TALK 2: THEORY X
PAVEL SAFRONOV
1. String theory in 5 minutes
In this talk we will be interested in two flavors of string theory: type IIA and type IIB
and M-theory.
The classical background for type II theories is a 10-dimensional Riemannian (more precisely, Lorentzian) manifold together with some other fields that we will ignore for now.
Similarly, the classical background for M-theory is an 11-dimensional Riemannian manifold.
All three theories have certain extended objects:
• Type IIA. Branes: D0, D2, D4, D6, NS5. F1 string.
• Type IIB. Branes: D(-1), D1, D3, D5, D7, NS5. F1 string.
• M-theory. M2 and M5 branes.
Here the number refers to the dimension of the object minus 1. E.g. a string has a 2dimensional worldvolume. The letter refers to the type of coupling one has with respect to
background fields:
• D-branes are electrically or magnetically charged under Ramond–Ramond fields,
• F1 string is electrically charged under the B-field,
• NS5 brane is magnetically charged under the B-field,
• M2 brane is electrically charged under the C-field,
• M5 brane is magnetically charged under the C-field.
R
Here “electrically charged under X” means we have a coupling
of
the
form
X while
R
magnetically charged means we have a coupling of the form Xd for ∗dXd = dX.
We have the following basic relations between the theories:
• M[X10 × S 1 ] = IIA[X10 ] and the radius of the circle gives the value of the dilaton
field. The brane mapping is easy to figure out by dimensions
• T-duality: IIA[X9 × S 1 ] = IIB[X9 × S 1 ] here the radii of the circles are inverted. If
a brane wrapped the circle, after T-duality it does not and vice versa.
2. Constructing theory X
2.1. Du Val singularities. Recall that finite subgroups Γ ⊂ SL2 (C) have an ADE classification. For instance, An corresponds to the cyclic subgroup
exp(2πi/(n + 1))
0
.
0
exp(−2πi/(n + 1))
Correspondingly, we have an ADE classification of du Val singularities (simple surface
singularities). These are given by C2 /Γ.
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PAVEL SAFRONOV
A nice way to construct these singularities is as follows. Let gΓ be the simple complex Lie
algebra with the same Dynkin diagram as Γ. Let N ⊂ g be the variety of nilpotent elements
known as the nilpotent cone. The open G-orbit Oreg is called the regular nilpotent orbit.
The open orbit Osub in N − Oreg is called the subregular nilpotent orbit. It has codimension
2 in N . The Slodowy slice to Oreg is therefore a surface; it has the singularity of type Γ.
Moreover, this picture can be deformed over the base h/W and this gives a deformation of
the singularity; it is smooth away from the root hyperplanes.
2 /Γ. Here C
2 /Γ
]
]
2.2. Theory X in flat space. Consider type IIB string theory on R6 × C
2 /Γ, C) ∼
]
is a smooth deformation of the du Val singularity. One can show that H 2 (C
= h.
One can take a certain limit where the volume of the cycles in H2 goes to zero and at
the same time α0 → 0. The claim is that in this limit the theory essentially splits into a
pair of non-interacting theories: type IIB supergravity and a certain theory on R6 near the
singularity. The theory on R6 is the six-dimensional theory X.
2.3. Theory X on a Riemann surface. Let C be a Riemann surface. We could con2 /Γ, but as C × C
2 /Γ is not Calabi-Yau, this
]
]
sider type IIB string theory on R4 × C × C
compactification will not have any supersymmetry.
Instead, one can proceed as follows. Recall that the Hitchin base of C is
Hitch(C) := Γ(C, KC ×Gm h/W ).
Diaconescu–Donagi–Pantev (arXiv:hep-th/0607159) construct (after Szendroi) a Calabi-Yau
3-fold X → C given a point in the Hitchin base of C. Essentially, X is a twisted family of
deformations of C2 /Γ.
Considering type IIB string theory on R4 × X gives what one might call theory X on
R4 × C. Note that we have a moduli space of theories parametrized by the Hitchin base
Hitch(C). In the next talk we will hear that this is just the Coulomb branch of the 4d gauge
theory obtained by compactifying theory X on C.
2.4. Lift to M-theory. We will consider theory X of type A in the flat space.
Consider the following hyperK¨ahler metric TNn on R3 × S 1 . Let x be the coordinate on
3
R and θ the coordinate on S 1 . The metric is
1
g = U dx ⊗ dx + (dθ + ω · dx)2 ,
U
where
n
X
1
1
U=
+ 2
|x − ai | R
i=1
and dU = ∗d(ω · dx).
Here ai ∈ R3 are parameters which will describe location of the branes in the dual picture.
One sees that at infinity the manifold looks like R3 × S 1 with the circle of radius R while
near the “nuts” ai the circle shrinks to zero radius. One can realize the natural projection
TNn → R3 as the hyperK¨ahler moment map of the U (1) action along the fibers.
The claim is that IIB[R6 × TNn ] in a certain limit still decouples into type An−1 theory
X on R6 and supergravity in the bulk.
TALK 2: THEORY X
3
Now, let us apply T-duality along the circle fibers. At infinity the circle has essentially
constant radius, so we expect to get type IIA string theory on R6 × R3 × S 1 with something
special happening near the location of the “nuts”. One can show that this corresponds to
inserting NS5-branes along R6 intersecting R3 at the nuts.
This system lifts to M-theory on R6 × R3 × S 1 × S 1 with n M5-branes along R6 .
Let us also comment on type A theory X on R4 × C from the M-theory perspective. The
configuration of M5-branes on R4 × C in R4 × C × R5 is not supersymmetric. One can
instead consider R4 × X4 × R3 , where X4 is a Calabi-Yau surface with a holomorphic curve
Σ ⊂ X4 and M5 branes that wrap R4 × Σ. For instance, we can take X4 = T ∗ C. In this case
we get a moduli space of theory X parametrized by curves Σ ⊂ T ∗ C which are n : 1 covers
of C. In this way we recover the spectral curve description of the Hitchin base Hitch(C).
Note that in terms of the 4d gauge theory the curve Σ is known as the Seiberg–Witten curve
while C is sometimes called the Gaiotto curve.
3. Properties of Theory X
In the previous section we have given a construction of a 6d theory X on R6 and R4 × C.
The theory depends on a choice of a simply-laced Lie algebra g. The theory has N = (2, 0)
supeconformal symmetry.
In this section we will derive some properties of the 6d theory from M-theory; thus, we
will only consider type A theories, but will also mention general expectations.
3.1. Abelian 6d theory. Consider a single M5 brane in R11 . The low-energy theory is
what we call the abelian theory X.
As shown in arXiv:hep-th/9510053, the bosonic field content of the M5 brane consists of a
self-dual two-form B and five scalars φa which describe the location of the M5 brane in the
transversal space. In fact, this is the tensor multiplet in the 6d N = (2, 0) supersymmetry
algebra.
A two-form can be thought of as a curving on a U (1)-gerbe. More precisely, U (1)-gerbes on
a manifold M with curving are elements in the hypercohomology group H 2 (M, U (1) → Ω1 → Ω2 )
similar to the interpretation of U (1)-line bundles with connection to be elements of H 1 (M, U (1) → Ω1 ).
We have a homomorphism
H 2 (M, U (1) → Ω1 → Ω2 ) → H 0 (M, Ω3 )
given by the de Rham differential. This associates to a U (1)-gerbe with curving its field
strength 3-form H. If the underlying gerbe with connective structure is trivial, then the
curving is a two-form B and the field strength is H = dB.
The action of a U (1)-gerbe theory is
Z
S ∼ H ∧ ∗H.
If we assume that H is self-dual, then H ∧ ∗H = H ∧ H = 0, so the theory is not
Lagrangian.
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PAVEL SAFRONOV
3.2. Compactification. Consider X[R5 × S 1 ] and take the limit as the radius of the circle
goes to zero.
In M-theory this corresponds to considering M[R5 × S 1 × R5 ] with N M5 branes along
R5 × S 1 . This compactification gives rise to type IIA theory on R5 × R5 with N D4 branes
along the first R5 . The low-energy limit of the theory on the D4 branes is the 5d N = 2
supersymmetric Yang–Mills theory with gauge group U (N ).
More generally, one can say the following.
Expectation. Compactification of type g theory X on a circle of small radius R1 corresponds to the 5d N = 2 supersymmetric Yang–Mills theory with gauge group G. The gauge
coupling is inversely proportional to R1 .
Remarks:
(1) One might wonder which gauge group G one obtains when one compactifies type g
theory X as we have many choices: adjoint group, simply-connected group, ... This
is a subtle question and we refer the reader to Witten’s review arXiv:0905.2720, in
particular Section 4.1.
(2) This explains how one obtains 5d Yang–Mills with simply-laced gauge groups. One
can obtain a non simply-laced Dynkin diagram from a simply-laced one by a process
known as folding. In the field theory language this corresponds to twisting the fields
along the circle by an automorphism.
(3) Here is an easy way to see the dependence of the Yang–Mills coupling on the radius
R1 . The action is
Z
1
S∼ 2
F ∧ ∗F.
gY M R5
Here ∗ is the Hodge star in 5 dimensions which depends on the metric g5 on R5 .
However, since the 6d theory is conformally-invariant, transformations g5 7→ λ2 g5
and R1 7→ λR1 should leave the 5d action invariant. This implies that gY2 M ∼ R1 .
Now let us further compactify on a circle S 1 of small radius R2 . Then we get the 4d
N = 4 Yang–Mills theory with gauge group G. There are two ways to see that. First, one
can consider the compactification of the 5d Yang–Mills on S 1 . The vector multiplet in 5d
N = 2 supersymmetry algebra contains a connection Aµ (for µ = 0...4) and five scalar fields
φa . Under compactification, we get a connection Aµ (for µ = 0...3) and six scalar fields: A4
and φa . This is exactly the vector multiplet in 4d N = 4 supersymmetry. Moreover, the
gauge coupling in 4d is
1
R2
∼
.
2
gY M
R1
Another way to get the same result is to work in string theory. We consider IIA[R4 ×S 1 ×R5 ]
with N D4 branes along R4 × S 1 with the radius of S 1 being small. Under T-duality, this
corresponds to IIB[R4 × R6 ] with N D3 branes along R4 . The low-energy theory is again
U (N ) 4d N = 4 Yang–Mills.
Expectation. Compactification of type g theory X on a product of circles S 1 × S 1 of
radii R1 and R2 corresponds to the 4d N = 4 supersymmetric Yang–Mills theory with gauge
group G. The gauge coupling is proportional to R2 /R1 . The compactification in a different
order results in an S-dual description of the same theory.
TALK 2: THEORY X
5
Let us now give a more detailed description what happens for abelian theory X under
compactification. As we explained in the last section, the tensor multiplet in 6d N = (2, 0)
6
(for µ, ν = 0...5) and five scalars φa .
supersymmetry consists of a self-dual two-form Bµν
6
Under compactification the self-dual form B splits into a vector A5µ = Bµ5
and a two-form
5
6
Bµν = Bµν (for µ, ν = 0...4).
The field strength H = dB 6 has the following expression in terms of the 5d fields:
H = dB 5 + dA5 ∧ dx5 .
The self-duality equation H = ∗6 H implies that
dA5 = ∗5 dB 5 ,
i.e. the two-form B 5 is the dual of the vector A5 . Thus, the bosonic field content consists of
a vector A5 and five scalars φa .
Next, after compactifying the 6d tensor multiplet on a torus to 4d, we get a two-form
6
6
6
6
4
. The
and a scalar φ4 = B45
and Cµ4 = Bµ4
(µ, ν = 0...3), two vectors A4µ = Bµ5
Bµν = Bµν
6
field strength H = dB is
H = dB 4 + dφ4 ∧ dx4 ∧ dx5 + dA4 ∧ dx5 + dC 4 ∧ dx4 .
The self-duality equation H = ∗6 H implies that A4 and C 4 are dual one-forms and B 4 is
dual to φ4 . Thus, the bosonic field content consists of a vector A4 and six scalars φa and φ4 .
If one exchanges the compactification circles, one switches the field A with its dual C just
like one expects from S-duality.
3.3. Defects. Defects in the 6d theory can be obtained as intersection of branes in M-theory.
For instance, both M2 and M5 branes can end on M5 branes. As shown in arXiv:hepth/9512059 and arXiv:hep-th/9701042, using charge conservation one can show that M2-M5
brane intersection has dimension 2 while M5-M5 brane intersection has dimension 4.
Thus, we expect surface and codimension 2 defects in theory X. Here is a classification of
the simplest kind of defects:
• Surface defects in type g theory X are labeled by dominant integral weights of g, i.e.
finite-dimensional representations of g.
• Codimension 2 defects are labeled by nilpotent orbits of g.
Note that these are only the simplest defects. For instance, theory X on R4 × C with some
codimension 2 defects we have described along R4 gives rise to a superconformal theory in
4d. For instance, to obtain pure SU (2) 4d N = 2 theory, one has to consider more general
kind of defects which give rise to irregular punctures in the Hitchin system on C.
The simplest kind of defect in 5d N = 2 Yang–Mills is the supersymmetric Wilson line
operator associated to a finite-dimensional representation of G. It is obtained from wrapping
the surface defect in theory X along the circle.