#### On the 6d origin of discrete additional data of 4d gauge theories

Yuji Tachikawa
0
0
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
Starting with a choice of gauge algebras, specification of a 4d gauge theory involves additional data, namely the gauge groups and the discrete theta angles. Equivalently, one needs to specify the set of charges of allowed line operators. In this note, we study how these additional data are represented in 6d, when the 4d theory in question is an N = 4 super Yang-Mills theory or an N = 2 class S theory. We will see that the ZN symmetry of the so-called TN theory plays an important role. As a byproduct, we will find that the superconformal index of class S theories can be refined so that it can give 2d q-deformed Yang-Mills theory with different gauge groups associated to the same gauge algebra.
Contents
1 Introduction
2 Basics of six-dimensional N=(2, 0) theory
2.1 Discrete three-form fluxes
2.2 The partition vector
2.3 Hamiltonian interpretation
3 Statement of the results
4 N=4 super Yang-Mills
4.1 SU(N ) and SU(N )/ZN
4.2 General case
5 Class S theories
5.1 Action of the center on the Tg theory
5.2 Discrete charge lattice of class S theories
5.3 Hamiltonian interpretation of a class S theory
5.4 The 3d Coulomb branch
6 Superconformal index and q-deformed Yang-Mills
7 Conclusions and future directions 1 3 3
1 Introduction
As is well-known, specification of a gauge theory requires a choice of the gauge group
and not just its Lie algebra. This is particularly true when the spacetime geometry is
nontrivial, or equivalently when general line operators are considered, which create holes
in the spacetime. The choice of the gauge group does not yet quite fix the quantum field
theory considered, and one needs to specify the set of allowed charges of line operators, or
equivalently discrete angles of the theory [1, 2].
The aim of this note is to study how these additional data are expressed when the 4d
theory considered is a theory of class S, namely, a theory obtained by compactifying the
6d N=(2, 0) theory on a Riemann surface C, and how the S-duality group acts on these
data. For simplicity, we will only consider Riemann surfaces without punctures.
The necessity of these additional discrete data arises from the fact that the 6d N=(2, 0)
theory does not have a unique partition function on a closed six-dimensional manifold.
Rather, it has a partition vector [39], valued in a finite-dimensional vector space with
multiple natural bases. When we just compactify this theory on C, we get a 4d theory with
a partition vector; its components with respect to a chosen basis correspond to partition
functions with fixed discrete fluxes through various cycles of the 4d spacetime. To obtain a
4d theory with a partition function with Hamiltonian interpretation, we need to sum over
the flux sectors in a consistent way. There are multiple ways to achieve this, and this is
where the additional discrete data come in.
Let us consider the 6d theory of type g, and denote by C the center of the
simplyconnected group Gsimp of type g. When the Riemann surface C is T 2, the corresponding
class S theory is the N=4 super Yang-Mills theory with gauge algebra g. In addition
to the choice of the precise gauge group G whose Lie algebra is g, we need to specify
a discrete theta angle when G is not simply connected. As discussed in [2], these data
are equivalent to the specification of maximal set of mutually local discrete electric and
magnetic charges of line operators, and is given by a maximal sublattice of CC compatible
with the Dirac quantization conditions. We identify it with a maximal isotropic sublattice
L of H1(C, C) C C, where the isotropy means that the charges are mutually local. We
will also find a formula for the 4d partition function in terms of L. This will reproduce the
Vafa-Witten formula [10] of the action of the S-duality on the partition function, in a way
manifestly generalizable to the whole theories of class S.
By using a Riemann surface C of genus g instead of the torus, we obtain a class S
theory. We assume C does not have punctures. Then the corresponding class S theory
has g3g3 as the gauge algebra, and has as matter contents 2g 2 copies of the so-called
Tg theory. Here, the Tg theory is the class S theory for a three-punctured sphere, usually
called the Tn theory when g = An1. The 6d analysis dictates that the additional discrete
data of the class S theory associated to the Riemann surface C are characterized by a
maximal isotropic sublattice of H1(C, C) C2g. We will see that this stems from the fact
that there is just one C global symmetry in the Tg theory itself, and that an arbitrary class
S theory inherits this global symmetry C.1
As a small application, we study how this global C symmetry affects the relation of
the superconformal index of class S theories and the q-deformed Yang-Mills on C. We
will see that by utilizing this global C symmetry of the class S theory, we can obtain
q-deformed Yang-Mills on C with arbitrary gauge groups belonging to the same g with
arbitrary discrete torsion.
The note is organized as follows. In section 2, we recall the general structure of
the partition function of the 6d N=(2, 0) theory. In section 3, we summarize how the
discrete data of a class S theory are encoded in the 6d language, and describe how the
4d partition function and the 4d Hilbert space are given in terms of the partition function
and the Hilbert space associated to the 6d theory. In section 4, we study the case of N=4
super Yang-Mills in detail, reproducing the results in [2]. In section 5, we generalize the
discussions to class S theories, by utilizing the global C symmetry of the Tg theory. We will
find that an arbitrary class S theory always have a global symmetry C. In section 6, we
1This last point was already found in [3] where they say this assertion cannot at present be tested
independently in any obvious way. It seems like an interesting probe of the inner nature of the still rather
mysterious (0, 2) theory. It is reassuring that after all these years we understood the nature of the 6d
theory at least slightly better.
study how this global C symmetry affects the equality of the superconformal index of class
S theories and the q-deformed Yang-Mills on C. We close the note with a brief discussions
in section 7. We assume that every four-dimensional manifold we deal with is Euclidean,
spin, and does not have torsion in its cohomology, for simplicity.
Basics of six-dimensional N=(2, 0) theory
Discrete three-form fluxes
Pick a 6d N=(2, 0) theory of type g, where g = An, Dn or E6,7,8, or a direct sum thereof.
Given g, let C be the center of the simply-connected group of type g, as tabulated in table 1.
There is a natural pairing
C C U(1).
For example, when C = Zn, the pairing of x, y mod n is given by e2ixy/n.
We put the 6d theory on a closed six-manifold X. The three-form of the 6d theory has
discrete fluxes valued in C through the three cycles of X. We might be tempted to consider,
then, the partition function Za(X) of the 6d theory, obtained by fixing the discrete flux to
be a given element a H3(X, C). It is known however that this is not possible, due to the
self-duality of the three-form. What can be done is as follows.
The pairing (2.1) on C induces an antisymmetric pairing
which is non-degenerate. We denote it as
for a, b H3(X, C). For example, when C = ZN , we have
We can not specify the value of the fluxes which have nontrivial pairing under (2.2).
Instead, we split
where A and B are both isotropic, in the sense that ha1, a2i = hb1, b2i = 0 for all a1,2 A
and b1,2 B. Then we can fix the discrete flux a A and consider the partition function Za
labeled by a. Here it is very important that these partition functions are defined relative to
H3(X, C) H3(X, C) U(1)
2 Z
ha, bi =
N
H3(X, C) = A B
the splitting (2.5). For example, given another splitting H3(X, C) = A B, Z0 computed
relative this splitting is different from Z0 computed relative to the splitting (2.5). To
better understand the situation, it is useful to introduce a vector space Z in which we have
a partition vector of the 6d theory. A splitting such as (2.5) equips Z with an explicit basis,
and Za are then the components of the partition vector with respect to this basis.
The partition vector
Let us implement this procedure concretely. For a H3(X, C), we define operators (a)
such that
This commutation relation characterizes the uncertainty relation of the discrete C fluxes of
the self-dual 3-form theory inherent in the 6d N=(2, 0) theory, and is a finite analogue of
the standard Heisenberg commutation relation.2 It is known that the operators (a) has
a unique finite-dimensional irreducible representation which we denote by Z, see e.g. [11].
The partition vector of the 6d theory takes values in this vector space Z.
Note that these operators (a) are not the operators in a 6d quantum field theory in the
ordinary sense. They are better thought of as operators in the 7d topological theory whose
Hilbert space on a six-dimensional spatial slice gives the space Z in which the partition
vector takes values. This is parallel to the situation of the 2d chiral CFTs and the Verlinde
line operators: the Verlinde operators are operators acting on the space of conformal blocks,
and the space of conformal blocks is the Hilbert space of the 3d topological theory (such
as the Chern-Simons theory) on a 2d spatial slice.
One way to construct Z and its natural basis is as follows: we split H3(X, C) as in (2.5).
Then (a) for all a H3(X, C) commute among themselves, and therefore we can find an
explicit basis of Z where (a) are simultaneously diagonalized. The basis vectors are
given by
Zb, b B
where we have the actions
When we vary the metric on X, the splitting (2.5) can be kept constant at least locally.
Therefore it makes sense to talk about the dependence of Zb on the metric. We denote
them by Zb(X), etc.
We can exchange the role of the sublattices A and B in (2.5). Then we have Za(X) for
a A where (b) are diagonalized instead. The bases {Zb}bB and {Za}aA are related
2More mathematically, we use the pairing (2.2) to define the finite Heisenberg group via the extension
1 U(1) H3(X, C) H3(X, C) 0.
Note that we use additive notation for a, b H3(X, C) but multiplicative notation for (a), (b) H3(X, C).
Strictly speaking, there is no section : H3(X, C) H3(X, C) defined on the whole H3(X, C), as is always
the case in quantum mechanics due to the operator ordering. We use this slightly wrong but common
notation in physics literature.
by a discrete Fourier transformation:
Za =
where we dropped a factor given by a power of |C|, which are not very important in our
analysis in the note.
Hamiltonian interpretation
If the 6d N=(2, 0) theory were an ordinary quantum field theory, it would associate a Hilbert
space to a given a five-dimensional constant-time-slice Xe . In our case, the situation is
slightly more complicated [9]. For simplicity, we only consider the case X = S1 Xe . we
have the canonical splitting
= H3
H2
The basis of Z where the elements of H2 Xe , C
are diagonalized is given by Zv for v
H3 Xe , C . Similarly, the basis of Z where the elements of H3 Xe , C
are diagonalized is
given by Zw for w H2 Xe , C . They are related by
Zw =
The standard conjecture concerning the 6d N = (2, 0) theory says that Zw is essentially
given by the path integral of the five-dimensional maximally supersymmetric Yang-Mills
theory with gauge group Gadj of the adjoint type on Xe , where the label w H2 Xe , C
specifies the Stiefel-Whitney class w = w2 H2 Xe , C of the principal Gadj-bundle
associated to 1 (Gadj) C.
As we will see below, to have a consistent Hamiltonian interpretation of class S theory,
we require that Zv with v H3 Xe , C , instead of Zw with w H2 Xe , C , to have the
Hamiltonian interpretation:
where is the circumference of S1, F is the fermion number, H is the Hamiltonian of the
system, acting on the Hilbert space Hv.
Statement of the results
Let us consider the class S theories obtained by compactifying this theory on a Riemann
surface C of genus g. The class S theories consists of 3g 3 copies of g N=2 vector
multiplets and 2g 2 copies of the Tg theory. We now need to specify additional discrete
data to fully specify the 4d theory. Instead of giving various arguments and then extracting
the results, we present our conclusions in a concise form in this section. The consistency
of the statements below will be checked in the sections that follow.
1. The additional data are the maximal isotropic sublattice L of H1(C, C). Here the
isotropy means that for any l1,2 L we have hl1, l2i = 0. The sublattice L specifies
the set of the allowed discrete charges of the line operators of the theory. Let us
denote such a fully specified theory by Sg(C, L).
2. Given a 4d manifold Y , the partition function of this class S theory is given by a
unique element ZL(Y C) of the space Z(Y C), specified by the condition that ZL
is invariant under the quantized discrete fluxes in H2(Y, C) L:
(v)ZL = ZL, for all v H2(Y, C) L.
3. A class S theory (for a Riemann surface C of genus g 2 without any puncture)
always has the global symmetry C. Therefore, on a four-dimensional manifold with
nontrivial 1, we can introduce a background flat connection for this global symmetry
C. Equivalently, we can gauge a subgroup of C.
4. Given a 3d constant-time-slice Ye , the Hilbert space HSg(C,L) Ye
theory is given by3
of the class S
Here, for concreteness, we considered Ye with trivial 1, and a class S theory whose
global C symmetry is not gauged. The spaces Hv,k on the right hand side are the
Hilbert spaces given by the 6d theory, and v measures the electric and magnetic
fluxes through the two cycles of Ye , and k specifies the charge under the global C
symmetry of the class S theory. Note that the 6d theory itself defines Hv,k for
arbitrary v H2 Ye , C H1(C, C), and we take the direct sum over a specific
subset determined by L.
These properties are stated without any reference to any decomposition of C into pants.
Therefore the action of the S-duality, i.e. the mapping class group of C, is completely
transparent in this formulation. But the relation to the choice of the gauge group and of
the discrete theta angles is made somewhat obscure. We will study these points in the rest
of the note.
Before proceeding, we stress that this additional data L H1(C, C) are specified
in addition to a decomposition into pants when we talk about a weakly-coupled duality
frame, although a decomposition into pants determines a natural isotropic decomposition
H1(C, Z) = A B and therefore a natural maximally isotropic sublattice A C and B C.
Surely L = A C and L = B C are two natural choices of the additional data given a
weakly-coupled frame; but they are not all.
3This statement needs a slight modification when H(Ye , Z) has torsion. In that case, there are subtle
shifts in v to sum over, presumably due to the discrete charge flux generated by the geometry, see [12].
Now, let us consider the case where C = T 2. Then the low-energy limit of the
fourdimensional system is described by the N=4 super Yang-Mills theory with gauge algebra
g. The group H2(C, C) = C C can be identified with the discrete electric and magnetic
charges of the Wilson, t Hooft or in general dyonic line operators of N=4 super Yang-Mills
theory. The pairing
H2(C, C) H1(C, C) U(1)
is the modulo one of the Dirac quantization pairing. An isotropic sublattice L of H1(C, C)
is such that for any l1,2 L we have hl1, l2i = 0, i.e. they satisfy the Dirac quantization law.
A maximal isotropic sublattice L is an isotropic sublattice to which we can add any more
element preserving the isotropy. Therefore, such an L can naturally be identified with the
allowed set of charges of line operators in a consistent theory.
For simplicity we consider g = AN1 and therefore C = ZN . Generalization to other
cases is immediate. Let us define a basis of H1(C, C) by saying that every element can be
written as fix a splitting
eW + mH H1(C, C)
N=4 super Yang-Mills
SU(N ) and SU(N )/ZN
where e, m = 1, . . . , N . This picks a particular duality frame, such that W is the Wilson
line in the fundamental representation and H is the t Hooft line with the minimal magnetic
charge. Consider a 4d manifold Y . For simplicity let us assume H1(Y ) and H3(Y ) are
trivial. Then, we can split H3(Y C, C) as follows:
H3(Y C, C) = H2(Y, C) W H2(Y, C) H .
The first term and the second term on the right hand side correspond to electric and
magnetic fluxes through two-cycles of Y , respectively. We use a basis of Z(Y C) where
the part H2(Y, C) W is diagonalized. Equivalently, we consider a partition vector whose
components are given by
Zv := ZvH (Y C), v H2(Y, C).
We identify Zv to be the partition function of N=4 super Yang-Mills based on a principal
bundle of Gadj = SU(N )/ZN , with the condition that the Stiefel-Whitney class of the
bundle associated to 1(Gadj) C is fixed to v.
When L = {eW }, the element in Z(Y C) invariant under the quantized action of
H2(Y, C) L is clearly just Z0. This is, up to a constant multiple, the partition function
of N=4 super Yang-Mills with gauge group SU(N ), because trivial Stiefel-Whitney class
0 H2(Y, C) means that the gauge bundle can be lifted to an SU(N ) bundle. We note
that {eW } is the maximal set of the charges of the allowed line operators of the theory
with gauge group SU(N ).
When L = {mH}, the element in Z(Y C) invariant under the quantized action of
H2(Y, C) L is clearly just
recall the explicit action of the Heisenberg group in (2.8). This is the partition function of
N=4 super Yang-Mills with gauge group SU(N )/ZN with zero theta angle. Indeed {mH}
is the maximal set of the charges of the allowed line operators of the theory with gauge
group SU(N )/ZN , and we summed over all possible Stiefel-Whitney classes v.
The choices L = {eW } and L = {mH} can be exchanged by changing the complex
structure of the torus as 7 1/ , and therefore we have
X Zv
1
.
This relation of the partition functions Z0 and Pv Zv goes back to [10]. The explanation
using discrete Fourier transform was already essentially given in [7]. Here, we identified
the partition function via the condition of invariance under H2(Y, C) L. This is also
essentially done e.g. in [3].
Let us now consider a general maximal sublattice L C C, specifying the set of charges
of mutually local line operators. We would like to determine the element ZL of Z(Y C)
invariant under the quantized action of H2(Y, C) L, as a linear combination of Zv where
v H H2(Y, C) H1(C, C) specifies the Stiefel-Whitney class of the gauge bundle. This
gives an explicit formula of the partition function of the N = 4 super Yang-Mills with the
set of line operators being L, in terms of a summation over the topological class of gauge
bundles with an explicit phase factor.
For concreteness, we again restrict to the case g = AN1 and therefore C = ZN . Let
L {eW }e=1,2,... = {kiW }i=1,2..., where k is an integer dividing N . Then an element in L
of the form mH + nW with minimal positive m has m = N/k from maximality of L. We
have the choice n = 0, 1, 2, . . . , k 1. Then L is generated by kW and mH + nW with
m = N/k.
We now need to construct (a) for all a H2(Y, C)L, such that (a+b) = (a)(b).
To do this, we need to specify (w kW ) and (w (mH + nW )). The representation of
the former is given already in (2.8),
The representation of the latter needs some more work. In (w (mH + nW )), we
can consider w as an element of H2(Y, Zk). We distinguish the pairing of H2(Y, Zk) by
writing it as hh, ii, defined as in (2.4), and keep using h, i for the pairing of H2(Y, ZN ).
Note that for w H2(Y, Zk), we have mw H2(Y, ZN ), and our normalizations are that
hmw, mwi = m hhw, wii. We find that the following satisfies all the requirements:
(w (mH + nW ))Zv = einmhw,vi+inhhw,wii/2Zv+mw.
Here, the phase
is more properly understood to be given using the Pontrjagin square P(w), if necessary;
we will continue to use our informal notation. In our convention, the quantity RY w w/2
on a spin manifold is well defined as an integer modulo k. For more on the Pontrjagin
square, see [8, 13].
The invariance under (4.7) means that only those Zv with kv = 0 modulo N appear
in ZL. Equivalently, we have v H2(Y, Zk) H2(Y, ZN ). The invariance under (4.8) then
requires that we have
ZL =
vH2(Y,Zk)
The restriction v H2(Y, Zk) H2(Y, ZN ) in the sum means that this is a partition
function of N=4 gauge theory with gauge group SU(N )/Zk, and the phase in hhv, vii /2 is
exactly the structure found in [2] for the N=4 gauge theory with the set L of line operators.
Class S theories
The discussion in the last section did not depend much on the particular choice of C = T 2,
and therefore, it can naturally be generalized to other C of general genus g 1. For
simplicity we assume there is no puncture and no outer-automorphism twist lines on C,
and let us consider a theory of class S obtained by compactifying the 6d N=(2, 0) theory
of type g on C.
The only point which needs a further discussion is the identification of the H2(C, C)
C2g as the possible label of the discrete charges of line operators. Before proceeding, we
note that for the class S theories of type A1, the analysis of the charges of all possible
line operators was carried out in a beautiful paper [14], and then the possible choices of
maximally mutually-local subset of charges were discussed in [1]. What is discussed below
is a natural generalization of their discussions.
Let us fix a pants decomposition of C. Correspondingly, the theory has a realization
as an N=2 gauge theory with gauge algebra g3g3 coupled to 2g 2 copies of the Tg theory,
i.e. the class S theory corresponding to a three-punctured sphere. To analyze the combined
system, we need to recall some properties of the Tg theory.
Action of the center on the Tg theory
The theory Tg has g3 flavor symmetry. The operator content of Tg is not fully known,
although its superconformal index has a rather-well-established conjectural form, given
in [15, 16]. There, it was found that every operator which contributes to the superconformal
index is in a representation R1 R2 R3 of g g g, such that the action of C on R1,2,3
are the same. Equivalently, the Tg theory has a flavor symmetry Gs3imp, where Gsimp is
the simply-connected Lie group with the Lie algebra g, but the center C3 does not all act
independently on the theory. Rather, there is a natural map
given by (a, b, c) 7 abc, and only Ctri-diag acts faithfully on the Tg theory.
It seems natural to assume that these statements on the center charges of the operators
of Tg theory hold including non-BPS operators. As we will see, this assumption leads to
a consistent interpretation of the properties of the class S theory, and it seems difficult
(at least to the author) to add non-BPS operators which do not satisfy this property, still
preserving the overall consistency. When g = A1, the TA1 theory is a theory of eight free
chiral multiplets Qiau, where i, a, u = 1, 2 are the SU(2) flavor symmetry indices for SU(2)3.
In this case, it is easy to see that the center Z2 of any of the three SU(2) just multiplies
Qiau by 1, confirming the assumption above. When g = A2, the TA2 theory is believed
to have an enhanced symmetry E6. The assumption above translates in this language that
all operators are representations of the adjoint form E6/Z3.
Before proceeding, we note that the statement that only Ctri-diag acts faithfully only
applies to point operators of the Tg theory. It is easy to consider external line operators
on which the full action of C C C can be distinguished: we just have to consider a
pure flavor Wilson line operator associated to one of the G symmetries. For example, in
TA1 theory, we can consider a Wilson line in the fundamental representation of each of
the three SU(2) symmetries. Similarly, we expect that there are external line operators
of the TA2 theory which do not transform under the adjoint form E6/Z3, but under the
simply-connected version E6 [17].
Discrete charge lattice of class S theories
Now, let us come back to the study of the class S theory one obtains by compactifying the
6d theory of type g on a Riemann surface C of genus g 2 without any punctures. We
have the g3g3 gauge multiplets, coupled to 2g 2 copies of the Tg theory.
If we have the g3g3 gauge multiplets in isolation, we have a lattice Enaive = C3g3 of
discrete electric charges and a lattice Mnaive = C3g3 of discrete magnetic charges. In the
last subsection, we argued that a single Tg theory has a single C global symmetry. With
2g 2 copies, we have the flavor symmetry C2g2. Then, we have a natural map
controlling how the center C3g3 acts on the copies of the Tg theory. A crucial point is
that the image of this action is C2g3. For example, consider the case g = 2. Let us take
a duality frame where three g couple to a diagonal combination of one g from one Tg and
another g from another Tg, see figure 1. Then, the map is given by
and therefore the subgroup
is not gauged.
Therefore, we have the sublattice
(a, 1) C2
characterizing the center charges of the dynamical operators coming from the copies of the
Tg theory. Due to the screening by the dynamical operators of charges g, the lattice of
the discrete electric external line operators is now
In order to satisfy the Dirac quantization condition with respect to the dynamical operators
of charges g, the lattice of the discrete magnetic external line operators is now
Therefore, we naturally have the identification
M =
H1(C, C) C2g E M.
Note that the relation (5.1) can be thought of describing the H1 of a sphere minus three
punctures, if we identify each C with the dual of S1 around one puncture. Then, the
splitting of H1(C, C) into E and M in (5.8) is the geometrically natural one associated to
the pants decomposition.
Now, we can repeat the analysis given in section 4 for the N = 4 super Yang-Mills
when C = T 2 almost verbatim. The maximal set of mutually local line operators is given
by a maximally isotropic sublattice L H1(C, C).4 The partition function of the 4d class
S theory on a four-manifold Y with trivial 1 is then given by the essentially unique vector
in Z, invariant under (v) for all v H2(Y, C) L.
Recall the action (5.2) of the center of the gauge groups to the global symmetries
C2g2 of copies of the Tg theory. The image is C2g3, and therefore we see that the class S
theory associated to a Riemann surface of genus g 2 without any puncture has the global
symmetry C, independent of the choice of the set L of allowed charges of line operators.
In a weakly-coupled frame, the action of this universal global C symmetry to act on one of
the Tg theory nontrivially, and acts on all the other copies of the Tg theories trivially. The
existence of global symmetry C for any class S theory was already pointed out in [3].
It is of course possible to gauge a subgroup of this global C symmetry further, and
regard the resulting theory as a new class S theory. In the rest of the paper, however, we
stick to the convention where this global C symmetry is not considered to be gauged; we
will still utilize background C gauge fields.
4For the 6d theory of type g = A1, this statement was originally found in [1]. There, the isotropy was
stated as the condition that the chosen one-cycles on C should have even intersection numbers.
A priori, a strongly-coupled theory such as the Tg theory can have intrinsic additional
discrete data corresponding to the choice of the allowed set of external line operators, just
as the N=4 super Yang-Mills theory with a given Lie algebra g had. The Tg theory has one
such additional data, which are rather trivial: we can gauge a subgroup of the tridiagonal
center symmetry Ctridiag acting on Tg. The partition function of the Tg theory with different
choices of will then be different on a non-simply-connected manifold. But this is a rather
trivial additional data, related purely to the flavor symmetry.
The Tg theory does not seem to have any more discrete data in addition to this. If it
really had such additional data, that would give an additional term on the right hand side
of (5.8), thus ruining the overall structure of the 6d N=(2, 0) theory recalled in section 2
and the discussions in section 4. Therefore, it is strongly likely that the Tg theory does not
have a choice of the allowed set of external line operators, except those coming from the
gauging of the center flavor symmetry C.
Hamiltonian interpretation of a class S theory
Let us now consider the Hamiltonian interpretation when the four-manifold Y has the
form Y = S1 Ye . We assume that Ye has trivial 1 for simplicity. To get the Hamiltonian
interpretation, we need to use the basis of Z associated to the splitting
H3 (Y C) = H3 Ye C
H2 Ye C
and simultaneously diagonalizing (x) in x H2 Ye C in (5.9). Therefore the elements
of the partition vector is given by Zv,k where
= H3 Ye C .
On this basis, we need to construct the quantized action of
H2(Y, C) L =
L
L .
Calling an element v w, the action (4.8) is now given by5
Then the combination which is invariant under these operators is
X
and therefore the Hilbert space associated to Ye is given by
M
This is the statement in (3.2).
5This is true only when H(Ye , Z) has no torsion. In general there can be a term in the exponent
proportional to the torsion part of w.
The additional summation over k H3 Ye , C C can naturally be thought of as a
summation over the charge sectors under the global symmetry C which exists for any class
S theory. This feature will be elaborated further in the next section.
The 3d Coulomb branch
Before proceeding, let us now discuss the 3d Coulomb branch of the moduli space of a class
S theory Sg[C, L] compactified on a circle S1. This problem was already analyzed in [1]
for the case g = A1 and there is no essential change in this general case.
Consider the moduli space M of the Hitchin system on a Riemann surface C of genus
g without puncture. We let the gauge group of the Hitchin system to be Gsimp, the simply
connected group associated to the Lie algebra g. There is the Hitchin fibration
where M4d Coulomb is the 4d Coulomb branch. The generic fiber is T 2grankg and is an
Abelian variety.
There is a natural action of H1(C, C) on M, which commutes with p, constructed as
follows. We identify an element l H1(C, C) with a flat C bundle L(l) over C. Then
we can tensor L(l) to the Gsimp-bundle in the the Hitchin system, by multiplying the
transition functions. This can be consistently done, because C is the center of Gsimp. This
operation clearly commutes with the Hitchin fibration p. Then M/H1(C, C) is the moduli
space of the Gadj-Hitchin system on C, where the topological class of the Gadj-bundle is
assumed to be trivial.
Given a maximally isotropic sublattice L H1(C, C), we can instead take the quotient
Then we identify M/L as the 3d Coulomb branch of the class S theory Sg[C, L].
Presumably, the fiber of M/L is a principally-polarized Abelian variety. More generally,
the fibration of the Donagi-Witten integrable system of an N = 2 supersymmetric theory,
once the maximal set L of the mutually local line operators is fixed, will be a
principallypolarized Abelian variety. This point needs to be studied in more detail.
Superconformal index and q-deformed Yang-Mills
The global symmetry C which exists for any class S theory can be used to refine the relation
of the superconformal index of class S theories and 2d q-deformed Yang-Mills originally
found in [15, 16]. We will see below that by utilizing C we can have 2d q-deformed
YangMills with non-simply-connected gauge group with or without discrete torsion.
The superconfomal index of a 4d N = 2 theory is the partition function on S3 S1,
with an appropriate choice of background fields to preserve supersymmetry. Here, we only
consider the simplest case with one parameter q, where we have
Z S3 S1 = TrH(1)F qER
where H is the Hilbert space on the S3, F is the fermion number, E is the energy (or the
scaling dimension of the operator under the state-operator correspondence), and R is the
SU(2) R-symmetry normalized to take 1/2 in the fundamental representation.
Consider a class S theory Sg[Cg, L] obtained by compactifying the 6d N = (2, 0) theory
of type g on a Riemann surface Cg of genus g 2 without any puncture. As S3 S1 does
not have two-cycles, the partition function does not depend on the choice of the maximal
isotropic sublattice L, and we have [15, 16]
ZSg[Cg,L] S3 S1 =
where Kg(q) is a certain prefactor, the summation is over the irreducible representations
of g, and dimq is the quantum dimension of . Up to a prefactor, this is the partition
function of the q-deformed Yang-Mills on Cg with gauge group Gsimp, where Gsimp is the
simply-connected one associated to g.
This formula can be derived from the superconformal index of the Tg theory
where (a, b, c) Gs3imp are the exponentiated chemical potentials for the flavor symmetry.
The numerator has factors (a) which is the character of a in the representation ,
and also prefactors K(a) which purely consists of characters of tensor powers of adjoint
representations.
Note that the tri-diagonal center symmetry Ctridiag, (5.1), is manifest. Let us define
the pairing (, ) of an element C Gsimp and an irreducible representation of
Gsimp by saying that acts by the multiplication by a phase e2i(,) on the irreducible
representation . Then
Then, we find
Now, the label runs over the irreducible representation of Gadj, i.e. the adjoint form
associated to the Lie algebra g. This gives the partition function of the q-deformed
YangMills theory with gauge group Gadj.
Now, the summation is over the irreducible representation of G = SU(N ) whose N -ality
is k mod N . This gives the partition function of the q-deformed Yang-Mills theory with
gauge group Gadj = SU(N )/ZN , with an additional phase factor
in the Lagrangian, where w2 H2(C, C) is the Stiefel-Whitney class of the Gadj bundle
over the Riemann surface C.
Note that by including the phase factor e2ikn/N on the left hand side, we are projecting
down to a subspace of the Hilbert space with a fixed charge k under the global symmetry
C = ZN . This corresponds to restricting to a single summand of k H3 Ye , C in the
general expression (5.14) for the Hilbert space and the partition function (5.13), where
Ye = S3 in this setup.
The appearance of Gadj as the 2d gauge group can be understood as in [18], where the
S1 direction is compactified first. In this approach, we have the 5d maximally
supersymmetric Yang-Mills on Xe = Ye C. As explained in section 2.3 and in particular (2.11),
computing the partition function fixing an element k H3 Ye , C means that weighing
the partitinon function of the 5d super Yang-Mills with gauge group Gadj with nontrivial
w2 H2(C, C), exactly with the weighting factor (6.9). Reducing the 5d Yang-Mills on S3,
we obtain a 2d Yang-Mills with gauge group Gadj.
So far we showed how to obtain Gadj as the 2d gauge group. It is clear that we can
extend the construction in this section to have the q-deformed Yang-Mills on C for the
arbitrary gauge group G whose Lie algebra is g together with arbitrary discrete torsion (6.9).
Conclusions and future directions
To fully specify a gauge theory, we need to specify the set of allowed charges of line
operators, or equivalently the global structure of the gauge group together with the discrete
theta angles, as first found in [1, 2]. In this short note, we studied how these data are
encoded in the 6d N=(2, 0) theory, when the gauge theory we consider is a theory of class
S in 4d. The results are summarized in section 3 and we do not repeat them here.
Many points need to be clarified further. We list some of them below:
to understand more fully the behavior of the partition vector of the 6d N = (2, 0)
theory of type g. It is known that the 6d theory of type U(N ) has an honest partition
function, and the rather subtle behavior of the partition vector of the 6d theory of
type SU(N ) arises from trying to decouple the Abelian U(1) part, see e.g. [5]. Also,
when the center C of the simply-connected group G associated to the Lie algebra g
is of the form C Zn2 , we can choose a natural maximally isotropic sublattice of
H3(X, C) for arbitrary closed six-manifold X by requiring the flux to be annihilated
by multiplication by n. Then we have a genuine quantum field theory in 6d.6 Starting
from a genuine 6d quantum field theory using these constructions might shed a new
light on the behavior of the 4d theories.
To extend the analysis in this note to the class S theory associated to the Riemann
surfaces with punctures. To do this, we need to understand the behavior of the
partition vector of the 6d N = (2, 0) theory on a closed 6d manifold together with
codimension-two defects. Currently, the properties of the codimension-two defects
are inferred by studying its behavior in lower dimensional compactifications, and it
seems difficult to directly study the behavior of the partition vector in 6d. Using the
holographic dual of the codimension-two defects found in [19] might be useful; after
all, the behavior of the partition vector of the 6d N = (2, 0) theory was first found in
this holographic context [3, 4].
To extend the analysis in this note to the spacetimes with torsion cycles and/or
non-Spin manifolds. Incorporating the case with torsion cycles will be important to
study the lens space index, i.e. the partition function on S3/Zr S1 [12, 20]. Some
supersymmetric theories can be formulated on non-Spin but SpinC manifolds. In the
case of 6d N = (2, 0) theory of general type g, there can be more possibilities. These
points would be important to study class S theories on CP2, for example.
To study the behavior of external line operators of class S theories in more detail.
They arise from external codimension-4 operators of the 6d N=(2, 0) theory, and can
be analyzed from this point of view. This will tell us not only the discrete charges of
the line operators, but also a more detailed structure, corresponding to the distinction
of the center C = {weight lattice}/{root lattice} and the weight lattice itself.
The author hopes to come back to some of the issues listed above in the future.
It is a pleasure for the author to thank A. Neitzke, K. Ohmori, S. Razamat, N. Seiberg,
N. Watanabe and B. Willet for helpful discussions. He would also like to thank O. Aharony,
G. W. Moore and N. Seiberg in particular for carefully reading the manuscript and giving
many illuminating comments, and J. Distler for pointing out an error in the v1 of the
preprint. The author is supported in part by JSPS Grant-in-Aid for Scientific Research
No. 25870159, and in part by WPI Initiative, MEXT, Japan at IPMU, the University
of Tokyo. The work was completed during the authors stay at the Aspen Center for
Physics and at the Simons Center for Geometry and Physics. The author thanks these two
institutions for their generous hospitality, and was partially supported by the NSF Grant
#1066293 during his visit at the Aspen Center.
6The author thanks O. Aharony and N. Seiberg for explaining this fact to him.
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