JHEP03(2018)105 Springer March 4, 2018 March 19, 2018 : : Bailey December 21, 2017 -root system and : n n A Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP03(2018)105 b,c = 1 quiver gauge theories with the gauge [email protected] , N . 3 and Vyacheslav P. Spiridonov quiver gauge theories and the 1712.07018 a The Authors. c Supersymmetric Gauge Theory, Supersymmetry and Duality
We study the integral Bailey lemma associated with the A , = 1 [email protected] N E-mail: Wiedner Hauptstraße 8-10, A-1040 Vienna,Laboratory of Austria Theoretical Physics, JINR, Dubna, Moscow region, 141980, Russia Laboratory of Mirror Symmetry, National6 Research Usacheva University str., Higher Moscow, School 119048 of Russia Economics, Institut f¨urTheoretische Physik, Technische Universit¨atWien, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: groups being products of SU(nconfirmation + 1), is we provide achieved evidence by forditions explicitly various holds. new checking dualities. that We Further the discussquantum ‘t chromodynamics a Hooft (SQCD), flavour symmetry and anomalyits by breaking matching manifestation making con- phenomenon in use for a of web supersymmetric of the linear Bailey quivers lemma dual we to SQCD indicate that exhibits full s-confinement. Abstract: identities for elliptic hypergeometric integralssuperconformal generated indices thereby. of Interpreting four-dimensional integrals as Frederic Br¨unner 4d lemma JHEP03(2018)105 ], relates two distinct the- 1 4 25 – 1 – 12 26 root system n SU dualities 24 ↔ 24 + 3 14 15 c 8 19 = N = 1 supersymmetry, and, as described in section 3, are cousins of f -confining quivers s 15 N 20 23 4 12 1 + 2 + 2 27 c c N = N > f f The discovery of dualities in supersymmetric gauge theories has led to a new way of 5.3 A set of fully 4.4 N 5.1 An example:5.2 N Extending the web: SU 4.1 Field content 4.2 Anomaly matching 4.3 N 2.1 The lemma 2.2 A recursion relation 3.1 The index 3.2 Supersymmetric QCD ories at an infraredThe conformal theories fixed possess point,quantum one chromodynamics, being and are weakly, therefore theMany referred other other to strongly dualities as supersymmetric for coupled. theories QCDered, (SQCD). with some various degrees close, of some supersymmetry further have been away on discov- the family tree of supersymmetric gauge theories. time geometries and the theory of special functions. understanding quantum field theory atcoupled the and nonperturbative level: hence not a theory accessibledescription that in with is terms perturbative strongly of methods aThe can weakly prototypical have coupled example, an nowadays theory equivalent, known with dual as entirely Seiberg different duality degrees [ of freedom. 1 Introduction The past two decadesglance have appear witnessed to remarkable be development completely in unrelated: two supersymmetric fields field that theories in at various first space- 6 Conclusion 5 A Bailey tree of quivers 4 A simple Bailey quiver 3 SQCD and superconformal index Contents 1 Introduction 2 The Bailey lemma for the A JHEP03(2018)105 ] ]. ], 2 7 9 , 8 ] for an 6 -series. A q ], which are at the top 5 – 2 – ], a topological quantity counting BPS states which we 4 , 3 = 1 Seiberg duality, a relation known as the elliptic beta integral was N This article deals with a technique from the theory of hypergeometric functions that, Considering the above, one might ask whether it would have been possible to reverse These fields have surprisingly made first contact in the work of Dolan and Osborn [ Almost simultaneously the theory of special functions has independently seen impor- The requirement for two theories to be dual to each other leads to important constraints in the present work. albeit well established in mathematics,of has physics. so far Starting not seengenerate from much infinite a application trees in simple or the known context chains seed of identity, identities, the e.g. Bailey for lemma hypergeometric allows series one or to the order of discoveries and predictidentity Seiberg duality, and starting from interpreting the both correspondingthe integral sides answer as is indices yes. ofcorresponds supersymmetric While to gauge an it index theories. is or a not,in Clearly one priori the may not hope certainly try clear of and if uncovering examine a evidence known integral given for relations elliptic dualities. hypergeometric This integral is precisely the approach we adopt In the context of used by Dolan andindices. Osborn Furthermore, to known prove physical dualities the ledtities equivalence to for new of conjectures elliptic the about hypergeometric unproven corresponding integrals, iden- and superconformal leading physics. to Many a explicit fruitful examples interchange dealing of with mathematics dual field theories can be found in [ is supposed to be equal onpergeometric both integral. sides This of has a opened duality, cansymmetric up actually the partition be possibility functions rewritten of from approaching as a the an novelof study elliptic point of hy- the of super- view. duality is As different, thethe field the theory content corresponding on of both integrals elliptic sides will hypergeometric also integrals look allows different. for Fortunately exact proofs of these equalities. identities relating integrals which at firstMany glance might such look identities rather different have fromdescribed found each below. other. applications in the context supersymmetric dualitieswho as have shown that the superconformal index (SCI), a quantity that as mentioned above present article has discovered theof elliptic hypergeometric the integrals hierarchy [ of hypergeometricgeometric functions functions in the and sense theirinteresting that q-analogs they results by generalize have plain an since hyper- exposition additional been to deformation derived the parameter. subject. on the Of Many particular theory interest of is these the fact objects; that see there [ is a large number of in general for dualon theories supersymmetry-preserving is background the geometries, supersymmetric itto can partition the be function. superconformal shown For to indexwill theories be [ describe placed equivalent [ in more detail in section 3. tant advances in the field of elliptic hypergeometric functions. One of the authors of the higher-dimensional theories. on the observables of bothpasses theories. is the An equality important of triangle check anomalies ofof corresponding the validity to that duality, global also Seiberg’s symmetries known example on as both ‘t sides Hooft anomaly matching. Another observable that matches In many cases these dualities can be understood from the point of view of symmetries of JHEP03(2018)105 ) ]. c = 1 19 N ) itself ], Bailey 2.7 12 ], which can 18 + 1. The absence ]. Similar concepts SU identities. Some n 17 ] it was generalized to ↔ = -confinement, and more s 11 c N ]. In [ 10 + 1 for c N = f N ]. – 3 – 16 ]. Combined with integral identities that lie out 14 ]. Remarkably, the Bailey lemma also generates the star- 13 Bailey tree a physical interpretation. The seed identity ( root system, an elliptic hypergeometric integral that was suggested n n ]. We consider the corresponding Bailey lemma and its applications to the derivation Superconformal indices of quiver gauge theories that exhibit This is precisely what we find. We interpret the integral identities as indication for While of mathematical interest in its own right, the goal of this article is to go fur- We will start by discussing the so-called “Bailey pair” corresponding to a seed integral integral relates Seiberg dual nodes of the quivers discussed in [ 15 n be extended to an infinite sequence of dual theories possessing product gauge groups [ of our results were already announced in [ generally Seiberg duality, have appearedA before in the literature.as Just in like the in present our work case, also the arise in the context of Berkooz deconfinement [ by s-confinement. As shownnontrivial. later, the In fact, flavour symmetry we find structureindicating that of a the the flavour symmetry symmetries Bailey breaking of quiverssuperpotential. two phenomenon is dual We that theories show need can this not explicitly beintegral agree, for on explained each a by side simple of the instance thefurther of presence duality. the by We of also tree ordinary a show involving Seiberg that only the duality one Bailey and tree of another quivers set is of extended known SU duality relations and after writingthis down the evidence field by content explicitly includingthe confirming R-charges, expressions we the containing reinforce ’t more Hooft thanlinear anomaly one quiver integration matching are gauge conditions. superconformal theories,gauge In indices where fact, symmetry, of and nodes relations correspond between to quivers vector of multiplets different of length an appear SU(N to be induced exists a description purelyreasonable in to terms assume of that gauge thesuperconformal invariant integrals indices degrees generated of of by supersymmetric the freedom.of Bailey gauge integrals It lemma theories, in also is and an correspond therefore that identity to can the be different explained numbers by s-confinement. ther and give thecan A be understood asSQCD the in equality the of the caseof superconformal of an indices integral s-confinement, of on where trivial, electric the in and right-hand accordance magnetic (magnetic) with side what indicates happens that in the the gauge case group of becomes s-confinement, namely that there of integral identities. Aexpressions notable characteristic with of different the numbers resultingidentity that Bailey of relates tree integrals an is expression to thatnone with each it at an relates other. all. arbitrarily high number In of particular, integrals to it one gives with an triangle relation, allowing for thethe construction highest of known solutions level of ofof the complexity reach Yang-Baxter for [ equation the of Bailey lemma, the tree growsidentity even on larger. the A in [ single-variable elliptic hypergeometric integrals,such ideas which to was integrals thehimself (as it was very became trying first known to applicationthat). from develop of a the A historical formalism investigation furtherroot to in extension systems integrals, [ was of but realized the he in Bailey could [ not lemma find to a elliptic use hypergeometric of integrals on concrete example are the Rogers-Ramanujan identities [ JHEP03(2018)105 = z (2.1) , k k z i π dz 2 , . ∗ ]. Even though it n =1 Y k C = 1 20 ) an integral operator k ∈ z p, q wz ; +1 =1 , ) Y k k n t , z C z ( 1 1 ∈ , − j M ) are symmetric holomorphic < ) ) z z | , z ( ( q 1 | f +1) f − , k , z ) n | ( z 1 p j j | . σ z p, q < n ; ) | Γ( 1 q p | − ; k , z +1 q , . . . , z ( j . Denote as n , + 1)! n n +1 ≤ ) ) (1) tw k k n j p q k ( σ ; j T n s +3 pq s ,S , v Γ , . . . , n n j – 10 – n = t t √ ) = +2 =1 +2 Y k +2 n =1 v , v n . Y k ( +1 n = 1 = +2 ] and it coincides with the equality ( s +1) n I +2 m +3 =1 m Y j n 15 pq n ( = , v +2 n + t s ) = I k n n 1 v ) by applying eq. ( s − +3 = = , . . . , vs n +3 1 , j n T 2.15 +1 1 s vs lie inside the unit circle, n 1 ( − j v − , . . . , v tx , . . . , t n and (1) 1 (1) n n 1 s I − I t 2 1 u, t = v ) ; × − j 2 +1 v on the right-hand side of eq. ( Γ( pq +3 n ( n − +2 =1 (0) (1) n ) n = ( Y k n I I k st , t ]. Denote ( denote the sets of parameters on which the integral operators act. s j × k , . . . , s 21 t t ) = pq 1 v s rsw √ ( ( +3 =1 I n k and , the result is actually identical to eq. ( = = (1) n 1 I Q s j − s +3 v It is important to realize that there exists a generalization of eq. ( It is convenient to denote Let us consider the simplest example that may arise from these relations. Substituting n s = 0 replacing = ˜ on the right hand side of eq. ( by Rains in [ Of course an analogousv calculation can beWith done the idea for of interpreting eq. theseremember ( identities that in one terms may of obtain physics in both mind, sides it of is this instructive to equation by starting with the expression where the arguments of arbitrary free parameter (thethe total derived number relation of can be free rewritten parameters in is the equal form to 2 and the subsequent identification This relation wasbecomes first evident derived after in the identifications [ where obtain the symmetry transformation where m at an expression with two integrals. Applying eq. ( and JHEP03(2018)105 ) +2) m p,q ( n ; (2.25) (2.26) (2.22) (2.23) (2.24) I j ) = 0, in z 1 m − i p,q t ; k t ). +1 1 ) m +2 n T ) , and first apply 2.18 p,q ) + ; l Γ( m k s p, q ( = 1 supersymmetric n ( z . . ; I +2 1 i k 1. Then we may write +1 ! i z − N m j s =1 =1 i 1 l i Y Y j m + < t π ,z dz ) +2 n | 1 2 +1 1 ) m k Γ( − k m + z p,q m =1 /s ; j n i S Y p,q +2 k z s ; =1 k t m +1 1 1 k Y j w + Γ( m j,k − j i n z . S π dw ) vw | +1 ,..., 2 n . +2 , ( m ) m | Γ( n ) = I n +1 k . 1 ≤ n 1 =1 1 ( + 0 ) Y k m m n s +1 − k =1 I Y +2 n /t ) Y j ) simply as ( n C s m 0 n S j w ) is given by − s l n ) and ( − p,q ) t ,...,t ; n w n , . . . , t Γ( j 1 ( m 1 − 1 t Γ( t I w ; v 2.19 − , and let all 2.17 ( +1 v ) ), i.e. let us rewrite eq. ( n × +3 Γ( +3 +1 m ≤ ( +1 1 n m m m I m ) +2 Y + + =1 2.18 l Y n n n j ∞ pq q x corresponding to the flavor symmetries of the Y − ( ( i,j j L ]. – 13 – y exp j = J and X 25 2 ) = ) exp ∞ =1 ∞ g p Y ) n − ( ) + p, q ; = E z dµ q, p ( x ; G ! = x Z adj ) Γ( ( n χ H q ) q ) = q − − − . n ∗ ) can be turned into combinations of q-Pochhammer symbols cor- p )(1 x n C )(1 p, q, y − ( p p 3.3 . The index can be evaluated explicitly following prescription of [ ∈ I − β − pq x 2 (1 (1 1 n ) is the character of the adjoint representation under which the gauge fields ) = z =1 ∞ denoting the Cartan generators of respective groups), and is independent of the ( X n j -grading operator because of which the index only receives contributions from 2 adj F Z χ exp p, q, y, z As a consequence, the whole machinery of elliptic hypergeometric integrals [ Dolan and Osborn discovered that the integral of eq. ( ( i and is defined for can be applied to the study of supersymmetric dualities. where Denotes a generalized q-Pochhammer symbol. The elliptic gamma function then arises as responding to elliptic gamma functions. This can be seen from the characters of thesingle corresponding particle index representations itself of is thebe counting holomorphic related gauge sections to and of a line flavor certain bundles, groups. index-character and [ as The such can elliptic hypergeometric integral.expressions The like main eq. observation ( is that plethystic exponentials of where transform, while the second term is a sum over the chiral matter superfields that contains The integral can be interpreted asof a the continuous geometry, sum as over all madestate flat index precise connections is by on the a supersymmetric S groups: quantity localization that [ depends on the characters of both the gauge and the flavor which tells us that it isexponential equivalent of to the a single-particle gauge-invariant integral state over the (or so-called single plethystic letter) indices (gauge invariant) states with index only depends onbackground the geometry, and fugacities the fugacities theory ( chemical potential is the JHEP03(2018)105 . c N − flavour . Both f i flavours ˜q B N f i N + 1: in this = c c Mq U(1) ˜ N N × ∝ f f ) = f f f f /N /N W R f ) ) ) for R c c N c /N /N /N c c c N N ˜ N SU(N ˜ U(1) and the vector multiplet N N N . The latter is coupled to − − U(1) = 1 SQCD, since it is im- × i f f M ) ˜q c f N N c N , and a gauge field in the vec- ˜ = 1 SQCD. B i N ˜ , there exists an infrared fixed = 1 SQCD. ) gauge group and N c / c ˜ , where the entries denote rep- arises for Q c / 0 N and N c N B 1 N i 3 ] for U(1) N 1 ( 1 q − and < − i U(1) f t Q ) f t ¯ ) < N 1 f f f 2 ¯ 1 1 ( f / SU(N c s-confinement – 14 – N SU(N s ) f . It is noteworthy that while the superpotential of symmetry and a SU(N s ¯ ) 1 SQCD with gauge group SU( f f 2 f R 1 ) SU(N c ˜ ) SU(N . The matter content of electric N ¯ magnetic 1 . The matter content of magnetic c f f adj 1 1 0 1 ¯ f f f SU( adj 1 1 0 1 , has a U(1) SU(N 0 i i Table 1 Table 2 ˜q q V M i i ˜ . This theory, which in the spirit of Montonen-Olive duality is commonly V Q Q electric V An important check of Seiberg duality is that global ’t Hooft anomalies as captured An interesting phenomenon known as , there now exists a fundamental mesonic degree of freedom 0 to confirm that the SCIs, as defined in the previous section, of both descriptions match. of the fundamental fields ofno the breaking theory. of Note chiral that symmetry. the quarks remain massless, i.e.by there triangle is diagramsalso match captures for information both about sides anomalies, of has the by theory. now become A standard more technology, is refined check, that the electric theory vanishes,theories that are of asymptotically the free and magnetic flow theory to is the given aforementionedcase by fixed point. the gaugegenerated group dynamically. of The the degrees magnetic of freedom side are is gauge-singlet trivial composite and states a made confining superpotential is exists a dual description: The flavour symmetries are themagnetic same quarks as and in squarks theV electric in theory. the In chiral addition multiplets the to magnetic fundamental quark multiplets throughof a nonvanishing this superpotential. theory The can matter be content found in table referred to as symmetry. The matterresentations content under is which summarized in thesymmetries. table fields Seiberg transform discovered that forpoint for at SU 3 which the symmetries, theory is and conformal. charges Furthermore he for found that U(1) at this fixed point, there In this section, weportant will for briefly the review remainder Seiberg ofconsists duality of the [ quarks paper. and SQCD squarkstor with in multiplet SU(N chiral multiplets 3.2 Supersymmetric QCD JHEP03(2018)105 . l ), s to and and S , i 1 2.14 s − l quivers ˜ t x quiver − I ˜ Q R , while ˜ 1 2 B ) pq = 1. Matching of ( i ! ˜ t ,l → 1 f l τ =1 N i s c +2. The fugacities of the +1 Q c N ) in terms of the variables N m = − + i U 2.15 ˜ s n x c f = =1 N N i f − ) Q c N is the fugacity for U(1) N − f x f ) ) N 2 k N ( ), will be identical. Their distinction will z pq c 1 N − j +1 and ) U 2.18 f 1 satisfying n ,z pq are hidden in the definition of the parameters 1 – 15 – − N ( t ˜ s = − ) k B f x z c c j are the R-charges of the chiral quark and antiquark Γ ), where the right-hand side gives the index of the N N ,z ) and ( ˜ ! Q U(1) k )+ ). The statements about other sets of parameters, as 1 1 f SU(N ,l R w 2.22 − × 1 N 1 τ 2.17 t × − − j 2.16 ) c s f +1 ) 1 and N c ,w f N 1 Q ), respectively, and ( − k (2+ − R f 1 w c SU(N U j ) and ( N N f 2 × w N -confinement as described above is not limited to SQCD, but can be s ˜ s 1+ s ) Γ( 2.15 c ) f c N f pq − , where N N l ( f ≤ ˜ s N x ) ] by realizing that the SCIs of electric and magnetic SQCD can be rewritten as Y Γ + j n z = ) T Γ S Z , . R-charges and baryon number can be restored by setting pq l ( n w +1 t Let us begin by rewriting the right hand side of eq. ( In this section we study the full physical interpretation of the expressions on the right This was the first instance of a long series of successful applications of techniques c =1 T l Y N Z are the fugacities of SU(N → quiver l i × × I ˜ only play a role once we consider more complicatedintroduced constructions in in later the sections. indicate previous that section, we will suggestively interpret denoting it as the a expression superconformal index: as symmetry. Edges connecting two nodesin stand general for also bifundamental be multiplets, nodes and corresponding there will to flavour symmetries. hand sides of eqs.written ( more generally in eqs. ( The phenomenon of generalized to occur alsoTheir in field a content classconsisting can of of be nodes supersymmetric and summarized edges. gaugeof in Nodes theories distinct denote so-called known gauge vector multiplets groups, as quiver in which the diagrams, means adjoint that representation which quivers are in general graphs admit more than one gauge but rather use it to predict new physical relationships. 4 A simple Bailey4.1 quiver Field content for elliptic hypergeometric functionslogic to to supersymmetric other gauge integrals as theories.superconformal well, By we indices will applying are determine this sections. the generated matter This by content way, we of the will the not Bailey theories use the whose lemma, mathematical apparatus as to discussed confirm known in dualities, subsequent and t multiplets (given in table t the SCIs is provenmagnetic through theory, which eq. can ( be confirmed by changing variables as described above. function counting states, shouldand be Osborn identical. [ A proofelliptic hypergeometric of integrals. this In was fact, theas first SCI can achieved of be the by electric seen Dolan theory after isflavour the given group identification by SU(N eq. ( This is to be expected, since for two dual descriptions the physical content, and hence any JHEP03(2018)105 , ⊂ σ ! a 1 flat a z 1) σ i − j (4.1) 1 z π dz 1 . − 1 2 − j f ,k 3 − 1 2 = 1 chi- U(1) w τ − 1 . =1 1 c ,l l Y × a − = 1 vector 1 − ˜ N s N c τ σ 1 N 1 − , SU(N j 1 N 1) − +1 f f 1 τ z c f l N N N N ˜ − s , respectively. In s f − Q σ U r c = ˜ 1 x ) N , U(1) l − 2 pq c σ , where we have iden- U τ ( N f 1 f , − 1) w N 1 − N 2 f ) ˜ s − j c and N − f z and U(1) 1 ) l c 1 N τ ) − l N , and SU(N pq ˜ t t ( ˜ +1+ ) pq Q − c f r ( f ) N , and a simple quiver diagram in Γ ˜ t , 1 . Furthermore, there appears to be 1 pq Γ 3 i ( and SU(N a − − c a c z w SU(N i +1 z N 1 N c =1 =1 ) Γ π l Y dz − Y , SU(N k c − N 2 ⊂ f f 1 f τ =1 N a and N ! l Y N τ a j w i c U i ! z w =1 N 1 Y π j dw + 1) 1 = – 16 – 2 − j c ) − c U(1) c z ,l k N 1 = 1, we see that the global symmetries are given, r 2 N z − l × τ 1 =1 c w σ Y a U 2 , − j c N l τ 1 1 c f ˜ t N 1 N ,z − ! N 1) 1 − 1 f j ˜ s +1 =1 1 z c − x k N l − U i c z N , by SU(N c f act as the fugacities of U(1) c σ j 1 N Q R N − 1 N z N + f ˜ s − U 1 = f N x 2) Γ( − s N − ,l − f c = f 2 ) f − N N τ f N ,l ˜ s ≤ N 1 1 x +2 τ − f and ˜ − + f f c and U(1) Y c N , c SU(N j = ,l = 1 . c . Black dots correspond to vector multiplets, while the arrow connecting them Γ s τ U 1 ) Γ Γ 1 f Figure 1 − c c +1 In order to see that the fugacities match with those of subgroups of the symmetries of We have summarised the field content in table This expression matches the generic form of a superconformal index after inserting =1 f c i Y =1 =1 N N Y Y =1 r N j eSQCD N l I × × × singlets do not appear in the diagram. SQCD, we decompose the superconformal index of the electric theory as figure represents the bifundamental field,antifundamental pointing representation. towards Chiral the multiplets gauge onlygiven group charged by under associated white one boxes, with gauge where the group anan are arrow arrow pointing away pointing from a away box indicates from the it fundamental, the antifundamental representation. Note that gauge in addition to U(1) and U(1) fact, these symmetryi.e. groups are SU(N regularSU(N subgroups of the global symmetries of SQCD, tified N multiplets, with corresponding holonomies highly nontrivial fieldral content multiplets thatQ transforming may in be representations described of as various global a symmetry combination groups. of From where all relevant charactersconnections into of the two plethystic distinct exponential gauge and groups integrating SU(N over the S JHEP03(2018)105 ) 2 ), ). f − N f ) 2 c − N (4.2) c c c − N f 2.17 2.18 R N c +2 +2 N N f − N f f f f f c − f N 2 f − − N f N N N N N Nf N N f f − N ( N c ( 1 distinct f U(1) c N N c N N N + 1, corre- N − − 2 c c c N 2+ N N 1 B c − c c 1 1 = − j N 0 0 2 f N N − − z − f 1 U(1) N ,l − 1 ! N 1 τ 1 +1 ,l 1 − , τ c 1 − 1 c c c +1 +1 1 1 +1 N c a − c c 1 N N σ N 1 0 1 c c a z N N N − i − N N − U(1) f c +1 c − − π dz − c 1 N N N 2 N U 0 1 1 ˜ σ Q − c − U r =1 1 1 c 1 1 c c f N x 1 1 Y ) a N N c N − N − U(1) N pq − i ( + z σ ) f c 1) Γ N N ) s − − f k f 1 1 1 1 1 f f +1 Q N z c =1 2 r N 1 l Y 1 ( ) 0 N c − j − j c N U z pq SU(N =1 1 ) f ( ). ,z N Y ,k j τ 1 − N 2 – 17 – ) ˜ t pq − τ k Γ 4.1 ( k x 1 z c z +1) . j − ¯ ¯ 1 c 1 1 0 f f f N x j c ,z − Γ j z N − 1 i ) k − − 1 f ,z ! σ f w 1 1 SU(N ,l 1 N Q 1 N 1 − − k τ − ). Explicitly rewriting it in terms of the variables in an 1 − j R σ f z c U 1 2 1) j N ˜ f N Q +1 flavours can be rewritten in terms of z ,w +1 ) we obtain r − 1 − N c c 1 given by eq. ( ) 2.16 = c s N Γ( (2+ − k N 0 S ˜ ¯ Q 4.1 1 1 1 1 1 1 1 , Q pq f c f c r w − ( r U N ) N j > N N f 2 f 1 ≤ w f pq − N Γ ˜ quiver t ( 1+ N Y 1 Γ( and ) I c j 2.15 n z = = c = T . The matter content of the quiver gauge theory whose field content corresponds to the Γ ¯ ¯ T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 adj 1 1 1 0 0 0 1 f f f f Z , adj 1 1 1 1 0 0 0 1 Q n w +1 r c =1 T SU(N Eq. ( l Y N Z quiver 1 2 1 1 2 2 12 1 2 × × I ¯ ¯ Q Q Q Q Q M B B V V We will therefore gothe on right and hand describe side theanalogous of manner quiver to eq. whose eq. superconformal ( ( index is given by linear quivers, where theof global the symmetries quiver of described additional above.and quivers flavours, It are is follows that given possiblyprinciple SQCD, by dual depending we subgroups to on may a the act largeand number with of number that the colours there of operation is linear on a quivers. a second different equation However, set we note may of that use parameters in in as the in iteration eq. procedure, ( namely ( with sponding to index of SQCD with Table 3 superconformal index JHEP03(2018)105 . l ⊂ ˜ t ! 1 1 j τ − z (4.4) (4.3) − j 1 f f . ,k w N N 2 ,l ˜ t 4 σ U(1) 1 1 σ j operations − × = z . c +1 τ ,l 1 a N l 1 T c 1 a z τ − N i 1) σ f π + dz N − c 2 +1 and 1 1 c f N 0− 1 and N S 0 − U 0− l =1 c x Y a U U N + Q f c r +1+ N ) 1 N c f ˜ t − . The line with two arrows − N i N f pq 2 f 1 ˜ z s ( 2 N N and SU(N f 1 1 ) ) − − N s c ˜ t Γ ) pq pq N f ˜ 1 Q ( ( r − +1 ) f , c =1 j l Y Γ N a N 0 z pq Γ a z ( 1 i SU(N +1 c ,k U c =1 and figure 2 =1 − π l dz Y N Y j c N σ 2 4 Γ ⊂ = N ) 1 =1 a ! k Y − j k − a σ c z w f 1 ,l i w 1 N 2 1 − N j 1 π dw − − j σ z c − 2 f ! 1 c U(1) N , N , z 1 j l – 18 – N − 1 i 1 z − ˜ s × τ =1 0− r c − k 0− Y a 1 2 z N w +1 U − σ 1 j U c 1 c f f Q 1 z c ! N N r 1 1) N N 1 0 N ) f ˜ t Γ( 0− − j − N ˜ − U Q z pq c ˜ t r U c c ( 1 1 x ) N N c f − N i ≤ N − N τ pq = ˜ t Γ ( − 1 − Y x 1 − j eSQCD , as fugacities corresponding to regular subgroups of the flavour symmetries. ( 0 and , as can be made apparent from the decomposition of the index of electric SQCD: Γ I f 2.27 t U 1 1 N Γ ) Γ ˜ t f − Figure 2 c c =1 f The fact that the indices of both quivers are equivalent to that of ordinary electric Again, the field content is summarised in table i Y =1 =1 N N Y r N Y j × × × constructed in a similar manner. SQCD indicates a nontrivialthat RG one flow of connecting the thegroup. gauge two This theories. is nodes supported of by One the each very interpretation derivation theory of is the s-confines, identities leading leaving to the only equivalence of one nontrivial gauge corresponds to a field transforming inWe the also fundamental draw representation the of both quiver gaugeof diagram groups. corresponding eq. to ( the combinationfigures of have SU(N SU(N where In complete analogywith to the previousThe case, structure is we precisely may the interprete same, this only quantities, the together original factors are interchanged. We now JHEP03(2018)105 ) 2 )) f − ST N f ) 2 c − N c c c − N f R 2.16 N c +2 +2 N N f − N f f f f f c − f N 2 f − − N f N N N N N Nf N N f f − N ( N c ( f U(1) c N N c N N N N − 2 c N 2+ B c c c 1 1 N N 0 2 0 1 N − − − − U(1) 1 +1 1 − c 1 c σ N c − N 1 1 c c +1 +1 c N 1 1 0 c c − + N N N − f N N c U(1) 1 N − N − ) (and analogously eq. ( − τ 1 c c 1 1 1 c − N N 1 1 2.15 f N − − − N U(1) σ 1) − ¯ f 1 1 1 1 f SU(N ). τ – 19 – 4.3 +1) ¯ c 1 1 0 f f f SU(N ), and all gauge anomalies vanish. Furthermore, we have τ ) to one of the integrals corresponding to one of the gauge 1) 4.3 − given by eq. ( c 2.7 N T , 1 1 1 1 1 1 1 f − f ) and ( quiver I 4.1 SU(N . w w ) itself is interpreted physically in terms of s-confinement, as the index ). The integration variables corresponding to the nodes (ordered from left to ) c ¯ 1 1 1 1 f f 2.7 and 2.27 SU(N z , z ) y . Diagrammatic representation of the gauge theory corresponding to the combined c . The matter content of the quiver gauge theory whose field content corresponds to the ¯ 1 adj 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 f f f f adj 1 1 1 1 0 0 0 1 SU(N However, the flavour symmetries of the quivers do not match with those of SQCD, 2 1 12 2 2 1 1 2 1 ¯ ¯ V V B B M Q Q Q Q Q ones where the symmetry is not broken. 4.2 Anomaly matching We have calculated thethe quivers coefficients of for eqs. all ( triangle anomaly diagrams corresponding to but corresponds tophysical regular interpretation. subgroups, Such a and mismatchsuperpotential this may which be adds explained breaks another by the thefollowing nontrivial original the presence RG symmmetry element of flow a group. to the nontrivial theorysymmetry the One eventually is reaches possibility a enhanced is point such where thenstudy that it that these one s-confines, and arrives theories flavour at in electric more SQCD. detail It to would see be whether interesting the to present quivers can be related to is derived by applyinggroups. eq. ( But ( on the left hand(magnetic) (electric) does side not. contains an integral, while the one on the right hand side Figure 3 quiver of eq. ( right) are superconformal indices: as explained in section 2.2, eq. ( superconformal index Table 4 JHEP03(2018)105 ) ) is 4.2 c c 2 2 f f f c c . The N N c 2.19 c N N N N N + 1 ) ) 1 2 − 1 N c N f f − − − − to indicate -quiver the N N N 1 c 4 c S + 1) 2 − f II N − − c N I N 2 N ( f + 1 + 1 − 2 τ σ c B B R R N c c are identical, while − 2 c c N = 0 in the language N N 2 R N N ) U(1) U(1) f U(1) U(1) U(1) U(1) m N × × × × × × + 1)( + 1)( 2 2 2 2 2 2 c − c c 1) 1) N 1) N N ( ( + 1) − + 1 + 1) − + 1) − f c c f f c c N ), which we denote ( ). The left-hand side is the super- SU(N SU(N SU(N SU(N SU(N SU(N 2.19 3 τ B R 3 R symmetry, while the right-hand side has 2.19 . To make this explicit, we write the su- B B c c c c U(1) 2 1 U(1) f U(1) U(1) c N N N N 1) flavour symmetry becomes trivial for both N N − × × + 2, corresponding to U(1) − − − U(1) c c 2 τ − 2 τ − ), this time to the other integral as compared to – 20 – c × c N × N ) N f − = N 2.15 U(1) U(1) f f − U(1) f N 2 1 (4.5) c 2 f 1 f × SU(N N 2 N 1) c − − N 2 ) itself. × c c 2 c N f − − 3 3 3 τ ) − B ) to eq. ( R N N f + 2, and as such its field content is given by table N N 1 U(1) f 1) 1) 2.15 c − 2.7 N × ( + 1) − − U(1) N U(1) U(1) = 1 electric SQCD as described in section 3.2, with the addi- f c c 1) − × × × N = c 2 − 2 2 N − f f N 1) 1) 1) f SU(N SU(N N − − − + 2 3 σ . In particular, there exists a dual description obtained from the right hand B R R R ). This can again be understood as in terms s-confinement, as eq. ( -quiver they read anomaly coefficients involving U(1) c c c c T SU(N N T N N SU(N × 2.19 U(1) − U(1) U(1) U(1) U(1) − − f and f = N f 1) × × × ), respectively. The results agree exactly for all coefficients. For the f S 2 σ 2 σ B 2 − f 4.4 SU(N We proceed with the interpretation of eq. ( SU(N SU(N U(1) U(1) U(1) right-hand side, however, isand more four complicated. of There themThe are are left-hand in side gauge total has singlets. SU(N eightSU(N chiral Furthermore, multiplets the flavourperconformal symmetries index are on different. the right-hand side of eq. ( side of eq. ( obtained by applying eq. ( what is used to prove eq. ( conformal index of tional constraint 4.3 N An interesting situation arisesof for section N 2. Inquivers this case the SU(N while for the all others differ by a minus sign. and ( nonvanishing ones are given by compared global anomalies with those of the subgroups of SQCD made explicit in eqs. ( JHEP03(2018)105 R c c +2 +2 +2 +2 +2 +2 +2 +2 j 2 2 2 2 4 4 (4.6) (4.7) N N c c c c c c c c z 2 2 l N N N N N N N N U(1) σ j z +1 1 l c B c c , σ N c c i 1 N N 0 0 0 i z − N N i − 1 c f − − f 1 π U(1) dz N 1) factors, with N N f 2 ˜ s − N 1 − c f ˜ c 1 s f − N 1 N x =1 c N ˜ t 1 i Y +1 τ + 1 N c c c f . − 1 f 1 1 0 c − c +1 i N N N N 1 1 N c i z N ) N − ) i 1 N − +1 − c U(1) − π 1 dz − pq c c pq N 2 ( ( N N f 1 1 − N − − k ˜ t Γ =1 c Γ i Y c f σ k 1 N 1 0 1 N τ +1 N 1 c 1 1 σ − ˜ s 1 1 c − j +1 1 − N x − 1 1 j c c c 1 − − z j N c 1 f − 1 0 1 z c 1 N N f z N f f N 1 N +1 − 1 N − 1 N N f c U(1) N − + − l − − ˜ c s l ˜ s N 1 N 1 τ f x τ 1 ˜ N t x c N 1 c 1 x + N ) are the fugacities of the U(1) factors, and c f − N 1 f f N − N τ f + pq N ) N − ( c f N ˜ t +1 1) c 1 ˜ f t N N c f x pq N N – 21 – ) N ( N − − ) Γ ˜ t ¯ ¯ f f 1 1 1 f f f pq are the fugacities of the SU(N c N 1 Γ ( pq and N ) f l 1 ( ˜ t f − N − j 1 1 pq ˜ s z N Γ + 2. ( − − j SU(N Γ 1 s f c c f 1 f z 1 N N f N f 1 N N k ) ˜ t Γ N − 1 − N σ σ ˜ t ˜ t 1 = − 1 pq 1 1 x 1) f f − ( = − c f − 1 =1 f − N f N c l f Y N l f − ˜ N t flavour symmetries. The explicit expression is given by N N N f τ N ¯ f 1 1 1 1 1 f f f N Γ ˜ t s c − N ) N ) 1 =1 x ˜ s N f ˜ s Y j c = 1, while ˜ − f 1 pq f =1 f N n and ( l 2 l − N Y N τ ˜ T t N − l ) Z 1 f f ˜ s c 1 2 Γ N N SU(N − 1 =1 n N pq Y ) SU(N j f ) ) − ( κ × c 1 =1 f × f n N l pq pq N T s N ¯ ¯ = ( ( 1 1 1 1 1 1 f f f f f ) Q s Z Γ adj 1 1 0 0 0 1 f 1 n = SU(N − = ˜ Γ Γ κ =1 c l Y ) in terms of these fugacities, which shows that they correspond to actual subgroups k labels the baryonic symmetry. One can rewrite the index on the left-hand side of l . The matter content of the theory corresponding to the SCI on the right-hand side of ). Note the relation N × × × σ SQCD σ x I 1 +2 +2 = c c − 2.19 We have again calculated the coefficients for all global symmetries and found them to 2.19 1 2 f 00 i i N N II 1 2 =1 N l I Q Q Q Q M M B B V match with those of regular subgroups of SQCD. The nonvanishing coefficients are given eq. ( of the SU(N where Q again the field content can be read off directly: Table 5 eq. ( that it is the second theory to arise from s-confinement of the original quiver, such that JHEP03(2018)105 ) × 1) (4.9) (4.8) fixed 2.15 trans- c c 2 c 2 2 − f c f N N f 2 f T N c c N N , + 1 + 1 + 1 N c N N N c c c N − − + 1) 2 c N N N N c 1 N quiver N − − − I ( the same is actually dual IR I , and 3 σ σ 3 σ B B R R R S , 1) . − + 2: U(1) U(1) U(1) U(1) U(1) U(1) U(1) T , f c quiver × × × × × I , 2 σ , 2 σ 2 σ 2 σ 2 σ )) = N II quiver c 1) 1) 1) SU(N f I I N − − − U(1) U(1) = f 4) f f S , − c N SU(N SU(N SU(N quiver I 2 f 3 + ( = c c N 2 − c 2 2 – 22 – ): f c II f ( N N c I 2 f N c c N N + 1 + 1 + 1 N 2 2.7 − − c N N N c c c N = − + 1) 2 − c N N N N c N N − − (4 + ( c − eSQCD N I + 2 to simplify expressions. integral ( c n 4 + N 3 τ 3 τ τ B B − R R R = 1) f U(1). We leave a detailed study of these deformations for the future. − U(1) U(1) N U(1) U(1) U(1) U(1) U(1) f × × × × × × × 2 τ 2 τ 2 τ 2 τ 2 τ 2 B )) and the A 1) 1) 1) U(1) 3 R SU(N − − − × U(1) U(1) U(1) f 2.16 f f 1) U(1) − A possible explanation for thepresence of equivalence a of nontrivial superconformal RG indicesall flow is connecting four of all theories of course them. the arepoint, It equivalent i.e. is descriptions it not is of clear,to however, not a that electric guaranteed theory SQCD. that at the one theory corresponding and to From the derivation oftheories the arise corresponding as indices s-confinings-confining it descriptions description seems of the of that two either theaccordingly. quivers. two of One non-quiver both may nodes, switch and to the obtain one of the two theories The field content offorms the in theories regular subgroups corresponding of to the the presence flavour symmetries of of a electric superpotential SQCD. breaking This suggests the larger symmetry. We have a setgroup, of four whose distinct superconformal theories,and indices ( two of match them due quivers to with the a product Bailey gauge recursions (( f SU(N SU(N SU(N This result again hints towards the presence of a superpotential deformation of the N • • • • SU( For now, let us summarise that we have, in the case of N where we have used original theory such that flavour symmetry is broken down to the subgroup SU( by JHEP03(2018)105 f N ) 2 f − ) 1 1) N 2) 2) f 1 f ) +2) − c c − − − − N N c c c − c j c c N N f − R N N N ( +2 N +2 N N f f − N f f c − − − w − f 2 − f − − +1 N N f f f N N Nf N c f N 1 f f N ( +2 N N c ( N ,l U(1) N c N N N ( c ( ( − ( ( N 1 N f c c f f Nc N − ) τ N N N N N 2 c +4 N 2 2+ c +1 N 1 c p, q 1 1 ; N 1 − − k − c c − B c c c ! c z N N c c c c ! N N N 1 N 0 2 0 2 0 1 1 − − N N N − N f f ,l − − k − f j U(1) 1 N N 1 N z τ U − − 1 , z f 1 1 1 − j c +1 1 − N − +1 c − 1 k − c N τ c ˜ s 1 w c c +1 +1 N z 1 1 N N c c − c c N N c 1 1 0 f − c j N N 1 1 N N − f − − N N N z U(1) f N − − c 1 ) N U N U − Γ( x c 1 pq c 1 1 c N ( − 1 N c N − N 1 1 − − f 1). This is reflected in the double 1 − − − 1 σ ) c c − f 1 N Γ 1 1 1 1 c c − f N N 1 1 N − − N c − − c N N − 1 − c − f U(1) c N 1 N ,l 1 − f +1 − − 1 N − 2 c =1 N f − N f N − l Y ! Y τ 1 2 f f f N N N ≤ j k ( N N N c U − σ w ˜ ! s σ f 1 N i f 1 j j ), which we rewrite in terms of the variables used f N 4.10 SU(N N U N p, q )+ Γ τ 1 U x ; f U − c k f ! − 1 N 2.26 c 1 +1) N 1 N f − w N ¯ ¯ 1 − c 1 1 1 f f f ,l 1 ˜ s c f 1 − N − 1 x N f τ N − f j c f N 1 +2) SU(N − N N N c +1 ˜ s 1 − , w τ − (2+ c + N c 1 1 ( 1) f N c +1 2) N − − − k c N 1 − given by eq. ( c N − f f − − 2 c N N w ). This leads to an additional set of theories connected to those f +2 f N N ( N S ¯ U j c 1 1 1 1 1 1 1 , f f N N ˜ f s − 1+ f N f w ) ) N N +2 1) 2 f 2.22 +4 ˜ f s − N Γ( pq pq 2 c c N c quiver 2 ( f ( SU(N c c N N I N N f ) N N w − − ) N − f c f ≤ Γ Γ 2 ¯ ¯ c pq 1 1 1 1 N f f f N 1 ( + 2 N m ) ) Y integral, although with different parameters, as it is equivalent to the right ( − j c ) n pq fugacities does not s-confine, but instead has a magnetic dual description in n pq ≤ z N ( Γ + 2 we are presented with the scenario that the sector of the quiver correspond- ( 1 N =1 κ pq 1) l Y j − 1 c ( z − m z f − c > = N T c Γ N Γ N ¯ ¯ Z 1 adj 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 f f f f . The matter content of the quiver gauge theory whose field content corresponds to the 1 N 1 S =1 Γ adj 1 1 1 1 0 0 0 1 − , > f Y k − − − n w +1 c generated by the electric side. At this point wethe should magnetic side mention of that originalby the Seiberg an duality, Bailey A as recursion the corresponding mayhand index side also is of be also eq. given applied ( to =1 f =1 f f c f =1 T i Y Y l =1 Y k N Y N N Z j N N • SU(N quiver 0 × × × × × × 2 1 2 1 I 12 2 1 2 1 ¯q ¯q V V B B M M q q q terms of nontrivial gauge group,integral given on by the SU(N right hand sideabove of as eq. ( 4.4 N For N ing to the Table 6 superconformal index JHEP03(2018)105 T (5.7) (5.8) (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) , lead (4.10) and , m , S (1) 0 . , I (1) 0 0 1 I and (0) (0) n 1 C 0 1 I I 0 C n 1 0 0 n 1 0 . T 1 S S a 1 . Again the fields 1 T 1 1 a z n 1 C 1 1 i 6 T T 1 C ! 1 π dz 1 1 1 2 T = = S − j a = z a w = (0) (0) n 1 i ,k I 2 I (1) 0 π dw 0 τ 0 (1) I 0 n 2 1 1 0 I 1 T T 1 − 0 1 C 1 1 n 1 c − 0 C 1 =1 c N S S 1 . This is due to the involvement 0 Y 1 a i S − N 1 S 1 f = = w 1 1 C N ! 1 C k 1 − 1 (0) (0) 1 n 1 z T I I 1 U S 0 0 1 − i n 1 = = − σ T T c 1 1 1 (0) n 1 1 N 1 − (0) 1 I T T − c I 0 f f 1 N 1 0 1 N = = N T − T ˜ s 1 f 1 1 – 24 – 1 1 N C (0) (0) n 1 C − 1 − ) can, depending on the choice of I I 1 c 1 1 c 1 0 0 1 n T + 3, i.e. we will consider possible ways of rewriting N S N − S S − c 1 f f = 1 1 f 1 n = 2.23 N N N S S ˜ s x (0) 1 (0) = N 1 1 I − = = I − + 3 f 0 1 c 0 1 1) 2) c c S N − S (1) (1) 1 n − N 1 1 c − 1 c 1 I I ), and ( C f N 1 1 C N 1 1 n N − 1 1 1 − = N f T T x . f , , T , S N + 2.18 f N ( ( = = = 1. This leads to (1) = f 0 (1) (1) = 0 0 c (1) 1) 0 2) I N I I + 1 parameters, and omit the arguments of corresponding N I − n 2 0 − 1 ), ( 0 0 1 1 c 0 (1) (1) (1) 1 1 1 n c (1) 1 ) C n C C N I I I C I N 0 0 0 1 1 1 − 1 0 1 1 1 n 1 1 1 1 − pq f C S T 2.17 ( f S S C S T N 1 1 1 1 1 1 N 1 1 1 ( 1 1 1 ( f = = c S T S S T T N Γ N = 2, i.e. 2 ======1 ) (2) c (2) 1 n − I I c N (0) (0) (0) (0) (1) (1) pq 1 1 1 1 1 1 N ( I I I I I I =1 Y r 0 0 0 0 1 1 − 1 1 1 1 1 1 f S S S T T T We summarize the field content of the corresponding theory in table Γ N 1 1 1 1 1 , making use of the aforementioned operations. For simplicity, we will consider the 1 1 1 S S T T × × (2) n with the additional equalities Seiberg duality nowconsider generically leads to many more relations. For simplicity, we will corresponding Bailey tree forI N action on the first operators. We then obtain The principle of interpreting integralindices that identities provide in evidence terms for new of dualitiesidentities equivalences that may be of may be generalised superconformal to formed arbitrarily from complicated the the building operations blocks as ( explained into section 2. very Combining large families of related physical theories. In the following, we will build the in an analogous manner as above. 5 A Bailey tree5.1 of quivers An example: N are organized in representations ofSQCD. regular The subgroups field of content is thethe presence of flavour of symmetries course two of similar, additional fields: electric apartunder one from the mesonic gauge different gauge symmetry charge singlet corresponding assignments and toof one and the field the fugacity that magnetic is charged description of a sector of the quiver. Anomaly matching may be verified JHEP03(2018)105 ↔ SU (5.9) (5.11) (5.10) , it is ↔ : (1) n , (1) I 1 ) 1 I n ) (with the + 2 and is X +3 c , , 0 n , , 2.27 = .... U(1). This is a + 1 + 1 + 3 + 3 = , . . . t × (1) n 0 1 n n I t ; (1) 1 ≤ ≤ K) I l < K l l < K l + 2), while that on the 1 +3 1 < K < N − . The flavour symmetry c 0 n l ) by X ≤ ≤ ≤ ≤ t 1 1 1 1 K l + 2 5.9 C 1 c 1 + 1 + 1 Q SU(N , . . . , s X 0 1 1 × 1 = s ( , with 0 C SU(N K K (1) n T = I + 2) × ) c i (1) 1 I and 1 1 T/t l j C s SU(K) 1 1 , t , K l , j ,K ,K l l X × l l t t 1 – 25 – 1 s s Q ). For brevity, we will refrain from listing the field 1 C 1 +1 +1 S/s 1 1 +1 +1 i n n = n n 3 ) ) U(1) = ) acts as a starting point for a new branch in the ) = 1 and are given by ) SU dualities , s K i K K K K t × operations is given for example by eq. ( +3 S (1) m j 1 n ↔ /T /S I , /T /S l T 1 , s 1 K) t K K ) is given by SU(N j K ]. In the language of the present article, the transformations K ) = t T S X i S − T ( ( 1 1 +3 ( 21 ( s 5.9 +3 n l and , . . . , t K K ) with n C 1 − − Γ( +1) +1) t +1) +1) + 2 Q S K K n n ; = n n +1 +1 c 2( 2( +3 2( n n 2( 2.14 = n +3 , . . . t ) ) ) ) ], there exists another set of dualities derived from mixed SU (1) 1 n 1 ≤ I t T j 9 1 ; , 1 ≤ , SU(N 8 l S/T T/S T/S S/T C s +3 Y × n = , . . . , s = ( = ( = ( = ( +3 1 integral ( 0 l 0 l 0 l 0 l n l i