Infinitely many Scft
with flavor symmetry
We present evidence that for each Lie group there is an infinite tower of 4D SCFTs, which we label as with , having (at least) flavor symmetry . For , coincides with the Argyres–Douglas model of type , while for larger flavor groups the models are new (but for a few previously known examples). When its flavor symmetry is gauged, contributes to the Yang–Mills beta–function as adjoint hypermultiplets.
The argument is based on a combination of Type IIB geometric engineering and the categorical deconstruction of arXiv:1203.6743. One first engineers a class of models which, trough the analysis of their category of quiver representations, are identified as asymptotically–free gauge theories with gauge group coupled to some conformal matter system. Taking the limit one isolates the matter SCFT which is our .
Infinitely many SCFT
with flavor symmetry
Sergio Cecotti***e-mail: email@example.com and Michele Del Zotto†††e-mail: firstname.lastname@example.org
One of the most remarkable aspects of extended supersymmetry is the possibility of constructing and studying in detail many four–dimensional SCFTs which do not have any (weakly coupled) Lagrangian formulation and hence are intrinsically strongly coupled. The prototype of such theories are given by the Argyres–Douglas models [AD], which have an classification; those of type () have a global symmetry which may be gauged [CV11]. Other important classes of SCFTs are the so–called class– theories [Gaiotto, GMN09], the models of [CNV] ( being a pair of groups), and their generalizations [arnold1, arnold2, arnold3].
Of particular interest are the SCFT with an exceptional flavor symmetry, . Here the basic examples are the Minahan–Nemeschansky (MN) models [MN1, MN2] (see also [seii, ben]); the flavor symmetry alone rules out any weakly coupled description; for instance, if we gauge the symmetry of the last MN model we get a contribution to the –function which is of an hypermultiplet in the minimal representation (the adjoint) [E8beta].
The purpose of this letter is to present evidence for the existence of infinitely many such SCFT. For each Lie group — in particular, for , , and — we have an infinite tower of models with (at least) flavor symmetry. For a given , the models are labelled by a positive integer . We denote these models as . When coupled to SYM, will contribute to the YM –function as
This implies that cannot have a Lagrangian formulation except for sporadic, very special, pairs . While these sporadic Lagrangian models are not new theories, they are quite useful for our analysis because, in these special cases, we may check our general results against standard weak coupling computations, getting perfect agreement.
In simple terms our construction is based on the following ideas (see ref.[CNV] for the general set–up). We start by considering the ‘compactification’ of Type IIB on the local Calabi–Yau hypersurface of equation
where stands for (the versal deformation of the) minimal singularity of type . Seen as a 2d superpotential, corresponds to a model with central charge at the UV fixed point equal to , where is the central charge of the minimal SCFT of type . Since , the criterion of the correspondence [CNV, GVW] is satisfied, and we get a well–defined QFT in 4D. For the theory we get is just pure SYM with gauge group [CNV]. By the usual argument (see e.g. [tack, CNV, CV11]) for all the resulting 4D theory is UV asymptotically free; in facts, it is SYM with gauge group coupled to some matter which is ‘nice’ in the sense of [tack], that is, it contributes to the YM –function less than half an adjoint hypermultiplet.
Taking the limit , we decouple the SYM sector and isolate the matter theory that we call . It is easy to see that this theory should be conformal. Indeed, the ‘superpotential’ (1.1) is the sum of two decoupled terms; at the level of the BPS quiver of the 4D theory, this produces the triangle tensor product [kellerP] of the quivers and (compare, for , with the pure SYM case [CNV, cattoy]). The decoupling limit affects only the first factor in the triangle product, so, roughly speaking, we expect
Modulo some technicality, this is essentially correct. Then, from the correspondence, it is obvious that the resulting theory is UV conformal iff ‘(something depending only on ’ is. This can be settled by setting . In this case is Argyres–Douglas of type [CV11, cattoy] which is certainly UV superconformal. Hence is expected to be superconformal for all and . (Below we shall be more specific about the first factor in the rhs of (1.2).) Alternatively, we can argue as follows: the gauge theory engineered by the CY hypersuface (1.1) has just one essential scale, ; the decoupling limit corresponds to a suitably defined scaling limit ; therefore we should end up to the UV–fixed point SCFT.
The construction may in principle be extended by considering the triangle tensor products of two affine theories, , which are expected to be asymptotically–free theories with non–simple gauge groups.
Technically, the analysis of the decoupling limit is based on the ‘categorical’ classification program of 4D theories advocated in ref.[cattoy]. In the language of that paper, our problem is to construct and classify the non–homogeneous –tubes by isolating them inside the light subcategory of the 4D gauge theory.
The rest of this letter is organized as follows. In section 2 we briefly review some material we need. In section 3 we analyze the 4D gauge theories of the form : we study both the strong coupling and the weak coupling. We also discuss some examples in detail. In section 4 we decouple the SYM sector and, isolate the SCFT, and describe some of their physical properties. In section 5 we sketch the extensions to the models. Technical details and more examples are confined in the appendices.
2 Brief review of some useful facts
We review some known facts we need. Experts may prefer to jump to section 3. For the basics of the quiver representation approach to the BPS spectra of 4D theories we refer to[CV11, ACCERV1, ACCERV2][cattoy].
2.1 Af gauge theories and Euclidean algebras
We shall be sketchy, full details may be found in [CV11] and [cattoy].
The full classification of the gauge theories whose gauge group is strictly and which are both complete and asymptotically–free is presented in ref.[CV11]. Such theories are in one–to–one correspondence with the mutation–classes of quivers obtained by choosing an acyclic orientation of an affine Dynkin graph. For () and () all orientations are mutation equivalent, while in the case the inequivalent orientations are characterized by the net number (resp. ) of arrows pointing in the clockwise (anticlockwise) direction along the cycle; we write for the Dynkin graph with such an orientation (). The case is different because there is a closed oriented –loop. The corresponding path algebra is infinite–dimensional, and it must be bounded by some relations which, in the physical context, must arise from the gradient of a superpotential, [ACCERV1, ACCERV2]. For generic , is mutation–equivalent to the Argyres–Douglas model [CV11, cattoy] which has an global symmetry. By the triality property of , the Argyres–Douglas model is very special: its flavor symmetry gets enhanced to — this exception will be relevant below.
One shows [CV11, cattoy] that these affine theories correspond to SYM gauging the global symmetries of a set of Argyres–Douglas models of type as in the table
where stands for the empty matter and for a free hypermultiplet doublet. The Type IIB geometry which engineers the model associated to each acyclic affine quiver in the first column is described in ref.[CV11]. For instance, for the geometry is
One also shows [CV11, cattoy] that the contribution of each matter system to the YM –function coefficient
is given by
Using this formula, one checks [CV11] that the models listed in (2.1) precisely correspond to all possible (complete) matter systems which are compatible with asymptotic freedom.
For our purposes it is important to describe the decoupling process of the matter from the SYM sector; it is described in terms of the BPS spectrum in ref.[cattoy]. The BPS states which have a bounded masses in the limit are precisely the BPS particles with zero magnetic charge. In terms of the representations of the acyclic affine quiver these light states correspond to the ones having vanishing Dlab–Ringel defect [RI, CB]. To describe the BPS states which remain light in the decoupling limit, one introduces the Abelian (sub)category of the light representations111 See also §. 2.3 below. [cattoy], which — in the affine case — precisely corresponds to the category of the regular representation [RI, CB]. This category has the form [RI]
where the are stable periodic tubes; for generic , is a homogeneous tube ( period 1) [RI]. This, in particular, means that for these affine models the light BPS states consists of a single vector–multiplet, the boson, plus finitely many hypermultiplets, which are the BPS states of the matter system (the matter spectrum at depends on the particular BPS chamber). It follows that the matter sector corresponds to the rigid bricks222 A representation is a brick iff , and it is rigid if, in addition, . of [cattoy]. The rigid bricks belong to the finitely–many tubes which are not homogeneous. It is well–known that for each affine quiver there is precisely one non–homogeneous tube of period for each matter subsystem in the second column of table (2.1). To show that the matter isolated by the decoupling process is the combination of Argyres–Douglas models in table (2.1) one may use either rigorous mathematical methods or physical arguments. Let us recall the mathematical proof [cattoy]. The quiver of the matter category associated to a tube of period is obtained by associating a node to each simple representation in the tube and connecting two nodes , by arrows. The is easily computed with the help of the symmetry of the periodic tube; the resulting quiver is then a single oriented cycle of length . The same results may be obtained on physical grounds as follows (say for the case ): in eqn.(2.2), stands for the scale set by asymptotic freedom, as specified by the asymptotic behavior of the complex YM coupling [tack]
The limit is . We may take this limit keeping fixed either or . These two limits correspond, respectively, to considering the local geometry of the hypersurface (2.2) around and , which are precisely the two poles of the with affine coordinate ; this is identified with the index set in eqn.(2.4) (and also with the Gaiotto plumbing cylinder [cattoy]). Now it is clear that as we get two decoupled physical systems described by the geometries
which (formally at least) correspond to and , respectively. The periodicity of the two periodic tubes then corresponds to
The cyclic quiver should be supplemented by a superpotential . The correct is easy to compute [cattoy]: is just the –cycle itself. The pair is mutation–equivalent to a Dynkin quiver [CV11], and hence the matter system consists of one Argyres–Douglas system per each (non–homogeneous) tube of period in the family (2.4). This gives table (2.1).
2.2 Triangle tensor products of theories
This subsection is based on [CNV, kellerP] and §. 10.1 of [cattoy]. Suppose we set Type IIB on a local CY hypersurface of the form
From the correspondence [CNV], we know that this geometry defines a good 4D QFT provided the LG model defined by the superpotential has at the UV fixed point. In this case the 4D BPS quiver has incidence matrix333 The incidence matrix of a –acyclic quiver is defined by setting equal to the number of arrows from node to node , a negative number meaning arrows in the opposite direction . is then automatically skew–symmetric.
where is the Stokes matrix encoding the BPS spectrum of the (2,2) LG model [CV92]. For superpotentials of the special form (2.7) the 2d theory is the product of two totally decoupled LG models, and hence the BPS spectrum of the 2d theory may be obtained as a ‘product’ of the ones for the decoupled models, . This gives the incidence matrix for
The corresponding operation at the level of quivers is called the triangle tensor product [CNV].
It is convenient to give an algebraic interpretation of this ‘product’ of (2,2) LG theories which fixes the associated superpotential [kellerP, arnold1, cattoy]. We assume that the quivers and of the (2,2) LG theories , are acyclic — hence, by classification [CV92, CV11], either orientations of Dynkin graphs or acyclic orientations of affine graphs. Let , be the corresponding path algebras. We can consider the tensor product algebra spanned, as a vector space, by the elements and endowed with the product
Let , (resp. ) be the lazy paths ( minimal idempotents) of the algebra (resp. ). The minimal idempotents of the tensor product algebra are ; for each such idempotent there is a node in the quiver of the algebra which we denote by the same symbol. The arrows of the quiver are444 Here and are the maps which associate to an arrow its source and target node, respectively.
However, there are non–trivial relations between the paths; indeed the product (2.9) implies the commutativity relations
In the physical context all relations between paths should arise in the Jacobian form from a superpotential. In order to set the commutativity relations in the Jacobian form, we have to complete our quiver by adding an extra arrow for each pairs of arrows ,
and introducing a term in the superpotential of the form
enforcing the commutativity conditions (2.11). The resulting completed quiver, equipped with this superpotential, is called the triangle tensor product of , , written [kellerP][arnold1, cattoy].
Examples. If both , are Dynkin quivers their tensor product corresponds to the models constructed and studied in [CNV]. If is the Kronecker (affine) quiver and is a Dynkin quiver of type , is the quiver (with superpotential) of pure SYM with gauge group [CNV, ACCERV2, cattoy].
Although mathematically the procedure starts with two acyclic quivers, formally we may repeat the construction for any pair of quivers, except that the last step, the determination of , may be quite tricky. When one factor, say , is acyclic there is a natural candidate for the superpotential on the quiver: is the sum of one copy the superpotential of per node of , plus the terms (2.13) implementing the commutativity relations.
2.3 The light subcategory and –tubes
Suppose we have a theory, which is a quiver model in the sense of [CV11, ACCERV1, ACCERV2] and behaves, in some duality frame, as SYM with gauge group coupled to some ‘matter’ system. We fix a quiver which ‘covers’ the region in parameter space corresponding to weak gauge coupling. Then there is a set of one–parameter families of representations of the quiver , , , which correspond to the simple –boson vector–multiplets of . Let be the corresponding charge vectors. The magnetic charges of a representations are then defined by [cattoy, half, nonsimply]
where is the Cartan matrix of the gauge group and the skew–symmetric integral bilinear form is defined by the exchange matrix of the quiver .
States of non–zero magnetic charge have masses of order as , and decouple in the limit. Thus the BPS states which are both stable and light in the decoupling limit must correspond to quiver representations satisfying the two conditions: 1) for all ; 2) if is a subrepresentation of , then for all . The subcategory of all representations satisfying these two conditions is an exact closed Abelian subcategory which we call the light category of the theory (w.r.t. the chosen duality frame).
If the gauge group is simple the light category has a structure similar to the one in eqn.(2.4); indeed [cattoy]
where the Abelian categories are called –tubes. Almost all –tubes in eqn.(2.15) are homogeneous, that is, isomorphic to the ones for pure SYM with group . The matter corresponds to the (finitely many) –tubes in eqn.(2.15) which are not homogeneous. Just as in §. 2.1, there is a finite set of points such that the –tube is not homogeneous, and we can limit ourselves to consider one such –tube at the time, since distinct –tubes correspond at to decoupled matter sectors ([cattoy] or apply the physical argument around eqn.(2.6) to the hypersurface (1.1)).
A very useful property of the light category , proven in different contexts [cattoy, half, nonsimply], is the following. Assume our theory has, in addition to , a decoupling limit (e.g. large masses, extreme Higgs breaking), which is compatible with parametrically small YM coupling , and such that the decoupled theory has support in a subquiver555 As explained in [half], this happens whenever the controlling function of the corresponding subcategory [cattoy] is non–negative on the positive cone in of actual representations. of . Then
a relation which just expresses the compatibility of the decoupling limit with . This fact is quite useful since it allows to construct recursively the category for complicate large quivers from the light categories associated to smaller quivers. The light category has a quiver (with relations) of its own. However, while typically a full non–perturbative category has a –acyclic quiver, the quiver of a light category has, in general, both loops and pairs of opposite arrows (see examples in [cattoy, half, nonsimply]). It depends on the particular superpotential whether the pairs of opposite arrows may or may not be integrated away.
3 The models
We consider the triangle tensor product where stands for an acyclic affine quiver (listed in the first column of table (2.1)), and is an Dynkin quiver. Since and , the total is always less than 2, and thus all quivers of this form correspond to good QFT models. If , the model correspond to pure SYM with group . In figure 1.1 we show the quiver (with superpotential) corresponding to the simplest next model i.e. , the general case being a repetition of this basic structure666 For , and we have an equivalent square product quiver without ‘diagonal’ arrows; for we may reduce to a quiver with just diagonal arrows.. We call the full subquiver ‘the affine quiver over the –th node of the Dynkin graph ’, or else ‘the affine quiver associated to the the –th simple root of the group ’; it will be denoted as , where .