2-Group Symmetries of 6d Little String Theories and T-duality
22-Group Symmetries of 6d LittleString Theories and T-duality
Michele Del Zotto † and Kantaro Ohmori (cid:93) † Department of Mathematics and Department of Physics and Astronomy,Uppsala University, Uppsala, Sweden (cid:93)
School of Natural Sciences, Institute for Advanced Study,Princeton, NJ 08540, USA (cid:93)
Simons Center for Geometry and Physics, SUNY Stony Brook,Stony Brook, NY 11794, USA
Abstract
We determine the 2-group structure constants for all the six-dimensional littlestring theories (LSTs) geometrically engineered in F-theory without frozensingularities. We use this result as a consistency check for T-duality: the 2-groups of a pair of T-dual LSTs have to match. When the T-duality involvesa discrete symmetry twist the 2-group used in the matching is modified. Wedemonstrate the matching of the 2-groups in several examples.————————–September 2020 a r X i v : . [ h e p - t h ] S e p ontents
1. Introduction 22. LST from F-theory: a lightning review 43. 2-group structure constants and the LS charge 64. 2-groups and T-duality 8 N = (2 ,
0) and its T-dual N = (1 ,
1) . . . . . . . . . . . . . . . . . . . . 104.3. LST for M5 branes along S × C / Γ and their T-duals . . . . . . . . . . 104.4. Heterotic instantons probing a C / Z k singularity . . . . . . . . . . . . . . 134.5. Heterotic instantons probing an E singularity . . . . . . . . . . . . . . . 14
5. An endpoint constraint for T-duality from (cid:98) κ P ,
1) LSTs . . . . . . . . . . 19
A. Proof that (cid:98) κ P is invariant under blow-ups 21
1. Introduction
In the past few years we have made lots of progresses about six-dimensional supersym-metric systems decoupled from gravity. Such theories can have either 16 or 8 conservedsupercharges, corresponding to (2 ,
0) and (1 , ,
0) supersymmetry respectively. In6D there are two types of UV behaviors: superconformal field theories (SCFTs) and littlestring theories (LSTs). This was conjectured based upon gauge anomaly cancellationreading between the lines of [1], and can be argued for using geometric engineering tech-niques in F-theory [2]. The hallmark of these models is given by the presence of stringsamong their excitations [3, 4]. SCFTs are characterized by the fact that such stringsbecome all tensionless at the conformal point. LSTs instead have an intrinsic built-instring tension, which entails these systems have T-duality and hence are not ordinarysix-dimensional local quantum field theories [5]. Famous examples of the above are ob-tained by carefully decoupling gravity from the worldvolume theories of stacks of NS5srespectively in IIA, IIB, and Heterotic superstrings [5, 7]. Some other early foundationalworks on the subject include [8–14] (see also [15] for a review). A generalization of QFT that can describe LSTs is proposed in [6]. A 2-groupglobal symmetry [22, 23] is a mixture of a 0-form (i.e. ordinary) global symmetry anda 1-form global symmetry (in the language of [24], see also [25]). In particular such2-group global symmetry in a LST always contains a universal subfactor of the form G (cid:98) κ P , (cid:98) κ R = (cid:16) P (0) × SU (2) (0) R (cid:17) × (cid:98) κ P , (cid:98) κ R U (1) (1) LST , (1.1)where P (0) is the 6d Poincar´e 0-form symmetry group, SU (2) (0) R is the global N = (1 , U (1) (1) LST is the 1-form symmetry associated to littlestring (LS) charge [16]. Here (cid:98) κ P and (cid:98) κ R are the 2-group structure constants, that deter-mine the mixture between higher form global symmetries of different dimensionality. Inparticular, the invariant background curvature 3-form H (3) LST satisfies a modified Bianchiidentity involving the background instanton densities consisting of P (0) × SU (2) R back-grounds and the constants (cid:98) κ P and (cid:98) κ R (see (3.3)). These quantities have been computedrecently for 6d (1,1) LSTs and for the 6d (1,0) LSTs on NS5 branes of the HeteroticSpin(32) / Z [16].The main purpose of this short note is to explore the 2-group symmetries of all theexamples of LSTs constructed via F-theory in [2], building upon the classification of 6dSCFTs [26, 27]. . Our results can be summarized in the following Claims: • There is a simple formula to compute the structure constants for the 2-groupsymmetry of all six-dimensional LSTs; • The structure constant for the mixture of the one-form symmetry and the R -symmetry is always non-zero (for an interacting unitary LST); • The 2-groups should be the same between T-dual pairs of LSTs, provided theT-duality does not involve twists; • For twisted T-dualities (i.e. T-dualities which involve symmetry twists), a slightlymore complicated relation exists. We substantiate the claims above by exhibiting several non-trivial examples below. Asthese examples demonstrate the 2-group structure can provide a useful consistency checkfor T-dualities among 6d LSTs, which is a useful criterion to exploit in the context of We refer our readers to the important foundational papers [17–21] where the crucial interplay amongthe Green-Schwarz mechanism and 2-groups (and string 2-Lie algebras) was originally derived. See also [28–30]. Equation (3.6). Notice that it is possible that the T-dual of an untwisted LST is a twisted LST. For example, theT-dual of the 6d N = (1 ,
1) supersymmetric Yang-Mills LST with a non-simply laced gauge groupis a twisted compactification of a 6d N = (2 ,
0) LST. See sections 4.1 and 6. The latter was the main motivation forus to publish this study.The structure of this letter is as follows. In section 2 we give a short coincise review ofthe geometric engineering of LSTs in F-theory to fix notations and conventions. Alongthe way, we also begin exploring the geometric engineering counterpart of the field theo-retical results of [16], in particular we find the origin of U (1) (1) LST within the defect groupof the corresponding LST. In section 3 the formulas for (cid:98) κ P and (cid:98) κ R are given. In section 4several examples are discussed of T-dual pairs of LSTs both with 16 and 8 supercharges.In all cases we find consistency. In section 5 we discuss a constraint on endpoints for T-dual pairs which arises from (cid:98) κ P . In section 6 the 2-group structure constant matching isgeneralized to twisted T-dualities and examples with 16 supercharges are demonstrated. Disclaimer.
To keep this paper short we are not pedagogical: this will benefit theexperts, but might make this paper hard to read for a novice. We refer the latter tothe first few sections of [35] or to the review [36] for the necessary background aboutgeometric engineering 6d theories in F-theory, as well as to the paper [16] for a beautifuldiscussion of 2-groups in the context of 6d theories.
2. LST from F-theory: a lightning review
In this section to fix notation and conventions we briefly review some aspects of thegeometric engineering of (isolated) 6d LSTs in F-theory [2] that are relevant for our dis-cussion below. The F-theory geometric engineering involves a 3-CY which is an ellipticfibration over a two complex-dimensional non-compact K¨ahler surface S . The case inwhich the 3-CY is a general genus-one fibration is relevant for twisted compactifications:the 6d F-theory dynamics is captured by the Jacobian of the genus-one fibration, in-equivalent genus-one fibration with the same Jacobian correspond to different twists ofthe 6d theory down to 5d. The strings of the 6d theory arise from D3 branes wrapping the compact curves inthe base, and the lattice Λ ≡ H ( S , Z ) is identified with the string charge lattice of thetheory. The lattice Λ is canonically equipped with a quadratic intersection pairing( · , · ) : H ( S , Z ) × H ( S , Z ) → Z (2.1) See the IAS seminar
Geometry and 5d N=1 QFT by Lakshya Bhardwaj, which is available online atthe url:https://video.ias.edu/HET/2020/0330-LakshyaBhardwaj where such applications were an-nounced [34]. Throughout this note for simplicity we will assume the backgrounds do not involve O + planes, whichwould require a slightly different formulation of the theory [30]. Our methods can be generalized tothat class and it would be interesting to do so. A Jacobian fibration on the base S is equivalent to the axio-dilaton backgound of type IIB string onthe base. The choice of a particular 3-CY realizing the given Jacobion corresponds to an additionalstructure, i.e. the twist on S , in the F-theory/M-theory T-duality [37, 38]. S is ruled and has a unique homology class of self intersection zero, which istherefore not shrinkable: its volume defines a scale for this geometry, that is identifiedwith the LST scale. We denote the corresponding curve Σ . Wrapping Σ with a D3brane one obtains the BPS little string of the model. Choosing a basis of generators Σ I for H ( S , Z ), the negative of the intersection paring η IJ ≡ − (Σ I , Σ J ) I, J = 1 , ..., r + 1 (2.2)is identified with the Dirac pairing among BPS strings. The integer r is the rank of thecorresponding 6d theory, i.e. the dimension of the corresponding tensor branch. Theunit little string charge ∆ ∈ Λ is given by the collection of (positive) integers N I suchthat Σ = r +1 (cid:88) I =1 N I Σ I gcd( N , ..., N r +1 ) = 1 ∆ = ( N , ..., N r +1 ) (2.3)and by construction it corresponds to the (unique) primitive eigenvector of η IJ in Λ withzero eigenvalue [2].In geometric engineering the vacuum expectation values of the scalar components inthe 6d tensormultiplets are identified with the volumes of the curves Σ I : (cid:104) Φ I (cid:105) ∼ vol(Σ I ) . (2.4)In particular, for a little string theory there is a constraint on the vevs of tensormultipletsscalars M s = vol(Σ ) = r +1 (cid:88) I =1 N I vol(Σ I ) = r +1 (cid:88) I =1 N I (cid:104) Φ I (cid:105) . (2.5)The matrix η IJ also encodes the positive semi-definite kinetic matrix for the tensormultiples on the tensor branch of a LST. We denote b (2) I the dynamical self-dual 2-formfields on the tensor branch. The linear combination B (2) LST ≡ r +1 (cid:88) I =1 N I b (2) I (2.6)is therefore non-dynamical and it corresponds to the superpartner of the scalar in (2.5):this is the background 2-form tensor field for the U (1) (1) LST higher form symmetry. Defect group for LSTs.
At this point it is also nice to remark that by an appropriate If the LST is obtained as the worldvolume theory on NS5 branes decoupled from the gravity sector,this background U (1) (1) LST is literally the NSNS B -field in the decoupled gravity supermultiplet, andthe corresponding little string is the fundamental string of the ambient string theory. D (2) = Z ⊕ p (cid:77) j =1 Z m j (2.7)where the integers m j > η IJ [39–46]. Notice that the first factor is the term corresponding to the zero eigenvaluegiving the LS charge: as remarked in [45] whenever the defect group has a factor Z indimension d , we expect to obtain a U (1) ( d − higher form symmetry. For the theorieswe are considering in this paper d = 2 and we obtain precisely the U (1) (1) LST higher formsymmetry we encountered above.
Remark.
In this note we are interested only in the fate of the 2-group structure constantsunder T-duality. The remaining factors of the defect group are slightly more subtle toanalyze because(a) upon circle reduction (and twist) factors in D (2) can also mix with factors in D (1) (associated to the global form of the gauge group of the 6d LST) in non-trivialways, and(b) if the reduction involves a twist, the corresponding discrete 0-form symmetry canact on the lattice Λ and can have a nontrivial interplay with D (2) .We plan to address these phenomena in more details in future work [47].
3. 2-group structure constants and the LS charge
In this section we derive general formulas for the 2-group structure constants (cid:98) κ P and (cid:98) κ R and we discuss several applications to untwisted T-dual pairs.Whenever one of the curves in the F-theory base is a part of the discriminant locusof the elliptic fibration, there is a corresponding non-abelian gauge group for which thestrings are BPS instantons. In such a case, the corresponding tensor multiplet has aGreen-Schwarz coupling, necessary for the cancellation of the gauge anomaly. In facts,all tensor multiplets can be given GS couplings involving background gauge fields forthe Poincar´e symmetry as well as for the other global symmetries of the theory (seee.g. [48–50]). As we shall see below, in the case of LSTs, the fact that each dynamicaltensor field has such Green-Schwarz coupling generates an interplay among backgroundsfor the fields entering in the anomaly polynomial and the 2-form background field for the U (1) (1) LST higher form symmetry. This is the origin of the 2-group symmetry for LSTs [16]. If we had a Z ( d ) symmetry, the background of it would be a Z -valued d -cocycle H ( d ) . Howeverit is natural to expect that such a background should actually be realized as the background fieldstrength H ( d ) = dB ( d − with a U (1) ( d − background B , in a continuum QFT. su ( k ) so ( k ) sp ( k ) e e e g f h ∨ k k − k + 1 12 18 30 4 9Table 1: Dual Coxeter numberesRecall that the dynamical two-form fields b (2) I have a Green-Schwartz coupling of theform [49] η IJ (cid:90) b (2) I ∧ X (4) J . (3.1)Here the 4-form X (4) J can be determined field theoretically for all those tensors thatare involved in the cancellation of gauge anomaly. This is always the case for tensorswith pairing η II ≥
3. The only tensors that are not paired to gauge groups in theF-theory construction must have η II = 1 or 2. We assume that in the former casethe corresponding model is an E-string, in the latter the N = (2 ,
0) theory of type a .As we have discussed above we are not considering a frozen F-theory geometry. Withthese assumptions, the GS term η IJ X (4) J for gravity and R-symmetry background fieldsis [49, 51, 52] (see also [16]) η IJ X (4) J = h ∨ g I c ( R ) + 14 ( η II − p ( T M ) , (3.2)where h ∨ g I is the dual coxeter number of the gauge group g I coupled with the I -thtensormultiplet, and we normalize h ∨ ∅ to 1 for the cases η II = 1 , The index I in η II should not be summed.The 2-group structure constants for the LST are captured by the modified Bianchiidentity for the 2-form background field of U (1) (1) LST [16]12 π dH (3)
LST ≡ (cid:98) κ R c ( R ) − (cid:98) κ P p ( T M ) . (3.3)In presence of GS couplings, all the tensor fields have modified Bianchi identities of theform 12 π dH (3) I = η IJ X (4) J = h ∨ g I c ( R ) + 14 ( η II − p ( T M ) (3.4)Now from (2.6) H (3) LST = r +1 (cid:88) I =1 N I H (3) I (3.5) These two cases in facts are better thought of as having gauge algebras sp and su respectively,which indeed have h ∨ = 1 [35] by continuation. We will use both notation interchangeably below. We have normalized our characteristic classes with the opposite conventions of [16] — compare ourequation 3.3 with their (1.18). (cid:98) κ R = (cid:80) r +1 I =1 N I h ∨ g I (cid:98) κ P = − (cid:80) r +1 I =1 N I ( η II −
2) (3.6)thus determining the universal 2-group structure constant for all 6d LSTs from F-theoryconstructed in [2].
Remarks:
1. We stress that by including the other global symmetry background gauge fieldsin η IJ X (4) J computing the structure constants for the other factors of the 2-groupassociated to background fields for the other global symmetries is straightforwardby the same method.2. For the theories with r = 0 the method here is not strictly speaking applicable, butin the absence of paired tensors the same formula can be derived from the mixedanomaly [16].3. By anomaly inflow from 6D to 2D [51, 52], the modified Bianchi (3.4) induces the’t Hooft anomaly on the little string. (cid:98) κ R and (cid:98) κ P are the ’t Hooft anomalies of SU (2) R and SO (4) symmetries on the worldsheet. What is special to the littlestring is that its charge is not gauged, and therefore the little string worldsheettheory and its anomaly in a LST can be directly compared with that of a candidateT-dual LST.
4. 2-groups and T-duality
Here we see examples of calculations of the structure constants for a few theories, andwe apply these as a consistency check for T-dualities: the 2-group structure constantshave to match for T-dual pairs of LSTs. This is simply the natural generalization ofsymmetry matching we do to check a proposed duality. Combined with the obviousconditions that the 5D rank (i.e. the sum of the 6D rank and the ranks of the gaugegroups in the tensor branch EFT) should match between a T-dual pair, the 2-groupstructure constants provides a strict condition for a pair of LSTs to be T-dual.
Before diving into examples, let us review (and slightly extend) the geometric version ofLST T-duality in F-theory [2]. We define a pair of 6d LSTs T and (cid:98) T to be T-dual iftheir (untwisted) circle compactifications give rise to the same 5d KK theory. We definea pair of 6d LSTs T and (cid:98) T to be twisted T-dual if they become equivalent 5d KK8heories upon compactification on a circle in which at least one of the two theories istwisted by the action of a (possibly discrete) symmetry. An analogous effect is well-known in the full heterotic string theory [53, 54]. The twisted T-dualities of 6d LSTs area rather less understood phenomenon.Let us denote the local 3-CY we are considering X . As we have reviewed in section2 the 6d physics is fully determined by the F-theory of the Jacobian, which we denote F/ J X , where J X is an elliptic fibration over the base S . We denote the corresponding 6dtheory T F/ J X . Upon circle compactification we obtain a 5d KK theory T M/ X . Geometryseems to suggest there are two cases to be considered • Case 1 : X is elliptically fibered over the base S and thus J X (cid:39) X : in this case,the theory T M/ X is just the circle reduction of T F/ J X at finite radius by the usualM-theory/F-theory T-duality [55]; • Case 2 : X is a genus-one fibration over the base S : in this case the theory T M/ X is a twisted circle compactification of T F/ J X [38]. LST T-dualities from geometry [2]. The physics of T M/ X is such that the 3-CY X and its resolutions correspond to a given chamber on the 5d Coulomb branch. Otherchambers are realized by flopping
X → X µ , where we have denoted with X µ the 3-CY obtained from X by a sequence of flop transitions µ . If the 3-CY X µ admits aninequivalent genus-one fibration, over a different base (cid:98) S it will give rise to an inequivalentJacobian (cid:98) J X µ and therefore to a different 6d theory T F/ (cid:98) J X µ obtained from the F-theoryof (cid:98) J X µ . Remark.
Notice that in the above discussion µ could also be the identity (correspondingto no flops): if that is the case X itself admits two inequivalent genus-one fibrations.This is often the case for LSTs of type K (in the terminology we introduce in section 5below).Now we can distinguish between the two cases(a) T-duality : If X and X µ have inequivalent elliptic fibrations, we have a T-dualitybetween the LSTs T F/ J X and T F/ (cid:98) J X µ (b) Twisted T-duality : If X and X µ have inequivalent genus-one fibrations of whichat least one is not elliptic, we have a twisted T-duality between the LSTs T F/ J X and T F/ (cid:98) J X µ Not many examples are known of twisted T-dualities: we discuss some for the case ofLSTs with 16 supercharges in section 6 below. See e.g. section 3 of [56] for a review of the geometric engineering dictionary in M-theory. .2. N = (2 , and its T-dual N = (1 , The N = (2 ,
0) LST of type g is T-dual to the LST which is the UV completion of the6d N = (1 ,
1) pure SYM gauge theory with gauge algebra g . For the N = (2 ,
0) case,the string Dirac pairing is identical to the Cartan matrix of the corresponding affine Liealgebra g (1) . The LS charge coincides with the minimal imaginary root of g (1) , in otherwords it is given by N I = d I (4.1)where d I are the Dynkin (co)marks for the algebra g ( d I for the affine nodes is understoodto be 1). The self-intersection number η II is 2 for all I . Therefore, we obtain (cid:98) κ R (type g N = (2 ,
0) LST) = (cid:88) I d I h ∨ ∅ = h ∨ g (4.2) (cid:98) κ P (type g N = (2 ,
0) LST) = 0 . (4.3)On the other hand, from the mixed anomaly argument of [16], the structure constantsfor the 2-group symmetry of the N = (1 ,
1) LST of type g is (cid:98) κ R (type g N = (1 ,
1) LST) = h ∨ g (4.4) (cid:98) κ P (type g N = (1 ,
1) LST) = 0 . (4.5)Clearly we have a match. Notice that this equality is valid because the 2-group symmetryof the N = (2 ,
0) LST has a non-trivial structure constant κ R even though the theorydoes not have any gauge field on its tensor branch. S × C / Γ and their T-duals As a first (1 ,
0) example we consider slight variaton on the theme in the previous example.Let us consider the LST living on a stack of K M5 branes with transverse space S × C / Γin M-theory. The latter is realized in F-theory by a geometry of the form g Γ g Γ · · · g Γ // · · · // (4.6)where the symbol // indicates that the K curves in the base form a closed loop. TheLST which is T-dual to (4.6) is geometrically engineered with a collection of -2 curvesintersecting along an affine g (1)Γ diagram, with fiber I d I K , where d I are the correspondingDynkin labels for each node.Whenever Γ (cid:54) = Z N the above geometry is still singular, and the generalized quivercontains minimal ( g Γ , g Γ ) conformal matter [57] at each collision of -2 curves. For all theabove geometries (cid:98) κ P = 0, while (cid:98) κ R is non-trivial. The case Γ is a cyclic subgroup of SU (2) . Let us consider the example Γ = Z N . In10hat case the LS charge is (1 , , , ...,
1) and the corresponding gauge groups are simply su N , therefore (cid:98) κ R = KN. (4.7)The symmetry of this formula is not a coincidence, and it is indeed expected: in this caseT-duality is precisely swapping the fiber with the base of the fibration in the F-theorygeometry su N su N · · · su N // · · · K // T ←→ su K su K · · · su K // · · · N // (4.8)A third T-dual has been proposed in [58] for this class of models, that has an F-theoryrealization su NK/(cid:96) su NK/(cid:96) · · · su NK/(cid:96) // · · · (cid:96) // (4.9)where (cid:96) = gcd( N, K ). It is straightforward to check that also this model share the same2-group structure constants, which gives a further consistency check to the proposalof [58].
The case Γ is a binary dihedral subgroup of SU (2) . If we take Γ to be the binarydihedral group of order 8, we have g Γ = so , and the corresponding resolved base is so sp so sp · · · so sp // · · · K K // (4.10)where the elementary cell 4 , K times. For this theory the LS charge is(1,2,1,2,...,1,2) and we have (cid:98) κ P = 0 (cid:98) κ R = 8 K. (4.11)The corresponding T-dual LST in this case is given by a base that consists of -2 curvesintersecting along an affine d (1)4 diagram, all supporting a fiber which is of I d I K typewhere d I are the Dynkin labels for d (1)4 . The latter also coincide with the correspondingLS charge (1 , , , , (cid:98) κ R = K + K + K + K + 2 · K = 8 K (4.12)as expected. For a general binary dihedral group we obtain g Γ = so n and the corre-sponding resolved base is so n sp n − so n sp n − · · · so n sp n − // · · · K K // (4.13)The LS charge is the same as the structure of the BPS string lattice is unaltered. There-11ore we obtain (cid:98) κ P = 0 (cid:98) κ R = 4( n − K. (4.14)In addition, the 5D rank r D of the theory is r D = 2 K − Kn + K ( n −
4) = K (2 n − − . (4.15)The T-dual configuration is a collection of -2 curves arranged along an affine d (1) n diagram,with fibers of I d I K type where d I are the Dynkin labels for d (1) n . The latter has rank r D = n + 4( K −
1) + ( n − K −
1) = K (2 n − − . (4.16)The LS charge in this case is (1 , , , , , ..., Remark . One might wonder whether there is a third T-dual, as it seems to be the casewhen Γ is cyclic. One can immediately see that there can be no nontrivial T-dualitiesamong the class of theories described by (4.13) from the constrains r D and (cid:98) κ R . However,when K = 1, the N = (2 ,
0) and (1 ,
1) LSTs of type so (4 n −
6) have the same r D and (cid:98) κ R as the theory (4.13). It is interesting to determine whether these LSTs are actuallyT-dual to each other or not. The case Γ is an exotic discrete subgroup of SU (2) . We can also consider moreexotic examples, for instance for Γ the binary tetrahedral discrete subgroup of SU (2),we have g Γ = e and the corresponding resolved base is sp su sp e sp su sp e · · · sp su sp e // · · · K K K K // (4.17)The associated LS charge is (3 , , , · · · , , , ,
1) and (cid:98) κ P = 0 (cid:98) κ R = 24 K (4.18)are the 2-group structure constants from our formula. The corresponding T-dual LSTin this case is given by a base that consists of -2 curves intersecting along an affine e (1)6 diagram, all supporting a fiber which is of I d I K type where d I are the Dynkin labelsfor e (1)6 . The latter also coincide with the corresponding LS charge (1 , , , , , , (cid:98) κ R = K + K + K + 2 · K + 2 · K + 2 · K + 3 · K = 24 K (4.19)as expected. 12 .4. Heterotic instantons probing a C / Z k singularity Let us consider another simple N = (1 ,
0) example, which is given by a stack of N Heterotic E × E NS5 branes probing the C / Z k singularity. The tensor branch geometryis ∅ ∅ su (2) su (3) · · · su ( k ) · · · su ( k ) su ( k − · · · su (2) ∅ ∅ · · · · · · · · · . (4.20)where the total number of the nodes (including the LST scale) is N + 1 and (for the sakeof notational simplicity) we assume N + 1 > k + 1. The LS charge for this geometry is(1 , , , ..., κ R = (2 + k ( k + 1) + ( N + 1 − (2 k + 2)) k ) = 2 − k + kN (4.21) κ = 2 . (4.22)The dimension of the Coulomb branch after circle compactification to 5d is r = (cid:88) i (rank( g i ) + 1) − − k + kN. (4.23)The T-duality is the well known E × E T ←→ Spin(32) / Z heterotic T-duality from [9]. The tensor branch structure of N Spin(32) / Z instantonsprobing the A k − singularity is, when k is even, sp ( N ) su (2 N − su (2 N − · · · su (2 N − k + 8) sp ( N − k )1 2 2 · · · . (4.24)as a first check, the 5d rank is r = 12 (cid:16) N − k + 8 (cid:17)(cid:16) k (cid:17) + N + N − k − kN − k + 1 . (4.25)the structure constants are κ R = 2 − k + kN (4.26) κ = 2 . (4.27) The tensor branch of the LST is hard to see in the heterotic string frame, but it becomes evident inthe heterotic M-theory frame and its reduction to superstrings of Type I’. su and sp groups which have the property h ∨ g = rank( g ) + 1 (4.28)Combined with the fact that the LS charge null vector N I = 1 ∀ I , the matching of r implies the matching of (cid:98) κ R , therefore the structure constant does not give an additionalconstraint in this case. E singularity As a more non-trivial example, we consider N heterotic instantons probing an E (binarytetrahedral) singularity. The tensor branch EFT of both cases are studied in [9] andexpected to be T-dual to each other. The E × E side is, for N = 11, [9, 57] su g f su e su e su f g su . (4.29)Here accounting for N is slightly more complicated due to brane fractionalization. Onetrick is to find the E-string LST with largest rank into which this theory has a Higgsbranch flow. For the case at hand there is a tensor subbranch, where the theory lookslike su g f e e f g su . (4.30)from there it can flow into the E-string LST with N = 11. For general N ≥
10, thetheory should have a tensor subbranch with effective description su g f e · · · e f g su · · · . (4.31)the full tensor branch structure can be obtained by the blow-up method in [26]. The 5drank is r = 12 N − . (4.32)The LS charge for (4.29) is(1 , , , , , , , , , , , , , , , , , , , , , , . (4.33)for higher N , the pattern 1 , , , (cid:98) κ R = 24 N − , (4.34) (cid:98) κ P = 2 . (4.35)14he Spin(32) / Z side is, from [9], sp ( N ) so (4 N − sp (3 N − su (4 N − su (2 N − , (4.36)when q ≥
8. The 5d rank of this theory is r = 4 + N + 2 N − N −
24 + 4 N −
33 + 2 N −
17 = 12 N − , (4.37)which is the same as (4.37). The LS charge is (1,1,3,2,1), and the structure constantsare (cid:98) κ R = ( N + 1) + (4 N −
18) + 3(3 N −
23) + 2(4 N −
32) + (2 N − N − (cid:98) κ P = 2 , (4.38)which are consistent with the E × E side, as expected.
5. An endpoint constraint for T-duality from (cid:98) κ P Our reader might have noticed that while (cid:98) κ R can take various values, (cid:98) κ P take only thevalues 0 or 2 in all the above examples. This is actually generally true, at least forthe LST constructible in the F-theory without O + . In [2] is found that the endpointconfiguration (which is the base after successively shrinking all (-1) curves) for an LSTof this kind is either a rational curve with self-intersection number 0, or one of the fibersin the Kodaira classification. Let us call an LST with the former endpoint an LST oftype O , while an LST with the latter endpoint (any out of the Kodaira classification)an LST of type K . Since the equation for (cid:98) κ P in (3.6) has a geometric meaning on thebase of F-theory and it is actually an invariant under the blowing-up/down procedure,we can compute the Poincar´e 2-group structure constant at the endpoint using equation(3.6) for the endpoint. See appendix A for the proof. For all the type O LSTs, we havethat η (endpoint) = 0 while the LS charge is 1. Therefore, (cid:98) κ P (all LST of type O ) = 2 . (5.1)On the contrary, for all LSTs of type K , the endpoints are such that the diagonalcomponents η II (endpoint) always equal 2. Therefore we conclude (cid:98) κ P (all LST of type K ) = 0 . (5.2)An immediate consequence is that the type is a T-duality invariant: an LST of type K must be T-dual to another LST of type K , and an LST of type O must be T-dual to15nother LST of type O . Remark.
The D (2) factor of the defect group of this geometry is a blow-up invariant aswell [41]. Therefore it depends only on the possible endpoints. We have D (2) = Z LST for all LSTs of type O Z LST ⊕ Z ( G K ) for all LSTs of type K (5.3)where for a given Kodaira type K we denote G K the universal cover group correspondingto its split form. For example, if the Kodaira type is I ∗ n +1 , we have G I ∗ n +1 = Spin (2 n +9)and D (2) = Z LST ⊕ Z .This prescription has to be modified for LSTs constructed from configurations with O + [39, 40], but we will not consider examples of that sort in this note.
6. Twisted T-dualities and 2-group structure
In this section begin a study of the behavior of 2-groups upon twisted T-dualities for 6dLSTs. We focus on LSTs with 16 supercharges as motivating examples. The formalismwe develop here extends to the case of LSTs with 8 supercharges.
In Section 4.2, we saw that the N = (1 ,
1) LST of type g , which is the UV completionof the N = (1 ,
1) SYM, is T-dual to N = (2 ,
0) LST of same type, when g is one of A k , D k and E , , . While in the N = (1 ,
1) side, we can consider a non-simply-laced gaugegroup, in the N = (2 ,
0) side the type is restricted to
ADE . Therefore we expect theT-duality for a non-simply-laced type should involve a discrete symmetry twist along S on N = (2 ,
0) side.The above expectation is confirmed by the geometrical version of LST T-duality inF-theory along the lines we discussed in section 4.1. The 6d (1 ,
1) LSTs with non-simply laced gauge groups are of type K , with an F-theory base that contains a genusone curve, i.e. the I Kodaira fiber. Since the gauge groups are non-simply laced, thecorresponding gauge fibers are non-split according to the Tate algorithm [59] (see also[60]). Because of the monodromies in the non-split fibers, swapping the I base with thefiber of the fibration in the non-simply laced case one must obtain a genus-one fibrationwith nontrivial multisections of order equal to the order of the outer automorphismfolding. Therefore geometry predicts we obtain a twisted compactification of a 6d (2 , S where the outer automorphism twist of the corresponding affine Dynkindiagram is acting as a permutation symmetry on the the tensormultiplets, along thelines discussed in section 3.3 of [38]. 16o test such a twisted T-duality using the method in this paper requires understand-ing the behaviour of the 2-group symmetry upon twisting by the action of a discretesymmetry acting on the string charge lattice non-trivially. The full 2-group associatedto the resulting 5d KK theory is much more complicated than the mere (continuous)6d 2-group, even restricting our attention to the 5d KK 2-group corresponding to theuniversal subgroup G (cid:98) κ P , (cid:98) κ R of equation (1.1) which is the focus of this note. Howeverthe latter is mapped to a well-defined subgroup of the 5d KK 2-group: in the followingsection we compute the structure constants for such subgroup. This requires developingthe formalism slightly, to which we now turn. Let us consider a general LST compatified on S with twist, and denote the groupgenerated by the twist P T . As illustrated in [38, 61], the effect of P T can be read offfrom the tensor branch EFT, and in general it acts as a combination of permutations ofthe tensormultiplets and outer automorphisms of the gauge groups and the flavor groups.An interesting effect on the 2-group symmetry occurs when P T involves a permutationof the tensormultiplets b (2) I . To understand this effect it is necessary to discuss the mapof the 6d BPS strings to the BPS strings and particles of the 5d KK theory. Mapping 6d BPS strings to the twisted 5d KK theory.
Let us denote with [ I ]the P T -orbit of tensor nodes including the node I , and with orb( P T ) the set of such P T -orbits. The 6d BPS strings of the 6d theory give rise to • BPS strings of the 5d KK theory . Since only P T -invariant combinations oftensormultiplets survive the twisting, the 6d string charges are mapped to only | orb( P T ) | integer string charges in the 5d theory. Strings whose charges belong tothe same P T -orbit are identified. Label the tensormultiples consistently with ourchoice of basis for the string charge lattice around (2.2) (in such a way that N I is the charge measured by the I -th tensor). A boundstate of 6d BPS strings withcharge ∆ = ( N , ..., N r +1 ) , corresponding to N J D3 branes wrapping the curve Σ J , maps to a boundstate of5d BPS strings with charge P P T ∆ = (cid:0) N twist[ I ] (cid:1) [ I ] ∈ orb( P T ) N twist[ I ] ≡ (cid:88) I ∈ [ I ] N I . (6.1) • BPS particles of the 5d KK theory . Wrapping a BPS string on the KK S gives rise to a 5d BPS particle. In the untwisted case, this establishes an embeddingof the 6d string charge lattice into the 5d KK theory particle charge lattice. Inthe twisted case, instead, not all possible strings can be wrapped on the KK S P T on the strings, only boundstates of 6dBPS strings that are left invariant by the action of P T can wrap the KK S in thiscase (this is an effect similar to twisted sectors in orbifold CFTs). This implies thatthe string charges of such states occur in closed orbits of P T and are therefore alsoparametrized by | orb( P T ) | integers. We define the charge of such BPS particles γ P T = (cid:16) Q wrap[ I ] (cid:17) [ I ] ∈ orb( P T ) (6.2)normalized so that the particle coming from the minimal set of strings belongingto [ I ] has charge Q wrap[ I ] = 1.In this section we focus on the subsector of the BPS spectrum of the P T -twisted 5dKK theory generated by the two kinds of BPS excitations above. In five-dimensionaltheories strings and particles can be mutually non-local, and this is indeed the case forthe subsector of interest, which follows by Mapping 6d Dirac pairing to the twisted 5d KK theory and U (1) (1) symmetry. The 6d Dirac paring among BPS strings, induces a non-trivial Dirac pairing between theBPS strings and particles we have discussed above: (cid:104) γ P T , P P T ∆ (cid:105) D = (cid:88) [ I ] , [ J ] ∈ orb( P T ) Q wrap[ I ] η [ I ][ J ] P T N twist[ J ] := (cid:88) I, [ J ] Q wrap I η IJ N twist[ J ] . (6.3)where Q wrap I is the same integer for all I ∈ [ I ] by construction. In other words, thepairing matrix η [ I ][ J ] P T , whose size is the number of P T -orbits, is η [ I ][ J ] P T = (cid:88) I ∈ [ I ] η IJ (cid:48) for an arbitrary J (cid:48) ∈ [ J ] , (6.4)which was also introduced in [38]. Note that (6.4) is no longer symmetric as it is apairing between objects of different dimensionalities.Now we can consider the defect group of the twisted 5d KK theory obtained from η P T . In particular, we have a U (1) (1) form symmetry corresponding to the string withthe minimal nonzero string charge ∆ P T = ( N P T [ I ] ) [ I ] ∈ orb( P T ) satisfying (cid:88) [ J ] ∈ orb( P T ) η [ I ][ J ] P T N P T [ J ] = 0 , (6.5)that is the primitive right-null-vector of η P T . The 2-group structure constants of the twisted KK theory.
The 2-group struc-ture constants (cid:98) κ P T R and (cid:98) κ P T P for the twisted compactified theory can be obtained by just We have a sum over I ∈ [ I ] because the particle of type [ I ] consists of all the strings of type I ∈ [ I ],as explained above. by ∆ P T in equation (3.6) (cid:98) κ P T R = (cid:88) [ I ] ∈ orb( P T ) N P T [ I ] h ∨ g I (cid:98) κ P T P = − (cid:88) [ I ] ∈ orb( P T ) N P T [ I ] ( η II − . (6.6) Twisting and fractionalization.
Note that the charge of untwisted LS in the com-pactified theory, P P T ∆ , is also a right-null-vector, and thus it is proportional to ∆ P T up to an integer constant F : P P T ∆ = F ∆ P T . (6.7) F being greater than one means that the twisted theory has a fractional little stringwith fractionality F . Correspondingly, the 2-group structure constants for twisted anduntwisted cases are also related by the fractionality index F : (cid:98) κ R, P = F (cid:98) κ P T R, P . (6.8)If two LSTs LST and LST are dual to each other with twist P T and P T respectively,the 2-group structure constant should match: (cid:98) κ LST , P T R, P = (cid:98) κ LST , P T R, P . (6.9)The fractionality constant F for each twist does not have to match: T-duality does notpreserve twisting. (1 , LSTs
Coming back to our motivating example. On the gauge theory side the computation ofthe structure constants is identical, whether the gauge group is simply laced or not. Theresult is (cid:98) κ R (type g N = (1 ,
1) LST) = h ∨ g (6.10) (cid:98) κ P (type g N = (1 ,
1) LST) = 0 . (6.11)In the case of the N = (2 ,
0) LST, the untwisted Dirac paring η is the affine Cartan ma-trix. The twisting with a permutation symmetry of the affine Dynkin diagram results in η P T , which is the (symmetrizable) Cartan matrix for the affine Dynkin diagram obtainedby the folding of the original affine diagram. For the relation between the diagrams andfoldings, see e.g. [62]. A natural guess is that the T-dual of the N = (2 ,
0) LST of type g A with a twist P T is the N = (1 ,
1) LST of type g B when the folded affine diagram is that This fractionalization and the rescaling of the 2-group structure constant should be a consequence ofthe detailed structure of the symmetry in the 6D LST. If the symmetry in 6D is the direct product of P T and the continuous 2-group, such a rescale does not happen. Therefore, we expect P T , continuous2-group, and other discrete (higher-form) symmetries to form a more general higher-group.
19f the untwisted (but not-necessary simply-laced) g B affine algebra. In the followingwe see that this conjuecture is consistent with the constraint (6.9) in some expamles.
The case of g B = g . We have that the 6d SYM structure constant is (see table 1) (cid:98) κ R = h ∨ g = 4 (6.12)We claim that the LST is a twisted T-dual to the (2,0) e LST. Indeed the collection ofcurves for the (2,0) e LST is organized along an affine e (1)6 Dynkin diagram2 (6.13)and the folding here is the Z action corresponding to the center Z ( E ) = Z which hasorbits (2 ) , (2 , , ) , (2 , , ) . (6.14)The corresponding twisted Dirac pairing is − − − − (6.15)Corresponding to the folding e (1)6 P T (3) −−−→ g (1)2 (6.16)The LS charge ∆ = (3 , , , , , ,
1) is mapped by such folding to P P T ∆ = (3 , , P T (3) = P P T ∆ = (1 , , N = (2 ,
0) model does not have any gauge groups, the correspondingstructure constant (cid:98) κ P T (3) R is (1 + 2 + 1) = 4, which matches with the N = (1 ,
1) side.
The case of g B = f . We have that the 6d SYM structure constant is (see table 1) (cid:98) κ R = h ∨ f = 9 (6.17)We claim that the latter is a twisted T-dual to the (2,0) e LST. Indeed the collectionof curves for the (2,0) e LST is organized along an affine e (1)7 Dynkin diagram, and the When the folded affine diagram is of twisted type, we expect the N = (1 ,
1) would involve an outerautomorphism twist on the gauge group. In addition, when the N = (1 ,
1) side is of sp type, thereis the discrete theta ambiguity. It would be interesting to study these points. e (1)7 P T (2) −−−→ f (1)4 (6.18)The LS charge ∆ = (4 , , , , , , ,
1) is mapped by such folding to (4 , , , , (cid:98) κ P T (2) R is (4 + 6 + 4 + 2 + 2) / N = (1 ,
1) side.
Remark.
The case of twisted T-duals for 6d LSTs with 8 supercharges is much richerand unexplored. The constraint (6.9) should be exploited to study the space of twistedT-dual LSTs.
Acknowledgements
MDZ thanks I˜naki Garc´ıa Extebarria and Paul-Konstantin Oehlmann for discussions onrelated topics. KO thanks Clay C´ordova and Lakshya Bhardwaj for discussions on thetopic. This project has received funding from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation programme (grantagreement No. 851931).
A. Proof that (cid:98) κ P is invariant under blow-ups Let us relabel the curves from 0 to r in such a way that the curve to be blown down isthe curve number 0. We will assume to have an LS charge ∆ = ( N , N , ..., N r ) withgcd( N , ..., N r ) = 1. Let us denote (cid:96) = gcd( N , ..., N r ) . We have that η = 1. In the argument below, for the sake of simplicity, we will assumethat η J is either 0 or −
1. For LSTs of sufficiently high rank this is not a restriction,but there are few exceptional cases that violates this assumption in very low ranks. Wewill comment about them at the end of the argument. With our assumptions, the Diracpairing for the blown down curve configuration is simply (cid:98) η IJ = η IJ − η I η J (A.1)where, slightly abusing notation, the indexes of (cid:98) η IJ run from 1 to r . The LS charge for η IJ is such that η IJ N J = 0 which entails that N = − r (cid:88) J =1 η J N J (A.2)21herefore 0 = r (cid:88) J =0 η IJ N J = η I N + r (cid:88) J =1 η IJ N J = − η I r (cid:88) J =1 η J N J + r (cid:88) J =1 η IJ N J = r (cid:88) J =1 ( η IJ − η I η J ) N J = r (cid:88) J =1 (cid:98) η IJ N J (A.3)We can take (cid:98) ∆ = ( N , ..., N r ) as the LS charge for the blown down configuration pro-vided (cid:96) = 1. Assume that that is not the case, then by (A.2) we have that N should bedivisible by (cid:96) as well, but this is in contraddiction with the fact that gcd( N , ..., N r ) = 1.Now we have that from equation (3.6) (cid:98) κ P = r (cid:88) I =0 N I ( η II −
2) = − N + r (cid:88) I =1 N I ( η II − r (cid:88) J =1 η J N J + r (cid:88) I =1 N I ( η II − − r (cid:88) J =1 η J η J N J + r (cid:88) I =1 N I ( η II −
2) (by our assumption on η J )= r (cid:88) I =1 N I ( η II − η I η I −
2) = r (cid:88) I =1 N I ( (cid:98) η II −
2) = (cid:98) κ P (cid:12)(cid:12)(cid:12) blow-down (A.4)Thus establishing that (cid:98) κ P = (cid:98) κ P (cid:12)(cid:12)(cid:12) blow-down for all LSTs that satisfy our assumptions. Byrecursively applying the above, we conclude that (cid:98) κ P can be computed from the endpointconfiguration.Let’s consider the exceptional LSTs that are such that η = 1 and there is an I suchthat η I (cid:54) = 0 ,
1. In fact, there is a single such case (without O + ), whose base is η = (cid:32) − − (cid:33) obtained by blowing up the a Kodaira node I at the node. The latter is type K . Theuniequness can be understood from the classification of endpoints. It is easy to see thatthis model has LS charge ∆ = 1 , (cid:98) κ P = 0. Again it is invariant withrespect to blow-down as expected. 22 eferences [1] N. Seiberg, “Nontrivial fixed points of the renormalization group insix-dimensions,” Phys. Lett. B (1997) 169–171, arXiv:hep-th/9609161 .[2] L. Bhardwaj, M. Del Zotto, J. J. Heckman, D. R. Morrison, T. Rudelius, andC. Vafa, “F-theory and the Classification of Little Strings,”
Phys. Rev. D no. 8,(2016) 086002, arXiv:1511.05565 [hep-th] . [Erratum: Phys.Rev.D 100, 029901(2019)].[3] A. Strominger, “Open p-branes,” Phys. Lett. B (1996) 44–47, arXiv:hep-th/9512059 .[4] N. Seiberg and E. Witten, “Comments on string dynamics in six-dimensions,”
Nucl. Phys. B (1996) 121–134, arXiv:hep-th/9603003 .[5] N. Seiberg, “New theories in six-dimensions and matrix description of M theory onT**5 and T**5 / Z(2),”
Phys. Lett. B (1997) 98–104, arXiv:hep-th/9705221 .[6] A. Kapustin, “On the universality class of little string theories,”
Phys. Rev. D (2015) 086005, arXiv:hep-th/9912044 .[7] A. Losev, G. W. Moore, and S. L. Shatashvili, “M & m’s,” Nucl. Phys. B (1998) 105–124, arXiv:hep-th/9707250 .[8] P. S. Aspinwall, “Point - like instantons and the spin (32) / Z(2) heterotic string,”
Nucl. Phys. B (1997) 149–176, arXiv:hep-th/9612108 .[9] P. S. Aspinwall and D. R. Morrison, “Point - like instantons on K3 orbifolds,”
Nucl. Phys. B (1997) 533–564, arXiv:hep-th/9705104 .[10] K. A. Intriligator, “New string theories in six-dimensions via branes at orbifoldsingularities,”
Adv. Theor. Math. Phys. (1998) 271–282, arXiv:hep-th/9708117 .[11] A. Hanany and A. Zaffaroni, “Branes and six-dimensional supersymmetrictheories,” Nucl. Phys. B (1998) 180–206, arXiv:hep-th/9712145 .[12] I. Brunner and A. Karch, “Branes at orbifolds versus Hanany Witten insix-dimensions,”
JHEP (1998) 003, arXiv:hep-th/9712143 .[13] K. A. Intriligator, “Compactified little string theories and compact moduli spacesof vacua,” Phys. Rev. D (2000) 106005, arXiv:hep-th/9909219 .[14] O. Aharony, M. Berkooz, D. Kutasov, and N. Seiberg, “Linear dilatons, NSfive-branes and holography,” JHEP (1998) 004, arXiv:hep-th/9808149 .2315] O. Aharony, “A Brief review of ’little string theories’,” Class. Quant. Grav. (2000) 929–938, arXiv:hep-th/9911147 .[16] C. Cordova, T. T. Dumitrescu, and K. Intriligator, “2-Group Global Symmetriesand Anomalies in Six-Dimensional Quantum Field Theories,” arXiv:2009.00138[hep-th] .[17] J. C. Baez, D. Stevenson, A. S. Crans, and U. Schreiber, “From loop groups to2-groups,” arXiv:math/0504123 .[18] H. Sati, U. Schreiber, and J. Stasheff, “ L ∞ algebra connections and applications toString- and Chern-Simons n-transport,” 2, 2008. arXiv:0801.3480 [math.DG] .[19] H. Sati, U. Schreiber, and J. Stasheff, “Differential twisted String and Fivebranestructures,” Commun. Math. Phys. (2012) 169–213, arXiv:0910.4001[math.AT] .[20] D. Fiorenza, U. Schreiber, and J. Stasheff, “ ˇCech cocycles for differentialcharacteristic classes: an ∞ -Lie theoretic construction,” Adv. Theor. Math. Phys. no. 1, (2012) 149–250, arXiv:1011.4735 [math.AT] .[21] D. Fiorenza, H. Sati, and U. Schreiber, “Multiple M5-branes, String 2-connections,and 7d nonabelian Chern-Simons theory,” Adv. Theor. Math. Phys. no. 2,(2014) 229–321, arXiv:1201.5277 [hep-th] .[22] C. C´ordova, T. T. Dumitrescu, and K. Intriligator, “Exploring 2-Group GlobalSymmetries,” JHEP (2019) 184, arXiv:1802.04790 [hep-th] .[23] F. Benini, C. C´ordova, and P.-S. Hsin, “On 2-Group Global Symmetries and theirAnomalies,” JHEP (2019) 118, arXiv:1803.09336 [hep-th] .[24] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized GlobalSymmetries,” JHEP (2015) 172, arXiv:1412.5148 [hep-th] .[25] E. Sharpe, “Notes on generalized global symmetries in QFT,” Fortsch. Phys. (2015) 659–682, arXiv:1508.04770 [hep-th] .[26] J. J. Heckman, D. R. Morrison, and C. Vafa, “On the Classification of 6D SCFTsand Generalized ADE Orbifolds,” JHEP (2014) 028, arXiv:1312.5746[hep-th] . [Erratum: JHEP 06, 017 (2015)].[27] J. J. Heckman, D. R. Morrison, T. Rudelius, and C. Vafa, “Atomic Classificationof 6D SCFTs,” Fortsch. Phys. (2015) 468–530, arXiv:1502.05405 [hep-th] .[28] L. Bhardwaj, “Classification of 6d N = (1 ,
0) gauge theories,”
JHEP (2015)002, arXiv:1502.06594 [hep-th] . 2429] Y. Tachikawa, “Frozen singularities in M and F theory,” JHEP (2016) 128, arXiv:1508.06679 [hep-th] .[30] L. Bhardwaj, D. R. Morrison, Y. Tachikawa, and A. Tomasiello, “The frozen phaseof F-theory,” JHEP (2018) 138, arXiv:1805.09070 [hep-th] .[31] L. Bhardwaj, “Dualities of 5d gauge theories from S-duality,” JHEP (2020)012, arXiv:1909.05250 [hep-th] .[32] F. Apruzzi, C. Lawrie, L. Lin, S. Schaefer-Nameki, and Y.-N. Wang, “Fibers addFlavor, Part II: 5d SCFTs, Gauge Theories, and Dualities,” JHEP (2020) 052, arXiv:1909.09128 [hep-th] .[33] L. Bhardwaj and G. Zafrir, “Classification of 5d N=1 gauge theories,” arXiv:2003.04333 [hep-th] .[34] L. Bhardwaj, “T-duality of LSTs and enhanced symmetries,” to appear .[35] M. Del Zotto and G. Lockhart, “Universal Features of BPS Strings inSix-dimensional SCFTs,” JHEP (2018) 173, arXiv:1804.09694 [hep-th] .[36] J. J. Heckman and T. Rudelius, “Top Down Approach to 6D SCFTs,” J. Phys. A no. 9, (2019) 093001, arXiv:1805.06467 [hep-th] .[37] V. Braun and D. R. Morrison, “F-theory on Genus-One Fibrations,” JHEP (2014) 132, arXiv:1401.7844 [hep-th] .[38] L. Bhardwaj, P. Jefferson, H.-C. Kim, H.-C. Tarazi, and C. Vafa, “Twisted CircleCompactifications of 6d SCFTs,” arXiv:1909.11666 [hep-th] .[39] L. Bhardwaj and S. Schafer-Nameki, “Higher-form symmetries of 6d and 5dtheories,” arXiv:2008.09600 [hep-th] .[40] F. Apruzzi, M. Dierigl, and L. Lin, “The Fate of Discrete 1-Form Symmetries in6d,” arXiv:2008.09117 [hep-th] .[41] M. Del Zotto, J. J. Heckman, D. S. Park, and T. Rudelius, “On the Defect Groupof a 6D SCFT,” Lett. Math. Phys. no. 6, (2016) 765–786, arXiv:1503.04806[hep-th] .[42] I. Garc´ıa Etxebarria, B. Heidenreich, and D. Regalado, “IIB fluxnon-commutativity and the global structure of field theories,”
JHEP (2019)169, arXiv:1908.08027 [hep-th] .[43] D. R. Morrison, S. Schafer-Nameki, and B. Willett, “Higher-Form Symmetries in5d,” arXiv:2005.12296 [hep-th] . 2544] F. Albertini, M. Del Zotto, I. Garc´ıa Etxebarria, and S. S. Hosseini, “Higher FormSymmetries and M-theory,” arXiv:2005.12831 [hep-th] .[45] M. Del Zotto, I. Garc´ıa Etxebarria, and S. S. Hosseini, “Higher Form Symmetriesof Argyres-Douglas Theories,” arXiv:2007.15603 [hep-th] .[46] C. Closset, S. Schafer-Nameki, and Y.-N. Wang, “Coulomb and Higgs Branchesfrom Canonical Singularities: Part 0,” arXiv:2007.15600 [hep-th] .[47] M. Del Zotto and K. Ohmori, “In preparation,”. .[48] V. Sadov, “Generalized Green-Schwarz mechanism in F theory,” Phys. Lett. B (1996) 45–50, arXiv:hep-th/9606008 .[49] K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, “Anomaly polynomial ofgeneral 6d SCFTs,”
PTEP no. 10, (2014) 103B07, arXiv:1408.5572[hep-th] .[50] K. Intriligator, “6d, N = (1 ,
0) Coulomb branch anomaly matching,”
JHEP (2014) 162, arXiv:1408.6745 [hep-th] .[51] H. Shimizu and Y. Tachikawa, “Anomaly of strings of 6d N = (1 ,
0) theories,”
JHEP (2016) 165, arXiv:1608.05894 [hep-th] .[52] H.-C. Kim, S. Kim, and J. Park, “6d strings from new chiral gauge theories,” arXiv:1608.03919 [hep-th] .[53] W. Lerche, C. Schweigert, R. Minasian, and S. Theisen, “A Note on the geometryof CHL heterotic strings,” Phys. Lett. B (1998) 53–59, arXiv:hep-th/9711104 .[54] E. Witten, “Toroidal compactification without vector structure,”
JHEP (1998)006, arXiv:hep-th/9712028 .[55] C. Vafa, “Evidence for F theory,” Nucl. Phys. B (1996) 403–418, arXiv:hep-th/9602022 .[56] C. Closset, M. Del Zotto, and V. Saxena, “Five-dimensional SCFTs and gaugetheory phases: an M-theory/type IIA perspective,”
SciPost Phys. no. 5, (2019)052, arXiv:1812.10451 [hep-th] .[57] M. Del Zotto, J. J. Heckman, A. Tomasiello, and C. Vafa, “6d Conformal Matter,” JHEP (2015) 054, arXiv:1407.6359 [hep-th] .[58] B. Bastian, S. Hohenegger, A. Iqbal, and S.-J. Rey, “Triality in Little StringTheories,” Phys. Rev. D no. 4, (2018) 046004, arXiv:1711.07921 [hep-th] .2659] M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, andC. Vafa, “Geometric singularities and enhanced gauge symmetries,” Nucl. Phys. B (1996) 215–252, arXiv:hep-th/9605200 .[60] S. Katz, D. R. Morrison, S. Schafer-Nameki, and J. Sully, “Tate’s algorithm andF-theory,”
JHEP (2011) 094, arXiv:1106.3854 [hep-th] .[61] L. Bhardwaj, “More 5d KK theories,” arXiv:2005.01722 [hep-th] .[62] J. Fuchs, B. Schellekens, and C. Schweigert, “From Dynkin diagram symmetries tofixed point structures,” Commun. Math. Phys. (1996) 39–98, arXiv:hep-th/9506135arXiv:hep-th/9506135