JHEP07(2020)012 Springer July 2, 2020 May 24, 2020 : : February 28, 2020 September 30, 2019 : : Published Accepted Revised Received Published for SISSA by https://doi.org/10.1007/JHEP07(2020)012 dualities in the literature. d gauge theories from S-duality . d 3 5 1909.05250 The Authors. c Duality in Gauge Field Theories, Field Theories in Higher Dimensions, Su-

We describe a general method to determine dualities between supersymmetric , [email protected] gauge theories. The method is based on performing local S-dualities in the geometry 17 Oxford St, Cambridge, MA 02138,E-mail: U.S.A. Department of Physics, Harvard University, d Open Access Article funded by SCOAP persymmetric Gauge Theory, Supersymmetry and Duality ArXiv ePrint: dualities. As an application,classified we by explicitly removal determine and a addition specialvastly of generalizes class edges many of of into irreducible a the dualities known Dynkin 5 diagram. This classKeywords: of dualities associated to the gaugematter theory. to We both find sidesirreducible that of dualities often which a cannot a more be dualitydualities obtained primitive can from then duality. more be are primitive This obtained obtained dualities.geometric by allows by More method adding us general adding described matter to in to define this both paper the sides allows notion of us of to an systematically irreducible construct duality. irreducible The Abstract: 5 Lakshya Bhardwaj Dualities of JHEP07(2020)012 denotes . θ θ ) n with a bifunda- π gauge theory with eight 4 d SU(2) . The notation SU(2) × k π SCFTs gauge theories can be dual to d . The same holds for Sp( d 5 18 , π 1 and 6 21 . Theories admitting more supersymmetry d d 32 22 ]. This paper is devoted to studying such 109 15 – 1 – , gauge theory ) with Chern-Simons level n 14 d 111 18 108 valued and can take values 0 = 1 supersymmetry in 5 2 108 109 Z gauge theory will refer to a supersymmetric 5 23 N 3 denotes SU( 44 d 37 ≥ gauge theories describing 5 n = 1 language. 54 25 d 52 N which is for with two flavors [ k θ 107 72 105 0 ) with other isomorphisms n 1 30 S + 1) dualities ) dualities A prime example is the duality between SU(2) SU(3) + 1) dualities r r ) dualities r r dualities 3 dualities ] for a sampling of recent work on this topic. dualities 2 ( ( (2 (2 2 r 4 13 sp so e f g su so – 1 Throughout this paper, a 5 See [ The notation SU( 4.4 Reducing irreducible dualities 4.1 Semi-simple to4.2 semi-simple Allowing extra4.3 edges Mixing 3.3 3.4 3.5 3.6 3.7 2.5 Adding matter 2.6 S-duality 3.1 3.2 2.1 Some geometric2.2 background Intersection matrix2.3 associated to a Prepotential geometry associated2.4 to a geometry Pure gauge theories 1.1 Dualities for 5 1 2 3 supercharges, or in otherwill words, also be treated in an SU(2) with theta angle It is well-known byeach now that other. different supersymmetric mental and dualities abstractly. 1 Introduction 4 Further generalizations 3 A special class of irreducible dualities 2 The geometry associated to a 5 Contents 1 Introduction JHEP07(2020)012 gauge SCFT d to each d dual gauge theory gauge theories, d of a Calabi-Yau d gauge theory then SCFT. gauge theory then d d d whose base can be shrunk 5 local piece supergravity theory admitting d supergravity theory admitting a d – 2 – gauge theories are referred as being d gauge theory description to another 5 d SCFT. d which carries this local piece. In terms of this local piece, the above- 4 SCFT compactified on circle. d rather than an elliptic fibration. gauge theory arises at low-energies in a 5 gauge theory describes a region (i.e. an open set) of the Coulomb branch of gauge theory describes a region of the Coulomb branch of a (possibly) mass- d d d gauge theory is non-renormalizable, thus its UV completion is not guaranteed. gauge theory describes a low-energy sector of a 5 d From the point of view of the local pieces associated to the two 5 d 6 5 a consistent UV completion. to a point, then thiselliptically piece fibered describes Calabi-Yau all threefold. thedescribes finite a volume The structure region corresponding of ofcompactified a 5 the on non-compact circle. Coulomb branch of a (possibly) mass-deformed 6 Calabi-Yau moduli space, thenof this a piece non-compact describes Calabi-Yaudescribes threefold. all a region the of The finite the corresponding volume Coulomb branch 5 structure of a mass-deformed 5 deformed 6 consistent UV completion. a mass-deformed 5 If the local piece embeds into a compact Calabi-Yau threefold, then the corresponding If this local piece can be expressed as an elliptic fibration If this local piece can be simultaneously shrunk to a point at a finite distance in the The 5 The 5 The 5 Now, it is often possible to change the parameters and moduli of the UV complete A 5 The existence of a section isExtra not care necessary. must be If taken there in is the no case section, when UV the completion fibration is is a referred supergravity to theory. as In a these cases, It is important to keep in mind that the existence of a Calabi-Yau threefold carrying this local piece 3. 2. 1. 3. 1. 2. 5 6 4 is not guaranteed.corresponding gauge In theory the cannot cases be UV where completed such ingenus-one a any fibration way Calabi-Yau and threefold hence does should be not discarded. exist,the we changes believe in parameters thattheory and the description moduli is should valid. be smaller than the energy scale below which the 5 description. In such aother. case, the twosuch 5 changes correspond tofirst a gauge geometric theory transition to the from local the piece local associated piece to associated the to second gauge the theory. theory to transition from one 5 threefold to any arbitrarythe gauge theory. gauge The theory correspondence isCalabi-Yau threefold between established the by local imagining piecementioned and three that kinds we of are UV completions compactifying take M-theory the on following form: a As discussed later in the paper, it is possible to associate a Three kinds of UV completions are especially interesting: JHEP07(2020)012 gauge d . We discuss a gauge theories, for more details. 3 appearing in this d irreducible duality . 7 4 2.6 leads to dualities between gauge theory descriptions of S contains a list of dualities d 1.1 1 curves living inside the Kahler − SCFT (the latter being compactified gauge theories which describe regions d d then consists of a collection of Kahler sur- gauge theory corresponds to resolving the or 6 8 d d – 3 – ], we do collect all the dualities SCFTs and little string theories. 15 d gauge theories in terms of the existence of a geometric d , we describe how a geometry is associated to any 5 . We discuss how implementing 2 S for the list of these dualities. 1.1 supergravity theories can admit curves of positive self-intersection, but such curves are gauge theories means that the two associated geometries are isomorphic to d d SCFTs. In section , we define a special class of irreducible dualities which are classified by adding d 3 We will also impose some further technical restrictions on the geometries toRest simplify of this paper is organized as follows. Section Moving onto the Coulomb branch of a 5 In this paper, we study dualities from this abstract, geometric point of view, without It is thus possible to forget about the underlying UV completion of 5 This does not mean that the other dualities appearing in this paper can be discarded, as the gauge For ease of communication, we refer to a resolved local piece as a “geometry” from hereafter. gauge theories. Another interesting point is that if two gauge theories are found to be and 6 8 7 d d theories participating in these dualitiesmight may suspect UV that complete only into the aUV gauge consistent theories completion supergravity which into theory. can a The UV theoryconstruction complete reader with into of gravity. a 6 However, theory this without isforbidden gravity may not in admit true. the For F-theory example, constructions the for base 6 in the F-theory each duality lives in awhich tower captures of dualities the built minimumsection on amount top of of what matter weand necessary removing call edges to an from the implement Dynkin the diagram of duality. a simple In gauge algebra. We explicitly work 5 theory. In this section, weas also a discuss geometric how the isomorphism S-duality5 of Type IIB string theorydual, appears then adding suitable matter on both sides of the duality preserves the duality. Thus, our analysis. These restrictions(partial) are list described of at generalizations the of beginning the methods of of section thisappearing paper in in this section paper which are relevant in the context of 5 between two 5 each other upto flopsurfaces. transitions associated to In compact thisthought paper, of as we implementing willinside S-duality the focus geometry of associated our Type to one attention IIB of on string the gauge a theory theories. on set See section some of of dualities the which surfaces can be of Coulomb branch of someon mass circle). deformed See 5 section associated local piece. Thefaces resolved local intersecting each piece other in a way consistent with the Calabi-Yau condition. A duality duality is unphysical and can be discarded. regard for whether ora not UV the completion. gaugeHowever, theories on The the on exploration basis two of of the sidespaper latter results which of of issue can [ the correspond is duality to left actually dualities to between admit 5 future papers on this topic. and define a duality betweentransition two 5 between local pieces ofries. Calabi-Yau If threefolds the associated local to piecesCalabi-Yau are the threefold, such two that then gauge they theo- the canlocal be corresponding pieces embedded duality in do a is not compact physical or admit a and a non-compact interesting. consistent completion If into the a Calabi-Yau threefold, then the JHEP07(2020)012 ] - d ]. n 15 for 15 (1.2) (1.1) 3 . π will denote ]. See below C gauge theories 15 F F d ) ) π π and will denote the m m S (2) (2) n − − Λ su su SCFTs (3 (4 , we close the paper by d 4 gauge theories with simple 0 0 d SCFTs (with the latter being (2) (2) d and 6 = 4, in which case it is su su number of hypers in fundamental as displayed above. Similarly, the d m n π n and 6 − d F ··· ··· ) 3 p ], implying that the former cannot be UV 3 F − 0 0 ) 15 SCFTs were constrained in the work of [ p ] only proposed necessary conditions for these − m (2) (2) d n − – 4 – 15 − su su n m − − and 6 (2) is relevant only if it carries no fundamental hypers d su 0 + 4 (2) F (2) n + (13 su p will denote the 2-index symmetric irrep, = 4, in which case it is su − will denote the adjoint representation. 2 (2) is relevant only when 2 p + (2 m S A su ) SCFT even though they satisfy the conditions of [ p n gauge theories describing 5 − 2 F F d will denote the fundamental representation, ) π π ) m − p d p ) will always be assumed to be non-negative. An edge between two F n (2) (2) ) contain the following dualities: or 6 − (7 − ( su su d n su su (4 ( 3.54 ]. It should be noted that [ m, n, p = = 15 ]. 12 – ) and ( 1 SCFT. In fact, as we will see, we find that some of the theories appearing in [ 3.44 ( Some of the dualities appearing below have appeared inIn the the literature following, before, particu- d denotes that the gauge algebra carries additional (2) is relevant only when F n representation. The theta angle for which are uncharged under anysu other gauge algebras.theta Thus, angle the for theta right-most angle for left-most Every integer gauge algebras denotes a bifundamental hyper and an edge between a gauge algebra and index antisymmetric irrep, the two spinor irreps, and completed into a 5 for such examples. larly in [ In other words, inappearing this in section, [ we collectgauge all theories, those thus dualities the which theoriesor involve a appearing 6 gauge here theory may notshould have be a dual UV to completion theories into not a appearing 5 in [ Here we collect all thosethat dualities can appearing describe in regions this ofcompactified paper Coulomb on branches which circle). of involve 5 5 with All a the simple dualities gauge algebragauge we on algebra study one which in side can this of describe paper the 5 duality, involve and a 5 gauge theory easy identification. We alsoof describe the the dualities, map thus between completelytop the specifying of Dynkin the the diagrams full irreducible ondiscussing tower two dualities of many sides dualities ways in in that this which can special the be considerations class. built1.1 of on this In paper section can Dualities be for generalized. 5 out all the irreducible dualities in this class and display them inside boxes in section JHEP07(2020)012 F ) π (1.7) (1.8) (1.9) (1.3) (1.4) (1.5) (1.6) m (1.12) (1.10) (1.11) (2) − F su ) π (1 π F m ) π (2) π (2) − m F 0 su ) (2) su π (2) (1 − m (2) su su (2) π π (2 su − π su (2) (2) (2 (2) F (2) 0 su F su ) (2) 0 π F su su m su (2) (2) (2) (2) F − su F = 0 is allowed. (2) su su su 2 (2 n su F F (2) (2) 0 2 (2) 3 F su su (2) F su (2) 0 su F F (2) F F (2) (2) su π ) 2 5 2 (2) p su su su su (2) π − F su F (2) π ) (2 F (2) 3 (2) p = 4 su F π su (2) su (2) F π ) − – 5 – p su = su (2) (2) F (2) (1 F − ) su su su = = m (1 F = ) = is non-negative, that is p − π n F = F − = F = (2) F ) m + (9 + 7 p + 4 su + 8 F 2 F ) − m 2 2 F 2 ) − p Λ Λ Λ 3+ m − m + 7 + = + 2 + (8 + 3 − (3) 2 2 − m 3 2 2 0 Λ su Λ − F (4) (4) (4) + (8 + 2 + (8 su + 2 2 su su + 8 3 2 2 Λ p 2 Λ + (10 − 2 Λ 3 (5) m + 2 Λ + 2 su + 2 2 m p + (4) 0 − 2 p su m − 2 (5) (6) m su (5) su (6) su su where we remind the reader that JHEP07(2020)012 F ) π (1.20) (1.16) (1.13) (1.14) (1.15) (1.17) (1.18) (1.19) m (2) − π π π su (2) (2 F (2) (2) F F (2) ) ) π π su su su F su m m ) π 0 (2) (2) F (2) m − − 4 su su (2) (2) su − (1 (2 0 su su (3 (2) (2) F F (2) (2) F 0 su su su su 0 0 (2) F 0 (2) (2) su su (2) (2) su 0 0 0 su 0 su (2) (2) (2) (2) p p F F su su su 2 2 ) ) su p F 2+ 4+ p p ··· 2 ) p F 5+ 2 F p ) − − 6+ p (3) (3) F 4 − ) F (3) 0 p 0 0 F (4 (3 p su su − p 2 (3) F ) − 2 (1 5+ su ) p − 3+ (2) (1 (2) (2) p su n – 6 – − su (3) m su su − (3) = = (1 su − = (3 su = F F ) ) 0 7 2 3 + 6 F p p F ) = n ) = (2) p F (3) − − (3) p su − + ( su su F m m − ) 2 F p − − Λ ) m p − + (8 F + − ) 2 = = 2 − p p − Λ + (8 + (8 − 2 p 2 2 − 2 + m + (7 F + n + (8 Λ Λ ) n 2 + (8 p 2 n Λ − 2 ( + 7 2 Λ − (3) 5 + 2 + 2 Λ 2 − su p p su + 2 Λ + 2 − − 2 2 (10 p + 2 (5) m m p 5. − 2 p + 2 3 − 2 2 su − − m ≥ 3 2 = (5) (5) n su su (6) (6) (6) (7) su su su su where JHEP07(2020)012 (1.26) (1.21) (1.22) (1.23) (1.24) (1.27) (1.29) (1.25) (1.30) (1.28) F F ) 3 ) π π π − 3 π m m 2 m (2) F (2) + (2) F (2) (2) − − π (3) π n 2 (2) 2 su su su su n su su (1 su (2) (2) (3) − su su su (9 0 0 0 0 0 0 (2) (2) (2) (2) F 0 0 (2) (2) (2) 0 su su su su su su su (2) (2) (2) su su su p 0 F 2 ) p 0 0 p F 4 F 2+ 2 p ) 2 ) (2) 4+ 4+ 2 p p 3 2 4 (2) (2) ··· (4) − Λ (4) su − − F (3) F su su (4) su (3) (6 (3) su 5 2 5 (2 su (4 su 0 su su – 7 – − F (2) ) n = 3 2 5 2 4 = p su = = F F = = − (4) (3) (3) 3 3 F F su ) su su m F F F ) F p ) p 0 − + 3 p − 2 + 7 − + 3 (2) − Λ = = = 2 m 2 Λ + (8 su Λ + − 2 3 + (7 F F Λ + (9 F + 2 Λ 2 + 3 3 Λ 3 2 0 + 7 + 7 + F Λ + 2 3 ) + 5 + (10 2 2 3 − p 2 p (5) + (5) 3 + 2 p Λ 3 2 Λ − − 2 Λ + p Λ p − su su n m − − 2 2 ) (6) 3 + + 2 + 2 n 3 + − n − 3 3 2 3 2 (3) p su ( 2 − − − 2 su (6) su m (5) (7) 6. (6) (9 (7) su su su su (4) contains a hyper in 2-index antisymmetric representation. ≥ (6) su su n = su where where JHEP07(2020)012 (1.39) (1.35) (1.36) (1.37) (1.38) (1.31) (1.32) (1.33) (1.34) 2 Λ 2 F F Λ 1 F ) F m − F 2 ) m 2 ) 2 m m 5+ m + 1) m n − F − m F (3) − ) m 2 − 2 ) F F 2 1) m m 1 2 2+ ) ) 2 2 1 n 1 4+ m (1 su 2 m − ( − 2+ 4+ − F m m 1) − − (3) n 7 F − n su (4) 1) 7 ( − − n − (3) (4) (2 su ( (5 su 1 − 0 n su + 1 su (7 su ( n − n F ( (2) ( su ) n p ( su su π F π ) π − π F π n (2) ) n (2) (2) F (2) p − (2) π su (2) − su F su 1 su − ) π su (8 su n (2) (7 F n (3) 4 (2) su − − = su = = su = = – 8 – (7 = (6 F ) F F = F p F F ) ) = = ) − m m + 7 m F + 8 2 ) F − − m 2 ) p − Λ F Λ ) − m − + m + 2 + (7 − + (8 2 m + (8 3 2 2 + 6 − 2 − 2 − Λ Λ − n Λ + (7 (4) + (4) + 2 2 + (7 + ( n + 2 su Λ su + (8 2 2 + 2 m 2 2 Λ 2 m Λ m 3+ ) Λ + 2 + n (5) + 2 ( m p (5) 2 + 2 − 2 su m su 3+ 2 su p m 3+ − ) 2 ) 4. 4. 4. 4. m n (6) ( n ) ) includes the following dualities: ( su ≥ ≥ ≥ ≥ n su ( su n n n n su 3.51 ( where where where where JHEP07(2020)012 (1.50) (1.46) (1.47) (1.48) (1.40) (1.41) (1.42) (1.43) (1.44) (1.45) (1.49) 2 F F m Λ F F ) 2 F F m 2 ) 2 2 F 2 3+ 2 m m ) 2 F 7+ m m ) 2 5+ m − (3) 2 5+ F m − m (4) ) Λ 2 − (4) (4 su F F 3 F m m F 5+ − m ) ) m (3) 2 2 (2 su m ) 2 ) 2 (4 su F 6+ 7+ 7+ m m 5+ (3) (2 su m 3 m − (3) − − su (3) (3) − − (4) (1 su 9 2 (3) p 2 7 F − (2 (1 su su 2 (1 su ) (3 su 3 4 F − (4) p 1 − − F − su − 7 2 (3) F p (3) (3) F − 5 − (3) π su 2 (4 π ) 5 su su p − F (2) (2) su F (3) 2 (2) (2) π − su su su (3) = su su = = (2 (2) su = su F = = = F ) F ) = F – 9 – ) = ) m p = m F F m ) ) F − = − F F − ) ) m m − F m ) + 2 m m − − p 2 F + (3 − − − + (4 Λ 2 − + (7 + 2 Λ 2 2 Λ + (4 + (4 m + 3 Λ + (8 Λ + 2 2 + (2 + (4 + 3 − 2 3 2 Λ Λ 2 2 − 3 Λ Λ + 2 Λ + 3 Λ Λ 3 + 3 + 3 + m (5) 2 + (3 + + 2 + 3 + 3 2 2 3+ m m m 2 su (5) p 2 2 m 2 Λ m m − 2 3+ 2 1+ 2+ su (5) m 3+ (6) (4) + 3 (6) su (4) (4) (5) su p (5) su su su su − 2 su su m (5) su JHEP07(2020)012 (2) in (1.57) (1.53) (1.55) (1.56) (1.51) (1.52) (1.54) sp F ) π F π π p ) π 2) F (2) (2) m − π − 3 (2) (2) n su n su ( − su (2) 2 su (2 (2) su − 0 su π 0 0 (10 (2) 0 (2) su (2) (2) (2) su π su (2) F F su su ) 2) su − m 5 5 n 5 ( − − F − F (2) ··· 2) − ··· 0 7 ) 2 ··· n n p sp n 2 − 0 − F (2) Λ ) − . F n 0 0 n ) p ( 0 (2) n sp p (1 2 (2 = (2) sp − (2) sp (2) − − su su m – 10 – su = F = − = + (7 = 14 + 7 2 F π F F ) p π Λ 2 π ) + (8 p ) is relevant only when it carries no fundamental hypers F Λ p 2 (2) n (2) + 7 − (2) + 2 ( Λ − 2 su + 2 su + 2 p su sp Λ − 2 m 3 2 F 3 + 2 2 ) F Λ − + (3 − ) − p p F + 2 ) 2 p ) − 2 0 n Λ 3 2 (4) − + 3 n m 0 ( − ) 2 ) 3 n su (2) n 2 + (8 (2) n n su − 2 + 3 ( (2) − ( 2 2 sp sp p − Λ su 2 sp su − (5) − (15 su + 2 (12 (5) (14 = p = = su − 2 m 4. ) n ) includes the following dualities: ) includes the following dualities: ≥ ( (2), the theta angle for n su 3.74 3.69 su ( ( where Like that are not charged underabove some duality is other relevant gauge only algebra. when That is, the theta angle of JHEP07(2020)012 (1.65) (1.63) (1.64) (1.58) (1.59) (1.60) (1.61) (1.62) F ) π F n 2 2 (2) (2) Λ π − 0 su sp F (2) (2) F (2) 2 (11 (2) su su su 2 (2) Λ su 0 sp π π F (2) ) 2) F F F ) (2) ) ) π 0 0 − F m su n ) 0 m 2 m m su ( 2 2 − (3) (2) (2) Λ m 2 Λ Λ π 2) − − − (2) 4 Λ sp sp sp − − (2) (1 sp (2 (3 F − 0 ) π (4 n 2 su n ( Λ m = = (2) (2 2 = (2) sp = − sp su F F ) ) F (2 π F ) ) = m m – 11 – m (2) m = − − = F − su − ) 2 m F Λ + (4 + (4 ) + (1 − + (4 2 2 = m 2 2 Λ Λ + 4 Λ Λ − 2 F + (7 + 3 + 3 + 3 2 (4) + 3 2 2 m m Λ + 3 + (5 2 m su m 2 2+ 1+ S S 3+ + 2 (4) (4) (4) m (5) ) contain the following dualities: 2 su su su 3+ su ) (9) + 3 (9) + 2 n 3.99 ( so so su 4. ) and ( ≥ n 3.96 ( where JHEP07(2020)012 (1.76) (1.71) (1.72) (1.73) (1.66) (1.67) (1.68) (1.69) (1.70) (1.74) (1.75) F F F 9 2 ) 5 ) ) m m m F (3) (3) 4 − − − F F F F m su su F m m m ) 2 ) ) ) (3) 2 2 2 2 (4 (3 (2 5+ 6+ 7+ 9+ m m m m su π − − − − (3) (3) (3) (3) 0 2 0 (2) (5 su Λ (4 (3 (1 su su su π π π (2) (2) 0 2 sp 2 2 (2) Λ (2) (2) Λ Λ su su (2) sp sp sp π su F F (2) (2) (2) F F (2) F 2 3 ) ) π π F (2) su su su su 0 3 m m su (2) (2) π F − (2) (2) − (2) F 2 su su = = = (2) su su (2 su = (3 = su – 12 – S S S ) ) ) = = S = = S ) = m m m = m S − − − + 2 S S S ) ) − S F ) + S + 2 m m F m + (4 + (2 + (5 F ) − − ) + (6 + 3 F F F − m m F F (7) + 4 − − + (5 + (4 so + (6 F F ) contain the following dualities: (7) + 3 (7) + 4 (7) + 2 F (7) + (9) + 3 so so so so so (9) + (6 3.109 (9) + (5 (7) + 2 (7) + 3 (7) + so so so so so ) and ( 3.106 ( JHEP07(2020)012 (1.94) (1.95) (1.83) (1.84) (1.85) (1.87) (1.88) (1.89) (1.90) (1.91) (1.92) (1.93) (1.77) (1.78) (1.80) (1.81) (1.82) (1.86) (1.79) should not 2 5 F Λ ) (3) m + 4 F ) su − 1 F F F ) ) − m F ) ) F m m F − ) m (4) ) + (8 m − − F F 0 m ]. 2 − F m su ) ) − ) Λ + 6 F − should cause the theory to 15 (2) − m m (2) 2 m 7 n + S + (4 + (4 F sp su − − + (2 − A 2 2 + (4 m + 2 F + (3 Λ Λ 2 + (4 2 + (2 F Λ + 2 F 2 Λ 2 2 +1+ +1 Λ S + (6 + (6 2 Λ 2: m n n + 2 + + (7 Λ + 3 + 3 0 π + 3 2 2 − − − 3 + 3 S S m m + 3 2 2 ≥ m m + 2 Λ + 3 m − − + 3 2 (2) 2 2 (2) 13 − − m n 4 4 6 2 1 2 m 3+ 2 − − − − m + 1) + 1) + 1) − su su (7) + 3 − − − n n n 1 (5) (5) so (7) + 5 (7) + 5 (4) ( ( ( (3) (3) (4) (3) + (4) (4) (4) su su so so su sp su su su su su su = su F (4) su su = ) π 2 – 13 – ======su m = = Λ 2 2 F F S S F F F F F (2) S A F S S = ) ) ) ) ) ) ) ) − Λ Λ SCFT, adding extra 3 ) ) su + 4 + S m m m m m m m m ) (2 m m d 1 + 8 π + 3 + 2 − − − − − − − − − m 2 − − 0 π (7) + 6 (7) + 7 Λ +1) − = (4) n so so ( (3) + 6 (2) ) + (3 + (1 + (4 + (5 + (8 + (8 + (6 su + (5 + (4 n sp n sp 2 3 2 S S S 2 F ( m F F (3) + Λ Λ Λ − + sp 3 sp + + (2 F (7) + (7 − + 2 ) π π (7) + π so (9) + 3 (9) + 4 (3) + m (4) (7) + 2 (7) + 3 so +1) +1) so so sp n (2) n − su so so ( ( SCFT despite it satisfying the conditions of [ ) ) already lifts to a 6 sp n n d ( ( A sp sp 1. 4. 3. 2. or 6 ) contains the following duality: ) contains the following dualities where ) contains the following two dualities: ) contains the following dualities: d ≤ ≤ ≤ ≤ (11) + (7 (7) + so m m m m so 3.146 3.125 3.140 3.113 ( ( ( ( have no UV completion.lift Thus, to according a to 5 the above duality, where Since where where where JHEP07(2020)012 ]. 15 F (1.96) (1.98) (1.99) (1.97) ) F (1.102) (1.100) (1.101) ) n n − − F F (2 ) ) (1 n n F 2, the left hand − , − 3 2 even though it , (3 n (2 n − F F = 1 2 2 p 0 ) ) − = 1 p 2 2 p p 2 F 1+ m ) Λ Λ Λ 3 2 (4) − − m 2 2 2 2 n (4) m 2 n 2 (4) su − Λ F (4) (1 (1 p − su ) 2 − ] for 2 1 su 2 p su Λ 1+ Λ ], the left hand side should not 15 2 − 2 (4) F ) F 0 15 (4) + (4 ) su F (1 0 m 4 ) 0 su F m SCFT for (2) Λ ) 0 m (2) − d − (2) su (4) to manifest the possible values for m su − (3 F (2) su su (1 ) − 0 0 (1 (4) + su or 6 m (1 (2) = sp d (2) = − = su = su (2 C = F C – 14 – ) ) C + ) p m p S = C = − SCFT even though it satisfies the conditions of [ − − + + d C F S ) C + (2 + ) + (5 m + (2 p or 6 S S + 4 ) ) S m d − − F ) n 2 n ]. ) n 11+ − − m 15 − − + (2 − S (5) ) contain the following dualities: ) + (2 + (3 (8) + (5 + (3 n su F F so ) ) F − ) m m 3.166 m (8) + (3 − − − so + (4 F ) and ( , which can at most be 1. (8) + (4 (8) + (4 p (8) we are only going to display results upto the action of triality. (8) + (3 3.162 (8) + 2 so so ( so so so and where we have not combined then fundamentals for For admit a UV completion into 5 side should not admitsatisfies a the UV conditions completion of into [ 5 Since the right hand side does not satisfy the conditions of [ Since the right hand side does not satisfy the conditions of [ JHEP07(2020)012 (1.112) (1.107) (1.108) (1.109) (1.110) (1.111) (1.103) (1.104) (1.105) (1.106) F ) F F ) π n ) π n π F n π π π − (2) ) π − (2) n − (2) 0 F su (2) (2 (2) (2) (2) su 2 (1 − su (1 su Λ (4) su su su 2 (2 su 2 1 2 n F 5 2 2 2 ) − 2 p F F 3 p 2 3 2 2 p Λ (3) ) 3 (4) 2 F (3) 2 Λ p 1+ F F π − su (3) 2 su (4) (3) (3) 3 2 (3) su − (3) su (2 su su (2) su su (4 su su F F ) π π ) π F F F m F ) π ) ) 0 m π π ) (2) π (2) F (2) m − π m m ) m − π su (2) su (2) (2) (2) (2) su − m (1 − − (2) − su (2 su (2) su su su (3 − su (1 (2 (2 su = – 15 – (2 π ======(2) = = C C su S C S C C + ) ) + 2 p C S + 2 S n ) S ) + 3 + 2 p ) − = F n − ) S S n + 2 − − m F − ) S + 2 + 2 + (3 − + (4 m F F S + (4 F ) ) ) + (3 + 3 ) ) contain the following dualities: − + (3 S n F m m F m F ) − ) − − + 2 m − m (8) + (5 3.173 F − − so + (3 (10) + 4 F (10) + (6 so so (8) + (3 (8) + (4 ) and ( (8) + (4 (8) + 2 so so so (8) + (4 so (8) + (3 (8) + 2 so 3.170 so ( so JHEP07(2020)012 F ) n (1.119) (1.116) (1.117) (1.118) (1.113) (1.114) (1.115) − (1 π F F (2) π π π ) F F ) ) su m (2) (2) (2) π m m − su su su (2) − − (2) (3 su 3 (1 (1 su (3) π π π π 2 F m su ) m 2 (2) (2) (2) (2) 5 2 3 m m 3+ m 2 2 1+ 2 su su 3+ 3+ su su 2 2 Λ F (3) − (3) Λ Λ 2 F (4) (5) 2 2 0 su (5) (5) su (2 su su F F su su (2) ) ) π π su m m (2) (2) 0 0 − − π 0 0 su su (2) (2) (2 (1 F (2) (2) ) F π (2) (2) su su – 16 – m su su su su = (2) = − su (2 π π = = = C = C (2) (2) = + 2 S + 2 su su S S S S ) + 4 S + 4 n + 3 + 2 F F = = ) F − ) F ) + 2 ) m m F m m ) S − − + (3 − − m F + 3 ) − F m − (10) + (2 (8) + (4 (10) + (2 ) contains the following duality: ) contains the following two dualities: ) contains the following dualities: (10) + (4 so so so so (10) + 4 (10) + (6 so so 3.189 3.184 3.181 (8) + (3 ( ( ( so JHEP07(2020)012 (1.129) (1.125) (1.126) (1.127) (1.128) (1.120) (1.121) (1.122) (1.123) (1.124) 0 F F 0 F ) 0 ) n 0 (2) 0 F n 0 0 (2) 0 ) − (2) su n − (2) π (2) su (2) (2) 0 (2) su (1 − su (1 su su su (2) su (2) (2) (2 su su π su F F (2) π (2) π ) π 2 (2) F F p sp ) sp F F π (2) (2) (2) sp (2) 3 2 π (2) − m sp sp sp (3) sp sp (2) F (2) − (4 sp F sp π F sp ) 0 (2 ) 0 F F F m F ) 0 (2) ) ) m 0 0 ) (2) 0 (2) m 0 − m m su m − π (2) su (2) (2) π π (2) su − (1 − − (2) − su (2 su su (2) su (3 (2) (2) su (1 (2 F (2 su ) 0 = su su – 17 – = m = (2) = = = − = su = C C (2 S 0 0 S C C + ) + 2 (2) (2) C S + 2 S n ) = S ) + 3 + 2 p su su ) F n − ) S S n + 4 − − m F S − ) + 2 + 2 − + (4 = = m F F + (4 + 2 F ) ) + (3 ) ) contain the following dualities: − + (3 S F m m F ) m S F ) ) − − m + 2 m − m (8) + (5 3.214 + 3 − F − F − so (10) + (2 so (8) + (3 (8) + (4 ) contains the following duality: ) and ( (8) + (4 (8) + 2 so so so (8) + (4 so (10) + 4 (10) + (6 (8) + (3 so so so 3.230 3.213 so ( ( JHEP07(2020)012 and (2.1) n SCFT SCFT (1.130) (1.131) (1.132) (1.133) (1.134) (1.135) F d d with the f or 6 or 6 d d and e . If the fibration 1 . P 1 P F ) m F F ) SCFT despite it satisfying ) − 2 m d m Λ − − + (6 or 6 + 3 2 m d 7 2 + (4 − F + (2 − 4 2 − F A Λ + 4 (5) ], it follows from the above duality that + 4 (3) n su f + 2 15 + 2 2 fibration over a base − su g π A cannot UV complete into a 5 = = 1 ]. gauge theory = = 0= 1 (2.2) (2.3) P = = ]. 2 F F + (2) 15 is associated to the fiber d 2 2 Λ f ]). 2 F F – 18 – e 15 + f ) ) · sp g f 1 cannot be UV completed into a 5 + 3 2 28 e , m m + 3 4 – = = Λ f π 2 − − F 25 and = 0 ) Λ + 3 1 (2) m 3 m P − + 3 + (6 + (3 sp − 2 π 9 m for g (5) 2 F (2) 13+ su ) + (6 − sp ] (see also [ 2 m g 24 − (3) – su ]. 16 cannot have a UV completion into 5 , 15 + (3 gauge theory in a particular phase. Many of the aspects discussed here F 11 m d 2 . Of particular interest to us are the holomorphic curve classes in + (valued in non-negative integers), the corresponding Hirzebruch surface is does not satisfy the conditions of [ n 13+ 2 F n F − Λ ) contains the following dualities: ) contains the following dualities: ) contains the following dualities: (3) is associated to the base + 4 + 3 4 e 3 su f 3.382 3.379 3.372 − ( ( ( Note that intersection numbers of complex curves inside complex surfaces are symmetric. 9 (5) where denoted by their intersection numbers. Thesefollowing intersection classes numbers are generated by two classes 2.1 Some geometricThis background local piece isother. described in The complex terms surfaces ofsurfaces relevant a to which collection the of are construction complex of definedhas surfaces gauge by degree intersecting theories specifying each are Hirzebruch a 2 The geometry associatedIn to this a section, 5 wethreefold describe to how a one 5 canare associate taken from a [ local resolved piece of a Calabi-Yau Thus, even though they satisfy the conditions of [ The above duality implieseven that though it satisfies the conditions of [ the conditions of [ Since su JHEP07(2020)012 . of C are , we i 11 (2.4) (2.5) (2.8) . An n x (2.12) (2.16) f F is the curve and S e to be in f ). C · blowups of 2.8 K b . The difference between n F . Thus, the intersection b n and F e · in h K of the form ( blowups on i and b x C 2 (2.9) i f γ , − e 2 (2.10) g i ij δ X βf − 2 (2.11) nf n − = 2 + − n + + = = 0= 1 (2.6) (2.7) = = 0= 0= 0 (2.13) (2.14) (2.15) C e = = 2 e f i i i βf · αe j · h · e x x x x f of a general curve ) – 19 – + · := · · · = h · · h C g , then the total transform of a curve i e h f h K C S x K αe + . Let us denote the exceptional curves created by the = K b n ( F C . We will use the convention that the total transforms , b are those mentioned above, and their intersections with is h n ··· F , and = 1 are denoted by the same names f i , h have genus zero determines the values of e ] to see how the full Mori cone of the surface can be determined from the data of gluing is a blowup of a surface 22 with and S e, f i has genus zero. The genus f x → , h e . ˜ S ˜ the resulting surface as S : 10 B ) in We also consider blowups of Hirzebruch surfaces. If we perform More general curves can be obtained by positive linear combinations of C It is possible to obtain many different surfaces by performing If ( 1 11 10 − these surfaces is being trackeda implicitly blowup in is, this since paper. everySee section Effectively, blowup the 2.5 is gluing of required curves [ tocurves track be by how applying as non-generic the generic above as genericity is criterion. allowed by thef existence of gluing curves. The possible holomorphic curves after the blowup can be decomposed as denote blowups by the curves numbers between The fact that in terms of which one can determine the genus of any can be obtained using the adjunction formula where K is the canonical class of the surface and does not depend on the identity of The curve important curve class in whose intersection numbers are JHEP07(2020)012 i 2 b n S S F (2.23) (2.24) (2.25) (2.26) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) and and its 1 n using the S I 2 by gluing a C 2 S is a non-generic ) of i 2 x and C ( 1 g S participating in the gluing i . i S 0 as a surface isomorphic to x ), we see that we can write the i i x − 6= 2.9 0 (2.27) e n > j and the genus X ≥ 2 (2.28) = 0 if there is no intersection between 1 g for − ij C b ) + with respectively. In principle, one can glue g C f i n − 2 ) = h f e x i b − 2 ) of S = 2 1 x F + 2 C ↔ ↔ ↔ ↔ to produce 2 ( 2 C = 8 h − i ( g e i j – 20 – h f e C 2 g x under special circumstances. The isomorphism is and e x f x + 1 can only be performed when K + 1 e ) = +1 ( 2 → → → → 1 S ≥ 1 b n . We define i i i . Such a gluing is allowed only if certain conditions − j C C curve F n 2 x x x ij x ( S = g n C − − → for − e in K f b n n F 2 I C is singled out. We will denote this isomorphism by . Using adjunction formula ( i Z x ). The gluing is sensible only if ∈ i as 2.9 γ b n are calculated in be denoted by F 2 j 2 to a curve S C . Notice that 1 0 and 1 S − n and ≥ I and i in . For such a collection of gluings to be consistent, we need to make sure that it j S 2 1 1 S C α, β C In a similar fashion, one can construct an intersecting configuration of surfaces One can manufacture a transverse intersection of two surfaces For ease of notation, we define a surface There is an isomorphism and i be relevant for geometries considered in this paper. by gluing curves inbetween different surfaces. LetS the curve inside to be Calabi-Yau, the following condition should be satisfied where multiple gluing curves between two surfaces, but only a single gluing (and no gluing) will are satisfied. We canadjunction compute formula the ( genus Moreover, for a local neighborhood of the resulting intersecting configuration of curve with the isomorphism given by where a special blowup inverse by blowup hitting the locus of the given by canonical class of from which we can compute where JHEP07(2020)012 . ) 1 i S S 2.30 (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35) (2.36) 4 S 42 is denoted near C and a surface j S 43 C C kj C · ki C kj 34 C C = · for the gluing to ji . The conditions for a collection of ki 3 i k C S C S · j C = 2 (2.37) C S 6= · = i · jk . · 32 in 2 f ji i i j ji i ji C C · S C f ij K C i C 6= = · C − . = K = = i j ik i jk i − = ij – 21 – S of that geometry. It is defined as S C S C C ij · = · · · and = 2 i i ii M ij C and C i K C K i ik M S 23 S to every geometry composed out of Hirzebruch surfaces, C C = = = , then the intersection is computed via · ij j j 3 k i 24 ij 2 S S S S C M · S C · 2 j i S S 21 · C i S . An edge between two surfaces denotes that there is an intersection 4 intersection matrix S ). That is, we must have and ). 1 2.31 , then the intersection is computed via . The equality of two different ways to compute the left hand side of ( i 12 S k to be consistent arise from the equality of different ways of computing the left S C 2.28 is the canonical class of 6= ij is the fiber of the Hirzebruch surface i 1 C i j S K , and f 3 6= lies in some other surface lies in S A diagonal entry of this matrix is Another intersection number inside a threefold is that of a curve Finally, we denote an intersecting configuration of surfaces in a graphical form. For i at the end of the edge between C C i where 2.2 Intersection matrixLet associated us to now a associate a geometry which matrix we dub as the If and between the two surfaces.S The gluing curve If denotes a configuration of four surfaces such that there are no intersections between for do not leadcondition to ( any new consistency conditions sinceexample, it the is graph equivalent to the Calabi-Yau where gluings hand side of ( be computed as leads to consistent triple intersections of the surfaces. The triple intersection numbers can JHEP07(2020)012 i ) k if ij S φ i α j φ φ must of 2.36 i (2.40) (2.41) (2.42) (2.38) (2.39) , then i as φ ji g f give rise α ij (1) gauge i u C S be the scalar i φ = 0 then gauge theory with d ij α . The pairing of ( g to give rise to a i S a . Then the off-diagonal entry of x be the coefficient of term i S ij,a . M5 branes wrapping ijk γ g c k a S X · as we now describe. Let j ij must be non-zero as well. Moreover, j + i S α of this abelian gauge theory is encoded in S i ji · S . It is possible that the phase under discussion does · − f i α 2 i i F φ ij 3 i S S = β S . The prepotential is a cubic polynomial in – 22 – i ij + S = = 3 = 6 i M e iii iij ijk ij c c c 6= 0 then α are associated to co-roots of i ij = S α ij correspond to roots of C i f curves of the Hirzebruch surface e, f on its Coulomb branch. M2 branes compactified on the fibers g are three distinct indices. the terms involving mass parameters. Let are the i . Thus, every geometry leads to an abelian gauge theory description in the far can be identified with the Cartan matrix of a semi-simple Lie algebra 12 d , f (1) gauge multiplet arising from i ij i, j, k e u . Then M We can see that for a geometry to give rise to a gauge theory, if If a geometry is such that the off-diagonal entries of its intersection matrix are such F We note that “ignoring” is not the same as setting the mass parameters to zero. It simply means that 12 where we truncate the prepotential tonot terms exist involving at only zero mass parameters. in 6 infrared irrespective of whethernon-abelian or gauge theory. not it Thethe prepotential can triple intersection be numbers described ofin the as the surfaces the Coulombwe branch ignore of a 2.3 Prepotential associated toThe a 3-form geometry gauge field offield M-theory in reduces 5 on each surface matter content of the gaugewill theory be is discussed encoded later. in the blowups on Hirzebruchbe surfaces zero. and On themust be other bounded hand, above if by three. M-theory compactified on the geometrygauge produces algebra a supersymmetriclead 5 to W-bosons andto thus monopole strings anddescends hence to the pairing of roots and co-roots and hence is indeed the Cartan matrix. The which is non-positive just like an off-diagonal entry ofthat a Cartan matrix. where the intersection matrix is just like the diagonal entry of a Cartan matrix. We can write the gluing curve JHEP07(2020)012 13   3 | using which (2.44) (2.45) (2.46) (2.43) f denotes F m 14 ) f ijk a h k d 2 0 1 − n F F with Coulomb is the same as . The notation − ). Each surface − f k 2 w ) mf mf F n + + 2.26 ( + (1 e h h φ su )–( · ) ) subfactor, f 2.23 n ( R e e ( ··· su w . Thus, | is a mass term for each full k k k ) for a half hyper − − f − -th R n n 1 2 3) subfactors of the gauge algebra R a − − ( ) parametrize weights of an irrep X ). One can see that interchanging 4 4 ∈ denotes the corresponding CS level. to ≥ e w f = n F F ( . f a f f k n R k k n h ( ( h h h su m − w 6 su − f − , X 2 n and g − F 2 can be obtained by exchanging the left and ≤ f 3 | k − ··· ··· . h 0: 0: – 23 – φ n n · ≥ ≥ r ≥ | k m m 2: r , , e e X denote various − e m m   k k a with 2 2 k 2 1 − − 4 4 k − ) − − 4 − − + n n n − ( n n n F F = 1 for a full hyper k,a F su φ − − f h h is said to have degree h h j,a n < k < n b n φ F = 1 = 2 − i,a k k φ , and f e e a ijk 3, 3, 3, 2 e d a k k ≥ ≥ ≥ k k − − 2 2 − are the roots of the gauge algebra n n n formed by the charged hypermultiplets, upto outer automorphism which is a symmetry of the pure gauge theory. 2 a − − , , , X n n denote the Coulomb branch parameters for the − r g k k k k n F F denotes the scalar product of the Dynkin coefficients of the weight − ) ) ) = F ) If the geometry admits a gauge theory interpretation, we can also compute 6 i,a of n n n Half-hypermultiplets do not admitHirzebruch mass surface paramters unless completed into a full hypermultiplet. φ n φ ( ( ( F · ( f 13 14 , 6 The geometries for pure right ends of the above two geometries for su the left and rightsu sides of the above diagram sends su We remind the reader that weare are defined also in using terms Hirzebruch of surfaces Hirzebruch surfacescorresponds of of negative to positive degree degree a via ( node of the Dynkin diagram of with the relationship ofgeometries intersection corresponding to matrix pure gauge to theories.without the They any Cartan are blowups. composed matrix, of We Hirzebruch we collect surfaces can them below. determine the su g the corresponding rank three invariant tensor, and 2.4 Pure gauge theories Using the above relationship between prepotential and triple intersection numbers, along where R hypermultiplet w branch parameters. The indices a one-loop calculation JHEP07(2020)012 0 (2.50) (2.51) (2.49) (2.53) (2.52) (2.47) (2.48) F f = 1 of the + 2 6 1 e n F F F 2 h e e 2 4 F e + 1) corresponds to n e h e e 2 (2 6 2 0 1 6 so F F F F F e e h h e e h 2 F +1) corresponds to the rightmost e n e (2 e e e so 8 2 1 8 F ) and the geometry cannot be determined F F F e n h ( h h h 0 1 0 2 sp F F F F e e ): – 24 – e π 2 ··· ··· ··· ··· which implies that there is a single Hirzebruch surface ): mod π 3 e ( 2 φ 2 π F e e = 8 e e mod 8 7 h F ( n n − − + 1) 2 2 n n F 2 2 F n nπ F F h h h h = ( = 3: e θ θ ≥ 4 ). F 4: 2, 2, n n ( (2) one finds 6 ≥ e ≥ ≥ e e e sp su = 8 implying that the Hirzebruch surface has no blowups. But, it does not fix n n n 6 5 +2 2 , , : − − +2 + 1), : n ), θ θ n n n + 1) in such a way that the fundamental representation of 2 π 0 ]. K 2 2 2 ) ) n n F In all the other cases that follow, the graph will always be oriented such that the left- n F F F n n 24 (2 (2 ( ( (2) (2) : (2 6 with the degree of Hirzebruch surface.results The for degree can be fixed byso taking the limit su For pure The relationship between the thetabased angle on of whatever weof have [ discussed so far. The relationshipsu is derived in appendix B.3 so e sp node. most node would correspondalgebra. to the fundamental representation of thesp corresponding Lie The surfaces are inso one-to-one correspondence withthe the left-most nodes node and of the the spinor representation Dynkin of diagram of JHEP07(2020)012 ) ) 4 f f F R R ( ( w e (2.57) (2.58) (2.59) (2.60) (2.55) (2.56) (2.54) w C in the ge- 4 -th Dynkin ) i f F h R ( e w 2 C F e . Said in the reverse ) -th Dynkin coefficient f i h R 1 ( F w 2 C F e e e 2 0 F F e e f e e ) for some particular weight m 0 ) 1 F e f F 2.58 i h f 2 i S 2 0 h + 2 · e e S F F e e − 3 ) · f ) 2 e f R F ( R w + (1 ( ), then we can find a holomorphic curve – 25 – h w f e C φ · C R e − ( 6 ) e ). One can cycle through various signs by changing f F w 8 R 2 F h ( e 2.43 F w 4 ). Then we can find a holomorphic curve h F f R h ( e w 8 e F ) is negative for 4 e ) is given by F f 6 2.58 h F R ( h w e ) is given by f 6 R e ( A phase transition occurs whenever the sign of ( F w : : : 8 : 2 8 7 4 F coefficient of is flipped. Geometrically this correspondsdirection, to all the a phases flop of of the the gauge curve theory can be generated by starting from the geometry of If the quantity ( in the geometry that is composed out of the fibers and blowups such that the we are in a phase in which the quantity is positive for the weight ometry that is composed out of the fibers and blowups such that the Once we add matter to adescribed gauge by theory, the different Coulomb prepotentials. branchgenerated is by divided These into absolute phases different values phases correspondthe in values to ( of different mass possible parameters signs and Coulomb moduli with respect to each other. Suppose g 2.5 Adding matter f e e JHEP07(2020)012 i ) 0 is d at f (3) 2.58 w (2.63) (2.61) su −∞ = f m is being taken e is the left surface 1 . The lower weights and respectively. Indeed, f S , then we perform i x C x d − . This is because making in the full geometry. We 2 f f + C 1 f is given by h f 1 and = 0 (2.65) w = 1 (2.64) F x i , we can reach a phase in which ( 1 ) (2.62) x r i S f e i − · ) f n 1 i since the CS level is changed by half-unit e 1 f · is the curve corresponding to the highest n − K F x − = 0. That is, the desired geometry is simply x h f − + f w k 1 2 = = ( – 26 – C marginally integrated in . The curve corresponding to such a weight = ( 1 curve appearing in the intersection product is the 2 0). Indeed we can compute ( curve of some Hirzebruch surface lead to phases which are g (3) S S f e e is , . − · e · . According to the above claim, the other weights 2 . Finally, we glue together all the blowups pairwise w su i 1 f P x x 1 1 S S F = 3 and − − n is obtained by simply adding a blowup onto the left-most F + indicates that the intersection product of ) for 1 2 2 is completely integrated out. The marginally integrated in phase S ) 2.44 f (3) e 15 · su -th Dynkin coefficient of the highest weight is x i and in particular the 1) are associated to the curves 1 − S , are the roots of gauge algebra i is the right surface) r 2 1) and (0 S As an example, consider marginally integrating in a fundamental to the pure We will focus our attention only on full hypermultiplets from this point on. By turning more negative leads to a final phase transition to the phase connected to , Flop transitions of curves involving the 1 curve of the Hirzebruch surface f 15 − ( not described by theand same are gauge referred theory. to Sometimesgeometry as such non-gauge-theoretic which phases is phases. cannot a be One local described such neighborhood non-gauge-theoretic by phase of any gauge is theory described by the and where the notation ( in the surface e According to theweight above given claim, by the Dynkinand coefficients blowup (1 upon integrating in aof fundamental. the The gauge geometry theory fornode of the the marginally geometry integrated ( in phase theory. This gives rise to the theory claim that the curve associatedcan to be the highest written weight as where m which point the hyper has a very simpletheory. Now geometric if construction. the number We of start blowups on withwith the the each surface geometry other, for so that pure all gauge of them lead to a single curve of fibers and blowups. on a large negative massis for positive a only full for hypermultiplet this the phase highest by weight saying and that negative the for hyper all the other weights. We refer to corresponding to a particular phase and performing flop transitions of curves composed out JHEP07(2020)012 ) A − 12 + C 1 1 x, x and and 0 f 2.63 S 2 x (2.74) (2.71) (2.72) (2.73) (2.70) − Λ (4) 2 − . In the since the + su x e F x, f 21 to C − + 2 e 2 2 f (4) + 1) before the phase 1 1 (2.69) su − f − , 1), and negative for the , = 0= (2.68) and the gluing curve 1 1 1 x )) on the left surface 1 and the genus is zero, so 3 1 2 0 − S S F F F − )) )) x x 2.22 is e e is changed from to 1 − − 1 0 1 (2.66) )–( 21 S to yield the gauge theory F f f − 0) and ( C ( ( 0 1 1 1 2 1 , = 1 (2.67) in · · F F F 2.19 1 e (4) x S ) = x K from - e e x 1) in this isomorphism frame are e su )) − + ( 1 e x h − + ( 1 − 2 S , f 2 S 1 − ) 1 1 0 S (0 f ) . This geometry corresponds to the phase in f ( F F f , 2 ( f · · – 27 – · · 1) S e e 1 , e e K x 1 K - e e on e − changing ( = = ( = ( = ( (see equations ( 1 0 1 0 0 , x 1 2 1 2 I 1 F F F . The blowup has to hit the gluing curve S S S S 0) − 1) from negative to positive. This corresponds to flop of 0 1 1 , · · · · , I F . In the second step of the flop, we perform a blowup on 1 theory. The resulting gauge theory is ) ) ) ) 0 x x x x e e 1 − F to − − − − 1 2 1 1 2 of (4) 1 1 1 1 F F F x f f f f e su ) is positive for the weights (1 ( ( + + 2 2 2.58 f f ( ( since that is the curve associated to the weight (1 was affected by the blowdown. Thus x 12 C − ) for the weight ( 1). The curves corresponding to the three weights are respectively. And the curve to be flopped has become the blowup to the curve 1 − f x x , 2.58 + − 2 e As a third example consider adding adjoint to As another example, consider marginally integrating in two fundamentals and an an- Other phases can now be accessed by doing flops. Suppose we want to now change the thus changing it from 2 for which the associated geometry is tisymmetric to the pure the corresponding geometry is after performing the isomorphism which the quantity ( weight (0 respectively, as the reader can easily check. the geometry after flop is which can be rewritten as x, x, f first step of thefrom flop, we blowdown S gluing curve by performing the isomorphism The curves corresponding to (1 sign of ( the curve transition. Notice that theindeed self-intersection this of curve can be flopped. To perform the flop, we first rewrite the geometry ( we can check that JHEP07(2020)012 (2.80) (2.75) (2.76) (2.77) (2.78) (2.79) near the right subfactor, thus x theory. That is, 1 1 − , so we should flip − ¯ F (3) ⊗ (3) su F su ⊕ and marginally integrate in 0 π (3) into the (2) su ¯ F sp 0 1 0 1 1 1 1 1 2 1 1 0 leads to F F F F F F F F 1 h f x x x S e e 2 - e e - + h f e 2 2 + e into pure 2 ¯ F to rewrite the above geometry as 2 ⊗ – 28 – subfactor and 3 S x F e e with a bifundamental marginally integrated in. To 0 e e e e e e with a bifundamental. 5 2 1 1 1 0 1 2 1 (3) 5 2 5 1 6 1 6 5 − F F F F x x F F F F − su (2). Let us start from (3) (3) sp su su ⊕ are flop equivalent to each other, and hence the theta angle is ⊕ 3 2 living in the left surface into the F 3 2 x F (3) + (3) − π su on the right surface ) su f n 0 ( leads to I . Similarly, one can show that the marginally integrated in phases for sp 2 0 S (2) (3) and rewrite the above geometry as in and sp su x F near the left surface denotes the blowup in the left surface, and + x 0 ) Let us now discuss some subtleties about CS levels and theta angles when matter As the last example, let us see how the theta angle becomes irrelevant when a funda- n ( Now we see thatwe we have have integrated integrated in in 3 yielding the theory the lower describes the theory see this, we have to remember that a bifundamental is defined as physically irrelevant. charged under a mixedFor representation example, of the multiple geometry simple gauge algebras is integrated in. which is precisely the geometryinto obtained pure when one marginally integratessp in a fundamental Now we perform Flopping Flopping the curve surface denotes the blowup in the right surface. mental is integrated into pure a fundamental. The corresponding geometry is where JHEP07(2020)012 ) as with (2.83) (2.81) (2.82) ) with 0 n changes 2.79 ( (2) F . In other sp ¯ su F ⊕ ⊕ ⊗ ) 1 theory. In this F m 1 (3) 16 (2 su (3) su as described in what fol- ¯ F su ⊕ and , and then we would have 0 ). F F n , and hence can identify the ( (3) 0 ⊗ . It is conventional to assume 0 su ¯ su F (2) ⊕ (2) su ) 1 1 1 1 1 2 su (2) leaves the theory invariant. This F m F F F ( su into e e e e su ) leaves the theory invariant if the theta ⊕ into x x F into pure m ¯ F (3) F (2 su ⊗ su , then we write the geometry as and 3 – 29 – F ¯ and 3 F ) with a bifundamental. For 0 n 0 ⊗ e e e e ( x (3) x ¯ F (3) sp 1 1 1 0 1 1 0 1 0 1 su F F F F F F x x with a bifundamental. That is, simultaneously flipping su ⊕ π ) to rewrite it as into into + 1) (2) ¯ F (2) does not change theta angle, while integrating in 2.79 F su m su (2 ⊕ (2), it is conventional to distinguish between 1 su su − into (3) in ( for ¯ F (3) with a bifundamental. That is, the overall sign of the CS levels is ¯ F su su 5 2 ' F (3) ) is left unchanged. n su ( ⊕ sp 3 2 − If we treat the bifundamental as Now, let us consider the geometry We could have also thought of the bifundamental as Even though (3) 16 is true for any general a bifundamental, changing theangle CS of level of lows. Now we are integratinggauge in theory 2 as the sign of CS level and the theta angle for that integrating in the theta angle. Inzero other mass words, changes bringing theta in anglethe a but gauge fundamental taking theory from associated mass negative to toa infinity the positive above bifundamental. mass infinity geometry to from can zero be does identified as not. So, The way the geometrywords, is we written, are we integrating are 2 treating the bifundamental as physically irrelevant. This is true for any general which describes marginal integrationway, we of find thatsu we can also describe the theory associated to the geometry ( flipped the upper JHEP07(2020)012 . h (2.89) (2.87) (2.88) (2.84) (2.85) (2.86) curve and e interchanges a W- curve. Application described by S f b 0 F curves, and thus rotates f → . h b 0 F and e 2 Λ 1 0 1 0 + F F F f f + e +2 i e 2 theory. The dual brane web in Type IIB is f e x (2) + to the Cartan matrix of a gauge algebra 0 in the geometry, the intersection matrix can g sp . Thus, we obtain a duality between a gauge → → → i h (2) i e S – 30 – f x su changes the intersection matrix associated to the e e give rise to a W-boson and an M2 brane wrapping S leads to the geometry b 0 1 6 1 6 F F F 2 on some S . in S S f on this geometry interchanges the . This surface has an automorphism b 0 and a gauge theory with gauge algebra S F g gives rise to an instanton. Thus the automorphism b 0 F on the right surface which manufactures the pure S 0 F curve in to describing the matter content for For example, consider the geometry for Notice that the application of Indeed, we can see this relationship by considering the geometry described by a single An M2 brane wrapping e g Applying go from Cartan matrixUnder for this a process, gauge the algebra for blowups on thetheory geometry with go algebra from describing the matter content two vertical compact branes areof the two automorphism different representatives ofthe the brane diagram by ninetyS-duality degrees, on which the is brane indeed web. the same effect asgeometry. the application Thus, of by applying where two horizontal compact branes are two different representatives of the duality, suggesting that it should be related to S-dualitysurface of Type IIB superstring theory. which we claim implementsdenote S-duality this automorphism of by Type IIB in M-theorythe language, and henceboson we with an instanton, which means that it naturally implements a strong-weak coupling Consider the surface 2.6 S-duality JHEP07(2020)012 . . 2 F R Λ (3) is + 2 (2.93) (2.94) (2.95) (2.90) (2.91) (2.92) (2.96) 5 su − (3) su (2) to F ) to the left hand sp n 2 + Λ n m and (3) and integrating in F su + (2 + m ) of n F F − ): Integrating in the complex conjugate m n n n ( + 2 ( 1 2 5 1+ 0 1+ 0 su gauge theories − 5 F F 5+ d − − f f (3) which cannot be obtained from another + (3) e +2 (3) ¯ su F F 2 e su su = → → = 2 = 2 F – 31 – Λ π Λ 2 Λ (3). e e + ) (2) F su n sp m m of 1+ 6 1+ 6 ¯ F F F (2) + + (1 + irreducible duality sp (2) is dual to integrating in F ) and thus can be recognized as the gauge theory sp 5 m of − F (3) su (2) + (1 + sp ) is into pure 2.90 17 has led to the following duality between two 5 (2) is dual to integrating in The possibility of obtaining new dualities from old dualities by adding matter allows Another feature one can notice is that as soon as one finds a duality, one can add By integrating in an antifundamental, we mean that we integrate in a fundamental by decreasing its sp S has opposite effect on the CS level when compared to the effect on CS level caused by integrating in 17 ¯ associated mass parameter fromas positive integrating infinity in to a a fundamentalgrated. does, finite that value. Similar is statements This the are CS has trueR level opposite for decreases other effect by representations half on of when the an CS antifundamental level is inte- us to define theduality notion by of adding an matterduality to ( it. For example, the irreducible duality responsible for the That is integrating in of This is possible becauseoperations the performed extra to matter obtain that the we duality. add The simply mapping sits of there matter unaffected from by the which is isomorphic to that is we obtain the duality matter to both sidesside of of the the duality. above For duality. example, The let corresponding us geometry add is where, throughout thisduality paper, between the equality two sign gauge theories. between two gauge theories will represent mental The CS levels arecanceled unchanged by since the the change changeof caused caused by by integrating integrating in in an a antifundamental. fundamental So, is the application which can be recognized as marginally integrating in a fundamental and an antifunda- JHEP07(2020)012 α S , the (3.4) (3.1) (3.2) (3.3) d must be equal to α ) does not satisfy this = 0. After performing 2.74 n (3) gauge algebra in 5 su on the left surface to obtain 2 2 2 b n b n b n S F F F = 0. Moreover, a a a are the blowups on the right surface. x x x a a a β δ δ δ a x + + + a x βf βf αf a + + + (3). δ αe αe βe su + – 32 – i i i αf x x x transformations are applied on the geometry, the i i i γ γ γ S + + + e f f 1 1 1 b 0 b 0 b 0 transformations on the surfaces, without any mixing with F F F S . Suppose a local portion of the geometry looks as follows g . The gauge theory on the other side of the duality is allowed to carry any g transformations. n 0, we have lost the gauge theory interpretation. To restore it, we must perform I are the blowups on the left surface and i without any extra edges. For example, the geometry ( x α > tions on the allowed matter content. the that all the representations haveby highest one. weights with The Dynkin reducible indices dualities bounded built above on top of an irreducible duality have no restric- requirement. Even though thegeometry geometry has gives an rise extra edge to betweennot the appear leftmost in and the the rightmost Dynkin nodes diagram which of does only by a sequence of algebra general semi-simple gauge algebra. geometry should take theg form of the Dynkin diagram of the simple gauge algebra The matter appearing on both sides of an irreducible duality is assumed to be such We also assume that the geometries on two sides of the irreducible duality are related Moreover, at the point when We require that the gauge theory on one side of the duality must carry a simple gauge Let us consider a geometry satisfying the above requirements and giving rise to a 4. 3. 2. 1. on the right surface as well. This means that we must have Since we are looking forrestore a a duality between gauge gauge theory theories,one, interpretation, at since thus this no point forcing curve we of must the be form able to Since S on the right surface we obtain where Since the above geometrymust is be a strictly piece positive. of Now geometry suppose admitting that a we gauge perform theory description, simple gauge algebra The irreducible dualities canthe be geometric approach explored reviewed systematically in the indeducing last a a section. bottom-up special We demonstrate class fashion this of by by systematically irreducible using dualities defined by the following requirements: 3 A special class of irreducible dualities JHEP07(2020)012 i ) 1 2 ij b b S C 3.3 , we (3.9) (3.8) (3.5) (3.6) (3.7) rather P i x i γ + e transformation on ) to be lying outside curve, then we must S 3.1 j is of the form j are exchanged and e ij f breaks the edge between C transformation on both j S 2 2 2 S b 0 b b and F e 1+ 0 , and to a node i F i 1+1+ 0 f i a S F P contains no x x S - a y f j - γ x S - . + f g f lying in + of nodes in the Dynkin diagram of a simple , on the other side of the duality also we can i e y j - P x - = – 33 – x f - . This is the case when ij and f 1 j b C i j f 1 transformation on 1. We can in fact completely fix the form of ( b transformation on as well. After performing and 1 S 1. 1+ 0 1+1+ 0 b 0 j S i F F F to another surface S ≥ α > on each surface participates in the gluing. The rest of the µ ij x if C for i i 18 x i γ + denote the first two blowups on the left surface. The rest of the µe y and transformation on -th node in the Dynkin diagram of . The edges between these nodes are deleted while going to the other side of j x , a gluing curve g S i S the geometry necessarily describes a gauge theory with semi-simple gauge algebra. j before the duality. After the is left invariant. In terms of Dynkin diagrams, the transition can be represented as If there is a singlehave edge a between single edge between where a single blowup blowups do not participate in the gluing. which is just obtained bydisplayed exchanging above. the left and right surfacesThe in final the allowed configuration configuration is where blowups on the leftblowups on surface the do right not surface do participate not in participate the in the gluing. gluing. Similarly, all the -th and S Thus each irreducible duality in the special class being studied in this paper is char- The moral of the above analysis is that if after performing an i This is also the reason why we chose the gluing curve in the left surface of ( • • • • 18 than the more general acterized by first choosingLie a algebra connected set the duality. For anhave edge the connecting following options: a node a surface perform an and In other words, such athe combined according to the conditions that mustOnly be the satisfied following by three gluing curves possibilities discussed are in allowed: last section. has a non-negative genus JHEP07(2020)012 ) 3.7 must (3.18) (3.16) (3.10) (3.11) (3.13) (3.14) (3.15) (3.17) (3.12) ), ( ik C 3.6 . Otherwise, ), ( i 3.5 j j j j , but this is a contradiction g = 0 while keeping the genus implements the following tran- ik as displayed in ( i i i i j C a a a a a x · x x x x x a a a a a -th node becomes shorter after the − ij γ γ γ γ γ and j f C i contains the single node + + + + + , f f f f f f P would imply that there must be an edge , , on the other side we can have two edges y + + + ik j + 3 + 2 e e e − C e e , and after the duality the gluing curve · x i – 34 – = = 3 = 2 = = 2 and ij − is of the form i ij ij ij ij ij C f ij C C C C C j j j j C joined to is of the form P ij in -th node in the Dynkin diagram of C i i i i k k . That is, we can have the transition j after duality takes the form to j i i -th and C j non-negative. A non-zero ij C corresponding to before the duality; since Dynkin diagrams for finite Lie algebrasThe do possibilities not with contain two any edges loops. are One can see thatof it is notbetween possible the to ensure before the duality. However, this transition is possiblethere is only another node if take one of the threerespectively. possibilities This is the case when after the duality. The last case involving asition single under node the between duality transformation before the duality and becomes pointing from indicating that the rootduality. This corresponding is to the the case when If there is a single edge between • • • JHEP07(2020)012 can (3.26) (3.27) (3.28) (3.29) (3.30) (3.19) (3.20) (3.21) (3.22) (3.31) (3.23) (3.24) (3.25) P . 2 g and hence 2 g j j = j j j j g i i i i i i a a a a a a a x x x x x ) originating from a a a a x x a a a γ γ γ γ 19 γ γ γ + + + + + + + f f f f f f f + + 2 + 2 + + 2 + 3 + e e e e e e e = = – 35 – = 3 = 3 = 3 = 2 = 2 ij ij ij ij ij ij ij C C C j j C C C C j j j j . Thus, these possibilities characterized by three edges i i i i i i i before the duality actually completely classify all the irreducible j and i only contain the single node between dualities (of the special class being studied here corresponding to before the duality. In these cases, the gauge algebra before the duality must be corresponding to before the duality; and corresponding to before the duality; corresponding to before the duality; before the duality. The possibilities with three edges similarly are before the duality; and corresponding to before the duality; corresponding to corresponding to From this point on, whenever we say irreducible duality, we always mean an irreducible duality belonging • 19 to the special class of irreducible dualities being studied in this paper. JHEP07(2020)012 , r C R are . We g by . (3.38) (3.35) (3.36) (3.37) (3.32) (3.33) (3.34) i +1) as S 1 and r − ( 1 r by − su r R r for all R R i , then we place Λ h by 1 i − R r . We denote 1 F . We denote . We denote r r − -th Dynkin coefficient equal F F i r by by 1 by 2 1 r 1 R − r R + 1) as 1 1 R ) as 4 . r r ) as − − 2 F r as as r r as ( (2 (2 r − 2 r ··· 4 r by so e f g so sp . 2 3 1 3 g R r 3 − ··· ··· r ··· – 36 – respectively. 1 2 3 3 A 2 , 3 3. We also denote 3 ··· 2. We also denote 1. We also denote Λ − , 1 − − 2 . r 2 2 r Λ r ¯ F 3 1 , 2 ≤ ≤ F ≤ i by i i . We also denote i r by ≤ respectively. 1 1 ≤ ≤ R 2 4 A 1 . such that the number of edges in the Dynkin diagram for , R S , F h for 1 3 and for 1 for 1 for all i by i i i R by F 1 Λ , Λ r Λ Λ 2 2 and R by R R by g by by by , , 1 i i 1 1 i i and R R R R R R R . 2 appearing on one side of the duality. If both sides of the duality have a simple C C g be the representation such that its highest weight has i by and Let us label the nodes of Dynkin diagram of Let us label the nodes of Dynkin diagram of Let us label the nodes of Dynkin diagram of Let us label the nodes of Dynkin diagram of Let us label the nodes of Dynkin diagram of Let us label the nodes of Dynkin diagram of Before moving on to the discussion of irreducible dualities, we collect our namings for In the rest of this section, we obtain all irreducible dualities in the special class discussed 2 R S − r We denote We denote We denote R by We denote We denote also denote We denote various irreducible representations. Let us label the nodes of Dynkin diagram of Let to one and all other Dynkin coefficients equal to zero. We denote above and organize them in various subsectionsalgebra according to the identity of thegauge simple gauge algebra greater than or equalthe to duality the in number the of subsection edges corresponding in to the Dynkin diagram for JHEP07(2020)012 x x - - f f e e + + e e (3.42) (3.41) (3.39) 1 0 3 1 0 3 F F F F transfor- x x - - h h S e f e e ··· ··· 5 5 y y F F - in number), that is - + 2) (3.40) e f n p h h 2 0 2 0 ( p p F F su x x - - p e ⊕ f ··· ··· ··· p ··· 2) y y - - − e e e f n ( (a total of 2 0 2 0 ⊕ +1 +1 F F P p p 2 2 x (2) x - - F F e f su x ⊕ x - - e f 1 0 1 0 n n F F + 2) ··· ··· x x – 37 – - - ) can be written as m ( f f + + on the surfaces corresponding to these nodes. For su e e 3.41 S ,( e e −→ P 3 3 F F + 1) h h p m ··· + m ··· e e n + 5 5 F F m ( h h su m m + 1) dualities ··· ··· r 2) is uniquely determined to be ( e e ≥ su +1 +1 2, the above transition acts non-trivially on the gauge algebra n The geometry corresponding to the right hand side of this yet to be determined duality m m 2 2 ≥ F F mations on surfaces contained in where, since we are searchingamount for of irreducible dualities, blowups we required have to included construct only the the minimum geometry. After performing the (for where the dashed ellipsewe encircles will the nodes perform in then the transformation set Let us consider performing the transition 3.1 JHEP07(2020)012 f ), e e = 1 1 1 F F f m 3.41 (3.44) (3.45) (3.46) (3.48) (3.51) (3.43) h h e 2 m ··· +2 3 p F +2+ e p 5 h + 2) − 0 ) (3.47) p n +2) ( 2 p ≥ +1 for the representation ( F ) p ] su ··· p + m su h 29 n 2 + e − e n,m 0 3 and n +1 + 3 ) theory when a full hyper ) p n ≥ 2 2 p m − (2) n 3)( + ( F n n A 1 2 m π su − π su F Λ +2+ + 0, n (2) m ( h + (2) ( ≥ +1 su 1)! − 4 (3.49) p su ··· ··· +1 + m − e − anomaly coefficient → n m ) to the gauge theory associated to ( → +2) + Λ n m = 1 (3.50) 1 2 for 0 ( + 1) +2 m − ,m + +2 = ( ,n n − m p 3.43 1 20 2 m F (2) – 38 – +1 su n ,n F A 2 m 4) − su h + 2) A + 2) A n − ( = p )( m ( 2 1 n ( m su 0 + ··· su +2 − + (2 m (2) m n e ( Λ − su 5 + p 1, we obtain the irreducible duality m,n,p = is also known as the − k n +1 = ≥ m m,n +2 p m,n Λ + 1) m A A +2 2 is integrated into the theory from the direction corresponding to p + m,n,p F m k + m m,p h 0 and Λ k n is the addition to the CS level of an + 2) ≥ + +3) e m m ( ) in the literature, and an explicit expression for it is [ m p m,n ( n 3 su + ( 3, A 0 and − su 0. The CS level 1 2 n m su ≥ = 2, ( 4 → = +2 → − n . The quantity of m su We will display all irreducible dualities inside boxes for the convenience of the reader. n m p m 2 2 20 m F For for and transforming in −∞ Λ as where the two gauge algebras. as where the edges between two gauge algebras denote bifundamental matter charged under Equating the gauge theorywe associated obtain to the ( following irreducible duality which is flop equivalent to JHEP07(2020)012 and + 1, (3.52) (3.56) (3.58) (3.57) (3.54) (3.55) F , m i Λ } ··· 1 i , d − { i p, + i k = 1 n Λ i i + d m p + 2) d + p n ( +1 ) (3.53) =1 ) leads to the following + p i =1 su m i +3 + 2) for +1 p P π p 3.44 P +2 + m +2 + ( (2) n i,p ,m + F i,m + su 1 0 A n su +2 − π 1 − + ) algebra is associated to n m i,m of − (2) i +2 m + i (2) A A N + Λ m i su − ( m hypers transforming in fundamental n d su + A d + + i p su +2 . Thus, +2) on the right hand side of the above d +2 m + ¯ F → m +3 π d n +1 m m p =1 ) Λ ( = 0. For this case, we obtain the following i +1 + X d 2 p m =1 ··· ··· + i X (2) +1 m =1 m p Λ su i p X ,m 1 2 1 2 su ) is provided by the corresponding transition + = 1 2 +2+ (2) are relevant only when there are no extra 0 +1 F m F + – 39 – m ( m su +3 (2) 4) − 3.54 A = m ( on the left hand side of ( + 2 + su d − 1 2 + 2 + i m,n,p n + 2) k m p Λ + i F d = = = m + 2) has extra ) and ( = 2 and m ( 0 F ); so we need to be consistent with that convention when adding } } } + (2 + 2 i i i P n m − su d d d } 0 ( +2 i { { { (2) 3.51 p d 3.54 m p, { su m, su (3) d k = k ), ( . The CS levels are su m,n,p, 0 m,n,p, k m,p k ) and ( 3.44 i k } (2) i − d { su 3.51 + 1) +2 m, p m k extra hypers transforming under ), ( Λ i i + ). However, for the precise determination of CS levels, we need to also d d n 3.44 (2). The theta angles of an (2) as + 2) (2) connected to +1 3.39 + su su which end of the Dynkin diagram of an m =1 i m ( 0 and su m 21 ( P su → su To determine the most general reducible dualities associated to an irreducible duality, We can add matter on both sides of the above irreducible dualities to obtain other This convention is already implicitly chosen while converting the geometry to gauge theory while claim- m of that = 21 as Now the only remaining case is irreducible duality with specify which end of the Dynkin diagram is associated to ing dualities ( extra matter. we simply needmap to to specify the howirreducible roots the dualities of roots ( the ofdiagram algebra the ( on algebra the on right the side left of side the of duality. the duality This information for F fundamentals, in which case thedisplayed theta angle must be zero, and this is the reason we have where extra edges attached to gaugeonly algebras under denote those the gauge extra algebras. matterequation For content instance, has transforming and the dualities. For example, adding more general reducible dualities JHEP07(2020)012 (3) in (3.59) (3.60) (3.62) (3.61) su p 2 F e ¯ ¯ F F ¯ ¯ F F p ··· p p h ··· ··· p p ··· ··· 2 F e y - x F F - ¯ F f + e 2 0 n n F ··· ··· y - does not matter in this case since the x - – 40 – ¯ F F f ¯ F F + e and e F 2 F ) acts trivially on the gauge algebra. The corresponding 1 is generated by the geometry h m m ··· ··· 3.39 m m ≥ ··· ··· ), the map of matter content can for instance be specified by ), the map of matter content is specified by ), the map of matter content is specified by e F F ) has CS level zero, which is left invariant by outer automorphism m, p ¯ F F 3.54 3.51 3.44 4 F 3.54 h m ··· (3). e su m = 1, the transition ( 2 Actually, the specification of the duality ( of For the duality ( For the duality ( For the duality ( F n • • • For irreducible duality for JHEP07(2020)012 x x - - +1 f f m + + e e Λ e e (3.67) (3.65) (3.66) (3.63) (3.64) 1 0 3 1 0 3 F F F F +2 x x h h - - f f m,p + 1) duality. k r e e ( 5 5 ··· ··· +2) su F F p y y - - h h + f f 2 0 2 0 m ( F F x x su - - ··· ··· p f f ··· p ··· e e y - f y - +1 +1 1 0 f +2 p p p F 2 2 x - + F F f m f duality rather than 2 q ,m F 4 m f y +1 - 1) x m - − 2 f A ) or 3. The geometry for the right hand side is r 2 0 ( + n n F ··· ··· ≥ 1 + ( sp 1 m – 41 – y - n F x = − - f h p 2 m + q e = 2 and produces a self-duality of + 1) or r S e e m,p (2 k 6 4 so F F h h m ··· m ··· 2. First consider e e 8 6 ≥ F F n h h ··· ··· e e = 1, it is regarded as a +4 +2 Now consider the transition n m m x 1 2 2 - f F F where This geometry is flop equivalent to For So we restrict to uniquely fixed to be matter on both sidesdualities, only but leads we to willself-dualities, self-dualities. include that those In is irreducible this the self-dualitiesthe nodes paper, which irreducible of we self-duality. lead the will We to will Dynkin ignore reducible discuss diagram such non- such are self- examples non-trivially later. exchanged under where This duality only interchanges the perturbative andwithout non-perturbative particles modifying in the wither theory the gauge algebra or the matter content. Moreover, adding This geometry is left invariant by JHEP07(2020)012 e e 1 1 1 F F (3.72) (3.69) (3.70) (3.71) (3.74) (3.73) (3.68) h h e ··· +2 p 3 F 0 e 1 F + 2) π 5 h ≥ f p − +1 ( n +1) p ¯ 2 F m + m su ¯ F F ( n ··· y - + h x (2) - f e p n,m ··· su p 0 ··· + 2 0 e = 0, the geometry describing F +1 m n 3 p (2) ), we obtain the following irre- y 2 p A + − - F 1 2 n su x π m - 2 2 f π Λ 3.66 F + +1) + 3 e + h m (2) 2 ( +1 F − p ··· su +1 n + = 2 and m n e and flops is equivalent to the geometry e ) and ( + 1) → Λ + n 0 1 4 S m ,m − n n F +2 ( ··· ··· 3.68 + 2 n (2) p 2 +1 h – 42 – sp F F su m 5) A h + 2) 0 − p = + e ( ≥ n 1 2 6 su mπ 2 p = 0 if we want to find a geometry satisfying all of the ··· F Λ − p 3, + (2 (2) h e m + 2 ≥ su − 5 n m m,n,p − 2 m p ··· k ) after the action of n m ··· 1, ··· = π +2 ≥ + 3) e m 3.65 + 1) F 2 +1) p m m F ( m m,n,p +2 ( + k m h su 2 n F + 1) + e = 2, we must have m m ( ( 3 n 0. The addition of extra matter content to both sides of the above irreducible − sp su n 5 → +2 For − = p n m 2 2 F which leads, in a similar way as above, to the irreducible duality for conditions, listed at thedualities beginning we of are this exploring section,right in defining hand this the side of special paper. the class transition For of is irreducible and as duality (to produce new reducible dualities) is governed by with Equating the gauge theoriesducible arising duality from for ( On the other hand, ( JHEP07(2020)012 y - 3: π π x - f ≥ +1) +1) + (3.80) (3.77) (3.78) (3.79) (3.75) (3.76) p e p ( e n 2 ( 2 0 4 F F y + 1) h - + 1) x - p p ( f ( sp e f sp 4 1 0 6 F F ≥ x h π +1 - ) n p f p pπ + + n m ( (2) + ··· ··· p su y ··· 1 and p (2) - n,m e ··· f + su ≥ 2 0 m F +2 F p 0 A 2 x - F f n m, p π + (2) + y m - su +1) +1 f Λ p m ( 1 0 + 4 F n + 2 + − f ··· + 1) n +1 ,m n n y m m - ··· ··· ( +1 x Λ 0 - – 43 – f m sp A (2) 2 0 + 2 F F su = y - 1 + 6) x - f m − ··· +3 − + m ··· e m n 2 m mπ Λ ¯ e F − (2) + 2 + (2 p 4 m su F ··· +1 m = ··· h m m,n,p 1, we obtain the irreducible duality Λ k e π ≥ m,n,p + 2 6 k + 1) ,p +1) F 3 p m m, p ( m, h k + n + 1) + 4) ··· + p m ( m e ( + = 3 and sp su We can also consider the transition m n +2 ( m 2 = su F For with CS level which gives rise to the following irreducible duality for for which there is a consistent geometry satisfying the required conditions only for with the addition of extra matter specified by JHEP07(2020)012 1 x - f ≥ +5) e + p p e (3.82) (3.81) (3.84) (3.83) 1 0 3 (2 F F so x - h f 3 and ≥ F e + 5) ) n ··· p p 5 0, y (2 F (2 - f ≥ so h ¯ F 2 0 m F x - p f ··· ··· p ··· n y p - e + ··· p (2) f ··· m 2 0 su Λ +1 F ). For both of the above dualities, p p + 2 2 x - F f ) +1 3.79 i 0 m x x - Λ P f (2) - + h 1 0 su 2( n n F F ··· ··· x 4) – 44 – - 3 n n f ··· ··· − e + − e ··· p n 6 F e + 2 0 n 3 (2) F su h can be computed using ( 1. The geometry for the right hand side of the transition is m ··· m ··· ,p + 3) + (2 ≥ e 3 m p ··· p m m, 5 ··· +2 F k F m + 2 h n 2 and + 2) + 2 ≥ + 1) dualities m fundamentals not charged under any other gauge algebra. As always we should ( ··· n r m p su (2 (2 0, e so ≥ so = +1 m m 2 F where the extra edge attached tocontains the 2 last node on the right hand side denotes that which can be seen to give rise to the following irreducible duality for for 3.2 Let us start with the transition extra matter can be added according to where the CS level JHEP07(2020)012 6 F e (3.91) (3.86) (3.88) (3.85) (3.89) (3.90) ) x - f + 5) 0 F + e p p n 2 2( ··· (2) (2 1 0 sp so F 2 Λ x - f 3 ≥ 7 2 ··· n + p + y p (2) - ··· p m f ··· su 2 0 n 0 and m π F ··· + (2) (3.87) + 2) ≥ m x - (2) su Λ m f m ( 0 su + → y su - (2) 7 2 → f +1 + su m 2 0 p +2 Λ F + = m 3 x m + – 45 – - n n f − F ··· ··· ··· +2 + 2) n 4) + 2) x m - (2) is not physically relevant. In both of the above f m Λ − m ( 0 ( su 0 1 F 1 we obtain the following irreducible duality n + su F su (2) ≥ x +1 - su p f m + Λ 0, e + ≥ + 3) + (2 F e n n ··· p 2. The right hand side of this transition can correspond to two m m ··· +2 m ··· 3 m ≥ F + 2 = 2, n m h ¯ F + 2) n (2 + 7) + 2 m p so ( ··· su 0 and + 2 0, as the theta angle of 0. For e ≥ m = → m → ··· (2 +1 Now, consider the transition m m m m so 2 F in this paper. The first geometry is which leads to the following irreducible duality for for geometries satisfying the conditions defining the class of irreducible dualities being studied as dualities, extra matter is integrated in according to with as remember the replacement JHEP07(2020)012 4 F 3 e (3.96) (3.97) (3.98) (3.92) (3.93) (3.95) (3.99) ≥ (3.100) = 0, in z n -2 m y - x - π f 0 2 n 0 and ··· +2 (2) Λ e (2) 2 ≥ sp 3 0 sp m F 2 F Λ y - x - π f 2 0, 4 π (2) Λ f F 7 2 → sp = 2 only if we have 1 0 (2) e + F n m su m x m π π - ··· S f z (2) (3.94) (2) (2) + 2) -2 + 2 su y 0 ¯ F 0 - su su m x ··· ( - +1 → (2) f (2) → → m su y 7 2 - su Λ +2 su f + e +2 +2 2 on the right hand side denotes a full hyper in + m 2 0 m m 3 3 0 F – 46 – 0 F = F = − x 4) - ··· + 2) y ) is consistent for (2) + 2) + 2) f - n x − S - m sp m m f ( x n ( ( - 0 3.95 f + 2 su su su +2 1 0 (2) and F m F 0 f su Λ x - 0 f + (2) F + 3) + (2 (7) + + e su +1 n so n m ··· e +2 Λ + 2 m 3 0, we obtain the following irreducible duality (2) (instead of a bifundamental). As F m sp ≥ h (2 + 2) ⊕ m + 7) + so m ( (2) m ··· su su (2 0. The second geometry ( 0. e so of = = 2 and m → 2 → ··· +1 The second geometry associated to this transition is n Λ m m m 2 ⊗ F which case it becomes and gives rise to the following irreducible duality with as F For where the edge between as which can be seen to give rise to the following irreducible duality for with For the above four irreducible dualities, the addition of extra matter is determined by JHEP07(2020)012 5 3) F ≥ F e + 5) ) (3.103) (3.104) (3.105) (3.102) (3.101) p p N ( (2 (2 x - su so f +2 e n ··· 1 0 F (2) x - f +3 su m F + 5) Λ ) p n p ··· + + (2 (2 0 y p m - +1 ··· p so Λ f ··· (2) m 2 0 + Λ m su F ··· 1 +1 x - + 2 ≥ 4 m f F Λ (2) − ··· y - 3 as the reader can construct in a fashion n su m, p f + 2 + 1) ≥ 2 0 F p 0 F n 5) = 3, x (2) – 47 – - − n n f n ··· ··· su p π x - +1) f + 2 + 9) + (2 m 1 0 p n ( 1 F mπ ≥ x + 2 2, the geometry for right hand side of the transition is - + 1) p f (2) m ≥ + e m su ( (2 n + 3) + (2 so sp e n p 4 and ··· m ··· m 3 ≥ ··· π = + 2 F 1. For n n +1) ≥ h 1, m ( n + 2 1, a geometry only exists for ≥ ). We don’t have to specify anything more since there are no ≥ m ··· m + 1) (2 0 and m so e ( m, p 3.101 ≥ m sp ··· +1 Now consider the transition For the transition m m 2 = F for algebras appearing in the above two dualities. The extra mattergram for the ( above two dualities is added according to the transition dia- and the following irreducible duality for having similar to geometries discussedduality for above. This geometry leads to the following irreducible JHEP07(2020)012 ¯ ¯ F F (3.113) (3.114) (3.115) (3.109) (3.110) (3.111) (3.112) (3.106) (3.107) (3.108) F F 5 n n n F 2 ··· ··· m (3) 3 F ) is 5+ su 3 e ≥ ¯ (3) F F n su 3.104 2 Λ i 0 x 0 and + 3 P (2) ) - 1 5 2 7 f ≥ su π − ≥ + 2 +2 m m m m π m ··· ··· ··· e m 5 m (2) ( m ··· S − 3 0 (2) su + F ¯ F F su + 3) 0 i ¯ F → x +1 + 2) m ) → (2) m ( P 5 2 - m Λ f su ( + su +2 + m + e m su ( – 48 – 3 = F − 4) − S ··· e + 2) n − = + 2) m 3 n ( F 0 m ( su h (2) su + 5) + 3 su S m + ··· + 3) + (2 (2 n +1 so n n e ··· ··· m +2 Λ + 2 0, we obtain the following irreducible duality m +1 m m ≥ 2 (2 F + 2) m so + 7) + m m ··· ( m su (2 0. The associated reducible dualities are obtained by adding matter according to 0. Additional matter content can be added according to so = 1, the geometry for the right hand side of the transition ( = 2 and m m m = → → ··· ··· ··· Now consider the transition n n m m Additional matter can be incorporated according to with which leads us to the following irreducible duality for For as For with as which gives rise to the following irreducible duality for JHEP07(2020)012 0 π 2 (2) (2) Λ (3.121) (3.118) (3.119) (3.120) (3.117) (3.116) 4 sp sp ≥ 2 Λ n 5 n ··· π (3) 3, in which case 1 and su ≥ (2) (2) ≥ n π su su 2 0 m (2) Λ 0 (2) sp 0 sp 0 (2) 2 su Λ (2) (2) +3 m su su 1 3 m π Λ ··· ≥ S − S n 4 4 + ··· +1) ). The first one leads to the following + n S + m m − − (2) ( m ··· ··· +1 + 2 n n Λ +1 su 0 m + 2 3.89 (2) Λ m +1 + 0 0 Λ (2) m su +1 1 Λ +1 m + 2 su (2) (2) ≥ = 3 and Λ + 2 m F – 49 – Λ su su + 2 n F π m + 2 F 5) 4 π + 2 F +1) − ≥ F m mπ 5) ( + 9) + +1) n n 5) m − mπ mπ ( (2) m 3. This geometry leads to the following irreducible duality + 9) + − = 3 and n + 1) (2 su (2) (2) ≥ n n m + 1) so m 1 and su n su (2 ( m ≥ + 3) + (2 so sp ( n n π ··· sp m = + 3) + (2 π + 3) + (2 π +1) 2. A geometry for this transition exists only for n + 2 = 3 m n ( ≥ +1) +1) m ≥ + 2 m m n (2 ( ( + 2 n m + 1) so m (2 m + 1) + 1) (2 ( so 1 and 1 and so m m sp ( ( ≥ ≥ m sp sp ··· = Now consider the transition m m = = there are two geometries justirreducible as duality for in the case of ( for and the following irreducible duality for for which a correspondingthis geometry section) (satisfying exists only the for conditionsfor detailed at the starting of The second geometry leads to the following irreducible duality for and the following duality for JHEP07(2020)012 ¯ 5 F F e (3.122) (3.123) (3.124) (3.125) (3.126) (3.127) (3.128) z F -2 y - x 0 is uniquely -2 f n n ≥ ··· ··· 2 +2 = 1 version leads F e g 3 m 3 0 m F y - 3 x - 3 and 1 f ≥ F ≥ n (2) f S e n su 1 0 + F n S x m m - 0 and + ··· ··· f i m x ≥ 0 Λ = 2 version leads to a duality between P m dualities later. The - + (2) m f ··· (7) + 5 4 f su so +1 +2 y e - ) allows a simple gauge algebra on the right m f 3 Λ = 5 0 2 0 F − 2. The + F – 50 – ··· S i , 3.120 n x F x - f 4) P = 1 - 0 f − x - + m (7) + 5 (2) e f n 2 so 1 0 su F . The geometry describing the right hand side of the above F x e - (7) whose Dynkin diagrams are mapped as f → + 1 so e + 3) + (2 S F +2 n n n ··· ··· m e and 3 + 2 F S + 2) m h m → (2 ( F so (3) is oriented such that extra matter is added according to su , and is treated in the subsection on ··· = 1 version of the transition ( 4 su f n ). = e m m ··· ··· +1 The Now, consider the transition 3.123 m (9) and 2 F which leads to the following irreducible self-duality In particular transition is fixed to be hand side of the transitionso only for to a duality between two where the which implies the following irreducible duality for The geometry for thedetermined right to hand be side of the transition for We emphasize that, in general, thewhich above are irreducible not self-duality self-dualities leads since to the reducibleto matter dualities ( on two sides of the duality is mapped according JHEP07(2020)012 4 ≥ 2 F g n (3.134) (3.135) (3.132) (3.133) (3.129) (3.130) (3.131) 1 and ≥ n n ··· ··· m (2) su F 2 F g 2 F g 5 0 F S (2) e + su 1 S m m π +3 ··· ··· 4 ≥ + m z (2) (2) Λ -2 n − (2) y m ··· - su + ¯ F su + n x su m -2 Λ → f +1 0 m = + +2 +2 Λ e (2) 3 m +1 = 3 and – 51 – su 3 0 m + 2 n F S Λ F π + 2) y - + x + 2 - 2 m +1) ( f Λ F m mπ ( su 5) + + 9) + F (2) − m f + 1) su n (2 0 m F ( so (7) + sp 3, leading to the following irreducible duality for n n so ··· ··· π ≥ = + 3) + (2 n = 0 admits a consistent geometry satisfying all the imposed conditions. +1) n m ( m + 2 + 1) m 1. This transition has a geometric representation satisfying all the required (2 m 0. Extra matter is added according to ( so ≥ sp = 2, only m m → ··· ··· The final transition we need to consider in this subsection is n m = m and the following irreducible duality for for properties only for and leads to the following irreducible duality For This geometry is as where JHEP07(2020)012 x - f e + e F 3 1 0 (3.138) (3.139) (3.137) (3.136) (3.140) (3.141) F F + 2) x - h + 7) p f ( p F sp (2 e ··· 0 + 2) + 7) 5 p ≥ y ( F p - f p sp (2 h 2 0 +3 F p x + - 3 and f m ··· 0 ≥ y p - e ··· p (2) n n + 2) f ··· + su π 2 0 0, m +1 m π F ( p Λ 2 ≥ (2) x su - (2) F + f su m su h 0 +1 x → - = m → f (2) Λ +3 1 0 p +2 +2 su F + i + m m x x F – 52 – m Λ - 3 n n P f - ··· ··· + + h − + 2) e 2 ··· + 3) + 2) n +1 m p ( m m e +7 ( 0 Λ p 0, we obtain the following irreducible duality su 2 1 + 2 3 su + F ≥ n (2) F F su h m, p + 7) 2. The geometry associated to the right hand side of the transition p e m ··· ≥ + 1) + (2 m 5 ··· +2 p n F m + = 2 and h n n + 2) + 3) + (2 0 and + p m ··· ≥ ( m ) dualities + ( r su ( 0. For m sp e ( m, p sp → sp = +1 m m 2 F as with which leads to the following irreducible duality for where must be Let us start with the transition 3.3 with JHEP07(2020)012 (3.144) (3.145) (3.146) (3.142) (3.143) F + 2) + 7) p ( p sp (2 0 (2) p 4 p ··· ··· p p − 0 n su m ··· ··· + m ≥ − m 3 ··· p Λ − ··· + + 2) n +1 m 3 and ( 0 m Λ su ≥ (2) n + 2 = su – 53 – 1, F n n n n ··· ··· ··· ··· ≥ mπ + 2) m ) for the pure gauge theory. This leads to the following p mπ F + 1) 2.49 m (2) + 2 ( n su sp 1 ≥ m m m π ··· ··· m m ··· ··· + 1) + (2 +1) p m ( + ¯ F 3 for the existence of a geometry satisfying all the requirements, and obtain n ≥ m + 1) + n m m ··· ( ( 1. The geometry corresponding to the left hand side of the transition is uniquely 0. Extra matter in the above two dualities is added according to the diagram sp sp ≥ → Finally, let us consider the transition Now let us consider the transition m = m for determined to be the geometryirreducible ( duality for We require the following irreducible duality for as JHEP07(2020)012 x - f e + e 1 0 3 (3.149) (3.147) (3.148) (3.151) (3.150) F F . Extra x - h f 2.6 F e ¯ F ··· + 6) 1 5 p + 1) ≥ y F - p (2 f p (2 h so 2 0 F x - 3 and f ··· ≥ 0 y p n - e ··· p n + f m ··· (2) m 2 0 0, Λ +1 +1 π 2 su F p p ··· F i 2 2 ≥ + x x - e (2) F f m P i +1 su - F x 0 h m x - Λ → P f - 2 (2) h 1 0 + +2 − su F F m n x – 54 – - 3) ) discussed for illustration in section e n n f ··· ··· − + 2 + 2) e ··· p F 2.96 m ( e + 2 0 su n 3 F (2) su h e + 4) + (2 m ··· p m 2. The right hand side of the above transition is governed by the 5 ··· +2 F m ≥ + 2 h n n + 2) m + 2 ) dualities m ··· r ( m ··· 1 and su (2 (2 e so ≥ so +1 = p m 2 F which leads to the following irreducible duality for for geometry Let us start by considering the transition matter is added according to 3.4 generalizing the irreducible duality ( with JHEP07(2020)012 F (3.154) (3.155) (3.152) (3.153) + 6) p + 1) p (2 (2 so +4 p + m p + 2) ··· p ··· m π ( su (2) su n n = ··· → ··· +4 +2 p m + Λ – 55 – m n n ··· ··· + 1, we obtain the following irreducible duality + 2) +1 m ≥ m F ( Λ p su + m ··· m F ··· 0 and + 1) ≥ p m m ··· m ··· = 2, ¯ F n + 8) + (2 p + 2 0. For 0. Extra matter is added in the above two dualities according to m → → (2 Now, consider the transition so m m as as with JHEP07(2020)012 y y w w - - - - x x z z - - - - y x f f - - e e + + f f e e (3.156) (3.158) (3.157) (3.161) (3.159) (3.160) 2 2 0 2 + + e e e e F F F 1 4 0 1 x = 0. It is F F F - 0 f y - 0 2 x (4) , m - Λ 2 f (4) Λ ··· su 2 su = 2 Λ y f 2 - f Λ n 1 0 3 2 0 F F x ≥ - x f - n f 0 and 0 +4 m y ··· ≥ - (2) f su y 0 and n 2 0 - π π + 2) f + , m F 2 and is shown below ≥ 3 m 2 0 m (2) (2) x - ( F Λ ≥ m f ≥ su su 0 x su n - + f x n - → → 2 (2) f +1 x - − su m 1 0 +2 +4 f Λ F n m m 1 0 – 56 – x + F - = f F x - ··· + 2) + 2) + f e 3) + m m e ( ( − 0 e n su su +2 e (2) 3 m 0, the above geometry gives rise to the following irreducible F 3 Λ su F ≥ + h h m +1 + 4) + (2 e m n e Λ +2 5 m 5 + F + 2 F F = 2 and h m h + 2) n (2 m so ( ··· ··· + 8) + 2. There are two consistent geometries that can describe the right hand side of su 0. The second geometry is valid for 0. For m e e ≥ → (2 = → n +1 +1 so m m m m 2 2 F F which leads to the following irreducible duality for this transition. The first geometry is valid for all with as duality with as presented below with JHEP07(2020)012 (3.165) (3.162) (3.163) (3.164) (3.166) 0 2 Λ (4) 2 su 0 2 Λ (4) 2 su π (2) su w w C 0 π - - z z - - + (2) y x (2) - - S f f su su 0 n + + + n e e ··· e e ··· 1 4 0 1 → (2) +1 F F F su m +2 y Λ - F m = ) becomes x – 57 – 3 - + f − F ··· + 2) n 3) 3.162 f m C ( − 0 0 n + su F (2) S m ··· m ··· su + F F ¯ F + 4) + (2 n +2 (8) + 2 3, it gives rise to the following irreducible duality m + 2 so = 0, the geometry ( ≥ m n + 2) m (2 m so ( su 0 and 0. Extra matter for the above three dualities is added according to ≥ = 2 and = → n m m and gives rise to the following irreducible duality For as with For JHEP07(2020)012 y - (8) x - f so y x - - + e f f e (3.169) (3.167) (3.168) (3.170) (3.171) 1 0 2 0 2 of F F F F x - f π ··· (2) y su - f 4 2 0 F ≥ x - 3 n f (3) y - (8). f su 0 and so 2 0 π S F ≥ (2) x n + 2 - n 0 m f su +1 (2) x ··· - m → ··· f su Λ 1 0 +2 + F m 3 F – 58 – x - − 5) f ··· n + 2) + e − m n 0 C ( e su (2) 3 (2) on the right side is chosen to correspond to su F m ··· m su ··· S h + 2) + (2 n e +2 F + 2 m 5 F m (2 h + 2) so m ( ··· su 3. The geometry describing the right hand side of the above transition is e = ≥ +1 Now, let us move on to consider the transition n m 2 F which leads to the following irreducible duality for for That is, the fundamentalon of the left side and thus partially fix a triality frame of Extra matter is added according to with JHEP07(2020)012 . C π (3.172) (3.173) (3.174) (3.175) (2) so we can su 22 +3 + 2) is equivalent to 2 m k (4 (3) so su S S + 8) = π 5 2 ¯ ¯ F F F F m − C C (2) m (2 n with negative contribution to CS level n − su is flop equivalent to partially integrating in so S +3 → ··· ··· 2 + 2) m 5 2 − m (3) ( m ¯ F F − – 59 – su su are complex conjugates of each other, of + 2) C F = m ( or and π su S S m m (2) ··· ··· m m ··· ··· + 2 su F of + 2), +1 ¯ 0, we obtain the following irreducible duality F F m k F ≥ Λ (4 m + so F + 8) = = 3 and + 8) + 0. Extra matter is added according to 0. Extra matter is added according to n m m (2 (2 → → (3). Similar comments apply to many of the following cases. For so so Geometrically, this means that partially integrating in m m su 22 integrating a hyper in of For choose whether integrating in a hyper in spinor of as as with JHEP07(2020)012 (3.179) (3.180) (3.178) (3.181) (3.176) (3.177) 0 π 1 0 0 F F (2) (2) x - F f su f su y w - - x z - - y x y f - - - x + f f 2 e - e Λ f + 2 0 2 e e F F 4 0 1 1) 2) ). F F − − y - m m m m x ··· ··· ( - w 2 2 - (2 5 2 f z 2 - + 3.179 2 ≥ ≥ e x Λ - Λ ¯ F 2 f + 3) + 3) m m + e e m m ( ( 2 su su e F – 60 – 1 h F = = h e 4 e F C S 2 to avoid overcounting. There are two geometries that can 3 h + S F ≥ + 2 S h m +1 ··· + 2 m C Λ F e ··· m 2 e F + 6) + + 6) + 1 − m m m 2 (2 (2 m m F ··· ··· so so Let us now consider the following transition and leads to the following irreducible duality for The second geometry is and additional matter content can be added according to into which additional matter is added according to ( and leads to the following irreducible duality for for which we can restrict describe the right hand side of this transition. The first one is JHEP07(2020)012 z y x x - - - - f f f f 1 0 3 0 1 0 F F F (3.183) (3.182) (3.184) (3.185) x - f ··· π y - (2) f su 2 0 F 4 x - ≥ f n π π y - f (2) (2) 2 0 su su F 0 and C π x - ≥ f (2) + 2 n n 0 m S su x - (2) f ··· → + 2 ··· 1 0 su F +2 +1 x m m 4 - – 61 – f Λ − + ··· e + n + 2) F m e 0 7) ( − su 3 (2) F n su m ··· m h ··· ) + (2 e n +2 5 + 2 m F m h + 2) (2 so m ( ··· su e 4. The corresponding geometry for the right hand side of the transition is = ≥ +1 Now consider the transition m n 2 F which leads to the following irreducible duality for for with JHEP07(2020)012 π (3.187) (3.188) (3.186) (3.189) (3.190) (2) su 1 0 F x - +2 F f m x - + 2) y x f - - f f 2 0 1 0 m ( F F su y - m m x ··· ··· - f + n 2 e n ¯ F π ≥ ··· e ··· m (2) 2 su F F – 62 – h e = 4 F m h ··· m ··· C ¯ F ··· + 2 e S 2 without loss of generality. The right hand side of the transition is m 2 ≥ F m + 6) + 2 m m m ··· ··· 0. Extra matter is added according to (2 → so For the transition m leading to the following irreducible duality for we can take described by the geometry as Extra matter is added according to JHEP07(2020)012 3 and (3.191) (3.192) (3.193) ≥ n F + 6) p + 1) p (2 (2 so 1 ≥ 0 p n + (2) p m ··· p su ··· Λ ). The first one leads to the fol- 3 and + 3 ≥ +1 − n ··· 3.155 m n Λ n 1, n ··· ··· 0 ≥ + 2 F m (2) 4) su – 63 – n n − ··· ··· p + 2 mπ n (2) m ··· m ··· su + 4) + (2 2. A geometry satisfying all the criteria exists only for p m π ··· m ··· ≥ + 2 +1) n m n ( + 2 + 1) m 1 and m (2 ( ≥ so sp Now, we start considering transitions which admit more than one edge between some Now, consider the transition m, p = for leads to the following irreducible duality for pairs of nodes. The first such transition we consider is which admits two geometric descriptions just like ( JHEP07(2020)012 0 2 Λ (4) (3.196) (3.197) (3.194) (3.195) 2 su 0 4 2 (4) Λ ≥ π su n 2 (2) Λ 0 su 2 Λ (4) 1 and 2 0 su ≥ 0 (2) m su (2) C n + su + 1 3 π m S ≥ Λ − 4 C ··· +1) + n + m m − + ( ··· n +1 n n ··· S +1 0 ··· m (2) m + Λ 0 Λ (2) 3 su +1 su ≥ + 2 (2) = 3 and + 2 m – 64 – F n Λ su F n 4) π + 2 − F mπ +1) 1 and n 4) m mπ ( (2) ≥ + 10) + 2 − m ··· m su ··· m (2) n m + 1) su (2 m + 4) + (2 so ( n π sp π + 4) + (2 +1) + 2 = m n ( m +1) m (2 ( + 2 + 1) so m m + 1) (2 ( so m sp ( sp Now, consider the transition = = lowing irreducible duality for The second one leads to the following irreducible duality for and the following irreducible duality for JHEP07(2020)012 w - z w - - y z - - x x - (3.200) (3.201) (3.199) (3.198) (3.202) - f f + + e e e e 2 2 4 0 1 F F F y - x - = 0 S (7) f 2 (7) denotes a full- so m so S f (7) 2 so 1 0 F S x π - f S (2) + su n ··· w - + 0 π z y - m w - - y f - Λ (2) z (2) - x on top of it near 2 0 - 0 x su + - f F su f S + x (2) + e - +1 e e e 2 → f m su 2 4 0 1 Λ F F F x +2 - f 3 = m + y - – 65 – 1 0 x F − - F ··· f n 3) x + 2) - (7). Extra matter is added in the above two dualities (7) with an f 2 − m + 0 so Λ f so ( e n ⊕ + su (2) 0 e S F su (2) 3 + su F 3. Thus, it leads to the following irreducible duality valid for F (2) and of + 4) + (2 h ≥ su S n n +2 ⊗ e m F (8) + 2 + 2 5 so m F = 2, we can find the following consistent geometry if + 2) (2 3 h n m so ≥ ( n su ··· 0. For e = → 0 and +1 m m ≥ 2 F leading to the irreducible duality as with which makes sense for m whose right hand side corresponds to the following geometry where the edge between hyper charged under JHEP07(2020)012 S (7) 2 (3.206) (3.204) (3.203) (3.205) so S π (2) su S (7) 2 so S 0 C S 4 (2) + ≥ su 1 n π +3 S S m ≥ 4 + +1) Λ n m m − ( + ··· + n n 1 and n n n ··· ··· m ··· ··· (2) +1 ≥ Λ m 0 su + Λ m F (2) = 3 and +1 – 66 – + 2 su n m F Λ π + 2 +1) F m mπ ( 4) m m ··· m ··· m + 10) + 2 (2) ··· ··· − + 1) su m n m F (2 ¯ F ( so sp π = + 4) + (2 +1) n m ( + 2 + 1) m (2 m ( so sp The transition = leads to the following irreducible duality for according to and the following irreducible duality for JHEP07(2020)012 π (2) (3.208) (3.207) (3.209) su 3 (3) su C 0 (2) su S 4 n S n − ··· n n ··· n ··· + 2 ··· ··· 4 and gives rise to the following irreducible 0 +1 ≥ m (2) n = 2. Extra matter in the above two dualities is Λ – 67 – su n + 2 F 6) − mπ n m ··· m ··· m (2) ··· m ··· su 4 ≥ n + 2) + (2 π n +1) + 2 1 and m ( m ≥ (2 m + 1) so m ( sp The transition = admits a corresponding geometry onlyduality for for added according to There is no corresponding geometry for JHEP07(2020)012 w - z - y - x - (3.212) (3.211) (3.210) f w - + z e - e f 2 f 2 0 4 0 F F F y - x - f f 1 0 F x - f S ··· ¯ F F y - C f n n 2 0 n n F x - ··· ··· ··· ··· f x - f – 68 – 1 0 F x - f + e e m m ··· ··· 3 m m ··· ··· F h e 5 4, the geometry describing the right hand side of the above transition is F ≥ h n ··· 3. For e ≥ Now, consider the transition +1 n m 2 F for into which extra matter can be added according to JHEP07(2020)012 0 (3.213) (3.214) (3.215) (3.216) (2) su π 0 (2) (2) sp 4 su ≥ n S π π (2) 0 and S su (2) C ≥ n n sp + 2 n n m +1 0 ··· ··· ··· ··· m Λ (2) su + 0 – 69 – F 4 5) (2) − − su ··· n n 0 = (2) m m ··· ··· m m ··· ··· su + 2) + (2 n S + 2 + 2 +2 F m m = 0, and the corresponding irreducible duality is (2 m so + 2) (8) + 2 m ( so su = 3, a geometry for the right hand side of the transition satisfying all the requirements For the transition = n which leads to the following irreducible duality for For exists only for Extra matter in the above two dualities is added according to JHEP07(2020)012 0 π (2) (2) (3.217) (3.219) (3.220) (3.221) (3.218) su sp π 0 (2) (2) su su w - z - 5, we obtain the following x - y - f ≥ 0 x S + - e f n e f (2) π 1 0 4 0 F F F su (2) w C - sp 1 and 5 z - n S n y ≥ − - ··· x n - m + 2 f ··· ··· S + 0 e π +1 2 m + 2 (2) +1) Λ – 70 – 4. For m +1 su ( m ≥ + 2 e Λ (2) F n 2 su 6) + 2 F − F mπ h n m (2) ··· m ··· π su e +1) 4 + 10) + 2 m ( F + 2) + (2 m π 1, we obtain the following irreducible duality n (2 + 1) ≥ +1) so + 2 m m ( m ( m sp (2 + 1) so = m ( = 4 and sp Now, consider the transition n = For irreducible duality a consistent geometry exists only for Its right hand side is described by the geometry Extra matter in the above two dualities is added according to JHEP07(2020)012 (3.222) (3.223) (3.225) (3.226) (3.224) 0 (2) su π (2) su S m m ··· ··· 2 + 5) F 4 S f 2 2 ≥ m n n (2 m so ··· ··· = on the right side both are equivalent to adding – 71 – = 4 f of A S S + 3 + 2 3 m (2) or ··· m S Λ ··· +1 su m of Λ F + C (10) + F so + 6) + m m m (2 ··· ··· so Now, consider the transition Now, consider the following transition on the left side of the above duality. which leads to the following irreducible duality This leads to the following irreducible duality for S Notice that adding an and additional matter can be added according to JHEP07(2020)012 π π π (2) +1) (2) (3.229) (3.230) (3.231) (3.227) (3.228) m su ( su (2) su 5 π ≥ π π n π +1) (2) (2) m π ( su su (2) +1) su (2) 1 and m ( su ≥ 0 m + 1) (2) m ( su 1 π m ··· sp ≥ +1) 5 C m m ( − ··· n + 2 S π + 1) 0 m +1) + 2 ( (2) m – 72 – ( sp su +1 m (2) Λ su = + 2 mπ F C 8) (2) − = su + 2 n S + 2 C ) + (2 π 1 we obtain the following irreducible duality +1 n + 2 m ≥ +1) Λ S + 2 m ( m m (2 + 1) m so dualities ··· m + 8) + 2 + 6) + 2 4. This leads to the following irreducible duality for ( r = 4 and e m m ≥ sp Finally, consider the transition n (2 (2 n = so so for 3.5 Let us start by considering the transition which leads to the following irreducible duality for For JHEP07(2020)012 4 F e (3.234) (3.233) (3.235) (3.232) h 2 F 6 e F e 5 i x 7 F P - e 7 h 0 6 6 F F 2 7 7 (2) e e 5 5 F F i e x su e P - π h (2) 2 π π 31 su h (2) (2) (3) 5 su su F su = – 73 – e n n ··· ··· = = = 2 Λ h 3 Λ + 3 3 2 + F F Λ Λ 2 e + + Λ + 8 F F 6 + e 3 leads to the following irreducible dualities F x + 8 + 6 - m f ≤ 8 7 ··· m + e e + 5 ··· e n 8 e 1 0 + F m x - f 2 and ≥ x - In a similar fashion, the transition f n 1 0 F for which leads to the following irreducible duality mined to be The geometry associated to the right hand side of the above transition is uniquely deter- JHEP07(2020)012 (3.237) (3.236) (3.238) S (10) 3 so S 0 (2) su 2 − ··· 4, it leads to the following irreducible n ≤ 0 n ≤ (2) n n su ··· ··· – 74 – π = 0 and 2 (2) m F su m = ¯ ··· F 4. For m ··· ≤ n n Λ + + m F n + 2 2 and n ≥ 4+ e Now, consider the transition n for dualities duality is added according to It is clear how to add extra matter to the first two dualities, while extra matter for the last JHEP07(2020)012 (3.240) (3.241) (3.239) S (10) 3 so S π C S (10) S 3 (2) (10) 3 so so su S S S n n ··· ··· 3 10 11 − n n − − ··· ··· – 75 – (3) (3) (4) su su su ¯ F = = = 4 4 3 m ··· m Λ Λ Λ ··· m + + + ··· m 2 3 2 ··· Λ Λ Λ + + + F F F F + 5 + 3 + 3 8 8 7 e e e 2, we obtain the following irreducible dualities , = 1 Now, consider the transition m For Extra matter in dualities associated to this transition is added according to JHEP07(2020)012 2 5 2 Λ (5) ≤ 2 5 2 su (3.244) (3.243) (3.242) n 2 (5) ≤ Λ su 5 2 2 Λ 2 π (5) Λ (2) su 2 su Λ 0 = 0 and 3 m (2) 5 2 su 5 2 0 2 0 2 (5) Λ 2 (2) (5) Λ (2) su − su su 2 ··· su Λ 2 n Λ 2 3 , 0 − ··· 0 n = 1 (2) 0 0 su (2) m 0 (2) (2) su (2) su su su ), there are two geometries which can describe – 76 – π (2) 3.155 3 4 3 π su (2) (4) (3) (3) su su su su 5 = ≤ n = n 5. As for ( Λ = = = ≤ C + ≤ F + 5 4 5 n Λ Λ Λ S 2) + + + + + − 3 2 2 F m n Λ Λ Λ = 0 and 3 2) + + + − m + (2 F F F n n 3 and + 5 + 3 + 3 3+ ≥ + (2 e 8 8 7 e e e n n 3+ e The second geometry leads to the following irreducible duality for the right hand side ofduality the for above transition. The first one leads to the following irreducible for and the following set of irreducible dualities for JHEP07(2020)012 (3.246) (3.247) (3.245) 2 2 Λ (5) 2 su 2 2 π 2 2 Λ (5) ¯ F Λ 2 (5) (2) 2 su su su π 0 π 2 , (2) F (2) (2) su = 1 su su m n n ··· ··· 3 3 4 – 77 – F (3) (3) (4) su su su = = = m m ··· ··· m m ··· ··· C C C + + + ¯ F S S S + + + 2 3 2 Λ Λ Λ + + + F F F + 5 + 3 + 3 8 8 7 e e e Now consider the transition Extra matter can be added into the dualities associated to this transition according to and the following set of irreducible dualities for JHEP07(2020)012 (3.250) (3.248) (3.249) (3.251) π (2) +5 2 su m 2 Λ (5) ¯ F su 4 2 Λ (4) su 3 ≤ 6 − m m F S − 0 ≤ + 2 (2) + 2) su n n +1 ··· ··· m m ( 3 Λ su – 78 – − + ··· ¯ F n F 5) = 0 − (2) n su +2 m + (2 m m Λ ··· ··· n m m ··· ··· + + 6 and leading to the following irreducible duality m ≤ +1 2+ F +2 m e n m Λ + + m + 2) F + m ( +5 su 4 and m e ≥ n = Now, consider the transition This leads to the following irreducible duality for 1 with Extra matter can be added according to JHEP07(2020)012 π (3.255) (3.254) (3.252) (3.253) (2) su +8 2 ¯ F m 3 (4) su ≤ m ≤ π F (2) 3 − su m − → n n ··· ··· +2 + 2) m – 79 – m ( F + 2) su m ( su = m m ··· ··· m m ··· ··· S + 2 ¯ F +1 m Λ + F + 0. Extra matter can be added according to +5 m → e Now, let us move to the transition m which give rise to the following irreducible duality for 1 as with JHEP07(2020)012 (3.257) (3.256) (3.261) (3.260) (3.258) (3.259) π π (2) su (2) ¯ F su m m ··· k 3 Λ 2) 3 2) F m + 4) Λ k − − 2 m m ( m 3 + 4) 2 4 Λ su m ( + 7)( + 3)( su – 80 – m m ( ¯ F = (3 = = m = m k k C + S m S ··· m ··· + 2 + 2 F F +1 + m Λ +4 m + e 4, there are two possible geometries. The first one gives rise to the following +4 ≤ m e m m ··· ≤ For the transition with 2 irreducible duality Additional matter content is integrated in according to where with The second geometry gives rise to the following irreducible duality JHEP07(2020)012 F (3.262) (3.263) (3.265) (3.264) 3 7 (3) su ≤ n + m π π ≤ (2) (2) su su m 2 ··· C π 4 and 5 + ¯ F (2) C 0 ≥ su n n + (2) S → su n ··· ··· + 2 +2 4 m – 81 – +1 − ··· m n Λ + 2) + 0 m ( F (2) su 7) su − n m ··· m ··· + (2 +2 +1 m n + m e + 2) m ( su m ··· = Now, consider the transition This leads to the following irreducible duality for Additional matter into the above two dualities is incorporated using with JHEP07(2020)012 F (3.267) (3.266) (3.269) (3.268) 2 m − 3 − ¯ F (3) su F ¯ F 5 2 + m F 4 m + 2) m ··· ··· , m ( = 3 ¯ F su m n n ··· ··· π – 82 – (2) F su = m ··· m ··· 2 C ¯ F + C + S + 2 m m ··· ··· 0, and additional matter content is added according to +4 m e → Consider now the transition m which leads to the following irreducible duality for as Extra matter is incorporated according to JHEP07(2020)012 π (2) (3.270) (3.271) (3.272) (3.273) su 0 8 (2) su ≤ n + m π π ≤ (2) (2) su su π (2) n n n n 5 and 6 C su 0 ≥ + 4 → n (2) S ··· ··· +2 ··· ··· su m + 2 – 83 – 5 +1 F + 2) − m ··· Λ n m ( + 0 su F 9) (2) − su m m n ··· ··· m m ··· ··· + (2 ¯ F n + +2 m m e + 2) m ( 0. Additional matter content is added according to su → Consider now the transition m = It leads to the following irreducible duality for as with JHEP07(2020)012 π 6 (3.277) (3.275) (3.276) (3.274) (3.279) (3.278) ≤ (2) n su + 7 2 − m (3) 0 su F 4 and (2) ≥ su n 7 2 (3) su m m +2 ··· ··· π m 2 4 C , (2) ¯ F 0 + + 2) su n n = 3 C (2) m → ( m + su ··· ··· su +2 +1 3 m m – 84 – Λ − ··· n + 2) + π F m 0 ( 5) (2) su (2) − su su n m ··· m ··· + (2 = +2 n +2 + m m e C + 2) + 4 m S ( su + 2 m m ··· ··· +4 m = Now consider performing the transition Now, consider e This transition gives rise to the following irreducible duality for into which matter is added according to the rule leading to the following irreducible duality for with JHEP07(2020)012 (3.282) (3.281) (3.280) 7 2 (3) su F 3 7 2 − ≤ m ¯ F − m ≤ (3) su ¯ F F n n 5 2 ··· ··· + m – 85 – F + 2) m ( su m m = ··· ··· m m ··· ··· ¯ F 2 C + C + F + 0. Extra matter is added according to +5 → m Now, consider the transition e m which gives rise to the following irreducible duality for 1 as JHEP07(2020)012 F (3.284) (3.283) (3.287) (3.285) (3.286) ¯ F F m k ¯ F (3) su F 2 4 Λ m m ··· ··· , 3  − = 3 m ¯ F F ¯ F 3 + 14 m 2 Λ m + 3) 4 m + 5 ( 2 su m – 86 –  F − = = m 2 k C + m ··· m C ··· + ¯ F +1 m Λ + +4 m e m m ··· ··· Now, consider the transition which leads to the following irreducible duality for Extra matter is added according to Extra matter is added according to where JHEP07(2020)012 π (2) (3.288) (3.289) (3.290) (3.291) su 0 7 (2) su ≤ n + m 3 (3) su 5 and π ≥ (2) ¯ F F n n n n n su C 0 → + 4 (2) ··· ··· +2 ··· ··· su +1 m m – 87 – 4 Λ F + 2) − + ··· n F m ( 7) 0 su − (2) n su m m ··· ··· + (2 m m ··· ··· +1 n ¯ F + m +2 e m + 2) m ( 0. Extra matter is added according to su → Now, consider the transition m = which leads to the following irreducible duality for as with JHEP07(2020)012 π (2) (3.295) (3.293) (3.294) (3.292) su 0 (2) su 2 m 3+ 3 m ··· , (3) = 2 su ¯ F F m 2 5 − m – 88 – − ¯ F + 2) m ( su m m = ··· ··· m m ··· ··· +1 F m Λ + C + 4 F m ··· + The transition Consider now the transition +5 m e with extra matter content being added according to the following specification which leads to the following irreducible duality for JHEP07(2020)012 (3.298) (3.299) (3.297) (3.301) (3.300) (3.296) S π π (2) (2) π 4 su su F C , (2) = 3 su m 0 π (2) (2) S su m m m su ··· ··· ··· 4 , + 6) ¯ F = 3 2 S Λ 3 m m 2 (2 1) − − so m m 2 ( 5 2 2 2 Λ – 89 – Λ 2 + 3) + 3) = m ( m ( su su 2 = C = + C C C + 4 + 4 +4 + 4 m S e +1 + m F Λ + + +4 +4 m m m 4 ··· ··· ··· m , m e e Now, consider the transition = 3 m has two geometric solutions. The first one leads to the following irreducible duality for This leads to the following irreducible duality for In both of the above dualities, extra matter is added according to where extra matter is added according to and the second one leads to the following irreducible duality for JHEP07(2020)012 (3.302) (3.303) (3.304) S (10) 3 so S π (2) su n n ··· ··· 0 – 90 – (2) sp = m ··· m 4 ··· Λ + 2 Λ + 2 F + 4 8 e Now, consider Now, consider the transition which leads to the irreducible duality There are two geometries which can describe the right hand side of the above transition. JHEP07(2020)012 2 (3.306) (3.307) (3.305) 2 Λ (5) 2 su 5 2 2 (5) Λ su 2 2 Λ 2 π 2 2 Λ (5) Λ 2 (5) (2) 2 su 5 2 ¯ F su su 5 2 0 2 2 (5) Λ (2) (5) Λ su su su 2 Λ 2 π Λ 0 π (2) (2) (2) F su 0 su su π π (2) (2) (2) su n n ··· ··· su su 0 0 π – 91 – (2) (2) (3) sp sp sp 0 0 π (2) (2) (3) sp sp sp = = = m ··· m ··· C C C = = = + + + S S S 5 5 4 Λ Λ Λ + + + 3 2 2 + + + Λ Λ Λ 2 3 2 Λ Λ Λ + 2 + 2 + 2 + 2 + 2 + 2 F F F F F F + 2 + 4 + 2 8 8 7 + 4 + 2 + 2 e e e 8 8 7 e e e The first one leads to the following set of irreducible dualities In all the dualities associated to this transition, extra matter is added according to The second geometry leads to the following set of irreducible dualities JHEP07(2020)012 (3.309) (3.310) (3.311) (3.308) S (9) 2 so 5 S ≤ n π + m (2) su C 3 and π + ≥ (2) n 0 n + su m (2) Λ → su n n n n + ··· ··· ··· ··· +2 3 +1 m – 92 – − m ··· Λ F n + 2) + 0 m F ( 3) (2) su − su n m m ··· ··· m m ··· ··· + (2 n + +2 ¯ F m m 3+ e + 2) m ( su 0. Extra matter is added according to → = Now, let us consider the transition m as with which leads to the following irreducible duality for JHEP07(2020)012 (3.313) (3.314) (3.312) S (9) 2 so S π S (9) S 2 (9) (2) 2 so so su S S π 0 π (2) (2) (2) su su su n n n n ··· ··· ··· ··· 0 0 π – 93 – (2) (2) (3) sp sp sp = = = m m ··· ··· C C C m m ··· ··· + + + 5 4 5 Λ Λ Λ + + + 3 2 2 Λ Λ Λ + 2 + 2 + 2 F F F + 2 + 4 + 2 8 8 7 e e e Now, consider the transition Now, consider which leads to the following set of irreducible dualities JHEP07(2020)012 (3.317) (3.318) (3.315) (3.316) F 4 f 2 5 ≤ n 0 + (2) m su S 3 and π + n 0 ≥ (2) + n m (2) su Λ su → + n n n n ··· ··· ··· ··· 3 +2 +1 m m − ··· – 94 – Λ n F + + 2) 0 F m ( 3) (2) su − su n m m ··· ··· m m + (2 ··· ··· n + +2 m ¯ F m 3+ e + 2) m ( su 0. Extra matter is added according to = → Now, consider the transition m which leads to the following irreducible duality for as with JHEP07(2020)012 π (2) (3.321) (3.320) (3.319) su F 4 f 2 4 π π (4) (2) (2) su su su 0 F 4 f F 2 4 (2) f 2 4 su 4 0 (4) (4) (2) su su su 0 π π (2) (2) (2) su 0 su su π π (2) n n ··· ··· (2) (2) su su su – 95 – 0 0 π (2) (2) (3) sp sp sp 0 0 π (2) (2) (3) sp sp sp m = = = ··· m ··· S S S = = = + + + 5 4 5 S S S Λ Λ Λ + + + + 2 + 2 + 2 2 2 3 2 2 3 Λ Λ Λ Λ Λ Λ + 2 + 2 + 2 + 2 + 2 + 2 F F F F F F + 4 + 2 + 2 + 4 + 2 + 2 Now, consider the following transition 8 7 8 e e 8 8 7 e e e e which leads to the following set of irreducible dualities which leads to the following set of irreducible dualities JHEP07(2020)012 (3.323) (3.322) (3.325) (3.324) 6 ≤ π n (2) + su m ¯ F 4 and 0 ≥ (3) n sp π F S (2) 0 su + 2 (2) → su n n n n +1 ··· ··· ··· ··· m +2 3 Λ m – 96 – − + ··· n F + 2) 5) 0 m ( − (2) n su su + (2 m m ··· ··· n m m ··· ··· + m 2+ +2 e m + 2) m ( su = Now, consider the transition from which we obtain the following irreducible dualities for Extra matter is added according to with JHEP07(2020)012 π (2) (3.328) (3.327) (3.326) su 0 π π (3) (2) (2) sp su su 0 0 0 (3) (2) (3) sp sp su 0 π π (2) n n n n ··· ··· ··· ··· (2) (2) su su su – 97 – F 0 0 π (2) (2) (3) sp sp sp m m ··· ··· m m ··· ··· = = = ¯ F S S S + 2 + 2 + 2 2 2 3 Λ Λ Λ + 2 + 2 + 2 0. Extra matter is added according to F F F → + 4 + 2 + 2 The transition 8 8 7 m e e e which leads to the following set of irreducible dualities as JHEP07(2020)012 3 (3) (3.331) (3.330) (3.329) su 7 ≤ π π n (2) (2) + 3 su su m (3) su 0 1 and 3 (2) ≥ ≤ su π m 2 m C 2 5, 5 π +1) ≤ C + m − ≥ ( (2) ··· C + n n n su C (2) + 0 S su + n ··· ··· S (2) + 2 = 4 and 1 su – 98 – + 2 +1 n m π +1 Λ m +1) Λ m + 2 mπ ( + 2 +5 (2) F m + 1) e su 8) m ··· m m ··· − ( n sp π + (2 = +1) m +1 ( n + m + 1) e m ( sp Now, consider the transition = and the following irreducible duality for This leads to the following irreducible duality for JHEP07(2020)012 (3.333) (3.332) (3.335) (3.334) 0 7 (2) ≤ sp n + ¯ F m 0 π ≤ (2) (2) F su su 2 4 and 5 π C ≥ (2) 0 + 2 n su n n S (2) → su n n + 2 ··· ··· ··· ··· +2 4 +1 m – 99 – m − ··· Λ n + 2) + 0 F m ( 7) (2) su − su n m m ··· ··· m m ··· ··· + (2 +1 +2 n + m m e + 2) m ( su = Now, consider which leads to the following irreducible duality for Extra matter in the above two dualities is added according to with JHEP07(2020)012 0 (2) (3.338) (3.337) (3.336) sp 7 ≤ 0 π n (2) (2) + su su m 0 1 and 4 (2) ≥ − su m n 2 4, C ··· ≥ + n n n C 0 + n n ··· ··· ··· ··· S (2) su + 2 – 100 – F +1 m Λ mπ + 2 (2) F su 8) m m ··· ··· m m ··· ··· − n π ¯ F + (2 +1) m +1 ( n + m + 1) e m ( 0, and additional matter content is added according to sp → Now consider the transition m = This leads to the following irreducible duality for as JHEP07(2020)012 0 π π (2) (2) (2) (3.343) (3.340) (3.339) (3.341) (3.342) su su su π π 8 π ≤ mπ (2) (2) (2) n su su (2) su + su m 0 3 0 (2) 1 and ≤ su mπ (2) m ≥ ··· m su 6 m 4 + 1) ≤ , n n − 6, C ··· m n ( = 3 ≥ + 4 sp n 0 m π S ··· ··· (2) π C +1) + 2 = 5 and 1 m su ( – 101 – (2) n + 4 +1 π su S (2) m Λ +1) su + 2 m ( mπ + 2 +1 F (2) (2) m Λ su su π 10) m ··· − + 2 m +1) ··· n m = ( +5 π m e + (2 + 1) +1) n + m m ( C ( m e sp + 4 + 1) S m ( = m ··· + 2 sp +4 Consider now the transition Now, consider m e = and the following irreducible duality for It leads to the following irreducible duality for leading to the following irreducible duality for JHEP07(2020)012 6 7 2 ≤ − n (3.344) (3.345) (3.346) + (3) m su 1 and 7 2 ≥ (3) m su F 4, ≥ n 0 ¯ F (2) su ¯ F F 4 2 C − n n n n ··· n + C ··· ··· ··· ··· 0 + (2) – 102 – +1 su m Λ + F mπ 5) − (2) m m n ··· ··· su m m ··· ··· + (2 π +2 n + +1) m m ( e + 1) m ( sp Now consider performing the transition = This transition gives rise to the following irreducible duality for Extra matter is added according to JHEP07(2020)012 0 π (2) (2) (3.347) (3.348) (3.349) su su 7 ≤ 3 n + (3) m su 1 and 0 ≥ (2) m su 5, 5 ≥ ¯ F F n n n n − n ··· n 0 C ··· ··· ··· ··· (2) + 4 su – 103 – +1 m Λ mπ + 2 F (2) 8) su m m − ··· ··· m m ··· ··· n π + (2 +1) +1 n m ( + m e + 1) m ( sp Now, consider the transition = which leads to the following irreducible duality for Extra matter is added according to JHEP07(2020)012 π π (2) (2) (3.350) (3.351) (3.352) (3.353) su su π 7 (2) sp ≤ n + m π (2) su 5 and π ≥ (2) n n n n n su 0 → (2) ··· ··· +2 ··· ··· C su m – 104 – + 4 5 F + 2) − +1 ··· n m m ( Λ 0 su + F (2) 7) su m m − ··· ··· m m ··· ··· n ¯ F + (2 +2 +1 m n + m + 2) e m ( 0. Extra matter is added according to su → Now, consider the transition m = which leads to the following irreducible duality for as with JHEP07(2020)012 π (2) (3.355) (3.356) (3.354) (3.359) (3.357) (3.358) su π π π π (2) (2) (2) (2) su su su su 1 F π π π e π (2) (2) (2) (2) sp su su su e 4 m ··· , 7 1 F = 3 i π π π x n n m (2) (2) (2) P S - h sp sp su 2 + 5) ··· ··· S 2 m x - – 105 – 0 π π f (2 + e so (2) (2) (2) su su su 1 0 F x - = f 0 0 π m ··· m (2) (2) (3) x C ··· - f sp sp sp 1 0 + 4 = = = F +1 m C C C Λ + 4 + 4 + 4 + 2 2 3 F Λ Λ Λ + dualities + 2 + 2 + 2 m 4 +4 F F F ··· f m e Finally, consider the transition Now, consider the transition + 2 + 2 + 4 7 8 8 e e e which leads to the following irreducible duality for This leads to the following set of irreducible dualities 3.6 Let us start by considering the transition The geometry for the right hand of the transition is uniquely fixed to be JHEP07(2020)012 (2) su (3.365) (3.366) (3.367) (3.363) (3.364) (3.360) (3.361) (3.362) (3.368) (3) and sp 0 F 1 0 1 0 F F π π f π x x - S - (7) f f 7 (2) (2) (2) so (2). The second geometry is su su y su - S su x - x x - - f ⊕ f f 3 3 0 1 0 1 0 (3) Λ F F F sp π z π 6 ) x -2 x - 3 of - y , and the edge between (3) (2) f - (3) f (4) 4 Λ x F f - + sp su +2 sp e su f e ⊗ 2( 3 +2 e Λ 2 – 106 – = = = = e e e 5 6 F F 4 2 F Λ h A A h A h + + + + 3 F F F Λ + + 8 + + 2 e e 4 4 4 4 e f f f f 7 8 F F 6 F denotes the adjoint representation of on top of it denotes a full hyper in A 3 Λ Now consider the transition Now, let us consider the transition giving rise to the following irreducible duality The right hand side of the transition is described uniquely by the geometry which leads to the following irreducible duality where with conditions listed at the beginning of this section. The first geometry is and leads to the following irreducible duality The right hand side of this transition can correspond to two geometries satisfying the which leads to the following irreducible duality JHEP07(2020)012 (3.369) (3.370) (3.371) (3.378) (3.376) (3.377) (3.373) (3.374) (3.375) (3.372) 1 1 ¯ F F F e e F i i x x P P - - f f 2 + + e 2 0 Λ e 2 S F i 3 0 5 0 x + 3 F F ¯ F F 7 2 P i i − participates. The first one arises via the -2 x x (9) + 5 f 2 P P (5) so g - - +3 e f f su 2 = +2 +2 – 107 – e e = F e F + 5 e e 4 6 + 3 f F 4 3 1 f F F h h e e 5 3 F F dualities 2 g Now, consider the transition Finally, consider the following transition which gives rise to the following irreducible duality whose right hand side is described by Extra matter is added according to whose right hand side is described by the geometry 3.7 There are two irreducibletransition dualities in which giving rise to the following irreducible duality The right hand side of the above transition is described by the geometry Extra matter can be added on both sides according to the following diagram JHEP07(2020)012 (4.4) (4.1) (4.2) (4.3) (3.379) (3.380) (3.381) (3.382) (3.383) 1 F − F (3) 1 0 su ¯ F F x - f ¯ F F 1 F (3) x - 2 f su Λ 0 F ¯ F F 1 0 + 2 F 7 f π − = +3 x e - (2) (3) e sp su π = (2) = – 108 – x - su e 2 F e g 1 0 8 + 2 F F 2 g x 0 - f (3) su x - f 1 0 F π (2) su The second irreducible duality arises via the transition giving rise to the following irreducible duality whose right hand side is described by the geometry Extra matter content can be added according to This geometry leads to the following irreducible duality whose left hand side can be described by the geometry generalized. 4.1 Semi-simple toWe semi-simple can consider transitions withconsider semi-simple the algebras transition on both sides. For example, we can 4 Further generalizations In this section, we discuss some ways in which the considerations of this paper can be Extra matter can be added into the above duality according to which leads us to the following irreducible duality JHEP07(2020)012 S (4.9) (4.5) (4.6) (4.7) (4.8) (4.11) (4.10) F (2), applying su (4). Let us instead ¯ F ⊕ f 4 su F 5 2 (3) e (3) su F su (4) gauge algebra but the num- f su ¯ F + e x x 2 0 2 0 - - 1 0 F F 5 2 f f F y y (3) - - x x x - e e e su to implement the dualities. For example, we 4 0 4 0 F F S = – 109 – x y y x x - - - - - e e e f e with 2 0 n x x 2 0 0 2 2 - - I F F F f f Λ y - y x - + x - F f + + e 0 (5) e transformations are applied. For example, consider the geometry su 2 S F with other isomorphisms on the lower node to rewrite the above geometry as S S (4). The above geometry leads to the irreducible duality Even though the geometrynaively does to lead all to thefirst intersection nodes perform matrix does of not lead to the intersection matrix of can describe the right hand side of the transition We can combine the isomorphisms claim that the geometry Extra matter can be added according to 4.3 Mixing The intersection matrix of theber above of geometry edges describes in thesu geometry are more than the number of edges in the Dynkin diagram of which can describe the left hand side of the transition We can allow more edgesin to the appear phase where in the geometry as compared to the Dynkin diagram 4.2 Allowing extra edges JHEP07(2020)012 (4.17) (4.15) (4.16) (4.12) (4.13) (4.14) , we obtain . We obtain x x, y, z , we obtain F (2) 3 x, y su not equal to x 2 0 x x 2 0 2 0 - y - not equal to x - 2 0 x , we obtain 2 0 F f - 0 - F e F f F x f F f w F y (3) not equal to y x x 2 y - x - - - - y y e - x e - e f f x z su e e 4 1 4 0 4 0 4 0 4 1 F (4). The precise irreducible duality obtained F F F F = su w y y y - z - – 110 – - - y - w y w e z y - - - e - f - y - e - e z z y x - - - x - y - y 2 x 2 0 x h - x - x 2 0 2 0 - - f - x - 2 0 x Λ 2 0 f F f f - f - F e F f + F f F + + f e + + e e h F ) is + 6 0 4.9 (4) su on the lower node using a blowup on all the nodes to obtain 0 ) with ( on the lower node using a blowup S I on the lower node using the blowup 0 1 I on the lower node using a blowup 4.16 − 0 1 I − 0 I by matching ( whose intersection matrix indeed leads to Now we perform Performing Now applying Applying Now we perform JHEP07(2020)012 ] (4.18) (4.19) (4.20) (4.21) (4.22) (2014) 116 12 dualities from d arXiv:1311.4199 [ JHEP , (2014) 112 F 03 2 F + 2 Λ 6 5 + 2 − JHEP − + 2 5-Brane Webs, Symmetry Enhancement 7 6 , π − − (3) (3) )! su (2) (3) (3) su = sp su su = 2 3.379 π = = = – 111 – Λ ) 2 F F (2) g + 2 sp + 2 + 2 π 3.382 2 2 g g (2) ), which permits any use, distribution and reproduction in ]. sp ), we find that SPIRE IN 4.21 dualities can be derived by performing some physical operations. ][ ). As noted there, we can add matter to the above duality to obtain CC-BY 4.0 d This article is distributed under the terms of the Creative Commons ) and ( 3.146 Duality and enhancement of symmetry in 5d gauge theories ]. 4.19 SPIRE IN arXiv:1408.4040 and Duality in 5d[ Supersymmetric Gauge Theory [ Thus, the irreducible dualities can be “reduced” further into more primitive dualities. O. Bergman, D. G. Rodr´ıguez-G´omezand Zafrir, G. Zafrir, [1] [2] Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. Acknowledgments The author thanks Hee-Cheolwork Kim is and supported especially by Gabi NSF Zafrir grant for PHY-1719924. useful discussions. This on the two sides ofseen, the if duality one are changessometimes kept gauge fixed to algebra while further by removing remove composing matter,new matter. but notions with as It of other we reducibility will dualities,which have in be then all just future interesting it the works to other is and incorporate 5 possible find this the and truly other primitive 5 which is precisely the irreducible duality ( The notion of irreducible dualities used in this paper is defined when the gauge algebras which can be extended to the duality Combining ( the following duality Now, note the irreducible duality ( It is possible to obtainand irreducible composing dualities them. by For adding example, matter consider to the other irreducible irreducible dualities duality following from ( 4.4 Reducing irreducible dualities JHEP07(2020)012 , , , ]. , ]. , ]. 07 (2015) Nucl. SPIRE JHEP , IN SPIRE , 12 IN ][ SPIRE ][ JHEP IN ]. (2015) 157 , ][ (1996) 195 12 JHEP , ]. 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