6d N=(1,0) theories on T 2 and class S theories: part I
Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, Kazuya Yonekura
aa r X i v : . [ h e p - t h ] M a r Prepared for submission to JHEP
IPMU-15-0028, UT-15-07 N =(1 , theories on T and class S theories: part I Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, , and Kazuya Yonekura Department of Physics, Faculty of Science,University of Tokyo, Bunkyo-ku, Tokyo 133-0022, Japan Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8583, Japan School of Natural Sciences, Institute for Advanced Study,Princeton, NJ 08540, United States of America
Abstract:
We show that the N =(1 ,
0) superconformal theory on a single M5brane on the ALE space of type G = A n , D n , E n , when compactified on T , becomesa class S theory of type G on a sphere with two full punctures and a simple puncture.We study this relation from various viewpoints. Along the way, we develop a newmethod to study the 4d SCFT arising from the T compactification of a class of 6d N =(1 ,
0) theories we call very Higgsable. ontents H and the central charges 176.3 Examples 24 In the last few years, we learned a great deal about the class S theories, i.e. thecompactification of 6d N =(2 ,
0) theory on general Riemann surfaces with punctures.By starting from the 6d N =(2 ,
0) theories, which have a simple ADE classification,this construction gives a vast variety of 4d N =2 theories, that come from the choiceof the Riemann surfaces and punctures.There is another way to construct 4d N =2 theories from 6d: namely, we canput 6d N =(1 ,
0) theories on T . In this second method, there are no choice of thecompactification manifold, but there are a great number of N =(1 ,
0) theories in 6das shown in a recent series of works [1–3], thereby giving rise to a plethora 4d N =2theories. A natural question, therefore, is how much overlap there is between thesetwo constructions. – 1 – ain objective. As a first step in this direction, in this paper we show that asmall but natural subset of 6d N =(1 ,
0) theories on T gives rise to a small butnatural subset of class S theories. Namely, we show that the 6d N =(1 ,
0) theoryon a single M5-brane on the ALE space of type G = A n , D n , E n , when compactifiedon T , becomes the class S theory of type G on a sphere with two full puncturesand a simple puncture. The 6d theories in question were called 6d ( G, G ) minimalconformal matters in [4], and the 4d class S theories can be called the generalizedbifundamental theories. Using these terminologies, we can simply say that the T compactification of the 6d minimal conformal matter gives generalized bifundamentaltheory in 4d.For G = SU( N ) this relation is in a sense very trivial: a single M5-brane on the C / Z N singularity is just a bifundamental hypermultiplet of SU( N ) , and the classS theory of type SU( N ) on a sphere with two full punctures and a single punctureis also a bifundamental [5, 6]. For G = SO(8), a single M5-brane on the C / Γ G singularity gives rise to the rank-1 E-string theory, as pointed out in [1, 4]. The classS theory of type SO(8) on a sphere with two full punctures and a single puncturewas studied in [7], and it was found that it gives the E theory of Minahan andNemeschansky. Therefore our objective is to show the relation in the other cases;but our analysis sheds new light even on the simplest of cases when G = SU( N ). Pieces of evidence.
In the rest of the paper, we will provide other pieces ofevidences: • In Sec. 2, we follow the duality chain to show that the T compactification ofthe 6d minimal conformal matter is a class S theory defined on a sphere withtwo full punctures and another puncture that cannot be directly identified withthe present technology. • In Sec. 3, we compute and compare the dimension of the Coulomb branch bothin 4d and in 6d. • In Sec. 4, we show that the Higgs branch of the 4d generalized fundamentals,when the G flavor symmetry is weakly gauged, is given by the ALE space oftype G . This is as expected from the 6d point of view. • In Sec. 5, we compare the Seiberg-Witten curve of the 4d generalized bifunda-mental of type D and that of the 6d minimal conformal matter of type D in acertain corner of the moduli space and show the agreement. • In Sec. 6, we develop a method to compute the 4d anomaly polynomial of thecompactification of a class of the 6d N =(1 ,
0) theories we call very Higgsable ,apply that to 6d minimal conformal matters and show that they agree withthe known central charges of 4d generalized bifundamentals.– 2 –e conclude with a short discussion in Sec. 7. These sections are largely independentof each other and can be read separately. In particular, the analysis given in Sec. 6 isquite general and applies to all 6d theories we call very Higgsable: these correspond,in the F-theoretic language of [1, 2, 4], to theories whose configuration of curves C canbe eliminated by a repeated blow-down of − C end is empty, and a further complex structure deformation makes the theory completelyinfared free without turning on any tensor vevs. In other words, the theory has acompletely Higgsed branch where no tensor multiplet remains. This explains ourterminology very Higgsable . Let us first try to follow the duality chain to show that the 6d minimal conformalmatter on T is a class S theory on a sphere with two full punctures and a simplepuncture. We will see that there is one step we can not quite follow, due to our lackof knowledge of the 6d N =(2 ,
0) theory.We start from a single M5-brane on the C / Γ G singularity. This gives a minimalconformal matter of type G weakly coupled to G gauge fields in 7d. By putting iton a torus, we should have a 4d theory with G flavor symmetry, which is weaklycoupled to G gauge fields in 5d.Let us say that the torus T has complex structure τ . By compactifying on oneside of T and taking the T-dual of the other, we have Type IIB string theory on R , × S × R × C / Γ G with axiodilaton given by τ , together with a single D3-branefilling R , . We now take the limit to isolate the low-energy degrees of freedom andignore the center-of-mass mode of the D3-brane. We have the 6d N =(2 ,
0) theoryof type G on S × R , and the tension of the D3-brane becomes effectively infinite.Therefore we should have a BPS defect of codimension-2. With the class S technologycurrently available to us, we do not see how to directly identify this defect; let uscall it X .We now take the limit where S is small. Then we have a localized degrees offreedom, that is the class S theory of type G on a sphere with two full punctures anda puncture X , coupled weakly to 5d G gauge fields coming from the 6d N =(2 , G on S × semi-infinite lines.Therefore we conclude that the 6d minimal conformal matter of type G , whencompactified on T τ , is a class S theory of type G on a sphere with two full puncturesand a puncture X . At present, the most we can say just using the duality chain isthat we know that the puncture X is the simple puncture when G is either SU( N )or SO(8), and that the only statement that naturally generalizes this is that thepuncture X is always the simple puncture for arbitrary G . The authors thank D. R. Morrison for the suggestion that led to this naming. – 3 –
Dimensions of the Coulomb branch
In this section, we compute the dimension of the Coulomb branch both in 4dand in 6d, and show that the results indeed agree.
First, we take the 6d point of view. In Sec. 2 we followed the duality chain tomap the 6d minimal conformal matter on T to the Type IIB string on R , × S × R × C / Γ G with axiodilaton given by τ , together with a single D3-brane filling R , .Instead of directly study the Coulomb branch in 4d, let us put the theory on another S R of radius R and directly identify the hyperk¨ahler structure of the 3d Coulombbranch. Take the T-dual of this S R , and call it ˜ S /R . Then lift the whole systemback to M-theory. Here we are following the analysis of Appendix A.3 of [8].We now have M-theory on R , × ˜ S /R × T τ × R × C / Γ G and a single M2-branefilling R , . The singularity has G gauge multiplet on its singular loci, and the M2-brane can be absorbed into an instanton configuration. We conclude that the 3dCoulomb branch of the 6d minimal conformal matter on S R × T τ is given by theone-instanton moduli space of gauge group G on ˜ S /R × T τ × R .This gives an interesting new perspective on the tensor branch of the 6d minimalconformal matter. We consider an instanton configuration on T × R . By restrictingthe gauge field to T at a constant “time” t ∈ R , we define the Chern-Simons invariant CS ( t ). In our case, a single M5 gives a single M2 that becomes one instanton. Letus say CS ( −∞ ) = 0, then we have CS (+ ∞ ) = 1.At t = ±∞ , we need a zero-energy configuration, so the three holonomies g , , around three edges of T should commute. We take them to be in the Cartan of G .By following the duality chain, we see that they can be identified with the originalWilson lines of G used in the compactification. It is known that the Chern-Simonsinvariant of this flat gauge field on T is 0 mod 1. For simplicity, let us set g , , = 1at t = ±∞ .The quaternionic dimension of the moduli space including the center-of-massmotion but with the holonomies at t = ±∞ fixed, is found by the Atiyah-Patodi-Singer index theorem [9] to be d T ,G = h ∨ ( G ) − rank( G ) (3.1)where h ∨ ( G ) and rank( G ) are the dual Coxeter number and the rank of G . Thenegative term is from the boundary contribution. Therefore, this is the dimension The theorem of [9] is valid if the gauge field approaches to the value at t = ±∞ exponentiallyrapidly. That condition is satisfied by instanton configurations when the holonomies g , , aregeneric so that the gauge group is broken to its Cartan. Then the equation (3.1) follows from thefact that the 3d Dirac operator at t = ±∞ for the adjoint representation has 2 rank( G ) zero modesand the η -invariant (excluding the zero modes) of flat connections is zero. By continuity, (3.1)should be valid even if we take g , , →
1, although a direct analysis of this case is complicated. – 4 –plus one, due to the center-of-mass motion) of the Coulomb branch of the 4d theorywe obtain by putting the 6d minimal conformal matter on T : d T , ( G,G ) min. conf. matter = h ∨ ( G ) − rank( G ) − , (3.2)Let us see these degrees of freedom in more detail below. These details can beskipped in a first reading. G = SU( N ) . When G = SU( N ), h ∨ ( G ) = N and rank( G ) = N −
1, and then d T ,G = 1. This corresponds to the fact that a single M5 on C / Γ singularity onlyhas the center-of-mass motion as the tensor branch degree of freedom. G = SO(2 N ) . Next, consider G = D N . Recall [4] that a single M5-brane on C / Γ D N singularity can split into two fractional M5-branes, and the emerging gauge groupbetween the fractionated branes is USp(2 N − d T ,G = N −
2, since h ∨ ( D N ) = 2 N − D N ) = N . So we want to identify these degrees of freedom in the instantonmoduli space.First, recall that for D N = Spin(2 N ) gauge group, there is a unique commutingtriple ( g ∗ , g ∗ , g ∗ ) that cannot be simultaneously conjugated into the Cartan; theycan be chosen to be in a common Spin(7) subgroup, see Appendix I of [10]. TheChern-Simons invariant is 1 / so (2 N − T × R : • For −∞ < t < t , the configuration on T is basically flat and given by( g , g , g ) = (1 , , CS ( t ) stays almost constant close to 0. • At around t = t , the gauge configuration suddenly changes to ( g , g , g ) =( g ∗ , g ∗ , g ∗ ) dressed with holonomies in the Cartan of the commutant, so (2 N − CS ( t ) jumps to 1/2. • Again, for t < t < t , the configuration remains almost constant. • And then at around t = t , it suddenly changes back to ( g , g , g ) = (1 , , CS ( t ) to jump to 1.In these configurations, we see two parameters t , in addition to the N − so (2 N − N − t and t as the positions of the two fractionalM5-branes, and the USp(2 N −
8) gauge group between the two fractionated M5-branes as the S-dual of so (2 N −
7) we find here. The reason is that, after T compactification, we have a 3d theory coupled to 4d N =4 super Yang-Mills on thesegment. We know that the S R and ˜ S /R are T-dual to each other, and thereforethe coupling constants of the N =4 super Yang-Mills in these two descriptions areinversely proportional to each other, and therefore the groups we see are related byS-duality. – 5 – = E n . The analysis is completely similar to the cases above, using the data in[11]. For G = E , we have the following commuting triples:value v of CS
13 12 23 commutant G v e ∅ su (3) ∅ . (3.3)Then the one-instanton configuration can go through these commuting triples. Thedegrees of freedom in the instanton moduli space are now the “time” of the jumpfrom one commuting triple characterized by CS = v i to the next CS = v i +1 , togetherwith the holonomies in the Cartan of G v . In total, the equality (3.1) is reproduced if h ∨ ( G ) = X possible value v of CS (1 + rank G v ) (3.4)and indeed this is satisfied. We also see that this is the sequence of gauge groupswhen the M5-brane gets fractionated on the E singularity found in [4].For G = E , the list of the commuting triples arevalue v of CS
14 13 12 23 34 commutant G v e ∅ su (2) usp (6) su (2) ∅ . (3.5)and for G = E , these arevalue v of CS
16 15 14 13 25 12 35 23 34 45 56 commutant G v e ∅ ∅ su (2) g ∅ f ∅ g su (2) ∅ ∅ . (3.6)In both cases, we can check that indeed the crucial equality (3.4) is satisfied, and thesequence of the groups are the S-dual of the ones that appear in the fractionation ofthe minimal conformal matter, see [4].Actually, we can do a refined check of the above picture. Consider instantonconfigurations in which the gauge field at t = −∞ ( t = + ∞ ) is given by a commutingtriple with the commutant G i ( G i +1 ) and Chern-Simons number v i ( v i +1 ). Thedimension of the moduli space of these configurations is given by the Atiyah-Patodi-Singer theorem as d i,i +1 = h ∨ ( G )( v i +1 − v i ) −
12 (rank( G i ) + rank( G i +1 )) , (3.7)where h ∨ ( G )( v i +1 − v i ) should properly be defined by the integration of the secondChern class in the adjoint representation. Using the above tables for the values of v i and the groups G i , one can check (and it was indeed proved in [11]) that we alwayshave d i,i +1 = 1 for adjacent commuting triples in the tables. This is interpreted asthe fact that a fractional M5-brane has only the center-of-mass degrees of freedom.Here, it is interesting to note that the equality (3.4) is exactly the one thatguarantees the equality of the Witten index of pure N =1 super Yang-Mills of gaugegroup G computed both in the infrared using the gaugino condensation and in theultraviolet using the semi-classical quantization. For more, see e.g. [12].– 6 – .2 Class S perspective Before moving to the class S theory side, let us recall the necessary notions ofthe nilpotent orbits. A nilpotent orbit for an nilpotent element e in g is the set ofelements in g that are G C -conjugate to e . We denote the nilpotent orbit containingthe nilpotent element e by O e .There is a one-to-one correspondence between homomorphisms ρ : su (2) → g ,up to conjugacy, and nilpotent orbits O e . The precise map is given by e = ρ ( σ + ).For simplicity, we denote the nilpotent orbit containing ρ ( σ + ) as O ρ . In the case of g = su ( N ), these homomorphisms are classified by Young diagrams as is well-knownin the class S theory of type A N − . In general, regular (and untwisted) punctures X i of the class S theory of type G are classified by these homomorphisms ρ i .One of the important ingredients in the relationship between the theory of nilpo-tent orbits and the class S theory is the Spaltenstein map d , defined for any simpleLie algebra g . This is a map d : { nilpotent orbits of g } → { nilpotent orbits of g ∨ } , (3.8)where g ∨ is the Langlands dual of g . For example, when g = su ( N ), this map isto send a Young diagram to its dual diagram. In this paper we only encounter the g = g ∨ cases, so in the following we will assume this. Note that the Spaltensteinmap is order-reversing, d ( O ) ≥ d ( O ′ ) if O ≤ O ′ where the standard partial orderingfor nilpotent orbits is defined so that O e ′ ≥ O e if ¯ O e ′ ⊃ O e .The maximal orbit under this partial ordering is called the principal orbit O prin and is equal to d ( O ), the Spaltenstein dual to the zero orbit O . The dimension ofthe principal orbit is dim C O prin = dim( G ) − rank( G ) . (3.9)The next-to-maximal orbit is called the subregular orbit O subreg and is equal to d ( O min ), where O min is the minimal nilpotent orbit. The dimension of the minimalorbit is dim C O min = 2( h ∨ ( G ) − . (3.10)With these notions at hand, we put the N = (2 ,
0) theory of type G on a Riemannsurface of genus g with regular and untwisted punctures X i which correspond tohomomorphisms ρ i : su (2) → g . The complex dimension of the Coulomb branch ofthe resulting 4d N =2 theory is [13] d class S = X i d ( ρ i ) + ( g − G ) , (3.11) This O min is defined by the homomorphism ρ : su (2) → g which is used to embed the SU(2)one-instanton minimally into the group G . The dimension (3.10) is the same as the dimension ofthe one-instanton moduli space of G minus the dimension of the center-of-mass of the instanton. – 7 –here d ( ρ ) is contribution from the punctures and is given by d ( ρ ) = 12 dim C d ( O ρ ) . (3.12)Let us apply this formula to the class S theory we are considering, namely, (2,0)theory of type G on a sphere with two full punctures and a simple puncture. The fullpuncture and the simple puncture are defined so that the corresponding nilpotentorbits are O and O subreg , respectively. Then, the Coulomb branch dimension is d class S =dim C d ( O ) + 12 dim C d ( O subreg ) − dim( G )=dim C O prin + 12 dim C O min − dim( G )= h ∨ ( G ) − rank( G ) − , (3.13)where in the last line we used (3.9) and (3.10). This result agrees with (3.2). As the Higgs branch should remain identical under the T compactification, the6d theory and the 4d theory should have the same Higgs branch. We will checkthis below, at the level of complex manifolds. It would be interesting to extend theanalysis to the level of holomorphic symplectic varieties or hyperk¨ahler manifolds. Type
SU( N ) . When the type G of the theory we consider is SU( N ), both the6d minimal conformal matter and the 4d generalized bifundamental of type SU( N )are just a bifundamental hypermultiplet of SU( N ) . It naively seems there is notmuch to see here. However, we can still have some fun in this case, as we will seemomentarily.Consider a single M5 brane on the C / Z N singularity. The 6d theory consistsof the bifundamental of SU( N ) , weakly coupled to the 7d vector multiplet on thesingular loci on the left and on the right of the M5 brane. The Higgs branch of thesystem should describe the motion of the M5-brane on the C / Z N singularity. There-fore, we should be able to obtain C / Z N as the Higgs branch of the weakly-gaugedbifundamental. Let us check this statement. In the 4d N =1 notation, the bifunda-mental consists of Q ai and ˜ Q ia , where a, i = 1 , . . . , N . The invariant combinationsunder the SU( N ) acting on the indices a and i are B = det Q, ˜ B = det ˜ Q, M = Q ai ˜ Q ia /N. (4.1)Note also that the bifundamental couples to the 7d gauge field via the moment maps µ ji = Q ai ˜ Q ja − M δ ji , ˜ µ ab = Q ai ˜ Q ib − M δ ab . (4.2)– 8 –hey satisfy an important relation tr µ k = tr ˜ µ k for any k .The C / Z N singularity has 3( N −
1) smoothing parameters, that can be naturallythought of as ( µ R , µ C ) ∈ su ( N ) R × su ( N ) C , restricted to be in the Cartan; µ R are theK¨ahler parameters for the resolution and µ C the complex deformation. Therefore wecan naturally identify this complex deformation parameter µ C with µ ∼ ˜ µ above.Let us first consider the singular case µ C = µ = ˜ µ = 0. Using the standardrelation det Q ai ˜ Q aj = det Q det ˜ Q = B ˜ B and 0 = µ ji = Q ai ˜ Q ja − M δ ji , we find B ˜ B = M N . (4.3)This is indeed the equation of the C / Z N singularity. More generally, when µ C = µ = ˜ µ = diag( m , . . . , m N ) , (4.4)we have Q ai ˜ Q ja ∼ diag( m + M, . . . , m N + M ). Therefore, we have B ˜ B = N Y i =1 ( M + m i ) , (4.5)which is again the equation of the deformed C / Z N singularity. General type.
Let us proceed to the general case. The 6d minimal conformalmatter of type G , with the G flavor symmetry weakly gauged, should have theHiggs branch of the form C / Γ G , where Γ G is the finite subgroup of SU(2) of type G . Since the Higgs branch should be independent under the T compactification,we should be able to check this using the class S description of the 4d generalizedbifundamental.The Higgs branch of the class S theory in general was studied e.g. in [14]. Asdiscussed there, the Higgs branch of the class S theory of type G on a sphere withtwo full punctures and a single regular puncture of arbitrary type is described asfollows. We start from the Higgs branch X G of the T G theory, i.e. the class S theoryof type G on a sphere with three full punctures. The hyperk¨ahler space X G hasactions of G , and correspondingly has three holomorphic moment maps µ , µ , µ taking values in g C . The hyperk¨ahler dimension of X G is [13]dim H X G = rank G + 32 (dim G − rank G ) . (4.6)A puncture is specified by a homomorphism ρ : su (2) → g . (4.7)Such homomorphisms up to conjugation is known to be classified by the nilpotentelement e = ρ ( σ + ) up to conjugation. Let f = ρ ( σ − ). We now define the Slodowyslice S e at e by S e := { x + e | [ x, f ] = 0 } ⊂ g C . (4.8)– 9 –hen the class S theory of type G , on a sphere with two full punctures and a puncturespecified by e , has the Higgs branch of the form Y e = µ − S e (4.9)where we regarded µ as a map X G → g C .In our case we take e to be the subregular element, since we want to have asimple puncture. The dimension is thendim H Y e = dim H X G − dim H O subreg = dim G + 1 (4.10)where we used (4.6) anddim C O subreg = dim G − rank G − . (4.11)We would like to study the Higgs branch where the G flavor symmetry is coupledto the G L × G R gauge multiplets in one higher dimension, associated to the C / Γ G locus from the left ( G L ) and the right ( G R ). Therefore the Higgs branch of thecombined system is Z e = Y e /// ( G L × G R ) . (4.12)where /// denotes the hyperk¨ahler quotient. On a generic point of Z e , G L × G R is broken to its diagonal subgroup G diag , since the C / Γ G locus is now connectedand not separated by the M5 brane. The breaking from G L × G R to G diag shouldeat dim G hypermultiplets. Subtracting this from (4.10), we find that dim H Z e = 1:this agrees with our expectation that this Higgs branch describes the motion of anM5-brane along C / Γ G orbifold. The question now is to see that Z e = C / Γ G .To see this, we use the following fact: Let us say the T G theory has G × G L × G R flavor symmetry, and let us call the respective moment map operators as µ , µ L and µ R . Then the Higgs branch operators of the T G theory, invariant under G L × G R are just polynomials of µ [15]. We also know that µ , µ L and µ R satisfy the crucialrelation tr µ k = tr µ kR = tr µ kL (4.13)for any k .Now consider the M5-brane on a singular C / Γ G . This corresponds to the sit-uation where the G symmetry on the singular locus is unbroken. This means that µ L = µ R = 0, which forces µ to be nilpotent via (4.13). Therefore the image of µ in g C is the variety N of nilpotent elements, and the final Higgs branch is therefore Z e = S e ∩ N . (4.14)– 10 –he simple puncture corresponds to e being the subregular element, and it is a classicmathematical fact by Brieskorn and Slodowy [16, 17] that this space is the singularity C / Γ G . More generally, let us consider the case when the C / Γ G is deformed to a smoothmanifold. Such a smooth deformation can be parameterized by a generic element h in the Cartan of g C . The Higgs branch describing the motion of the M5-brane isthen ( O h,L × Y e × O h,R ) /// ( G L × G R ) (4.15)where O h,L and O h,R are two copies of the orbit O h of elements of g C conjugate to h ,and parameterize the vevs of the adjoint scalars in the 7d vector multiplets on theleft and the right. Since G L × G R is now broken to U(1) rank G , the dimension of theresulting Higgs branch is2 dim H O h + dim H Y e − (2 dim G − rank G ) = 1 , (4.16)again the expected answer.Obtaining the Higgs branch itself is equally straightforward: we now have µ ∈ O h , and the Higgs branch is now S e ∩ O h . (4.17)Again, it is a classic result of Brieskorn and Slodowy [16, 17] that this space preciselygives the deformation of the C / Γ G singularity by the parameter h . In this section, we compare the Seiberg-Witten curve of the 4d generalized bi-fundamental and that of the 6d minimal conformal matter on T when the type is D n . In principle we should be able to analyze the curves of arbitrary type G in auniform fashion, but the authors have not been able to do that.The 6d conformal theory of type D N , on the tensor branch, becomes USp(2 N − N flavors. Therefore, we should be able to reproduce the 4d curve of thistheory by giving a suitable Coulomb branch vev to the 4d generalized bifundamentalof type D N . There are many mathematical ways to connect the simply-laced groups G = A n , D n , E n , thefinite subgroup Γ G of SU(2), and the singularity C / Γ G . Probably the hyperk¨ahler quotient con-struction of Kronheimer [18] is more familiar to string theorists. But this result of Brieskorn andSlodowy was found much earlier in the mathematics literature. More precisely, they arise as follows. To obtain supersymmetric configurations of the 7d gaugefield, we have to solve Nahm’s equations on the half space x > x <
0) for G L ( G R ) as in [19],where x is the direction perpendicular to the M5-brane. The solution (at the complex structurelevel) is that a complex scalar Φ at x = +0 ( x = −
0) is in the orbit of Φ at x = + ∞ ( x = −∞ ).These Φ( x = ±∞ ) are just the vev of the field given by h Φ i = h . So the degrees of freedom fromthe 7d gauge field are given by Φ(+0) and Φ( −
0) which are in O h . – 11 –here is also another limit in which we can check the curve. Instead of goingto the 6d tensor branch, we can first reduce the 6d minimal conformal matter to 5dand add B-fields to the ALE space. This makes the system one D4 brane on the D N orbifold, which is given by the quiver of the form11 2 2 2 11 (5.1)where a circle enclosing i stands for an SU( i ) gauge symmetry, and the edge betweentwo gauge groups stands for the bifundamental. In the figure we used the case N = 6for explicitness. Adding B-fields corresponds to giving mass terms to the D N × D N flavor symmetries. Thus the 4d generalized theory should also realize this quiver bythe mass deformation.The relation of these two theories considered in 5d, namely the USp(2 N − N flavors and this D -type quiver theory, was called “a novel 5d duality”in [4], and is the type D version of the “base-fiber duality” of [20]. What we findhere is that the corresponding 4d theories are both a deformation of a single class Stheory, providing a 4d realization of these dualities. The curve of the generalized bifundamental.
The generalized bifundamentalof type D N is a class S theory of type D N on a sphere with two full punctures anda single simple puncture. Therefore, it has the following Seiberg-Witten curve0 = λ N + φ ( z ) λ N − + · · · + φ N − ( z ) λ + φ N ( z ) (5.2)where λ = xdz/z is the Seiberg-Witten differential, and φ k ( z ) is a k -differential. Wealso need the single-valued-ness of ˜ φ N ( z ) defined by φ N ( z ) = ˜ φ N ( z ) .Let us put the full punctures at z = 0 , ∞ and the simple puncture at z = 1.Writing t = z −
1, the condition at the simple puncture is, according to [7, 21] φ ∼ v t dt, φ ∼ ( v ) t dt, φ k> ∼ v k t dt, ˜ φ N ∼ ˜ v N t dt. (5.3)From this we find that the curve is given by1 z N Y i =1 ( x − m i ) + 2 c + z N Y i =1 ( x − ˜ m i )= 2 x N + M x N − + M x N − + U x N − + U x n − + · · · + U N − x (5.4) In terms of the Hitchin system, these rules are simply understood. The Seiberg-Witten curveis det( λ − Φ) = 0, where Φ is the adjoint field of the Hitchin system on the Riemann surface. Thecondition φ N ( z ) = ˜ φ N ( z ) comes from det( − Φ) = (Pfaff( − Φ)) for so (2 N ). The poles (5.3) comefrom Φ ∼ e/t , where e is in the nilpotent orbit corresponding to the minimal embedding of su (2)into so (2 N ) as su (2) ⊂ su (2) ⊕ su (2) = so (4) ⊂ so (2 N ). In particular, one can check the relation ofthe coefficients of φ and φ in (5.3). – 12 –here m i and ˜ m i are mass parameters, c = Q i ( − m i ˜ m i ) so that φ N ( z ) = ˜ φ N ( z ) issatisfied, M and M are quadratic and quartic polynomials of m i and ˜ m i such that(5.3) are satisfied for φ and φ . The Coulomb branch parameters are from U to U N − . The
USp theory.
Let us next recall the curve of USp(2 n ) with N f + N ′ f flavors:Λ n +2 − N f z N f Y i =1 ( x − m i ) + 2 c + Λ n +2 − N ′ f z N ′ f Y i =1 ( x − ˜ m i )= x ( x n + u x n − + u x n − + · · · + u n ) (5.5)where c = Λ n +4 − N f + N ′ f ) Q N f i =1 ( − m i ) Q N ′ f i =1 ( − ˜ m i ). The differential is λ = xdz/z .This curve in a hyperelliptic form was first found in [22]. The form given abovefollows easily from the brane construction, see e.g. [23].Setting 2 n = 2 N − N f = N ′ f = N , the curve becomes1 z N Y i =1 ( x − m i ) + 2 c + z N Y i =1 ( x − ˜ m i )= Λ x ( x N − + u x N − + · · · + u N − ) (5.6)where c = Λ c .Coming back to the curve of the class S theory (5.4), we consider the regime m i , ˜ m i ∼ O ( ǫ ), Λ := U ∼ O (1) and U k := U u k − ∼ O ( ǫ k − ). Then the firstthree terms of the right-hand-side of (5.4) can be neglected , and becomes (5.6).The identification of U with some power of Λ is natural since the vev of the tensormultiplet in 6d is proportional to the gauge coupling of the USp(2 N −
8) gauge group.This is consistent with the guess that this class S theory is the T compactifi-cation of the minimal conformal matter of type D N . Also, we learn that the tensorbranch scalar becomes U , of scaling dimension 6, independent of N , and is thecoupling constant of the USp theory. The D -type quiver. This is a completely different limit than the above USp limit.Note first that the D -type quiver (5.1) is in fact just the standard linear quiver withSU(2) N − gauge group, whose curve is well known.We start from the curve (5.4) of the class S theory, and focus on the neighborhoodof the simple puncture at z = 1, by setting z = (1 + t ) / (1 − t ), where t is very small.The curve is given, up to terms of O ( t ), by0 = t ( x N + c x N − + c x N − + c x N − + · · · + c N − x + c N )+2 t ( µ x N − + µ x N − + µ x N − + · · · + µ N − x + µ N )+( µ ) x N − + U ′ x N − + · · · + U ′ N − x + b N (5.7) This scaling limit is a little subtle due to the fact that our USp theory is not asymptoticallyfree. For example, we throw away the term x N but retain both zx N and z − x N . – 13 –here we have defined x N + c x N − + · · · + c N = 12 N Y i =1 ( x − m i ) + N Y i =1 ( x − ˜ m i ) ! ,µ x N − + · · · + µ N = − N Y i =1 ( x − m i ) − N Y i =1 ( x − ˜ m i ) ! ,U ′ k = − U k + c k , b N = ( − N Y i m i − Y i ˜ m i ! . The differential is λ = xdz/z ∼ xdt ∼ tdx . One can check that the above curve isachieved in the scaling limit t ∼ O ( ǫ ) , x ∼ O ( ǫ − ) , , m i + ˜ m i ∼ O ( ǫ − ) , m i − ˜ m i = O (1) , U ′ k ∼ O ( ǫ − k +2 ) and ǫ →
0. A similar limit was also considered in class Stheories of type A N − [24], and as in there, the parameters m i − ˜ m i may correspondto the masses of hypermultiplets in the quiver and m i + ˜ m i may correspond to gaugecouplings in 5d.The coefficients of the terms tx N − and x N − are constrainted by the nonlinearrelation of the pole coefficients at the simple puncture (5.3). This nonlinear relationis called a c-constraint in [7].Now, rewrite the curve as( ξ N + c ξ N − + · · · + c N ) | {z } = p ( ξ ) λ + 2 ( µ ξ N − + · · · + µ N ) | {z } = q ( ξ ) ( dx ) λ + (( µ ) ξ N − + U ′ ξ N − + · · · + U ′ N − x + b N ) | {z } = r ( ξ ) ( dx ) = 0 (5.8)where we introduced ξ = x . In the Seiberg-Witten curve of type D , ± x needs to beidentified, and therefore this is a natural choice.Let us put it in the Gaiotto form by defining ˜ λ = λ + q ( ξ ) dx/p ( ξ ), for which wehave ˜ λ + ϕ ( ξ ) = 0 . (5.9)We can check that ϕ ( ξ ) = ( dξ ) ( p ( ξ ) r ( ξ ) − q ( ξ ) ) / ξp ( ξ ) has second-order poles at N zeros of p ( ξ ). Thanks to the special forms of the coefficients of tx N − and x N − in (5.7), ϕ ( ξ ) is finite at ξ = ∞ . To study the behavior at ξ = 0, recall the scalinglimit described above. In that limit, we get [ c N b N − ( µ N ) ] / ( µ N ) → ϕ at ξ = 0 disappears in the scaling limit. This is a consequence of thecondition φ N ( z ) = ˜ φ N ( z ) . Therefore, we see that the curve is indeed that of theSU(2) N − quiver drawn above. With a little further effort, it can be checked thatthe residues of the double poles of ϕ are proportional to ( m i − ˜ m i ) , so m i − ˜ m i areindeed proportional to the hypermultiplet masses of the quiver.– 14 – Very Higgsable theories and the central charges
In this section, we study the T compactification of the class of 6d SCFTs thatwe call very Higgsable. We will determine the structure of the part of the Coulombbranch of the T compactification that comes from the 6d tensor branch, and showin particular that there is a point where one has a 4d SCFT. We will also show thatthe central charges a , c and k of the 4d SCFT can be written as a linear combinationof the coefficients of the anomaly polynomial of the 6d SCFT. Since the 6d minimalconformal matters are very Higgsable, we can apply the methods developed here toprovide another check of our identification.Let us summarize the contents of this section. In Sec. 6.1, we introduce the classof the 6d SCFTs of our interest, namely the very Higgsable theories. In Sec. 6.2, werecursively prove that • the T compactification of a very Higgsable theory gives a 4d SCFT, and • the central charges of the resulting 4d SCFT can be written as a linear combi-nation of coefficients of the anomaly polynomial of the 6d theory.In 6.3, we compute the central charges of the minimal conformal matter on T byusing the relationship with the anomaly polynomial of the minimal conformal matter.We will see that the resulting central charges indeed agree with the known centralcharges of the class S theory involved. Let us first define the class of 6d very Higgsable theories. In terms of the F-theoretic language of [1, 2, 4], a 6d SCFT can be characterized by the configuration C of curves on the complex two-dimensional base. We define a 6d SCFT to be veryHiggsable if successive, repeated blow-downs of − C empty, or equiv-alently the endpoint C end is empty. Then a further complex structure deformationremoves the singularity completely. In other words, there is a Higgs branch wherethe tensor multiplet degrees of freedom are completely eliminated, thus the wordvery Higgsable. As examples, the 6d ( G, G ) minimal conformal matters and the gen-eral rank E-string theories are very Higgsable, whereas the N =(2 ,
0) theory and theworldvolume theory of
Q > • Free hypermultiplets are very Higgsable. • An SCFT is very Higgsable if – 15 – it has a one-dimensional subspace of the tensor branch on which the low-energy degrees of freedom consist of a single tensor multiplet, one or morevery Higgsable theories, possibly with a gauge multiplet G , – such that the Chern-Simons coupling S CS of the self-dual two-form fieldof the tensor multiplet B , and its associated Green-Schwarz term I GS inthe anomaly polynomial is S CS = 2 π Z B ∧ I , I GS = 12 I , I ⊃ p ( T ) + 14 Tr F F −
14 Tr F G , (6.1)where the term Tr F F / F G / − F G / π ) − is absorbed into F G . Therefore, this means that the instanton-string has charge 1 under the tensor multiplet, which is the minimal consistent valueunder the Dirac quantization condition. The p ( T ) etc. are the usual Pontryagindensities of the background metric.We would like to study the T compactification of a very Higgsable theory.Consider a tensor multiplet scalar φ associated to a − u comes from the scalar φ , combined with the zero mode of theself-dual 2-form on the torus b = R T B : u ∼ exp( φ + 2 πib ) , (6.2)where b ≃ b + 1 due to the invariance under the large gauge transformation. Theclassical description in (6.2) is valid in the region where φ is large compared to thesize of T ; the moduli space can be significantly modified near φ ∼ u with all the other Coulombbranch variables. However, in the case of the scalar u for a − H of the Coulomb branch parametrized by it. This is becauseif a gauge multiplet is present on the minimally-charged tensor branch, the 4d gaugecoupling of the gauge field is infrared free, as we will prove below.Before proceeding, let us see two examples of this infrared freedom: • First, the one-dimensional tensor branch of the ( D k , D k ) minimal conformalmatter for k ≥ k − k , and therefore the system is infrared free as a 4d gauge theory. It may not be completely rigorous to write a Lagrangian like (6.1) for the self-dual 2-form B .But we will only need dimensional reduction of that Chern-Simons term under the compactificationon T given by 2 π R bI where b = R T B . – 16 – Second, the F-theory realization of the ( E , E ) minimal conformal matterhas three curves with self-intersection − − −
1. The middle − − − − E theory of Minahan and Nemeschansky. One copy has theflavor current central charge k E / G , it is meaningful to talk about the origin ofthe Coulomb branch of G even quantum mechanically. This determines the subspace H . H and the central charges6.2.1 Properties to be recursively proved Now, we use the mathematical induction to prove the following properties ofvery Higgsable theories: • The topology of H is always the same as that of the rank-1 E-string theory,namely, there are three singularities. Here, – two of them are the points where a single hypermultiplet becomes mass-less, and – the third of them is a point at which the non-trivial SCFT appears, withthe R-charge of the Coulomb branch operator u being 12. We call theresulting 4d SCFT as T d . • Writing the anomaly polynomial I of the 6d theory T d as I ⊃ αp ( T ) + βp ( T ) c ( R ) + γp ( T ) + X i κ i p ( T ) Tr F i , (6.3)the central charges a, c and flavor central charges k i of i -th flavor symmetry ofthe 4d theory T d are a = 24 α − β − γ,c = 64 α − β − γ,k i = 192 κ i . (6.4) Our normalizations and notations of 6d anomaly polynomials follows those in [26]. – 17 – .2.2 Rough structure of the proof
As the discussions will be rather intricate, here we provide the schematic struc-ture of the inductive proof. The first step is to check the relations (6.4) for the freehypermultiplets. In addition, we can check that free vector multiplets and free tensormultiplets both satisfy the relations (6.4).The inductive step is to study the system of a minimally-charged tensor mul-tiplet, with a very Higgsable theory. There are two subcases: i) when there is nogauge multiplet, and ii) when there is a single gauge multiplet G . The subcase i)corresponds to the appearance of an E-string, for which the structure of H was stud-ied long time ago [27]. In the subcase ii), the vev u ∈ H controls the dynamical scaleΛ( u ) of the gauge group G . Since the coupling of G is infrared free, Λ( u ) is the scaleof the would-be Landau pole. From holomorphy, we expect at least one point on u ∈ H where Λ( u ) is zero. This is where we should have a nontrivial 4d SCFT T d .From this, we will show that there will be two and only two additional singularitieson H , and that these are points where one massless hypermultiplet appears.In both subcases, we see that the structure of H is the same. Once this is known,we can employ the method of [28] to determine the central charges a , c and k of T d in terms of the 6d anomaly polynomial. This then confirms the general relation (6.4),completing the inductive process. H Now let us start the full discussion of the inductive step. We first would like toestablish the singularity structure of H . When there is no gauge multiplet on thetensor branch, we have the E-string theory, for which the structure of H is known[27]. There is a point where we have a 4d E theory of Minahan and Nemeschansky,where the R-charge of the Coulomb branch operator u is 12 and therefore the scalingdimension is 6. This is true for higher-rank E-string theory too.Let us next consider the case with a gauge multiplet with gauge group G . Denotethe very Higgsable theory on this tensor branch by S . The low energy theory onthis branch consists of S , the non-abelian gauge multiplet G , and a U(1) (or tensor)multiplet containing u , and we want to show that there is a point at which they arecombined into a single strongly interacting superconformal theory T .The theory S has flavor symmetry H (not necessarily simple), and its subgroup G ⊂ H is gauged by the non-abelian gauge group. The commutant F of G in H is the flavor symmetry of the total system. The term proportional to Tr F G p ( T ) inthe total 6d anomaly polynomial is given by I S + I tensor + I gaugino + I GS ⊃ ( κ S d G − h ∨ G −
116 ) Tr F G p ( T ) . (6.5)The gauge group G is anomaly free in 6d, therefore48 κ S d G − h ∨ G = 3 . (6.6)– 18 –sing the inductive assumption (6.4), we see that k S d G − h ∨ G = 12 > G gauge couplingin the 4d theory is infrared free. This guarantees that we can isolate the subspace H as we repeatedly emphasized above.In addition, away from the singularities on H , we can safely introduce the expo-nentiated complexified coupling η ( u ) := Λ − e πiτ G ( u ) (6.8)of the 4d G gauge field, defined at an arbitrary (but sufficiently small) renormalizationgroup scale Λ , where − h ∨ − k S d G / − πb ·
14 Tr F G (6.9)after the compactification, at least for large values of φ . Then − πb can be identifiedas the theta angle of the gauge group in 4d, Re( τ G ) = − b . Together with thedefinition (6.2) of u and holomorphy of τ G ( u ), we can see that in the region | u | →∞ , the G coupling behaves as η ( u ) ∼ u − . We expect η ( u ) to be a single-valuedmeromorphic function on H . We do not expect any zeroes of η ( u ): if there is azero, the gauge coupling of G becomes extremely weakly coupled there, but we donot know of any physics to explain it. A single valued meromorphic function withthe asymptotic behavior η ( u ) ∼ u − must have just a single simple pole. We definethe coordinate origin of H so that the pole of η ( u ) is at u = 0. This is the stronglyinteracting point where T d appears. The 4d theta angle of the G gauge multiplet at u = 0 is just given by the phase of u , globally on H .Slightly away from this point u = 0, the infrared physics is the theory S d coupledto the G gauge multiplet. The U(1) R G is anomalous by the amount (6.7). At theSCFT point this U(1) R symmetry should be restored and it must be anomaly free.By the anomaly matching, the Nambu-Goldstone boson of the spontaneously broken In the case in which η ( u ) can have multivalued behavior, there must be a duality transformationrelating those multi-values of the coupling constant. For example, in Seiberg-Witten theory of amassless U(1) field, a free U(1) has an electric-magnetic dual description which changes the couplingas τ → − /τ , and this was crucial for the multivalued behavior of τ [29]. However, in our case, η ( u )is properly understood as the position of the Landau pole of the infrared free gauge field, and inparticular it is a dimensionful parameter. There seems to be no duality transformation which sendsone value of the Landau pole to another, and hence η ( u ) is single-valued. However, if the gaugegroup were conformal rather than infrared free, we could have multivalued coupling constant onthe moduli space due to S-duality of the conformal gauge group. Such a situation indeed appearsin other theories and will be discussed elsewhere. – 19 –(1) R at u = 0 must contribute −
12 to the anomaly U(1) R G via the coupling (6.9),where 2 πb should be interpreted as the phase of u in the small u region. This can bedone by assigning the U(1) R charge R [ u ] = 12 (6.10)to the u near u = 0. Then the total U(1) R G anomaly is cancelled.We now show that there are two more singularities on H and that these twopoints are associated with an additional massless hypermultiplet. The proof goes asfollows: consider the Seiberg-Witten curve on H given by y = x + f ( u ) x + g ( u ) . (6.11)This curve is for describing the effective action of the U(1) gauge field coming fromthe 6d tensor multiplet, and it should not be confused with τ G which is the couplingof the non-abelian gauge group G .Using the special coordinate on H related to the curve (6.11) via dadu = Z A dxy , da D du = Z B dxy , (6.12)where A and B are the two independent cycles of the torus, the metric on H is ds = Im ( da ∗ da D ) = Im (cid:18) dadu ∗ da D du (cid:19) | du | . (6.13)The complex structure τ = da D /da is constant at | u | → ∞ since it is given by thecomplex structure of the T used in the compactification from 6d to 4d. Then f and g should behave as f ∼ u n and g ∼ u n for some n for large u . Furthermore, themetric on H at | u | → ∞ is the cylindrical one ds ∼ dφ + (2 πdb ) ∼ | d log ( u ) | sinceit just comes from the compactification of a free tensor multiplet. Substituting theasymptotic behavior of f and g to (6.11), (6.12) and (6.13), we obtain n = 1.Next, let us consider the singularity at u = 0. We set the asymptotic behaviorof f and g at u = 0 as f ∼ u p and g ∼ u q . Then, the R-charge of x and y in (6.11)is R [ x ] = 2 rR [ u ] , R [ y ] = 3 rR [ u ] , (6.14)where r = min ( p, q ). The R-charge of the Seiberg-Witten differential λ , which is thesame as the R-charge of u ( ∂λ/∂u ) = udx/y , is fixed to 2 since its scaling dimensionis 1. Using (6.14), the relation (1 − r ) R [ u ] = 2 (6.15)holds. The fact R [ u ] = 12 at u = 0 leads to r = q = 5 / p ( > r ) is 1. Thuswe obtain f ∼ u and g ∼ u near u ∼
0. Therefore the behavior of f and g on H is f ∼ u , g ∼ u + u . (6.16)In particular, examining the discriminant ∆ = 27 f + 4 g , there are two more sin-gularities other than u = 0 and that they are massless hypermultiplet points.– 20 – .2.4 Central charges from measure factors Before proceeding, let us very briefly recall the method of [28] to compute thecentral charges a , c and k of 4d N =2 SCFTs from their topologically twisted cousins;we almost follow the conventions used in that paper. We put an N =2 supersymmetricfield theory in 4d on a curved manifold with a non-trivial metric and a backgroundgauge field for the flavor symmetry F via the twisting of the SU(2) R R-symmetrywith one of the SU(2)’s of SU(2) × SU(2) ≃ SO(4) of the tangent bundle. In thefollowing we assume that F is nonabelian. We denote the Euler characteristic ofthe 4-manifold by χ , the signature by σ and the anti-instanton number for F by n .We also denote by u a set of gauge and monodromy invariant coordinates on theCoumlomb branch.The path integral of the twisted theory is given as follows Z = Z [ du ][ dq ] A χ ( u ) B σ ( u ) C n ( u )exp( − S low energy ) . (6.17)Here [ du ] and [ dq ] are the path integral measures for the massless vector multipletsand other massless multiplets on the generic point of the Coulomb branch. The A ( u ), B ( u ) and C ( u ) are factors induced by the non-minimal coupling of u to thenon-trivial background which are given, up to coefficients, as R log A ( u ) tr R ∧ ˜ R , R log B ( u ) tr R ∧ R and R log C ( u ) tr F F ∧ F F in the effective action on the Coulombbranch. Supersymmetry requires that they are holomorphic. See [30] for details.On a singular point on the Coulomb branch, we can have nontrivial superconfor-mal field theory. Then there must be an enhanced U(1) R symmetry at each of thesepoints, although U(1) R need not be defined globally on the Coulomb moduli space.The coefficients of the anomaly of U(1) R under background fields are related to thecentral charges a, c, k by supersymmetry as Z d x∂ µ j µ U(1) R = (4 a − c ) χ + 3 cσ + kn, (6.18)where the term χ is due to twisting SU(2) R . By using the same anomaly matchingwhich was used to derive (6.10), the central charges a , c and k are obtained as [28] a = 14 R [ A ] + 16 R [ B ] + a generic , (6.19) c = 13 R [ B ] + c generic , (6.20) k = R [ C ] + k generic (6.21)where R [ A, B, C ] are the U(1) R -charges of the measure factors A ( u ) , B ( u ) , C ( u ), and( a, c, k ) generic are the central charges at a generic point on the Coulomb branch. Theterms proportional to R [ A, B, C ] are the contributions from U(1) R Nambu-Goldstonebosons near each superconformal point. For the gauge group G , what we have foundin the previous subsection may be rephrased as k | G = 0, k generic | G = k S d G − h ∨ G = 12, C | G ∼ exp(2 πiτ G ( u )) ∼ u − and R [ C | G ] = − R [ u ] = − .2.5 Central charges6d anomalies. Suppose that the 4-form appearing in (6.1), now including thesecond Chern class c ( R ) of the SU(2) R background field, is given by I = dc ( R ) + 14 p ( T ) + 14 Tr F F −
14 Tr F G (6.22)The explicit value of d can be determined by the method explained in [26] but it isnot important here. The contribution to the 6d anomaly polynomial from (6.22) is12 I ⊃ dc ( R ) p ( T ) + 132 p ( T ) + 116 p ( T ) tr F F . (6.23)Therefore, the changes in the coefficients α, β, γ, κ of (6.3) are δα = 132 , δβ = 14 d, δγ = 0 , δκ = 116 . (6.24)
4d central charges.
We now would like to determine the changes in a , c , k in 4d.To do this, we use the method of [28] recalled above. Putting the theory on a curvedmanifold via twisting leads to the path-integral (6.17).As before, we denote by u the coordinate of H . We have one singularity at u = 0giving the 4d SCFT of our interest, and there are two additional hypermultipletpoints at u = 1 , λ where λ is the function of the complex moduli τ of the torus onwhich we compactify the 6d theory. We denote by R , ,λ , the R-charge of u near u = 0 , , λ . Then, the measure factors A , B and C transform under ( u − p ) → exp( iR p α )( u − p ) (where p = 0 , , λ ) as A χ B σ C n → exp[ i { (4 δa p − δc p ) χ + 3 δc p σ + δk p n } α ] A χ B σ C n (6.25)where δa p , δb p and δk p are differences of a , b and k between the theory on u = p andthe theory on a generic point of H . This is just the anomaly matching of the U(1) R anomaly (6.18) discussed above.Next consider very large | u | region. In this region, H looks like a cylinder log u ∼ φ + 2 πib . By the dimensional reduction of (6.1), the b has a coupling 2 πbI . In thetopologically twisted theory, the I of (6.22) becomes I = − d χ + 34 (1 − d ) σ + n F − n G (6.26)where we used the fact that c ( R ) = − χ − p ( T ) due to the topological twist, and σ = p ( T ) /
3. We abuse the notation for χ, σ and n to mean the densities of theEuler number, signature and anti-instanton number as well as their integrals, e.g., n = Tr F . Using (6.26) and noting that 2 πibI should be completed as log( u ) I due to holomorphy, we can determine the factor A χ B σ C n as A χ B σ C n ∼ exp (cid:20)Z log( u ) I (cid:21) = ( u − d ) χ ( u (1 − d ) ) σ u n F (6.27)– 22 –nd in particular, the phase shift under u → e iα u is given as A χ B σ C n → exp (cid:20) iα (cid:0) − d χ + 34 (1 − d ) σ + n F (cid:1)(cid:21) A χ B σ C n . (6.28)Now consider a circle S going once at a large value of | u | . The phase shift isgiven by (6.28) with α = 2 π . Then we shrink this circle so that it becomes smallcircles around each of the singular points u = 0 , , λ . The phase shift around eachcircle is given by (6.25) with α = 2 π/R p .It is known that B and C are single valued functions of u [30]. Then the phaseshift around the large circle should be the same as the sum of the phase shifts aroundthe singular points. First, for C we get1 = X u =0 , ,λ δk u R u = δk R (6.29)where we used the fact that δk ,λ = 0 because at u = 1 , λ only an additional hy-permultiplet appears which is not charged under the non-abelian flavor group F .Therefore we can determine the change in the flavor central charges: δk = R = 12 = 192 δκ, (6.30)where δκ is given in (6.24).Next, for B we get 34 (1 − d ) = X u =0 , ,λ δc u R u . (6.31)The δc at u = 1 , λ comes from a free hypermultiplet and it is given as δc = c hyper =1 /
12. The U(1) multiplet containing u is IR-free at u = 1 , λ and hence the R-chargeis that of the free vector multiplet, R ,λ = 2. Therefore we get δc = 2 − d = 64 δα − δβ − δγ. (6.32)where δα, δβ and δγ are given in (6.24).Finally, we consider A . In this case, A is not a single valued function [30].However, the nontrivial monodromy of A is fixed by the Seiberg-Witten curve of theU(1) multiplet of u . The equation (6.16) implies that the Seiberg-Witten curve iscompletely the same as that of rank-1 E-string theory on T . Therefore, the ratio A ( u ) /A E ( u ) is single-valued, where A E ( u ) is the A -factor of the rank-1 E-stringtheory on T .This A E ( u ) is known to behave as u / around u ∼ ∞ as can be seen from theanalysis of the E Minahan-Nemeschansky theory [28] or from the fact that the studyof the 6d anomaly polynomial gives d = − A ( u ) /A E ( u ) ∼ u − ( d +1) / . (6.33)– 23 –urthermore, the hypermultiplet contributions cancel out in the ratio A ( u ) /A E ( u )at u = 1 , λ . Therefore (6.33) is actually valid over the whole H . We get − d + 12 = δ (4 a − c ) − R [ A E ] R (6.34)where R [ A E ] is the R-charge of A E at u = 0. It is given as A E ( u ) = ( ∂u E /∂a E ) / [28] and hence [ A E ( u )] = 5. Thus δ (2 a − c ) = − d −
12 = − δα − δβ − δγ. (6.35)Combining (6.30), (6.32) and (6.35) with the assumption of the induction, the proofof (6.4) is completed. E theories As first examples of our general analysis, let us first consider the E-string theoryof general rank. When put on T , this is known to reduce to the general-rank versionof the E theory of Minahan and Nemeschansky. The central charges a , c and k ofthese theories were found in [31]: a = 32 Q + 52 Q − , (6.36) c = 32 Q + 154 Q − , (6.37) k E = 12 Q, (6.38) k SU(2) L = 6 Q − Q − , (6.39)where Q is the rank.The anomaly polynomial of 6d higher-rank E-string theories was obtained in[32]. The relevant coefficients in the anomaly polynomial are α = 7(30 Q − , β = − Q (6 Q + 5)48 , γ = 1 − Q . (6.40)and κ E = Q , κ SU(2) = 132 Q − Q − T As second examples, let us consider the central charge of the 6d (
G, G ) minimalconformal matter on T . The anomaly polynomial of that theory was obtained in– 24 –26]. The relevant coefficients in the anomaly polynomial are α = 75760 (1 + dim( G )) , β = 148 (dim( G ) − χ Γ | Γ | ) ,γ = − G )) , κ G = h ∨ G . (6.42)where | Γ | is the number of elements of the discrete group Γ used in the orbifold C / Γ,and χ Γ := 1 + rank( G ) − / | Γ | . From (6.4), we obtain the central charges as a = 124 (1 + 6 χ Γ | Γ | − G )) , c = 112 (1 + 3 χ Γ | Γ | − G )) , k G = 2 h ∨ G . (6.43)Then, we compute the central charges of the class S theory of type G on a spherewith two full punctures and a simple puncture. The relevant formula [13] is a = a simple + 2 a full − h ∨ G dim( G ) −
524 rank( G ) , (6.44) c = c simple + 2 c full − h ∨ G dim( G ) −
16 rank( G ) , (6.45) k G = k full , (6.46)where a simple and a full are the contribution from the simple and full puncture, respec-tively. The contributions from the punctures are given by [13] a simple = 124 (6 | Γ | χ Γ + 1) , a full = 124 (4 h ∨ G dim( G ) −
52 dim( G ) + 52 rank( G )) ,c simple = 112 (3 | Γ | χ Γ + 1) , c full = 112 (2 h ∨ G dim( G ) − dim( G ) + rank( G )) ,k full = 2 h ∨ G . Substituting these equations into (6.44), (6.45) and (6.46), we obtain the same centralcharges as (6.43). This provides a non-trivial check both for the central chargeformula in (6.4) and the duality between the minimal conformal matter on T andthe class S theory. In this paper we found that the world volume theory of a single M5-brane onthe tip of an ALE space of type G = A, D, E , namely the 6d (
G, G ) minimal con-formal matter, gives a type G class S theory with a sphere accompanied by two full-punctures and a simple puncture, namely 4d generalized bifundamental, by meansof T compactification.We have given several evidences on this statement. We provided the match-ing of coulomb branch dimensions and the Higgs branch geometry, and we checked– 25 –he agreement of the Seiberg-Witten curve in the case of type D in a certain cor-ner of the moduli space, by exhibiting the “base-fiber duality” indicated by the 6dbrane construction at the level of the 4d Seiberg-Witten curves. We also developeda new method to study the central charges of the T compactification of a class ofthe 6d SCFTs that we call very Higgsable, and applied this technique to the mini-mal conformal matters. We again found agreement with the central charges of theclass S theories. With these checks, we find that our proposed identification is wellestablished.Let us discuss some of the future directions. Other very Higgsable theories
There are many very Higgsable theories whichare neither (
G, G ) minimal conformal matters nor higher-rank E-string theories. For T compactifications of all of those, we showed that the formula (6.4) holds.Some of these theories can be obtained by considering “fractional M5-branes”on ALE singularities: • The ( E , SO(7)) minimal conformal matter, namely a “half M5-brane” on topof E singularity, • the ( E , G ) minimal conformal matter which is a “third M5-brane” on E singularity, • and the ( E , F ) minimal conformal matter which is a “half M5-brane” on E singularity.For the ( E , SO(7)) minimal conformal matter, we can find a candidate of thecorresponding 4d theory in the list of E tinkertoys [33]. Conbining the method of[26] and the formula (6.4), we find the central charges of T compactified ( E , SO(7))minimal conformal matters are a = 1198 , c = 352 , k E = 24 , k SO(7) = 16 . (7.1)Those numbers are exactly the same as the conformal central charges of E fixturewith punctures E ( a ), 2 A and the full puncture, where the notation of the punc-tures are of [33].Similarly, the candidates for the ( E , G ) and ( E , F ) minimal conformal mattermight be found in E or E fixtures. But the list of E and E fixtures are not yetavailable.Another natural series of very Higgsable theories can be found by consideringtheories on M5-branes on the intersection of an end-of-the-world brane and an ALEsingularity locus. In contrast to the minimal conformal matters, the theories areendpoint-trivial for all integer numbers of M on very Higgsable theories The worldvolume theories on multiple coincidentM5-branes on an ALE singularity locus, are not very Higgsable. Thus the approachof this paper cannot be directly applied and new methods need be introduced toinvestigate such theories.The N =(1 ,
0) SCFTs which are defined by the F-theory with Hirzebruch’s sur-face F n as its base are other cases recently studied in [34]. Although the structureof the base F n is very straightforward, in that it contains just one − n curve, ourmethod cannot be applied to these when n ≥
3. It would be interesting to devise amethod that can be applied to the T compactification of any 6d SCFT. Compactification with general Riemann surfaces and punctures
Our ulti-mate goal would be to study compactifications of 6d N =(1 ,
0) theories with generalRiemann surfaces with punctures giving 4d N =1 theories rather than 4d N =2.Although there clealy is a N =1 theory defined by compactification of a N =(1 , g ≥ N =(2 ,
0) case, the theory on the tube isalready non-trivial, preventing us from studying on S-dualities between compactifiedtheories. The T compactified theories studied in this paper might be a clue to findout the tube theories if one can find an appropriate boundary conditions at the endsof the tube.The authors hope to come back to these questions in the future. Acknowledgments
KO and HS are partially supported by the Programs for Leading GraduateSchools, MEXT, Japan, via the Advanced Leading Graduate Course for PhotonScience and via the Leading Graduate Course for Frontiers of Mathematical Sci-ences and Physics, respectively. KO is also supported by JSPS Research Fellowshipfor Young Scientists. YT is supported in part by JSPS Grant-in-Aid for ScientificResearch No. 25870159, and in part by WPI Initiative, MEXT, Japan at IPMU,the University of Tokyo. The work of KY is supported in part by DOE Grant No.DE-SC0009988.
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