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

Journal of High Energy Physics, Mar 2015

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 \( {\mathbb{Z}}_q \) gauge field, both in the non-supersymmetric case and in the \( \mathcal{N}=1 \) 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.

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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. 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Yuji Tachikawa. Magnetic discrete gauge field in the confining vacua and the supersymmetric index, Journal of High Energy Physics, 2015, 35, DOI: 10.1007/JHEP03(2015)035