f = 3. Note, these numbers ofcolors and flavors obey N f = N c + 1 and also N f = N c .Since N f = 3, this theory has a total of 6 chiral multipletsthat transform as doublets under the SU (2) gauge group.We choose the following assignment of integer R -charges,which satisfy the anomaly cancellation condition (7): R = (1 , , , , , . (9)With these R -charge assignments, the spectrum of thetheory on S consists of one SU (2) vector multiplet andfour fundamental chirals. In other words, it is preciselyour candidate for the simplest non-abelian 2d N = (0 , N = 1 SQCD with N c = 2 and N f = 3. Since N f − N c = 1, the dual theory is a Landau-Ginzburg model already in four dimensions. It has 15“meson” fields M ij interacting via a cubic superpoten-tial: W = Pf( M ) . (10)The R -charges of M ij compatible with (9) can be eas-ily deduced from the relation M ij ∼ ǫ ab q ai q bj , where q ai are the fundamental “quarks” of the original 4d N = 1SQCD. Specifically, we have R ( M ) = 2 , (11) R ( M i =2 ) = R ( M j =1 ) = 1 , (12)and the six components M ij with i and j not equal to 1 or2 all have R ( M ij ) = 0. Upon topological reduction on a2-sphere, the latter give rise to 2d N = (0 ,
2) chiral mul-tiplets Φ i , i = 1 , . . . ,
6. The mesons with R = 1 do notcontribute to the spectrum of 2d theory at all, whereasthe component M gives rise to a Fermi multiplets Ψ.Moreover, the 4d superpotential interaction (10) reducesto the 2d Pfaffian superpotential (2). All in all, this isprecisely the Landau-Ginzburg model that was proposedas IR dual to 2d N = (0 ,
2) SQCD. Here, we related theproposed IR duality in two dimensions to a more familiarSeiberg duality in 4d.As further evidence for the proposed duality, one cancompare elliptic genera of 2d N = (0 ,
2) SQCD and theLandau-Ginzburg model using the theta function iden-tity (4.7) from [20]. In the next section we match evenstronger invariants on both sides of the duality.
THE CHIRAL ALGEBRA
2d (0 ,
2) theories are known to have a sector of BPSoperators, defined by passing to the cohomology of Q + ,whose OPE has a structure of a vertex operator alge-bra, or chiral algebra [21, 22]. The exact chiral algebrais known to be an RG invariant [23], hence it can serveas a useful check of dualities, whenever it is possible tocompute it. This philosophy was utilized in the litera-ture before [4] mainly through the study of chiral rings(regular subsectors in chiral algebras) in (0 ,
2) NLSMs[24–26], which can be applied to gauge theories throughtheir low-energy sigma model descriptions.Here we are going to apply the chiral algebra ma-chinery to our gauge theory and its Landau-Ginzburgdual directly, rather than through the looking glass ofNLSMs. We begin on the gauge theory side, explainingthe method from [27] along the way. First of all, we workunder the hypothesis that the algebra is perturbativelyexact. While instantons are known to sometimes drasti-cally change the answer [28–31], we believe it does nothappen in our dual pair. The theory at hands admits nofamiliar vortex solutions, and moreover, results on theLandau-Ginzburg side confirm our assumption.On the gauge theory side, the chiral algebra can becomputed in perturbation theory in the gauge coupling e , which is essentially the spectral sequences method ap-plied to the Q + cohomology. In the zeroth order in e , wesimply impose the Gauss law constraint on the productof chiral algebras of free multiplets. The free (0 ,
2) chiralmultiplet (valued in a representation R ) contributes a βγ system of conformal weights ( h β , h γ ) = (1 − λ, λ ) (valuedin the same representation), which we denote ( β, γ ) ( λ ) .Parameter λ is not fixed and is related to the true R-symmetry of the IR CFT. The chiral algebra of a freevector multiplet is given by a small algebra of the bc ghostsystem valued in the adjoint of the gauge group; we call it ( b, c ) small . Its conformal dimensions are ( h b , h c ) = (1 , b ( z ) and c ( z ) can only contain derivatives of c ( z ) butnot c ( z ) itself. So the zeroth order approximation to thechiral algebra is given, after imposing gauge invariance,by: E = (cid:16) ( β, γ ) ( λ ) ⊗ ( b, c ) small (cid:17) G . (13)We then include all higher-order correction to Q + andsee that the way it acts on E coincides precisely withthe BRST operator. Namely, the BRST current is: J BRST = dim G X A =1 c A ( J A m + 12 J A gh ) ,J A m = iβT A γ, J A gh = − if ABC c A b B c C , (14)and it defines a Q BRST operator in the usual way. Onecan check that it is nilpotent, Q BRST = 0, precisely whenthe anomaly cancellation condition (1) holds, so it is con-sistent to study its cohomology. The perturbative chiralalgebra is then simply given by a BRST reduction of the“ungauged” chiral algebra [27]: H pert ( Q + ) ∼ = H BRST ( E ) . (15)Those familiar with the work of [32] may find a strik-ing similarity between this procedure and the one for thechiral algebra of Lagrangian 4 d N = 2 SCFTs. Theirprescription is given by a BRST reduction of the symplec-tic boson valued in the matter representation. Note thatsymplectic boson CFT is the same thing as the βγ systemat λ = (this λ is different from the one in [33], whereit referred to twisted sectors). Moreover, the BRST co-homology problem does not depend on the value of λ .Therefore, our prescription is not just similar to [32], itis exactly the same.This simple observation, whether it has any deepmeaning or not, helps us to avoid a lot of technicalities.It shows that our chiral algebra for 2d SU (2) gauge the-ory with four chirals coincides with the chiral algebra of4 d N = 2 SU (2) gauge theory with N f = 4. There was alot of evidence in [32] that the latter is given by an so (8)current algebra at level −
2, so our answer is: H pert ( Q + ) ∼ = so (8) − . (16)Even though this algebra does not depend on parame-ter λ , its value does not get washed out completely. Itenters in the choice of the stress-energy tensor, which co-incides with the physical left-moving stress-energy tensorof the IR CFT. Only for λ = it would coincide withthe Sugawara tensor. For different values of λ , we have: T = T Sug + ∆ T, ∆ T = (1 / − λ ) ∂ ( βγ ) . (17)Now we would like to approach this problem from theLandau-Ginzburg side, where we use the methods of [23].To compute the cohomology of Q + , we identify the co-homology of a superspace derivative D + acting on super-fields and then take their lowest components. One has totake into account the operator equations of motion: D + ∂ −− Φ ij = − i Ψ ε ijkl Φ kl ,D + Ψ = − , (18)where ∂ −− equals the holomorphic derivative upon Wickrotation. Since Φ ij and Ψ are annihilated by D + , theyare in the cohomology, and equations of motion implyrelations Pf(Φ) = 0 and ΨΦ ij = 0 in the cohomology.Assuming that the fields Φ ij have R-charge α in the IR,we can further identify the stress-energy tensor in thecohomology: T = X i>j h ∂ −− Φ ij ∂ −− Φ ij − α ∂ −− (Φ ij ∂ −− Φ ij ) i + i ∂ −− Ψ − i ∂ −− ΨΨ + i (1 − α )2 ∂ −− (ΨΨ) , (19)as well as u (4) currents: J ij = i Φ ik ∂ −− Φ kj + δ ji ΨΨ . (20)Composite operators should be defined with care: wealways have to subtract singularities appearing in collid-ing their constituent elementary fields. This cannot lift T and J ij from the cohomology because their singular-ities only have numeric coefficients, but can affect morecomplicated composite operators. We can identify suchcomposite operator in the classical D + cohomology:2 i ΨΨ ∂ −− Φ ij − ∂ −− Φ ni Φ np ∂ −− Φ pj . (21)To properly define quantum operators, we have to renor-malize them by subtracting singularities. The secondterm is actually non-singular, but we still need to splitpoints in order to correctly evaluate the action of D + onit. We define renormalized operators:[ΨΨ ∂ −− Φ ij ] ren (0)= lim ǫ → (Ψ( ǫ )Ψ(0) ∂ −− Φ ij (0) − iǫ ∂ −− Φ ij (0)) , [ ∂ −− Φ ni Φ np ∂ −− Φ pj ] ren (0)= lim ǫ → ∂ −− Φ ni (0)Φ np ( ǫ ) ∂ −− Φ pj (2 ǫ ) . (22)A very accurate computation at finite ǫ shows that now: D + n i [ΨΨ ∂ −− Φ ij ] ren − [ ∂ −− Φ ni Φ np ∂ −− Φ pj ] ren o = 4 ∂ −− ( − i Ψ ε ijkl Φ kl ) . (23)In an analogous situation in [23], similar computationwas used to argue that the cohomology class was lifted at the quantum level. In our case, though, the right-handside is actually D + -exact due to the equation of motion,being equal to 4 D + ∂ −− Φ ij . We conclude that in ourcase, quantum effects renormalize the cohomology class,and the correct one is given by: B ij = 2 i [ΨΨ ∂ −− Φ ij ] ren − [ ∂ −− Φ ni Φ np ∂ −− Φ pj ] ren − ∂ −− Φ ij . (24)This composite operator groups together with Φ ij and J ij to form the so (8) − current algebra, precisely match-ing the dual gauge theory result, including the OPE. Onecould call this an accident in the terminology of [34]. De-noting γ i · γ j = ε αβ γ iα γ jβ , where α, β are indices in thefundamental of su (2), we have: γ i · γ j ↔ Φ ij ,β i · γ j ↔ ∂ −− Φ ik Φ kj − iδ ij ΨΨ ,β i · β j ↔ B ij , λ = α. (25)The Landau-Ginzburg side has an extra fermionic oper-ator Ψ in the cohomology, which might look puzzling.However, our duality suggests that it simply decouplesas a free field along the flow. The way it enters the alge-bra shows that it is completely consistent to make sucha truncation, so that the interesting part of the chiralalgebra is indeed so (8) − . We can also conjecture thatthere are no other independent composite operators thathave to be taken into account. CONCLUDING REMARKS
Some of the facts observed in this note call for furtherinvestigation. In particular, the match of our chiral alge-bra with the one from [32] sounds intriguing and raises aquestion whether it is a pure coincidence or a manifesta-tion of some deeper connection.But most importantly, a lot of standard techniques arenot applicable to our theory due to the lack of normaliz-able vacuum. One example is the c R -extremization [10],equivalent to c L -extremization (due to c L − c R being fixedby the gravitational anomaly, equal to − U (1) currents are primary [23]. Applying it to our the-ory gives λ = 1 / c L = −
14. This is not in conflict withunitarity because of the lack of normalizble vacuum. Butfor the same reason, this value of c L does not have to becorrect because c -extremization also fails. On top of thatwe should add that in a theory with normalizable vac-uum, decoupling of Ψ along the flow would immediatelyimply that its dimension is at the unitary bound. Thisdoes not have to hold in our case either. Overall, we arelacking one extra handle on the dynamics of our modelto say more about its IR physics.Somewhat related, the fate of singularity at the originof the moduli space is not entirely clear as well. For thispuzzle, we can make a guess: if topological reduction on S from 4d to 2d commutes with the RG flow, we cansimply look at the N = 1 SU (2) gauge theory in 4d with N f = 3, which is a parent of our 2d theory. Accordingto [35], the classical moduli space in such a theory doesnot receive quantum corrections, and singularity at theorigin carries some massless degrees of freedom, so wecould guess that the same happens in our case. Further-more, the relation Pf(Φ) = 0 in the chiral algebra givesfurther evidence for this claim. Finally, we should notethat this problem might be amenable to the methods of[36] extended to 2d gauge theories.We thank M.Fluder, J.Heckman, D.Jafferis,D.Kutasov, N.Nekrasov, P.Putrov, J.Song for use-ful discussions. This work was supported by the WalterBurke Institute for Theoretical Physics and the U.S.Department of Energy, Office of Science, Office of HighEnergy Physics, under Award No. DE-SC0011632. Thework of MD was also supported by the Sherman FairchildFoundation. SG gratefully acknowledges support fromHarvard University, where some of the research for thispaper was performed during the fall 2017, as well aspartial support by the National Science Foundationunder Grant No. NSF PHY11-25915 and Grant No.NSF DMS 1664240. [1] A. Gadde, S. Gukov and P. Putrov, JHEP , 076(2014) [arXiv:1310.0818 [hep-th]].[2] B. Jia, E. Sharpe and R. Wu, JHEP , 017 (2014)[arXiv:1401.1511 [hep-th]].[3] A. Gadde, S. Gukov and P. Putrov, arXiv:1404.5314 [hep-th].[4] J. Guo, B. Jia and E. Sharpe, JHEP , 201 (2015)[arXiv:1501.00987 [hep-th]].[5] M. Honda and Y. Yoshida, arXiv:1504.04355 [hep-th].[6] A. Gadde, Phys. Rev. D , no. 2, 025024 (2016)[arXiv:1506.07307 [hep-th]].[7] A. Gadde, S. S. Razamat and B. Willett, JHEP ,163 (2015) [arXiv:1506.08795 [hep-th]].[8] S. Franco, S. Lee and R. K. Seong, JHEP , 020(2016) [arXiv:1602.01834 [hep-th]]. [9] S. Franco, S. Lee, R. K. Seong and C. Vafa, JHEP ,106 (2017) [arXiv:1609.01723 [hep-th]].[10] F. Benini and N. Bobev, Phys. Rev. Lett. , no. 6,061601 (2013) [arXiv:1211.4030 [hep-th]].[11] G. Tian and S.-T. Yau, “Complete Kahler ManifoldsWith Zero Ricci Curvature II,” Inv. Math. (1991)27.[12] S. Gukov, C. Vafa and E. Witten, Nucl. Phys. B ,69 (2000) Erratum: [Nucl. Phys. B , 477 (2001)][hep-th/9906070].[13] O. Aharony, S. S. Razamat, N. Seiberg and B. Willett,JHEP , 056 (2017) [arXiv:1611.02763 [hep-th]].[14] K. Hori and D. Tong, JHEP , 079 (2007)[hep-th/0609032].[15] N. Seiberg, Nucl. Phys. B , 129 (1995)[hep-th/9411149].[16] A. Johansen, Nucl. Phys. B , 291 (1995)[hep-th/9407109].[17] M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Nucl.Phys. B , 166 (1995) [hep-th/9501096].[18] C. Closset and I. Shamir, JHEP , 040 (2014)[arXiv:1311.2430 [hep-th]].[19] S. Gukov, arXiv:1707.01515 [hep-th].[20] P. Putrov, J. Song and W. Yan, JHEP , 185 (2016)[arXiv:1505.07110 [hep-th]].[21] E. Witten, Int. J. Mod. Phys. A , 4783 (1994)[hep-th/9304026].[22] E. Silverstein and E. Witten, Phys. Lett. B , 307(1994) [hep-th/9403054].[23] M. Dedushenko, arXiv:1511.04372 [hep-th].[24] S. H. Katz and E. Sharpe, Commun. Math. Phys. ,611 (2006) [hep-th/0406226].[25] J. Guffin and E. Sharpe, J. Geom. Phys. , 1581 (2009)[arXiv:0801.3955 [hep-th]].[26] I. V. Melnikov, JHEP , 118 (2009)[27] M. Dedushenko, unpublished notes (2015).[28] E. Witten, Adv. Theor. Math. Phys. , no. 1, 1 (2007)[hep-th/0504078].[29] N. A. Nekrasov, hep-th/0511008.[30] M. C. Tan and J. Yagi, Lett. Math. Phys. , 257 (2008)[arXiv:0801.4782 [hep-th], arXiv:0805.1410 [hep-th]].[31] J. Yagi, Adv. Theor. Math. Phys. , no. 1, 1 (2012)[arXiv:1001.0118 [hep-th]].[32] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelliand B. C. van Rees, Commun. Math. Phys. , no. 3,1359 (2015) [arXiv:1312.5344 [hep-th]].[33] M. Fluder and J. Song, arXiv:1710.06029 [hep-th].[34] M. Bertolini, I. V. Melnikov and M. R. Plesser, JHEP , 157 (2014) [arXiv:1405.4266 [hep-th]].[35] N. Seiberg, Phys. Rev. D , 6857 (1994)[hep-th/9402044].[36] C. Beasley and E. Witten, JHEP0501