Frozen singularities in M and F theory

Journal of High Energy Physics, Jun 2016

We revisit the duality between ALE singularities in M-theory and 7-branes on a circle in F-theory. We see that a frozen M-theory singularity maps to a circle compactification involving a rotation of the plane transverse to the 7-brane, showing an interesting correspondence between commuting triples in simply-laced groups and Kodaira’s classification of singular elliptic fibrations. Our analysis strongly suggests that the O7+ plane is the only completely frozen F-theory singularity.

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Frozen singularities in M and F theory

Received: May Frozen singularities in M and F theory Yuji Tachikawa 0 1 2 0 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 University of Tokyo , Bunkyo-ku, Tokyo 113-0033 , Japan 2 Department of Physics, Faculty of Science We revisit the duality between ALE singularities in M-theory and 7-branes on a circle in F-theory. We see that a frozen M-theory singularity maps to a circle compactication involving a rotation of the plane transverse to the 7-brane, showing an interesting correspondence between commuting triples in simply-laced groups and Kodaira's classi - cation of singular elliptic brations. Our analysis strongly suggests that the O7+ plane is the only completely frozen F-theory singularity. F-Theory; M-Theory; String Duality - 1 Introduction and summary 2 Frozen singularities in M-theory and their F-theory duals 3 Frozen F-theory 7-branes and their M-theory duals 4 Discussions 1 3 6 where g is a nite subgroup of SU(2) corresponding to a simply-laced algebra g produces a 7d super Yang-Mills theory with gauge algebra g at the singular locus. It might be less-known that there are (partially) frozen variants of such a singularity, still preserving 16 supercharges, characterized by a non-zero value of r := Z S3= g n d C = mod 1 (1.1) around it [1{3]. Here, C is the M-theory 3-form and d is a label on the nodes of the Dynkin diagram of g and gcd(n; d) = 1. On such an M-theory singularity, we have a gauge algebra hr. We list possible values of r, hr for all g in table 1.1 Let us now consider a situation where such a singularity is in a singular ber of an elliptic bration Xg over the complex plane2 Cz, such that there is an SL(2; Z) monodromy g around z = 0 acting on the elliptic ber. Possible conjugacy classes of monodromies were classi ed by Kodaira, see table 2. The central ber can be appropriately blown up so that the whole geometry is smooth, but we mainly consider the case that there is a singularity there. When the monodromy is diagonalizable, the supergravity solution can be taken so that the base Cz is conical, with a metric of the form dR2 + R2d 2 with a nontrivial periodicity + g. We call g the opening angle in this note. Now let us rst consider standard singularities which are not (partially) frozen. At each point on a base, we can reduce along an S1 of the ber and take the T-dual of the other S1 to have a type IIB setup on S1. The monodromy around z = 0 now acts on the axiodilaton of the type IIB theory, therefore we are now in an F-theory con guration: we have a 7-brane at z = 0, compacti ed on S1. As is well-known, when we shrink the elliptic ber on the M-theory side, the S1 on the F-theory side opens up. We see that a 7-brane characterized by a monodromy g, obtained in a manner outlined above, has a gauge symmetry g on it [4, 5]. { 1 { hr so(2k + 8) 1 2 sp(2k) 1Our convention is sp(2) = su(2). 3Without counting mirror images. 4Our convention is that O symmetries. brations over the base Cz. Here, g is the type of the singularity and g is the opening angle. The equation of the elliptic x3 + a(z)x + b(z) where z is the coordinate of the base, and = 4a3 + 27b2. ber is given by y2 = It is known, however, that the monodromy alone does not completely characterize a 7brane in F-theory. For example, the singular ber of type I4 can be realized in perturbative type IIB string theory by putting 8 D7-branes3 on top of an O7 plane,4 and also by just an O7+ plane. The former has so(16) gauge algebra on it but the latter does not have any, and therefore they are clearly distinct. Correspondingly, the former I4 singularity can be deformed, but the latter I4 singularity is somehow completely frozen, probably due to an e ect of a discrete ux [6]. This begs a natural question: are there other (partially) frozen variants of F-theory 7-branes? Clearly I4+k singularities have two versions: one given by an O7 plane plus 2In the following the subscript on C denotes the symbol for its coordinate. planes give orthogonal symmetries and O+ planes give symplectic with a d1 shift along S1. this note is to argue that there are no other (partially) frozen half-BPS 7-branes. We approach this question by rst studying an M-theory con guration on an elliptic bration Xg over Cw with a singularity of type g at the central ber, now with a nonzero value of r = nd de ned in (1.1). In section 2, we will see that, a ber-wise duality to the type IIB description, we have an F-theory con guration on (Cz S1)=Zd where w = z d with monodromy gd around z = 0, such that the quotient is given by z 7! e2 in=dz together We then run the arguments in reverse in section 3. Namely, we take a putative (partially) frozen half-BPS 7-brane in F-theory, and compactify it on S1. We will take a theory with gauge algebra hr which is given by the Langlands dual of the subalgebra of g commuting with this at bundle, or equivalently the commuting triple.5 If hr is empty the singularity is completely frozen. If hr is nonempty, the singularity is only partially frozen and can be deformed to a completely frozen one, whose type is a minimal one compatible with a given value of r. r = 12 dual is hr = sp(2k). For example, take g = e8 and r = 13 . The minimal algebra compatible with this value of r is e6, whose nontrivial commuting triple is in fact contained in f4. The commutant of this f4 is g2, whose Langlands dual gives hr = g2. As another example, take g = so(2k + 8) and . The minimal algebra compatible with this r is so(8), whose nontrivial commuting triple is in fact contained in so(7). Its commutant within g is so(2k + 1), whose Langlands Take now an elliptic bration Xg over Cw with a singularity of type g in the singular ber at w = 0. We remind the reader that g stands for the SL(2; Z) monodromy at w = 0. We furthermore put a nonzero value of r to (partially) freeze the singularity. We would like to construct an F-theoretic dual description of this setup. and the last subsection of [2]. 5These facts can be understood via the fractionation of M5-branes and its relation to instantons on R, see e.g. [7] and section 3.1 of [8]. Readable accounts on triples can be found in the last section of [9] { 3 { frozen with r = 12 . It is easiest to start with the case of the Ik ber with a Dk+4 singularity, (partially) When we reduce it to type IIA, this becomes an O6+ plane plus k D6-branes [6, 10]. Since the monodromy preserves an S1 of the elliptic ber up to a multiplication by 1, we can reduce the whole setup globally along this S1 to a genuine type IIA con guration on (Cz SI1IA)=Z2 with an O6+ plane plus k D6-branes on one of the xed points. On the other xed point, we should have an O6 plane, which is known to lift to a smooth con guration in M-theory. Note that we have w = z2, since the two points z on Cz are identi ed by the orientifolding action. Now, take the T-dual to obtain a type IIB con guration. We have a so-called shiftorientifold on (Cz SI1IB)=Z2, where the orientifold action on Cz is accompanied by a 12 shift of SI1IB with 2k D7-branes on the locus z = 0, see e.g. [11]. Note that there is su(2k) gauge algebra locally on the D7-branes, which is broken to sp(2k) by the compacti cation on S1 involving the orientifolding action, that acts as an outer automorphism of su(2k). The orientifolding action z 7! z when we go along SI1IB can also be understood as follows: we had RS3= g C = 12 around the singularity in the M-theory description. This 3-cycle can be deformed into a 3-cycle T given by a large Sb1ig In the type IIA reduction, this means that there is RSb1ig SI1IA B = 12 , which turns into a 12 Cz times the elliptic ber. rotation in the type IIB setup. Let us summarize what we have found in an F-theoretic language. We started from the Ik ber with monodromy g = further compacti ed on S1 such that z 7! an F-theory con guration on Cz=w1=2 with the I2k 0 1 k 1 on Cw, with r = 12 in M-theory. The result is ber whose monodromy is g2 = z when we go around S1. Now, from table 2, we nd that the local singularity has the form st = z2k where s, t are suitable combinations of x and y. The action z 7! z can be lifted to (s; t; z) 7! (s; t; z), which is known to correspond to a Z2 outer automorphism of the su(2k) gauge algebra of the 7-brane of type I2k. elliptic bration Xg with a compatible choice of r = nd by a 7-brane on (Cz SI1IB)=Zd whose monodromy around z = 0 is gd, where the Zd action is given by z 7! e2 in=dz together with a d1 shift along SI1IB. The results are tabulated in . Namely, the F-theory dual is given We nd that the opening angles g of Cw and gd of Cz satisfy the relation gd = d g, as it should be for the metric to be consistent. We also see that the action z 7! e2 i=d corresponds to the outer automorphism of a required order, as can be checked using the explicit equation of the elliptic brations given in table 2. This reduction of the gauge symmetry due to the outer automorphism is known in F-theory con gurations on complex surfaces. A small novelty here is that the base is real three-dimensional. Before proceeding, we pause to mention that the rotation of the phase of z by 2 nd on the plane Cz transverse to the 7-brane on the F-theory side can be derived by a further compacti cation on S1. Let us start from the M-theory side. We compactify the whole setup on S1. This is a type IIA con guration on an elliptic RT C(3) = nd , where, as before, T is a big circle Sb1ig bration Xg with a non-zero Cw times the elliptic ber, and C(3) { 4 { e6 e6 e7 e7 e7 e8 e8 e8 e8 e8 1 2 1 2 1 2 1 2 I alg. outer xed 1 1 1 0 1 1 0 0 0 0 0 0 0 0 now the RR 1-form.6 This lifts to a new M-theory con guration on is the RR 3-form. We then take a double T-dual along the elliptic ber. This is again a type IIA con guration on an elliptic bration with a non-zero RSb1ig C( 1 ) = nd , where C( 1 ) is (an elliptic bration Xgd over Cz) S 1 =Zd such that the generator of Zd is the rotation of the z plane by 2 nd together with the d1 shift of the M-theory circle.7 Clearly this is the S1 compacti cation of the F-theory con guration described above. 6Such backgrounds were rst considered in [12]. 7On the one hand, close to the singularity in the singular ber, this is essentially the con guration studied in [13]. On the other hand, if we replace the elliptic bration Xg by a compact K3, such a background was rst considered in [14]. { 5 { Frozen F-theory 7-branes and their M-theory duals Let us now study (partially) frozen F-theory 7-branes, by running the argument of the previous section in reverse. Again, it is easiest to start with the case which has a perturbative type IIB realization. Let us consider an O7+ plane. It has a monodromy of type I4 . We compactify the whole system on S1, and take the T-duality. We now have a type IIA system on (Cw S1)=Z2, with O6+ planes on both xed points. Therefore, its M-theory lift has two frozen singularities of type D4. Note that these two singularities are both on the same singular ber of the elliptic bration. Since the sum of two Dynkin diagrams of type D4 is contained in an a ne Dynkin diagram of type D8, we see that the singular ber has the type I4 , as it should be. times the elliptic any rotation. the quotient of S3 around each of the singularities. The three-cycle T given by Sb1ig Note that each of the D4 singularities has RTa C = 12 mod 1 around it, where Ta=1;2 is Cz ber is their sum T1 + T2, and therefore has RT C = 0 mod 1. This is compatible with the fact that the S1 compacti cation on the F-theory side does not involve With this warm-up, let us consider a general (partially) frozen half-BPS 7-brane at z = 0 of Cz, with monodromy g around z = 0. We assume it preserves 16 supercharges. Compactify it on S1 and take the M-theory dual. This operation should be possible away from z = 0 ber-wise. We then have an M-theory con guration of an elliptic bration away from z = 0, with the same monodromy g. Given that it preserves 16 supercharges, it is strongly likely that the M-theory geometry is given by an elliptic bration with singularities no worse than orbifolds of C2 by nite subgroups of SU(2). Let us say there are m singularities of type g1, . . . , gm at the central ber. At least two out of these m singularities should be (partially) frozen; otherwise we can change the Kahler parameter to have just zero or one frozen singularity, and we know those cases do not correspond to (partially) frozen 7-branes. Therefore, at least two of g1, . . . , gm are of type D or E, thus with three prongs. We also know that the sum of Dynkin diagrams of type gi is contained in an a ne Dynkin diagram whose type is determined by the monodromy g. Now, by direct inspection, we can easily see that the only a ne Dynkin diagrams that can contain more than one nite Dynkin diagrams with three prongs are of type D8+k with k 0. The two nite Dynkin diagrams are necessarily of the type D4+k1 and D4+k2 , with k1;2 0. By our assumption both are (partially) frozen. This is a type IIA con guration on (Cw S1)=Z2, with an O6+ with k1 D6-branes on one xed point, and an O6+ with k2 D6-branes on another. Taking the T-dual, we have a type IIB con guration on (Cw=Z2) S1 = Cz S1. We have an O7+ with k1 + k2 D7-branes at z = 0, and the whole system is further compacti ed on S1 with a Wilson line around it, so that sp(2k1 + 2k2) is broken to sp(2k1) sp(2k2). We conclude that a half-BPS (partially) frozen 7-brane is necessarily an O7+ plane, possibly with an integral number of D7-branes on top. Note that our analysis does not allow a stuck 12 D7-brane on top of an O7+, thus precluding the existence of half-BPS Of7 . This is consistent with the + analysis in [1]. { 6 { Discussions In this short note, we argued that there is a duality between M-theory con gurations on an elliptic bration on Cw with monodromy g around w = 0 with a (partially) frozen singularity with RS3= g C = nd mod 1, and F-theory con gurations of a 7-brane on Cz=w1=d with monodromy gd around z = 0, further compacti ed on S1 so that z is rotated as z 7! e2 in=dz when we go around S1. We then argued that, using the same logic, a (partially) frozen half-BPS 7-brane is necessarily a combination of an O7+ plane plus an integral number of D7-branes. HJEP06(21)8 Note that in our argument, we assumed that table 1 exhausted the list of (partially) frozen half-BPS codimension-4 singularities of M-theory. Therefore, we can state our conclusion in a slightly di erent way: if there is a (partially) frozen half-BPS 7-brane other than the O7+ plane plus D7-branes, there should also be a new, hitherto-unknown (partially) frozen half-BPS codimension-4 singularity in M-theory. The author considers this extremely unlikely. F-theory has been used in various di erent constructions in the string theory literature. Very often, it is implicitly assumed that the holomorphically varying axiodilaton corresponds to an elliptic bration with a section and that 7-branes are not (partially) frozen, and it was not clear how serious the unintended consequences were. In the last two years, genus-one brations without a section have been actively investigated, starting with [15], but there are very few works on the O7+ plane in the recent years, a notable exception being [16]. It may be the time to start investigating F-theory setups with (partially) frozen singularities seriously. The author hopes that this short note is useful as a rst step in that direction, by showing that there is no other frozen 7-brane than the O7+-plane. Acknowledgments The author thanks K. Hori and D. R. Morrison for discussions, and M. Esole for valuable comments on an earlier draft of this note. The work is supported in part by JSPS Grantin-Aid for Scienti c Research No. 25870159, and in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [hep-th/0103170] [INSPIRE]. Math. Phys. 6 (2003) 1 [hep-th/0107177] [INSPIRE]. [2] M. Atiyah and E. Witten, M theory dynamics on a manifold of G2 holonomy, Adv. Theor. { 7 { Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE]. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE]. 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Yuji Tachikawa. Frozen singularities in M and F theory, Journal of High Energy Physics, 2016, 128, DOI: 10.1007/JHEP06(2016)128