#### Magnetic discrete gauge field in the confining vacua and the supersymmetric index

Received: January
Magnetic discrete gauge field in the confining vacua and the supersymmetric index
Open Access 0 1 2 3
c The Authors. 0 1 2 3
0 Institute for the Physics and Mathematics of the Universe
1 University of Tokyo , Bunkyo-ku, Tokyo 133-0022 , Japan
2 Department of Physics, Faculty of Science
3 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan
It has recently been argued that the confining vacua of Yang-Mills theory in the far infrared can have topological degrees of freedom given by magnetic Zq gauge field, both in the non-supersymmetric case and in the N = 1 supersymmetric case. In this short note we give another piece of evidence by computing and matching the supersymmetric index of the pure super Yang-Mills theory both in the ultraviolet and in the infrared.
Supersymmetry and Duality; Discrete and Finite Symmetries; Global Sym-
1 Introduction and summary 2 3 4
With G = SU(2) and G = SO(3)
With G = SU(N )/ZN
With G = SO(N )
With general connected gauge groups
Introduction and summary
There have been many ideas proposed to explain the mechanism of the color confinement.
One influential idea has been the monopole condensation [1, 2], that the color is confined
due to the condensation of magnetically charged objects.
Let us for a moment consider a case where a U(1) gauge symmetry is broken by a scalar
field of electric charge q. In the infrared, there still is an unbroken Z
q gauge symmetry.
Such a discrete gauge field is locally trivial, but has a subtle physical effect globally. For
example, in a conventional superconductor, the Cooper pairs have charge 2, and therefore
there is a Z2 gauge symmetry.
systems, the confinement proceeds in two steps [3, 4]: the gauge group is effectively broken
to its maximal Abelian subgroup by the strong dynamics, which is then confined by the
condensation of monopoles. We see that, if the monopoles have charge greater than 1,
there can be a magnetic Zq gauge symmetry in the confining vacuum.
The appearance of magnetic Zq gauge symmetry is a much more general phenomenon,
independent of whether the confinement proceeds as above. For example, we argued in [5]
that pure (non-supersymmetric) Yang-Mills theory with gauge group SU(N )/ZN , with the
the infrared confining vacuum.
A rough argument, using the monopole condensation picture, goes as follows. Say, in
gauge field broken by a condensate of magnetic charge 1. In the SU(N )/ZN theory, the
periodicity of the magnetic U(1) gauge field changes by a factor of N . Stated differently, the
magnetic U(1) is now broken by a condensate of magnetic charge N , thus giving a magnetic
1Note that the instanton number of SU(N)/ZN gauge fields on a nontrivial spin manifold is 1/N times
discrete version of the Witten effect [6], and the condensate has the magnetic charge N
and the electric charge k. Therefore, what remains in the infrared is a Zgcd(N,k) gauge
field. The same conclusion can also be reached without using the monopole condensation
It can be said that the confining vacuum of the Yang-Mills theory is a version of
symmetry protected topological phase, if the reader allows the author to use a more fashionable
terminology used these days. This point of view is further studied in e.g. [79].
The aim of this short note is to provide another piece of evidence to the existence
pure Yang-Mills theory with various gauge group G. We start by studying the simplest
super Yang-Mills theory with arbitrary connected gauge groups in section 5.
It should be remarked at this point that all of the difficult gauge-theoretic computations
that are required for the analysis of this note have already been performed in [10, 11], and
what will be presented below is just a translation of the result (3.31) in [11] in the language
of [5]. Therefore, there is nothing new in this note, except for a possibly new viewpoint
that emphasizes the magnetic Zq gauge field in the confining vacuum.
With G = SU(2) and G = SO(3)
briefly recall the analysis performed there.
Z(L) the Witten index of the system
where H is the Hilbert space, F the fermion number, and H is the Hamiltonian. Using the
by-now standard argument, we know Z(L) is independent of L.
When L is very, very big, we can compute Z(L) using the structure of the infrared
360 rotation of the space. Therefore, there are N vacua related by the action of the
R-symmetry, with the gaugino condensate
By putting the system on a large T 3, these N vacua give N zero energy states. They
|ZSU(N)(L)| = N
When L is very, very small, the system is weakly-coupled, and the index Z(L) can be
computed reliably using semi-classical methods. To have zero energy states, the holonomies
The system is then effectively described by a supersymmetric quantum mechanics with
under the Weyl symmetry SN . The zero-energy states are then given by
|0i , (tr 12) |0i , (tr 12)2 |0i , . . . , (tr 12)N1 |0i
In the end we find
|ZSU(N)(L)| = 1 + rank T = N,
This is consistent with what we found in the infrared, (2.3).
The case with general N will be considered momentarily; let us first study the simplest
case N = 2.
We begin by considering when the system size L is very very small. As before, we need
to analyze the supersymmetric quantum mechanics based on three commuting holonomies
torus. This still gives N = 2 states as before.
But this is not all. We can take, for example, three matrices
g1 = diag(+1, 1, 1),
g2 = diag(1, +1, 1),
g3 = diag(1, 1, +1)
that mutually commute but cannot be in the same Cartan torus. In fact this is isolated
and its gauge equivalence class cannot be continuously deformed. This gives one
zero
Lifting from SO(3) to SU(2), we find that the holonomies g1,2,3 lift to Pauli matrices
the following, Cij is the T 2 formed by the edges in the i-th and the j-th directions of T 3.
In general, the possible choices of w2 are (1, 1, 1). The commuting triples in the
class (+1, +1, +1) are the ones that can be simultaneously conjugated to the Cartan torus
of w2, there is one isolated commuting triple, that gives one zero-energy state.3 In total,
|ZSO(3)(L)| = 2 + 7 = 9
2This is the w2 of the gauge bundle. In this note we only consider tori with trivial spin structure.
two states we found earlier. To see this, let us consider the partition function on a small T
Therefore, we should find the same when L is very, very big. But how? There are still
2 gauge symmetry while
the other does not [5]. More precisely, the theory has a line operator with nontrivial Z2
charge, coming from the t Hooft line operator in the ultraviolet. In the first vacuum it
has a perimeter law, and in the second vacuum it has an area law.
Thus, on a very big T 3, the first vacuum gives 23 states due to the choice of the
holonomies on T 3, and the second vacuum gives just 1. In total,4 we find
|ZSO(3)(L)| = 23 + 1 = 9
This is again consistent with the computation in the opposite regime (2.7).
With G = SU(N )/ZN
can be found easily. There are N vacua in the infinite volume limit, and as discussed in [5]
and recalled in the Introduction, the k-th vacuum has magnetic Zgcd(N,k) symmetry. Each
vacuum with Zq symmetry gives rise to q3 zero energy states in a large T 3. Therefore the
Witten index is5
|ZSU(N)/ZN (L)| = X gcd(N, k)
3 =
X (N/m)3(m),
where m|N denotes that N is divisible by m, and (m) is Eulers totient function, i.e. the
number of positive integers less than m and relatively prime with m.
To perform the computation in the opposite regime, we need to understand the moduli
space of commuting triples of SU(N )/ZN . First, the topological class of SU(N )/ZN bundles
on T 3 is labeled by its generalized Stiefel-Whitney class w2. In the case of a flat bundle
We then lift each element to SU(N ) and call them h1,2,3. Then they should commute up
to the center of SU(N ), i.e.
hihj = mij hj hi
of w2 along the temporal-spatial directions. When w2 is nontrivial along the spatial T
be mapped to each other by exchanging the time and the space directions. Therefore, we should have
(1)F0 = (1)F1 .
C := H1(T 3, Z2), given by tensoring the gauge bundle by another Z
2 bundle. The charge under C is
(1)F = (1)Fb with zero charge in C. Now, the two states with zero charge in C
are the same two
where mij is an N -th root of unity. This mij is w2 evaluated on the face Cij . The
topological class of the bundle is then given by
(w2(C23), w2(C31), w2(C12)) = (m23, m31, m12) Z3N .
that satisfy (3.2) can be conjugated to
. .
h(N) = diag(e2i/N , e22i/N , e32i/N , . . . , eN2i/N ),
for some integer k. The corresponding elements g1(N,2,)3 do not depend on k. Therefore, there
an SL(3, Z) transformation. To see this, consider the finite subgroup of U(1) generated
by m23, m31 and m12. This subgroup consists of l-th roots of unity for some l | N .
therefore we can perform a further SL(3, Z) transformation so that the magnetic flux is
Then the commuting holonomies h1,2,3 can be put to the standard form
sa Tl SU(l)
where Tl is the Cartan torus of SU(l). Again, h(N/l) has N/l choices, but they all project
1
down to the same element in SU(N )/ZN . Quantizing the supersymmetric quantum
mechanics based on s1,2,3, we get 1 + rank Tl = l states.
We now need to count the number of triples (m23, m31, m12) such that they can be
integers mod N such that gcd(x, y, z) = l. This is given by
|ZSU(N)/ZN (L)| =
X (N/m)3(m)
result above (3.1) of the computation in the infrared.
With G = SO(N )
G = Spin(N ), first studied in the appendix I of [13].
infrared, distinguished by the gaugino condensate
The commuting holonomies (g1, g2, g3) can be put into either of the following standard
where T is the Cartan torus of Spin(N ), or
where g1(7,2),3 is a lift to Spin(7) of the following SO(7) matrices
|ZSpin(N)(L)| = N 2,
ga T Spin(N )
ga = ga(7)sa
diag(+1, +1, +1, 1, 1, 1, 1),
diag(+1, 1, 1, +1, +1, 1, 1),
diag(1, +1, 1, +1, 1, +1, 1),
N 7
The former component gives 1 + rank T zero-energy states, and the latter component
gives 1 + rank T zero-energy states. In total, we find
|ZSpin(N)(L)| =
= N 2,
discrete theta angle, so there are two theories SO(N ), see [5]. As argued there, in the
all vacua have just Z1 gauge symmetry. Therefore, in the infrared, we find
logical type of the bundle is given by the Stiefel-Whitney class evaluated on the faces,
(m23, m31, m12) {1}3.
projecting the Spin(N ) commuting holonomies down to SO(N ). Then, these give (1 +
rank T ) + (1 + rank T ) = N 2 zero-energy states as before.
holonomies are either of the form
where g1(7,2),3 is the following SO(3) matrices
ga = ga(3)sa
ga = ga(4)sa
where g1(4,2),3 is the following SO(4) matrices
Quantization of the zero modes then give
(1 + rank T ) + (1 + rank T ) = N 2
In total, we find
in the ultraviolet computation, agreeing with the infrared computations.
We can similarly perform the check for SO(N )/Z2 or Spin(4N )/Z2 that is not SO(4N );
the explicit descriptions of almost commuting triples in [14, 15] are quite useful in this
regard. Instead of describing this, let us move on to a general analysis.
In fact we can give a uniform argument that the computations of the Witten index in the
ultraviolet and in the infrared always agree, given the facts derived in [10] and [11], once
the basic properties of the discrete theta angle and the magnetic gauge fields given in [5]
are taken into account.
connected and simply connected and K is a subgroup of the center C of G. The elements
of C label the discrete electric charge.7 The magnetic and the electric line operators in a
trivial, 1 U(1).
the discrete theta angle is zero, the magnetic charge m and the electric charge e of an
conditions on the allowed charges in (m, e) C C to be
m K,
computed for all G in [11].
is no longer a symmetry. But there is an additional symmetry that is useful. To discuss
7As abstract groups C and C
are the same, but it is useful for the author to distinguish them to make
it, let us consider for a moment a general situation where the spatial slice to be a
three
In the ultraviolet, note that the topological type of a G/K bundle on X is specified by
H1(X, K) given by
g |mi = hg, mi |mi
where h, i : H1(X, K) H2(X, K) U(1) is the natural pairing.
is partially confined to some subgroup as we will see below. There is no matter charged
g |ai = |gai
where we use multiplicative notation for the group structure in H1(X, K).
on X. When we compare the index in the ultraviolet and in the infrared, we should be able
In fact, it is easier to do so than to count the total index itself, as we will see soon.
Ultraviolet. Let us first perform the ultraviolet computation. When the gauge group
T 2 face in the direction ij. For each given m, the supersymmetric quantum mechanics on
The details of the computation depended on the choice of G, but in the end it was found
that there are always h states in total.
k and e simultaneously. For our purposes it is convenient to decide to measure the
Rin C3. Then what was found in [11] concerning the zero-energy states in a given sector of
m C3 can be summarized as follows, see (3.30) and (3.31) in that paper:
Every zero-energy states have the same (1)F = (1)rank G.
In total, there are indeed h states in that sector.
With this result it is easy to count how many vacua there are when the gauge group
given m K3 is
h
#{k = 1, . . . , nm | (k + disc)m = 0 (K)3}
Next, let us perform the infrared computation. There are h vacua, with the
gaugino condensate given by
In a theory with G/K gauge symmetry, we have t Hooft line operators whose magnetic
charges are valued in K. They give infrared line operators. On a spatial slice X, these lines
are labeled by H1(X, K). Let us say that the condensate is purely magnetic in the zero-th
vacuum when the discrete theta angle is zero. Then all these infrared line operators have
perimeter law. We can say that in the infrared, there is a gauge field with finite gauge
finite group K.
angle. As recalled above, the two effects can be combined, by changing the discrete theta
In the infrared description we adopted above, where the line operators are electrically
in the language of [17].
be naturally identified with the charge of the line operators.
0, they are confined as argued above and therefore they cost non-zero energy.
(k + disc)m = 0 they are not confined, and indeed there are |Kk|3 such states.
Let us now count the number of vacua with a given m, varying k. This is of course
which rotates these states. Therefore, there is exactly one state for each possible R-charge
the zero-energy states thus found have the same value of (1)F .
What we found in the ultraviolet (5.4) and in the infrared (5.6) are clearly
equal, including the charge under the R-symmetry. This is as it should be, since the Witten
index is independent of the size of the box, in each of the charge sector under the global
symmetry H1(X, K) of the system on X = T 3.
At this point we see that we did not add almost anything compared to the
understanding already given in [11], as already mentioned at the end of the Introduction. In the
previous sections we compared the total Witten index, that looked complicated. But in
fact it is easier and more trivially related to what was done in [11] to compare the index in
the ultraviolet topological class m H2(T 3, K).
The author would like to thank Particle Physics Theory Group at Osaka University, for
inviting him to give a series of introductory lectures on supersymmetric gauge theories;
it was during the preparation of the lectures that he noticed the question addressed in
this short note. It is also a pleasure for the author to thank O. Aharony and N. Seiberg
for helpful comments on 1-form global symmetries, and K. Yonekura and E. Witten for
of YT 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.
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any medium, provided the original author(s) and source are credited.
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