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SCFTs in 6D David R. Morrison Superconformal field theories in six dimensions Introduction N=(1, 0) SCFTs Strings No anomalies David R. Morrison Examples F-theory University of California, Santa Barbara Recent Progress in String Theory and Mirror Symmetry Brandeis University 7 March 2015 Quivers Classification Finite subgroups of E8 SCFTs in 6D David R. Morrison Superconformal field theories in six dimensions Introduction N=(1, 0) SCFTs Strings No anomalies David R. Morrison Examples F-theory University of California, Santa Barbara Recent Progress in String Theory and Mirror Symmetry Brandeis University 7 March 2015 Based on work done with M. Bertolini, M. Del Zotto, J. J. Heckman, P. Merkx, D. Park, T. Rudelius, and C. Vafa arXiv:1312.5746, arXiv:1412.6526, arXiv:1502.05405, arXiv:1503.????? Quivers Classification Finite subgroups of E8 Introduction I I I The maximum spacetime dimension in which a superconformal field theory is possible is six (Nahm). The degrees of freedom in such a theory are not described by particles, but the theory is a local quantum field theory (Seiberg and others). The worldvolume quantum field theory for a (stack of) M5-branes is a six dimensional superconformal field theory (with maximal supersymmetry). I Compactification of the maximally supersymmetric theory has led to a host of interesting theories in lower dimesions (very active area of research since 2009). I In this talk, we will focus instead on the minimally supersymmetric theories, that is, theories with N = (1, 0) supersymmetry. SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 N = (1, 0) superconformal field theories SCFTs in 6D David R. Morrison Introduction I The conformal symmetry of these theories is so(6, 2). I The superconformal algebra is described with 8 supersymmetry generators Qi and 8 superconformal generators Sj . I The theory has an su(2) R-symmetry. I These theories typically have nontrivial global (flavor) symmetries. N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 N = (1, 0) superconformal field theories SCFTs in 6D David R. Morrison Introduction I The conformal symmetry of these theories is so(6, 2). I The superconformal algebra is described with 8 supersymmetry generators Qi and 8 superconformal generators Sj . I The theory has an su(2) R-symmetry. I These theories typically have nontrivial global (flavor) symmetries. Multiplets in a 6D supersymmetric theory: I + , fermions) Gravity multiplet (gµν , Bµν I − , fermions) Tensor multiplet(s) (S, Bµν I Vector multiplet(s) (Aµ , fermions) I Hypermultiplet(s) (φR , fermions) N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 Strings and their tension SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs I I I I Having Bµν ’s implies that there are strings in these theories, and a BPS lattice Λ ⊂ R1,T . The scalars S1 , . . . ST are naturally parameterized by a space whose universal cover is SO(1, T )/SO(T ). The expectation values hSi i control the string tensions. When hSi i = 0 we get a tensionless string, and we expect a local superconformal field theory to describe it. Non-zero expectation values of Si parameterize the Coulomb branch of the theory, and this is where the theory is typically studied in detail. Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 Anomaly-free gauge fields SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs I I I On the Coulomb branch, the gauge fields are potentially subject to an anomaly. The Green-Schwarz-West-Sagnotti mechanism specifies modified Bianchi identities for gauge fields in these theories, which gives the possibility of an anomaly-free theory: potential anomalies from fermions in matter multiplets are cancelled by the gauge field anomaly which follows from the modified Bianchi identity. In practice, this requirement is a severe constraint on the gauge groups and matter representations. Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 Anomaly-free gauge fields SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs I I For example, one solution of the anomaly constraints allows an arbitrary number of copies of su(N), labeled by an integer j = 1, . . . , p, together with (N, N) bi-fundamental matter charged under su(N)j and su(N)j+1 , with an extra N fundamentals for su(N)1 and an extra N fundamentals for su(N)p . In fact, these theories also have a global symmetry SU(N) × SU(N) which can be thought of as occupying the 0th and (p+1)st positions in the chain, so that all of the matter sits in bi-fundamentals. (Sometimes it’s gauge-global bi-fundamentals.) Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 Other anomalies I I I The global symmetry group G of such a theory is typically anomalous, and cannot be directly gauged. However, in many cases one can add additional (free) hypermultiplets charged under G to render the combined theory anomaly-free. For example, in the su(N)⊕p theory described above, the global symmetry SU(N)p+1 is anomalous unless an additional N fundamentals for this group are added to the theory. Then the combined theory has an anomaly-free global symmetry, which can be gauged (using the GSWS Bianchi identity), extending the chain by 1. One can also ask if the theory can be coupled to gravity, a necessary condition for which is that the gauge-gravity anomaly vanish (using the modified Bianchi identity for the graviton as well). This was studied in some nontrivial cases in [Del Zotto, Heckman, Morrison, Park]. SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 Examples SCFTs in 6D David R. Morrison Introduction I I I The N = (2, 0) theories have an ADE classification. N=(1, 0) SCFTs A stack p of M5-branes gives one model for the N = (2, 0) theory of type Ap−1 . Strings The heterotic e8 ⊕ e8 string compactified on a local K3 surface with a point-like instanton with instanton number k provides a family of examples of N = (1, 0) theory. I Using the Hořava–Witten model of the heterotic string, the previous example can be viewed as p M5-branes dissolved into the boundary M9-brane, creating a point-like instanton in real codimension 4. I More generally, one can put p point-like instantons at the singular point of an ALE space C2 /Γ. No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D 1 David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D 1 David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 These examples can be studied in F-theory, using F-theory/heterotic duality [Aspinwall-Morrison ’97]. 6D SCFTs from F-theory I I F-theory is a quantum gravity theory which allows a nontrivial axio-dilaton profile, exploiting the SL(2, Z) symmetry of the type IIB string. The data of an F-theory compactification is provided by a base space B, a complex line bundle L on B, and sections f and g of L⊗4 and L⊗6 , respectively, determining a Weierstrass equation y 2 = x 3 + fx + g . I I If B has complex dimension 2 and if the total space of the elliptic fibration can be blown up to a Calabi–Yau threefold, we get a 6D supersymmetric theory. The strings arise from wrapping D3-branes along complex curves in B, and their tensions are supplied by the areas of these curves. Thus, to get a tensionless string limit, we need a configuration of algebraic curves which are contractible to a point. SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D I More generally, we can study a local model B which is a neighborhood of a curve collection {Σj }. The key condition for contractibility is that the intersection matrix (Σj · Σk ) be negative definite. This matrix also defines the lattice of BPS string charges. David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies I The key quantity for understanding these models is the discriminant locus ∆ := {4f 3 + 27g 2 = 0}. This typically contains some or all of the Σj ’s as components with some multiplicity. The Kodaira classification determines the type of singular fibers and also (when supplemented by Tate’s algorithm) the gauge algebra associated with each Σj . I There can be additional, non-compact components of ∆ in B. These are associated with global symmetries. Detailed investigation: [Bertolini, Merkx, Morrison]. Examples F-theory Quivers Classification Finite subgroups of E8 Pointlike instantons on C2 /Γ, as studied by [Aspinwall-Morrison ’97]. SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs 12 + n II∗ C∞ Strings No anomalies Examples ∗ II 12 − n F-theory C0 Figure 2: Point-like E8 instantons in the simplest case. Quivers Classification Finite subgroups of E8 re the curly line represents the locus of I1 fibres. This will be the case in all subsequent rams. The overall shape of this curve is meant to be only schematic. (In particular, we e omitted the cusps which this curve invariably has.) The important aspect is the local metry of the collisions between this curve and the other components of the discriminant ch we try to represent accurately. This is the F-theory picture of the physics discussed in [6] that each point-like instanton s to a massless tensor in six dimensions (here represented as a blowup of the original e Fn ). We also see that 12 − n of the instantons are associated to one of the E8 factors the other 12 + n are tied to the other E8 [14]. After blowing up the base however, one blow down in a different way to change n. Thus after blowing up, it is not a well-defined n this section we are going to force a “vertical” line of bad fibres (along an f direction) the discriminant so that it has aon transverse “horizontal” line of II∗ SCFTs in 6D Pointlike instantons C2 /Γ, intersection as studiedwith by the [Aspinwall-Morrison along C0 without any additional local contributions to the collision from the rest of the David R. Morrison ’97]. minant. One may show [34] that such intersections of curves within the discriminant correspond to fibres with the same J-invariant. In this section we require J = 0 which Introduction sponds to Kodaira types II, IV, I∗0 , IV∗ , and II∗ . In each case, the order of vanishing N=(1, 0) SCFTs 12 + s twice the order of vanishing of b, with an playing no significant II∗ rôle. Thus, to" analyze Strings ∞ divisor B . = 0 cases we need only concern ourselves with the geometry of C the No anomalies or example, let us consider the case illustrated in figure 4 in which we add a vertical of II∗ fibres along the f direction. To do this, we must subtract 5f from B " which Examples es that what remains can only produce 7 − n and 7 + n simple point-like instantons of F-theory II∗ ype we discussed above. It is therefore clear that, whatever else C we 0 may have done to Quivers − ten n of the instantons. Note uce this extra line of II∗ fibres, we have had to “use 12 up” Classification B " intersects f twice, producing collisions between the I1 part of the discriminant and ∗ Finite subgroups of ertical line of II fibres as shown. E8 Figure 2: Point-like E8 instantons in the simplest case. 7+n II∗ C∞ re the curly line represents the locus of I1 fibres. This will be the case in all subsequent rams. The overall shape of this curve is meant to be only schematic. (In particular, we e omitted the cusps which this curve invariably has.) The important aspect is the local metry of the collisions between this curve and the other components of the discriminant II∗ C0 ch we try to represent accurately. 7 −[6]n that each point-like instanton This is the F-theory picture of ∗the physics discussed in s to a massless tensor in sixIIdimensions (here represented as a blowup of the original e Fn ). We also see that 12 − n of the instantons are associated to one of the E8 factors the other 12 + n are tied 4:to 10 theinstantons other E8 [14]. blowing up the base however, one Figure on anAfter E8 singularity. blow down in a different way to change n. Thus after blowing up, it is not a well-defined SCFTs in 6D David R. Morrison 7 + n II∗ Introduction C∞ N=(1, 0) SCFTs Strings II∗ 6−n ∗ II No anomalies C0 Examples F-theory Blow up Quivers # II∗ ! ! ! ! ! " " ! ! " " " " " II∗ Classification 7 + n II∗ II∗ 6−n C∞ C0 Figure 5: 11 instantons on an E8 singularity. Finite subgroups of E8 SCFTs in 6D The intersection of two E8 branes (Kodaira type II ∗ ) is David R. Morrison associated to a Weierstrass model whose blowup is a Calabi–Yau threefold X whose map to the base B has a Introduction (complex) two-dimensional fiber over the intersection N=(1, 0) SCFTs point. Strings I In other words, when we compactify on a circle to get No anomalies Examples an M-theory model, we find an infinite tower of light F-theory states (from wrapping an M2-brane over arbitrary Quivers algebraic curves within the two-dimensional fiber). This Classification is another signal of conformality in the parent The elliptic fibration of figure 4 is quite singular and requires many blowups in the base Finite subgroups of six-dimensional before it becomes smooth. Fortheory. example, the degrees of (a, b, δ) for II∗ fibres are (4, 5,E10) 8 espectively. two such curves and we up the point of I OnThus, the ifother hand, by intersect blowingtransversely up the base B, blow we can ntersection, the exceptional divisor will contain degrees (8, 10, 20). As in section 3, this e → e are ensure that all fibers X B one dimensional. This ndicates a non-minimal Weierstrass model, and when passing to a minimal model, L is the that Coulomb branch thethese theory. adjusted inisa way subtracts (4, 6, 12)offrom degrees and restores KX to 0. We are hus leftIwith an exceptional of degrees (4, 4, in 8), detail which isfor a curve of IV∗ fibres. This Aspinwall and Icurve worked this out the collision new curve will intersect the old curves of II∗ fibres and these points of intersection also need ∗ of two II fibers. blowing up. Iterating this process we finally arrive at smooth model (i.e., no further blowups I need to be done) when we have the chain ! ! II∗ ! I0 ! " ! " ! " ! " ! " ! I∗0 " ! IV∗ " ! I∗0 " ! II ! II ! "IV ! "II ! "II ! "IV ! I0 ! ! II∗ . ! Quivers SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies I The su(N)⊕p field theory examples can also be realized by constructions in heterotic M-theory and in F-theory. I They are given by a stack of p M5-branes at an AN−1 singularity. Quivers I They also have a quiver description as above. Finite subgroups of E8 I They are realized in F-theory by a chain of −2 curves with Kodaira fibers of type IN over each one. Examples F-theory Classification SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D Variants David R. Morrison Introduction I Other types of quivers can realized by variants of this construction. N=(1, 0) SCFTs Strings No anomalies I For example, putting flavor branes IN and IN+pk through an Ap−1 singularity leads to: Examples F-theory Quivers Classification IN IN D1 D2 A(p-1) (p-1) IN+pk IN+pk with gauge algebra Finite subgroups of E8 D(p-2) ... D Lp−1 j=1 su(N + jk). Classification of 6D SCFTs from F-theory I I I I A key ingredient is non-Higgsable clusters [Morrison-Vafa ’96, Morrison-Taylor ’12]: curve configurations with a minimal gauge algebra. They are: 3, 4, 5, 6, 7, 8, (12), 32, 232, 322. The classification is largely bottom up, relying on field theory and anomaly cancellation. One hopes to eventually understand it purely in those terms. Classification result #1: contracting {Σj } leads to a B of the form C2 /G with G ⊂ U(2). Classification result #2: all 6D SCFTs in F-theory can be described by quivers, whose links may themselves be SCFTs. (Nodes and links are explicitly classified.) SCFTs in 6D David R. Morrison Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 A test: finite subgroups of E8 The basic heterotic M-theory example SCFTs in 6D David R. Morrison Introduction 1 N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 has an important variant: the instanton solution may be twisted by a Wilson line. Such Wilson lines correspond to homomorphisms Γ → E8 up to conjugacy. A test: finite subgroups of E8 The basic heterotic M-theory example SCFTs in 6D David R. Morrison Introduction 1 N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 has an important variant: the instanton solution may be twisted by a Wilson line. Such Wilson lines correspond to homomorphisms Γ → E8 up to conjugacy. The E8 global symmetry breaks to the commutant of the image of Γ. SCFTs in 6D David R. Morrison I I Cyclic subgroups of E8 (up to conjugacy) were classified by Victor Kac. For example, there are two cases for Z2 , with commutants (E7 × SU(2))/Z2 and Spin(16)/Z2 , respectively. Certain other subgroups of E8 (up to conjugacy) were classified by D. D. Frey. This includes some dihedral groups as well as SL(2, F5 ) and A5 . All of these are relevant for ADE subgroups of SU(2). I On the other hand, we can use the F-theory classification to ask what are the infinite chains, and how an infinite chain can end. I There is an almost perfect match! Introduction N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 SCFTs in 6D David R. Morrison Group ΓAn−1 = Zn ΓDn = Dn−2 ΓE6 = T ∼ = SL(2, F3 ) ΓE7 = O ΓE8 = I ∼ = SL(2, F5 ) Order n 4(n−2) Generators ζn ζ2n−4 , δ 24 ζ4 , δ, τ Quotient groups Zk if k | n Z2 , Dih2k if k | (n−2), D` if ` | (n−2) but 2` 6 | (n−2) Z3 , A4 48 120 ζ8 , δ, τ −(ζ5 )3 , ι Z2 , S3 , S4 A5 e 2πi/n N=(1, 0) SCFTs Strings No anomalies Examples F-theory Quivers Classification Finite subgroups of E8 Finite subgroups of SU(2), where ζn ≡ Introduction , δ≡ −1 e −2πi/n 2πi/5 1 e + e −2πi/5 ι ≡ 4πi/5 1 e − e 6πi/5 e −2πi/8 e 10πi/8 1 . 2πi/5 −2πi/5 −e −e 1 1 , τ ≡ √ 2 e −2πi/8 e 2πi/8 SCFTs in 6D E7 × SU(2) cases David R. Morrison Introduction N=(1, 0) SCFTs Strings su2 [E7 ] 1 2 su4 su4 2 [SU(2)] No anomalies ... 2 [SU(4)] Examples F-theory su2 su4 [E7 ] 1 2 2 su2 [E7 ] 1 2 su6 [SU(2)] so7 3 [SU(2)] Quivers su6 2 ... 2 [SU(6)] Classification Finite subgroups of E8 so9 sp1 so11 sp2 so13 sp3 so15 1 4 1 4 1 4 1 spn−4 4 ... 1 [SO(2n)]