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 2778583 , Japan
1 University of Tokyo , Bunkyoku, Tokyo 1130033 , Japan
2 Department of Physics, Faculty of Science
We revisit the duality between ALE singularities in Mtheory and 7branes on a circle in Ftheory. We see that a frozen Mtheory singularity maps to a circle compactication involving a rotation of the plane transverse to the 7brane, showing an interesting correspondence between commuting triples in simplylaced groups and Kodaira's classi  cation of singular elliptic brations. Our analysis strongly suggests that the O7+ plane is the only completely frozen Ftheory singularity.
FTheory; MTheory; String Duality

1 Introduction and summary
2 Frozen singularities in Mtheory and their Ftheory duals
3 Frozen Ftheory 7branes and their Mtheory duals
4 Discussions
1
3
6
where g is a nite subgroup of SU(2) corresponding to a simplylaced algebra g produces
a 7d super YangMills theory with gauge algebra g at the singular locus. It might be
lessknown that there are (partially) frozen variants of such a singularity, still preserving
16 supercharges, characterized by a nonzero value of
r :=
Z
S3= g
n
d
C =
mod 1
(1.1)
around it [1{3]. Here, C is the Mtheory 3form and d is a label on the nodes of the Dynkin
diagram of g and gcd(n; d) = 1. On such an Mtheory 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 Tdual 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 Ftheory con guration:
we have a 7brane at z = 0, compacti ed on S1. As is wellknown, when we shrink the
elliptic ber on the Mtheory side, the S1 on the Ftheory side opens up. We see that
a 7brane 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 Ftheory. For example, the singular ber of type I4 can be realized in perturbative
type IIB string theory by putting 8 D7branes3 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 Ftheory
7branes? 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 halfBPS 7branes.
We approach this question by rst studying an Mtheory 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 berwise duality to the
type IIB description, we have an Ftheory 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 halfBPS 7brane in Ftheory, 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 Ftheoretic dual description of this setup.
and the last subsection of [2].
5These facts can be understood via the fractionation of M5branes 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 D6branes [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 D6branes 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 Mtheory. Note that we have w = z2, since the two
points
z on Cz are identi ed by the orientifolding action.
Now, take the Tdual to obtain a type IIB con guration. We have a socalled
shiftorientifold on (Cz
SI1IB)=Z2, where the orientifold action on Cz is accompanied by a 12
shift of SI1IB with 2k D7branes on the locus z = 0, see e.g. [11]. Note that there is su(2k)
gauge algebra locally on the D7branes, 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 Mtheory description. This
3cycle can be deformed into a 3cycle 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 Ftheoretic language. We started from
the Ik
ber with monodromy g =
further compacti ed on S1 such that z 7!
an Ftheory con guration on Cz=w1=2 with the I2k
0
1 k
1
on Cw, with r = 12 in Mtheory. 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 7brane of
type I2k.
elliptic bration Xg with a compatible choice of r = nd
by a 7brane 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 Ftheory 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 Ftheory con gurations on complex
surfaces. A small novelty here is that the base is real threedimensional.
Before proceeding, we pause to mention that the rotation of the phase of z by 2 nd on
the plane Cz transverse to the 7brane on the Ftheory side can be derived by a further
compacti cation on S1. Let us start from the Mtheory 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 nonzero
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 1form.6 This lifts to a new Mtheory con guration on
is the RR 3form. We then take a double Tdual along the elliptic ber. This is again a
type IIA con guration on an elliptic bration with a nonzero 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 Mtheory circle.7 Clearly this is the S1 compacti cation of the Ftheory
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 Ftheory 7branes and their Mtheory duals
Let us now study (partially) frozen Ftheory 7branes, 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 Tduality. We now have a type IIA system on
(Cw
S1)=Z2, with O6+ planes on both
xed points. Therefore, its Mtheory 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 threecycle 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 Ftheory side does not involve
With this warmup, let us consider a general (partially) frozen halfBPS 7brane at
z = 0 of Cz, with monodromy g around z = 0. We assume it preserves 16 supercharges.
Compactify it on S1 and take the Mtheory dual. This operation should be possible away
from z = 0
berwise. We then have an Mtheory 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 Mtheory 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 7branes.
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 D6branes
on one
xed point, and an O6+ with k2 D6branes on another. Taking the Tdual, we
have a type IIB con guration on (Cw=Z2)
S1 = Cz
S1. We have an O7+ with k1 + k2
D7branes 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
halfBPS (partially) frozen 7brane is necessarily an O7+ plane, possibly with an integral
number of D7branes on top. Note that our analysis does not allow a stuck 12 D7brane on
top of an O7+, thus precluding the existence of halfBPS Of7 . This is consistent with the
+
analysis in [1].
{ 6 {
Discussions
In this short note, we argued that there is a duality between
Mtheory 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
Ftheory con gurations of a 7brane 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 halfBPS 7brane is
necessarily a combination of an O7+ plane plus an integral number of D7branes.
HJEP06(21)8
Note that in our argument, we assumed that table 1 exhausted the list of (partially)
frozen halfBPS codimension4 singularities of Mtheory. Therefore, we can state our
conclusion in a slightly di erent way: if there is a (partially) frozen halfBPS 7brane other
than the O7+ plane plus D7branes, there should also be a new, hithertounknown
(partially) frozen halfBPS codimension4 singularity in Mtheory. The author considers this
extremely unlikely.
Ftheory 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 7branes are not (partially)
frozen, and it was not clear how serious the unintended consequences were. In the last
two years, genusone 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 Ftheory 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 7brane 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
GrantinAid for Scienti c Research No. 25870159, and in part by WPI Initiative, MEXT, Japan
at IPMU, the University of Tokyo.
Open Access.
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Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[hepth/0103170] [INSPIRE].
Math. Phys. 6 (2003) 1 [hepth/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 [hepth/9602114] [INSPIRE].
Phys. B 476 (1996) 437 [hepth/9603161] [INSPIRE].
[hepth/9712028] [INSPIRE].
(2015) 054 [arXiv:1407.6359] [INSPIRE].
HJEP06(21)8
class S theories: part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
Phys. 5 (2002) 841 [hepth/0006010] [INSPIRE].
[hepth/9708118] [INSPIRE].
08 (2001) 021 [hepth/0107153] [INSPIRE].
[arXiv:1401.7844] [INSPIRE].
transitions  Part II: string theory, JHEP 10 (2013) 006 [arXiv:1307.1701] [INSPIRE].
[1] J. de Boer et al., Triples , uxes and strings, Adv. Theor. Math. Phys. 4 ( 2002 ) 995 [3] D. Morrison , Half K3 surfaces , talk given at Strings 2002 , July 15 { 20 , Cambridge U.K. [4] D.R. Morrison and C. Vafa , Compacti cations of Ftheory on CalabiYau threefolds . 1, Nucl . [5] D.R. Morrison and C. Vafa , Compacti cations of Ftheory on CalabiYau threefolds . 2, Nucl . [6] E. Witten , Toroidal compacti cation without vector structure , JHEP 02 ( 1998 ) 006 [7] M. Del Zotto , J.J. Heckman , A. Tomasiello and C. Vafa , 6D conformal matter , JHEP 02 [8] K. Ohmori , H. Shimizu , Y. Tachikawa and K. Yonekura , 6D N = (1; 0) theories on T 2 and [10] K. Landsteiner and E. Lopez , New curves from branes , Nucl. Phys. B 516 ( 1998 ) 273 [14] S. Kachru , A. Klemm and Y. Oz , CalabiYau duals for CHL strings , Nucl. Phys. B 521 [15] V. Braun and D.R. Morrison , F theory on genusone brations , JHEP 08 ( 2014 ) 132 [16] I. Garc aEtxebarria, B. Heidenreich and T. Wrase , New N = 1 dualities from orientifold