JHEP11(2017)115 . We 8 E Springer the USp(2) October 27, 2017 : November 10, 2017 November 20, 2017 ymmetric Gauge : : flows in the IR to a els are connected by a m Received Accepted Published ost generic superpotentials, metric field while preserving n particular, we argue that a s of the Weyl group of Published for SISSA by https://doi.org/10.1007/JHEP11(2017)115 [email protected] , b ) supersymmetric models with eight fundamental fields and n . and Gabi Zafrir 3 a 1709.06106 The Authors. U(1) symmetry. We also discuss an infinite number of duals for c × Duality in Gauge Field Theories, Global Symmetries, Supers
We discuss USp(2 7 , E [email protected] ) model with the addition of singlet fields and even rank m orbits of IR dualities 8 Kavli IPMU (WPI), UTIAS,Kashiwa, the Chiba University 277-8583, of Japan Tokyo, E-mail: Physics Department, Technion, Haifa 32000, Israel b a Theory, Supersymmetry and Duality ArXiv ePrint: theory with eight fundamentals and no superpotential. Keywords: USp(2 make also several curious observations about such models. I CFT with Open Access Article funded by SCOAP Abstract: Shlomo S. Razamat a field in thecoupling antisymmetric pairs representation. of Turning fundamental on fieldsan the to R m powers symmetry, of we thelarge give network antisym evidence of dualities for which the can statement be that organized the into orbit mod E JHEP11(2017)115 4 7 n 1 2 6 8 X 1 when elates Q 2 ] orbits m Q 10 X , 1 9 Q 2 ing is whether , a field in the Q i Q ], one can discover rticular setup. We 1 int is rather lacking at e that turning on most IR. This phenomenon is non trivial map of opera- twork of dualities and this llowing [ flavor symmetry of the gauge class. An understanding why iversality in statistical physics. derstood by constructing dual four cular there are dualities relat- ion together with permutations mple possessing different gauge duality web forms [ re added. This duality transfor- in universality class. ) model with superpotential m ent looking compactifications of a six dimensional ) model with superpotential n – 1 – ] as a property of integrals which imply equality of 9 , and possibly a superpotential with gauge singlet fields. X ]). . Here we discuss yet another duality transformation which r 8 7 – 2 E ) gauge theories with eight fundamental chiral fields m An interesting question one can ask to aid such an understand U(1) surprise orbits of dualities 1 × 7 7 E E In this short note we will discuss such a structure in a very pa There is though growing evidence that such dualities can be un 2.1 3.1 Duals of SU(2) SQCD with four flavors 1 certain singlet fields and superpotentialsmation involving was them considered a implicitly in [ is in the same universality class as a USp(2 theories with different rank. In particular a USp(2 general superpotentials of themodels form but preserving above, R breaking symmetry, this all duality the transformat of the Weyl group of the supersymmetric index of the two dual models. We will argu two different looking theoriesthe in moment. the UV flowthere to is the any same structure relating fixed different po theories in aconsider certa USp(2 various examples of seemingly very differentgroups, models, which for exa nevertheless reside in the same universality dimensional theories as geometrically equivalentmodel but differ (see for examples [ Different QFTs can flow tousually the referred same to conformal as field IR theory dualityConsidering in in supersymmetric high the energy theories physics or in un four dimensions fo 1 Introduction 3 Rank changing duality 4 The duality orbit Contents 1 Introduction 2 The It so happens thatnetwork these has models intriguing are grouping interrelated theoretic by models structure. a with large fixed Intors/symmetries ne rank between parti but various sides different of superpotentials the and duality. This a antisymmetric representation JHEP11(2017)115 . ) 8 n E with (2.4) (2.1) (2.3) (2.2) ] and of ], a 13 14 Q , 12 . e [ . The charges ymmetric rep- 8 ) ) and USp(2 E n ( 0 m T al description preserv- h a review of dualities surprise of Dimofte and r equals one case. We can 7 eory but with a collection 1 2 0 1 r E n 6). Then together with the . yl group of l 1 2 U(1) 0 e finish by combining the two y manifest in a similar manner e X y . U(1) gh the superpotential [ ] for 1 1 − 1 4 → − − 4 15 − n y l n 2 +1 e − 1 X n U(1) − 1 j U(1) q i q U(1) symmetry. In the particular case of . We argue that this model with a partic- ,X 8 1 28 × l m n SU(8) i,j/y 8 1 U(1). We call this model SU(8) − ) gauge theory with four flavours, 7 1 M n × E 1 – 2 – − ij/n orbit. We also make a curious observation about − i,j ) X 7 ) M 1 1 ) E n =1 ) n − → y X n − l ) with even = 2n 2n ( X m ( j n 1 2n W USp(2 2n n Q USp(2 i j Q does not exist and the model is SU(2) gauge theory with four Q X i,j/y j e orbits the full duality web is that of orbits of the Weyl group X M q X ) models on an 7 E m in the fundamental representation, and one field in the antis . This model was studied by various authors, in particular se orbits of dualities } X 7 8 ]. In section three we discuss a duality which relates USp(2 , E 11 = 1 the field ··· n , 1 The model above has multiple known dual descriptions. The du The note is organized as follows. We start the discussion wit ∈ { are summarized in table 2.1. This is a generalization of Intriligator-Pouliot duality [ The map between the operators is, When references below. The symmetry is SU(8) The charges of the fields are in table 2.3. ing manifestly the most symmetryof is singlet given fields by coupled the to same gauge gauge invariant th operators throu build many other duality frames which will have less symmetr dualities generating of the eight quarks generates orbits of the Weyl group of SO(1 fundamental flavors. models with superpotentials on bothtypes sides. of dualities In and section explain four how w they build orbits of the We resentation, 2 The The basic theory we consider is a USp(2 a a special property of the USp(2 transforming USp(2 USp(4) this model sits on the same conformal manifold as the Gaiotto [ ular superpotential flows to a SCFT with JHEP11(2017)115 ) 4 ( (2.8) (2.9) (2.5) (2.6) (2.7) ¯ 4 , two sets of dual r r r , ) 1 2 1 2 1 2 1 2 0 1 1 0 1 2 1 2 0 1 1 s an analogue of the − U(1) U(1) l U(1) l l e l 1 1 X on an ordered set of fu- 1 1 4 4 − − 1 1 − − symmetric square of 4 4 − − − n n of thirty five duals is then +4 4 4 − − i n n The matter content is then n n q 1 − − 2 2 +1 +1 2 U(1) +1 1 − − ). To construct these dualities n n − − 1 . U(1) n +4 U(1) 7 he symmetries as follows using b j b b A q +4 ( j 1 q 1 2 1 1 i 0 U(1) − ′ i,j/l 1 q 0 U(1) − 1 − 2 ] Seiberg duality. Another thirty five /W U(1) − 0 0 ˆ 1 2 ) 1 M 2 − 7 2 l + E ˜ ( X 1 SU(4) 1 ¯ 4 1 − W l 1 SU(4) 1 ¯ 4 6 1 SU(4) 1 4 1 ¯ 4 i,j/l 1 e X 1 1 j M – 3 – q i q =1 4 4 1 1 SU(4) X 4 1 1 i,j ¯ 4 1 1 SU(4) 1 6 SU(4) 4 i,j/l 1 1 n ˆ =1 1 l M X ]. − ( − ) − j 10 = ) 1 ) 6= ) 1 ,i 1 ) 4 n − ) W n − X =1 n − . This is the analogue of [ i,j 2n ( 6 2n 2n ( ( 2n 2n n n =1 USp(2 2n 2n n l X 2n n 2n 1 1 USp(2 1 USp(2 4 8 4 8 = 8 4 ,..., ,..., ′ /l /l 1 5 ,..., ,..., /l ,..., ,..., 1 5 W ˆ ˆ 5 1 Q Q X M M q q X q X M q ] discussed by Csaki, Schmaltz, Skiba, and Terning. The last 16 = 1 case. The number of duals is 72 = n It is convenient to encode the dualities as transformations with the matter content given in table 2.9. is the real representation The“baryons” map toduality the [ singlets anddescriptions mesons were to considered in mesons. [ This i duals have the superpotential given by the following, with superpotential given here, We have thirty five choicesgiven to by the perform following such matter splitting. content, One set The mesons map to singlets and “baryons” to “baryons” as anti to the we split the eightwritten fundamental in fields table into 2.5. two groups of four. gacities for the different symmetries. We will parametrize t JHEP11(2017)115 the con- map 1 (2.10) 4 r − X n . Let us − 7 t i E h , n, t . We define , n, t 4 1 × to be general 2 8 2 − ) ) ) theory to be − u u k 8 u n . Then the three qp , qp , n, t u u j + 8 7 2 − u u u u otential can be ob- u , = ( = ( , − + =5 and i − u u 8 j 6 j 2 − 7 u u . The powers of t , u u u l u 7 Q rpotential maps to itself + field of R charge j e will now discuss here a , , the theory has a point − ces to U(1) u u i 5 = 2 − trum is invariant under th q , Q − + j u st eight terms generate the u 2 u u ij/n − q e non trivial identification of nsidered. Note that under all 4 − of the USp(2 , i . 6 − , u u 4 M i 6 n 2 + u Q + 2 u j x u u s of this set, t u i symmetry. This fact was related , Q − + , . It is then plausible that on some and X 7 3 u u 2 + 7 − j u , being the fugacity of the U(1) sym- u E 5 E u u ] that combining two copies of USp(2) 5 m =2 , u i + X u h 2 2 + 11 u =1 − + u 4 j + u , is for the U(1) symmetry, and fugacities u u , − Q , t 1 2 + 4 4 u il/j u u u u = + ] with M u − + – 4 – 4 + 1 18 , u u u − , j , − → → → maps to itself or to 3 3 u ) ) ) X 1 u 17 u i + − + − l u Q )[ l u u , , n, t , n, t , n, t h X , Q 8 8 8 j 2 r − 2 2 ) 2 u u Q ). Denote , u , u , u i u =1 even. Turn on superpotential, + 7 7 7 = 1). We can take the parameters + − qp j X m/ u Q u u (( k , , u , u , u m , e , n, t 6 6 6 h 1 − 8 u 1 u , u , u , u =1 u , u 5 5 5 + − + 8 k 7 u u u , u , u , u Q , u 4 4 4 6 and thus forms representations of . , u , u , u → → 7 7 , u 3 3 3 th quark. Here the fugacity 5 i E and assume E , u , u , u ) , u 2 2 2 U(1) surprise 4 m 0 ( , u , u , u , u 1 1 1 × T 3 u u u 7 ( ( ( , u E 2 ] to a statement that the two copies of USp(2) with that superp It has been observed by Dimofte and Gaiotto [ 8 for SU(8) (then , u 1 j u analyze one example in detail. with one constraint coming from anomaly cancelation, point on the conformal manifold of this model the group enhan dualities we discussed imply the following transformation under the three dualities.symmetries. The only These effect imply ofWeyl that the group for duality of is example th the protected spec to the same powers on the dual side. This implies that the supe We claim this model is selfthe dual dualities under it the is dualities either we have that co an ordered set of fugacities parametrizing the gauge sector ( tributes to the index as Γ h metry under which the field transforms. We will then denote by These transformations together withWeyl group permutations of of the fir on the conformal manifold in the IR with, at least, 2.1 theories by coupling the gauge invariant operators as tained by compactification ofgeneralizations the of E the string former theory fact on to a higher torus. rank. W Consider weight for the in [ supersymmetric index nomenclature. Remember that a chiral JHEP11(2017)115 l e kl qp are oup M l L (2.12) (2.11) in the Q ψ i being an , fourth is in the Q qp forms the X, O 70 k j X f the chiral j Q of SU(8). In Q j i Q with i ¯ Q ψ l is then obtained .... O Q ndex we see that 336 Q O and R charge half i M lk + φ That is the super- and have vanishing 70 + ¯ ψ − X qp and the fields 2 qp i 378 1 Q . The operators lar as the superconformal − ,Q 63 we also need the lly also in the gauge indices. appearing at order flipped. lm 7 133 7 , in which case the F-term conditions ion of SU(8). This arises as to 2 ). The operator of USp(4) is a singlet, and so the M E x e at order i E X − r there will be no conformal manifold. 4 Q . For recent discussion of some aspects of x . 2 j x 1 r in t ¯ ψ arge 2 and so this is a marginal deformation + surprise model. This suggests that the M, X, x t 7 E gauge singlet fields. The superconformal R ,Q 1463 x lm – 5 – is forbidden by both the F-term and D-term conditions. , is in the representation and coupling them to a theory as ] assigns R charge zero to , in addition to the M 2 8 O X ) anomalies coincide with two copies of SU(2) the- and ) + 21 X M ij φ are fermionic partners of p n ¯ ψ being one of ( i a, c Q M + ψ U(1). The index is given by, k L q surprise model if the mesons are flipped. We thus expect ( Q × j 2 1 7 7 ) ’s antisymmetrically in the flavor index, but, since these ar with Q ]). The conserved currents multiplet contributes at order . It is worth verifying what are the operators giving such a E i symmetry somewhere on its conformal manifold. The model E with Q qp x 22 7 Q ( X, 2 L 1 2 E 133 m . The operator which gives us the required t X Giving such an expectation value Higgses the USp(4) gauge gr giving it a vacuum expectation value takes us on the conforma − Q l 70 56 4 + 3 Q X , xX ]. + + ml M X ji 20 U(1) and to construct the adjoint of 2 1 ¯ ¯ , ψ ψ ) M × 378 then cannot be made gauge invariant. The operator m 19 pq is zero, ( + Q l X 1 2 cancel each other in the index computation as a consequence o t X Q , ij SU(8) symmetry visible in the Lagrangian and if we study the i 336 j 56 . This generates the × M 2 Q = i is the gaugino. The fields ¯ ψ λλ , M 1 + km λ ¯ 28 ψ Consider the USp(4) theory with the mesons and This will cease to be the case for higher rank, and in particula Flipping, that is introducing chiral fields More specifically, an expectation value for + 1 of SU(8) × 3 2 4 particular this lacks theget rank it 4 one has antisymmetric to representat contract the 28 bosonic fields, they mustHowever then the fourth be totally contracted antisymmetric product antisymmetrica of the product with ring relation. The operator and completely antisymmetric power of the 63 charge under the U(1) symmetry, fermionic partners of fields Here has U(1) contribution. Let us list all the operators which contribut the USp(4) model to have symmetry derived by a maximization [ to the quarks. The superconformal ( the symmetry is enhanced to operator to be removed inthis the procedure IR, see is [ a standard technique in CFT as a fermionic operator, manifold of that model. to SU(2) potential is ories glued together with a superpotential, the expansion of the index (see [ In particular we see explicitly the conserved current of R charge of USp(4) model sits on the same conformal manifold. In particu To turn it on we must deform the superpotential by a term linea forces an expectation value.in This the operator SCFT. has conformal R ch JHEP11(2017)115 . 7 , E n (3.1) (3.3) (3.4) (3.2) (3.5) × under , + is dual ) j 1 2 m ) + − n n 1 ( m has R charge − − which form T j r at at least one m m n e ij t apparent on − X e 1 3 4 3 r 1 0 2 2 X 2 l − M q 1 3 q 1 0 n U(1) i 1 n symmetries are 2 q q U(1) X 1) . 8 → → 1 Q 2) − 7 j 1 − y k Q − − X j 6 e l X the superconformal repre- Q m X + 3 Q 5 1, assuming that the theory is bigger than i 2 2) Q − ij/k − m Q 4 Q m 2 1 n − Q M 1 3 m j (2 n − 2 2 1 q 133 ) side are detailed in table 3.3. Q 1 6 i , n n n Q q − − − ( − ial 9 1 3 U(1) and the R-symmetry. When we . y . Both operators have charge − are constructed from 6= 1 m U(1) 1 − − m n × (2 28 1 2 X 1 6 56 X