On skein relations in class S theories

Journal of High Energy Physics, Jun 2015

Loop operators of a class S theory arise from networks on the corresponding Riemann surface, and their operator product expansions are given in terms of the skein relations, that we describe in detail in the case of class S theories of type A. As two applications, we explicitly determine networks corresponding to dyonic loops of \( \mathcal{N}=4\;\mathrm{S}\mathrm{U}(3) \) super Yang-Mills, and compute the superconformal index of a nontrivial network operator of the T 3 theory.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:


On skein relations in class S theories

JHE On skein relations in class S theories A. 0 1 2 WA 0 1 2 j × T 0 1 2 0 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 University of Tokyo , Bunkyo-ku, Tokyo 133-0022 , Japan 2 Yuji Tachikawa 3 Department of Physics, Faculty of Science Loop operators of a class S theory arise from networks on the corresponding Riemann surface, and their operator product expansions are given in terms of the skein relations, that we describe in detail in the case of class S theories of type A. As two applications, we explicitly determine networks corresponding to dyonic loops of N =4 SU(3) super Yang-Mills, and compute the superconformal index of a nontrivial network operator of the T3 theory. Extended Supersymmetry; Supersymmetric gauge theory; Duality in Gauge 1 Introduction and summary 2 3 2.1 Examples: A2 and A3 Loop operators in 4d and skein relations in 2d Sums and products of loops Non-commutativity and the angular momentum Networks and skein relations in 2d Restriction of labels Canonical junctions and removal of digons Canonical junctions Removal of digons Crossing resolutions Reidemeister moves More simplifying relations The R matrix General crossing resolutions The intersection number and the powers of q SL(2, Z) action on the torus Triangle contraction relations Rectangle decaying relations Networks for N =4 Yang-Mills The product of Wilson loops and ’t Hooft loops Analysis in the Liouville/Toda theory Examples of products of loops in A2 Classification of networks on T 2 for A2 Some examples in the superconformal index The superconformal indices without lines 5.2 4d Wilson lines A network in the T3 theory A Examples of OPE and charge/network dictionary A.1 Pure dyonic loops A.7 An OPE of complex dyonic loops Introduction and summary (SCFTs) on Riemann surfaces C with punctures [1, 2]. These 4d theories are now called class S theories, and they are closely associated to certain 2d theories defined on C. For example, class S theories on S4 are related to the Liouville/Toda CFT on C [3, 4] and when considered instead on S3 × S1 they give rise to 2d q-deformed Yang-Mills on C [5–7]. How do supersymmetric defects1 appear under these 4d-2d duality relations? In this 4d side should come from a codimension-four defect on a codimension-one defect on the For class S theories of type SU(2), or equivalently A1, dyonic charges of loops on the 4d side is in one-to-one correspondence with topologies of non-intersecting loops on the 2d side [8]. When the 2d theory is the Liouville/Toda theory, these loops on the 2d side have an explicit realization in terms of the Verlinde operators [9] as shown in [10, 11]. When the 2d theory is the q-deformed Yang-Mills, these loops are instead realized as 2d Wilson loops.2 non-crossing ones: We only have to consider non-intersecting loops on the Riemann surface for the A1 case, thanks to the existence of a skein relation resolving each crossing into a sum of two = q−1/2 However, in the higher rank cases, namely for SU(N ) with N > 2 or equivalently lines inevitably appear. This results in the networks3 of lines on the Riemann surface, as special emphasis on the SU(3) case. In [14] the analysis was mainly carried out using the 1In this paper we use the words defects and operators interchangeably. 2This fact was not explicitly mentioned in the literature to the authors’ knowledge, but apparently it has been known to many practitioners in the field. We spell this out in section 5.2. 3The same objects are also called as webs or spiders. In this paper, we only use the terminology networks. done mainly in the framework of Toda CFTs for general q. These analyses gave rise to the skein relations that had been discovered in the context of mathematics before [16]. Our aim in this paper is to describe the skein relations of the networks in the general SU(N ) case. We have, for example, the relation = q N1 −1 in [17] in the context of knot invariants. The guiding principle for us is that the representation theory of the quantum group SUq(N ) underlies these networks and their skein relations. The relation of the loop operators of 2d CFTs and the quantum group has been known for quite some time, mainly in the context when q is a root of unity, see e.g. [18]. In the case of 2d q-deformed Yang-Mills, the relation of their loops and the quantum group is very direct, because the q-deformed Yang-Mills is a gauge theory whose gauge group is the quantum group [19, 20]. We have two applications of the skein relations. The first concerns the dyonic loop classification was performed in [21]. When the electric charge and the magnetic charge of a dyonic loop are parallel in the weight system, there is an obvious realization of such dyonic loop in the class S language as a loop wrapping the torus. When they are not parallel, it was expected that they are represented by networks on the torus. We will give a complete description in the case of SU(3). The second application is the study of a particular loop operator of the T3 theory. The TN theory is a strongly-coupled class S theory that does not have a useful Lagrangian description, and there is no direct method to describe a loop operator using the path integral language. Still, using the class S language, it is easy to see that there is a loop operator described by a nontrivial network on a three-punctured sphere. We compute the superconformal index of the T3 theory in the presence of this loop operator, and confirm that it indeed has an enhanced E6 symmetry, as expected from the fact [1] that the T3 theory is in fact the E6 theory of Minahan and Nemeschansky [22]. The rest of the paper is organized as follows: in section 2, we start by reviewing the relation of the operator product expansion of the loop operators on the 4d side and the skein relation on the 2d side. We emphasize the universality of the skein relation, independent of the choice of the 4d spacetime. Next, in section 3, we describe the skein relations explicitly for general SU(N ). We also explain how they can be understood in terms of the representation theory of quantum groups. In section 4, we turn to an application on 4d a pure ’t Hooft loop in the general SU(N ) theory. We also describe all the dyonic loops in the SU(3) theory in terms of networks, and point out a relation to the brane tiling. In section 5, we first explain how the 4d Wilson loop is mapped to the 2d Wilson loop in the γB ↔ LγALγB 6= LγB LγA ↔ context of the q-deformed Yang-Mills. We then use one of the skein relations to compute the superconformal index of the T3 theory in the presence of a loop operator corresponding to the simplest nontrivial network. In the appendix A, we have more examples of the realization of dyonic loops by networks in the general SU(N ) theory. Loop operators in 4d and skein relations in 2d In the Liouville theory, a Verlinde loop operator is defined in terms of monodromy actions see from the concrete calculations. See figure 1 for an illustration. Under the 2d-4d correspondence, the Verlinde operators map to loop operators of the 4d theory. Therefore, there should be a concept of ordering of loops on the 4d side, such that the product becomes non-commutative. In this section we review how this ordering arises, following [23–26]. See also the recent reviews [27, 28]. Sums and products of loops We consider the 4d setups where some kind of localization computations is possible. Typically there is a supercharge preserved in the background, whose square involves a linear combination of two isometries k1,2. Supersymmetric loops wrap along the direction of k1 and sit at the fixed point of k2. In the neighborhood of the loop, we can approximate the geometry as S1 where k1 shifts the coordinate along S1 and k2 is the phase rotation of C. The loop now can place multiple loops L1,2,... on x1,2,... preserving the same supercharge. This gives an intrinsic ordering of loops on the 4d side, and furthermore, the expectation values are unchanged under infinitesimal changes of the positions xi. The choice of the local supersymmetric background at a loop can be characterized by a single parameter which we denote by q. These statements can be explicitly checked in the case of the localizations on S1 × R3 [26] and Sb4 [29]. We define a formal sum L1 + L2 of two loops by networks in 2d operators on H loops in 4d where the ellipses stand for other operator insertions. The product L1 · L2 of two loops are now defined by h· · · (L1 · L2)(x)i := h· · · L1(x1)L2(x2)i where we demand x1 > x > x2 so that L1 and L2 are the loops closest to x from the left and from the right. Since the expectation values depend only on the order but independent of the relative distance, this gives a consistent definition. At this stage, we can make sense of the operator product expansion for defects. Suppose that there is a set L1,2,... of loops which cannot be decomposed into any sum of other simpler ones. We then have the following expansion of correlation functions that can be written succinctly as h· · · Li · Lj (x)i = Li · Lj = The OPE coefficients cikj (q) are asymmetric under the exchange i ↔ j due to the intrinsic ordering along R. However, we can simultaneously flip the R direction and S1 direction in the local S1 × C × R geometry to obtain another supersymmetric background. The background is parameterized by q as remarked before, and we use the parameterization cikj (q) = cjki(q−1). There is also an order for the defect networks on the two dimensional geometry side. In order to make the relation between two orders, let us consider the case when the 4d side is Sb4 and the 2d side is the Liouville/Toda theory [3, 4]. In this case the Verlinde operators associated to the networks on the 2d side act on the space Hconf of conformal blocks, and the loop operators on the 4d side act on the space Hhemi of holomorphic Nekrasov partition functions defined on a hemisphere [10, 11, 15, 30]. Since Hconf and Hhemi are naturally isomorphic, we can see the relations among three orderings as shown in figure 2. Corresponding to (2.4), there should be the operator product expansions of defect networks which are indeed skein relations as resolutions of the crossings. In section 4, we will see that we can calculate many OPE coefficients in terms of defect networks. Let us recall the origin of the non-commutativity when the geometry is globally S1 × C × R, following the discussions in [24, 26]. Considering the S1 as the time direction, the partition function of S1 × C × R is given by Ω = TrH h(−1)F e2πiλJ3 i Suppose now that we have a U(1) gauge theory, that the first loop L1 is purely electrically charged with electric charge e and that the second loop L2 is purely magnetically charged with magnetic charge m. Then there appears the Poynting vector carrying the pendent of the distance between two particles but the sign depends on the ordering, and therefore we have L1 · L2 = q−2emL2 · L1. In the classical limit (q → 1), this product becomes commutative. Networks and skein relations in 2d In this section we discuss possible types of networks in class S theories of type SU(N ) and their skein relations. Our guiding principle is that they are described by the structure of the and that the skein relations are universal under an appropriate parameter identification. The skein relations exhibited below already appear in the mathematical works [17, 31, 32], up to the overall factors and the changes in conventions. Skein relations introduce equivalence relations among all possible networks, and it would be extremely useful if we can pick a natural representative element out of a given equivalence class of networks allowing linear combinations of networks. In this paper we at least give a general method to simplify a given network: • In principle, edges in a network can carry arbitrary representations of SU(N ). We for k = 1, . . . , N − 1. These are the fundamental representations in the mathematical terminology,4 and within • Then we rewrite all crossings in terms of linear combinations of junctions that are at most rectangular. The concrete formulas are given in (3.36). We call this process crossing resolutions. In the A1 case, these procedures eliminate all the crossings and no junctions remain, thus reproducing the classification in [8]. In the A2 case, we will see that all digons 4Contrary to the standard physics usage, we do not restrict the fundamental representation to be the defining N -dimensional representation in this paper. and rectangles can be eliminated, and we will find a natural representative for a given equivalence class of networks. We will detail this process in section 4.4. Hereafter we use a version of the standard quantum number defined as and the factorial defined as q − q−1 h0i! = 1, hni! = hnihn − 1i!. As this section is rather long, let us pause here to explain the organization: in secare labeled by representations and they can have junctions corresponding to the invariant tensors. In section 3.2, we describe how an arbitrary representation can be rewritten in meet. In section 3.4, we show how a crossing of two edges can be rewritten in terms of junctions. We start from the crossing of two edges labeled by and then describe the general case. In section 3.5 we summarize the Reidemeister moves that are fundamental equivalence relations guaranteeing the isotopy invariance. In section 3.6 we note other useful skein relations that can be used to simplify networks. Finally in section 3.7, we k . In section 3.3, we then explicitly display the skein relations for A2 and A3. In general, it would also be important in the class S theory to study skein relations with full and other punctures in [1] or networks ending on other punctures. We do not, however, consider such objects in this paper. Before proceeding, let us first recall the fact that a codimension-4 operator of the 6d a multiple number of such operators can be joined. group SU(N ). On the 4d side, then, we can consider the Wilson loop operator in a representheory wrapped around the cylinder. Then this codimension-4 operator also needs to be labeled by a representation R. When we multiply two parallel Wilson loops with representations R1 and R2, we get On the 4d side, three Wilson loops in representations R1,2,3 can be joined at a point there is an invariant tensor in this triple tensor product. The number of independent ways to join them is given by the number of linearly independent invariant tensors. Then, three Since this should be an intrinsic property of codimension-4 operators of the 6d theory, we can join three codimension-4 operators along a one-dimensional loop on the 4d side. This gives a junction of three edges labeled by R1, R2, R3 on the 2d side. Using many such junctions, we end up having networks on the 2d side, which are our main interest in this paper. Restriction of labels First, note that Wilson loop on the 4d side in a representation R in a direction can be thought of as a Wilson loop in the representation R¯ in the opposite direction. This feature networks on the 2d side. This can be represented diagrammatically as As it is cumbersome to use arbitrary representations R as labels, we next rewrite representation R can be specified by a Young diagram (`i) where `i is the number of boxes is represented by (1 1 1 . . . 1) = (1k). Note that any symmetric polynomial of N variables x1, x2, . . . , xN , under a constraint symmetric polynomials, this means that any representation R can be decomposed as the For example, we have the equalities 3 χ(3) = χ(1) − 2χ(12)χ(1) + χ(13), χ(22) = χ(12)χ(12) − χ(1)χ(13) which we can diagrammatically depict, in the case of closed loops, as − 2 These relations are locally applicable on parallel edges. Therefore, we can insert some punctures or networks inside the circle, for example. a = b + c canonical junctions b = a + c N − b − c N − b − c non-canonical junctions Canonical junctions and removal of digons Canonical junctions k , k = 0, 1, . . . , N − 1. We can just use the integer k to label the edge, and an edge labeled by 0 can be removed. labeled by k has charge k under the center of SU(N ), and therefore we call these integer labels as the charge.5 For each trivalent junction, the sum of three inflowing charges must equal to zero from ∧ ⊗ ∧ to ∧ single invariant tensor in ∧ ⊗ ∧ distinguish the possible invariant tensors. Sometimes these labels k are then taken to be defined modulo N as in [14, 15], but it write down the invariant tensor rather explicitly using the quantum group representation theory when the net inflowing charge to a junction vanishes in Z. We call such a junction canonical. We call a junction non-canonical if the net inflowing charge vanishes only in ZN . See figure 3 for examples. Let us now describe the invariant tensors associated to the canonical junctions. Let be spanned by the vectors e1, . . . , eN . We define the q-deformed wedge product by the rule ei ∧ ej = −qej ∧ ei, (i ≤ j) to ∧ π1,1→2 : ei ⊗ ej 7→ ei ∧ ej . Furthermore, ∧2 can be naturally embedded within ι2→1,1 : ei ∧ ej 7→ −qei ⊗ ej + ej ⊗ ei, by the rule More generally, we associate to any canonical junction that combines labels a, b to a + b the projection 5Note that it is a special property of Ak that there is the one-to-one correspondence of the set of the fundamental representations ∧ including the trivial one and the charge under the center. or equivalently Removal of digons Now, we can check that any digons can be removed as and to any canonical junction that splits the label a + b to two labels a, b the map ιa+b→a,b : ek1 ∧ · · · ∧ eka+b 7→ (−q)ab i1<···<ia, j1<···<jb where we assume k1 < · · · < ka+b, the sum is over the disjoint split of indices {k1, . . . , ka+b} = {i1, . . . , ia} t {j1, . . . , jb}, and n(i, j; k) is the minimal number of adjacent transpositions to bring the sequence i1, . . . , ia, j1, . . . , jb to k1, . . . , ka+b. These maps are described in more detail mathematically in [33]. diagrams together with the ones with reversed arrows: hi1i!hi2i! . . . hili! where P` a=1 ia = k. Note that in the classical limit q → 1 the prefactor becomes (−1)Pa<b iaib follows from the classical epsilon symbol. This somewhat unusual sign is however necessary to match with the known skein relations in the Liouville/Toda theory, and it also simplifies the signs appearing in the general crossing resolutions (3.36). In the q-deformed Yang-Mills theory it would be more in section 5. When the sum of il is N , we can use the rule to evaluate a network with two trivalent hii!hji!hki! More simply, we can evaluate a closed loop with label k by considering it as a digon with edges labeled by k and N − k: convention in the q-deformed Yang-Mills. We also see at this point that, to compare with the skein relation of the Toda theory or the q-deformed Yang-Mills theory, we need to use hki!hN − ki! the relation Crossing resolutions The R matrix R = A(Q + q−1I ⊗ ). Let us first discuss the best-known case: the R-matrix for of SU(N ), which is Here, IV is the identity operator on a vector space V , Q is an operator Q = i6=j where eij is a matrix whose only non-zero entry is 1 at the i-th row and j-th column and A is the overall normalization which we will be fixed later. The action of Q on the base ea ⊗ eb of Q(ea ⊗ eb) =  eb ⊗ ea − qea ⊗ eb, eb ⊗ ea − q−1ea ⊗ eb, (a > b) (a = b) . The inverse of the R-matrix R is that we represent as and the natural embedding ι2→,1,1 : ∧q2 Note also that Q satisfies This is a special case of the digon elimination. We can now represent the R-matrix R diagrammatically as Q2 = −(q + q−1)Q = h2iQ. Below, we call the crossing (3.26) as positive and the crossing (3.28) as negative. The A1-case: in this case, ∧2 is the pseudo-reality condition is the trivial one-dimensional representation and there which we can diagrammatically write as Then the general equation (3.26) reduces to + q−1 −1 = A−1(Q + qI ⊗ ) = A−1  = A  = q−1/2 where we have set A = q1/2. We then have q−1/2 − q1/2 = (q−1 − q) These reproduce the standard skein relations of the Liouville theory found in [10, 11] under The relation Q2 = h2iQ shows = h2i = −χ (diag(q, q−1)). ature on the superconformal index. The minus sign here is a convention common in the Liouville/Toda literature, i.e. the definition of a loop in the representation differs by an overall minus sign between the Liouville theory and the q-deformed Yang-Mills. The A2-case: here we have = ∧2 , and therefore we have As we now only have one type of the label , we can drop it altogether. The general R-matrix (3.26) then becomes = q1/3 + q−2/3 General case: the analysis so far suggests that we should take General crossing resolutions Now let us move on to the crossing resolutions in the general case. The expression was found in [17] up to an overall factor, which we quote here: A = q N overall factor, this equality is invariant with a reversal of an arrow and the rotation of already discussed. Let us introduce the names to the fundamental objects on the right hand side of (3.36): Qab(i) := a + b − i a + b − i a − i b − i b − i a − i Note that the number of possible choices of i matches with the number of irreducible these Qab(i) cannot be further decomposed as parts of networks. ⊗ ∧ b . We expect that all The intersection number and the powers of q Let us briefly discuss the significance of the prefactor q aNb in (3.36). In general, two loop operators in a class S theory of type SU(N ) can be mutually nonlocal, and the nonlocality can be measured in terms of the Dirac pairing that takes values in ZN [34]. In terms of the 2d networks realizing the 4d loop operators, the Dirac pairing is given by their intersection number. We can define it by assigning a local intersection number to a crossing as follows: : − ab. ming the contributions from all the crossings: sign(c)a(c1)a(2) c ∈ ZN In the Liouville/Toda setup, we expect the expectation value of any network without the superconformal index, the expectation value of a loop operator on the 4d side is a non-locality, and we can think of the prefactor q aNb in (3.36) as encoding the difference in the local intersection number between the left hand side and the right hand side, to keep track of this non-locality. The difference in the powers of q among different summands in the resolutions of the crossings should be integral, and the relation (3.36) indeed satisfies this requirement. We can define the operation I which we call the inversion by reversing all the arrows simultaneously. This is an involution that we can identify with the charge conjugation on the 4d side. When the network is on a torus, we can also consider the action of SL(2, Z) on the networks. Two basic actions are the T -action and the S-action. T corresponds to sending 1-cycles on the torus. put on the torus and open edges are connected to the opposite ones. Reidemeister moves In knot theory, a projective representation in two dimension of knots and links in a three dimensional space is not unique, and any different representations can be mapped to each other by a combination of three so-called Reidemeister moves, see e.g. [35]. The move I straightens a twist in an edge, the move II slides one edge over another edge to two parallel edges, and the move III changes the order of three crossings. In the presence of junctions, we need to add another move, where we move an edge over a junction. We call this as the Since we expect that the charge of a loop in the 4d theory is determined by the isotropy class of networks, we would like to require that a network is invariant under these moves. This is indeed possible for the moves II, III and IV, but the move I results in a q-dependent factor. In the context of 3d Chern-Simons theory, this can be understood from the change in the framing of the link [36]. Let us describe these moves explicitly below. of the R-matrix R corresponding to the positive crossing. R-Move III: this is the Yang-Baxter equation which the R-matrix R should satisfy. R-Move IV: this is the additional move for the networks with junctions. R-Move I: finally, this move involves a nontrivial factor. = CV (q) = CV (q−1) hN−k+ii! A direct understanding of this coefficient in 4d or 6d would be an interesting problem. More simplifying relations Let us list various other skein relations that can be used to simplify networks. All relations except (3.48) are known in [31, 32] and references therein. Triangle contraction relations We have rules to remove triangles. In order to express the rules, we first map all the junctions so that they are canonical. Then, there are four possibilities up to the mirror images: Note that whether three vertices are totally ordered by arrows or not changes the look of the factors. hN − ai! hN − ii!hji! b hN − ai! hii!hN − ji! b q− N k − 1 k − 2 The rectangles Q(aib) that we had in (3.37) can not be further simplified, but there are many other rectangles that are equivalent to sums of simpler ones. Let us show one class: ` − j h` − ki! These relations assure that any network constructed from only rectangles around a tube can always be decomposed into a sum of closed loops around the tube. For example, using (3.47) recursively, we can see k + i k − i where the two horizontal thin parallel lines signify that they are to be identified so that the network is on a tube. There are various other relations. Here we just note one example: = q−k X (−1)iqi2+i = h3i k + 1 i=−k k − 2 The basic skein relation was (3.34), which we copy here [14–16]: = q1/3 + q−2/3 Let us record the A2 case as a summary. We have two types of junctions: Examples: A2 and A3 The skein relations of the A1 case and the A2 case have already been described in the litk − 1 Let us now discuss the next nontrivial case of A3. Note that the label 3 can be traded with 1 by reversing the arrow, and since 2 is a real representation we do not have to exhibit the direction for edges labeled by 2. In this case, there are also two types of junctions as we There are three types of crossing resolutions: The following two relations are useful to simplify the networks: = h2i There are three decaying relations for one rectangle: = q1/4 = q1/2 = q = h2i = h2i + q−1 As the fundamental aspects of defect networks in the two dimensional theories have been 2d side and loops on the 4d side. In this section, we restrict ourselves to the most familiar in [21]. It is not easy to construct the corresponding networks for the general Ak case, but we will see that the skein relations allow us to describe and classify the networks for A2 concretely. Before proceeding, let us quickly recall the possible charges of the loop operators of N -dimensional representation. They are explicitly given by hi = 1 − N , . . . , 1 − N , − N , . . . , − N − N , . . . , − N , 1 − N , − N , . . . , − N Note that ω1 = h1 and ωN−1 = −hN . Let us consider a Wilson loop labeled by an irreducible representation R. We can For a dyonic loop operator, we need to specify a pair of electric and magnetic charges The product of Wilson loops and ’t Hooft loops It is well known how pure Wilson loops and pure ’t Hooft loops are represented as loops on the torus: where we identify each pair of parallel opposite edges to make the parallelogram the torus. Here WR is the Wilson loop in the representation R, and we identify an irreducible representation and its highest weight vector. We use a similar notation for the ’t Hooft loop. the S transformation on the torus is naturally identified with S duality transformation of N =4 gauge theory. As seen in [37] and [30], Wilson loop WR and ’t Hooft loop TR are written in the form of matrix model: hWRi = hTRi = iRN−1 iRN−1 [da]Z(a)∗ ducible representation R. T (R)(a) are some functions of a related to the character χR via a λ In general, any loop operator is expected to be represented as hXi = iRN−1 [da]Z(a)∗ X Xν (a)Z(a − bν) T (`). One way to insert is h· · · T (`)W (k)Xi = [da]Z(a)∗ · · · where the ellipsis represents further insertions of other loops. labeled by m = 1, 2, . . . , min(k, `): h· · · [T W ](m`,k)Xi := [da]Z(a)∗ · · · iRN−1 detail of which we do not need either. The additions of WR and TR in the ordering of loops seen in section 2.1 are written as follows: hWRXi = [da]Z(a)∗WR hTRXi = [da]Z(a)∗TR [da]Z(a)∗ X Xν (a)χR(A(a − bν))Z(a − bν), [da]Z(a)∗ X Tμ(R)(a − bν)Xν (a)Z(a − bν − bμ). We then have (μ, λ) ∈ Π(∧` ) × Π(∧k ) hμ, λi = m − N h· · · T (`)W (k)Xi = Note that the decomposition of T (`)W (k) is independent of the ellipsis . . . and X assuring that this expansion is local and represent the product as T (`) On the other hand, the insertion in the opposite order is h· · · W (k)T (`)Xi = [da]Z(a)∗ · · · iRN−1 and we also have where we use h· · · W (k)T (`)Xi = 2 kN` −m [T W ](m`,k)Xi 2 kN` −m exp[2πibha − bν, λi]. In summary, we have found the relations × W (k) = [T W ](m`,k), × T (`) = 2 kN` −m [T W ](m`,k). Comparing the product expansion (4.20) and the graphical expansion (4.4) we find the following identification: ↔ [(μ(`m), λ(km))] ↔ q−hμ(`m),λ(km)i[T W ](m`,k) Let us see how T transformation of the SL(2, Z) duality action acts on these loop q−hλ,λie2πibhλ,aiZ(a)∗.Z(a − bλ) The Witten effect on the partition function can be re-expressed in the loop operators which is accompanied by and matches with the T transformation on the torus. Examples of products of loops in A2 Let us focus on the A2 case and perform some explicit computations. The examples in the general Ak case will be given in appendix A. We will see the geometric SL(2, Z) action on the torus is nicely mapped to the SL(2, Z) action on the electric and magnetic weight systems. as θ shifts by 2π. Summing it up over Π(∧` ), 2 T (`) −→ q−`+ `N [T W ](``,`) = D[(ω`,ω`)] = D(``,`) under θ → θ + 2π. Since D[ω`,ω`] = D(``,`) is given by Example 1. The simplest case is W a = 1 and b = 1: × T , which corresponds to the equation (4.4) with × T = q1/3D(10,1) + q−2/3D(11,1) in [14, 15] in the context of class S theory. The dyonic loop D(11,1) is obtained from the ’t Hooft loop T by an application of the T operation. In particular this loop can be mapped to a Wilson loop in some duality frame. The object D(10,1) cannot be mapped into a network localized on any one cycle by the torus modular transformations. In the language of charges, this means that the electric weight and the magnetic weight are not parallel. We can now work out how the SL(2, Z) transformations act on this particular network and the pair of weights, see figure 4. T −1 T −1 T −1 expressions of a pair of weights are equivalent via some Weyl reflections. Red weights correspond to electric weights and green’s to magnetic ones. Example 2. The next example is W + q−1 + q−2 In this example, the first term on the right hand side is a network that cannot be mapped by SL(2, Z) to any of the networks we already studied explicitly. It is natural to posit the following expansion the angular momentum. Then we can identify + (loops with lower weights) (4.29) networks and charges of the dyonic loops up to the contributions from lower weights.7 7The complication comes from two sources. One is common with what we encountered in section 3.2: Example 3. The third example is WAdj × T : the skein relation gives us Adj = q + q−1 while from gauge theory we expect = qD[(ω1,h2−h1)] + D[(ω1,h1+2h2)] + q−1D[(ω1,2h1+h2)] + (loops with lower weights). For a graphical representation of weights involved, see figure 5. The first term and the third term can be obtained by SL(2, Z) transformations on D(10,1). The second term is a new type: Example 4. Our final example is WAdj × TAdj. The skein relation gives + q−1  = 4(2 + q2 + q−2) + q−2  + q  irreducible representations are linear combinations of networks even in the Wilson loop case. Another is related to the bubbling effect of the monopole moduli space. See the related works to this subject [30, 38–40]. while the gauge theory computation yields WAdj × TAdj = q−2D[(λAdj,λAdj)] + q2D[(λAdj,−λAdj)] + (loops with the lower weights) 4d gauge theoretic computations. Classification of networks on T 2 for A2 We have seen some basic examples of products of loops and the identification of the charge and the network. Here we establish the general mapping between the networks and the Mills. This is a minimal extension of the dictionary of Drukker, Morrison and Okuda [8]. Let us first classify the possible A2 networks on the torus purely in terms of the skein relation. First, recall that all networks with crossings are resolved into those with junctions only. In particular, for the A2 case, there are only two types of junctions, namely the one where the heads of three arrows meet and another one where the tails of three arrows meet. Therefore the networks are bipartite [14] and there appear only polygons with degree-even vertices. We now use the skein relations we discussed so far. Recall the basic conventions we discussed in section 3.7. All digons can be contracted, and all rectangles are resolved to two pairs of curves, as we discussed in (3.52). At this point, the network might contain several disconnected components. If there are no vertices at all, the network consists of parallel loops wrapping the same one-cycle on T 2. Assume now there is at least one vertex. Pick a connected component. It has the topology of either a disk, an annulus or a torus. Now, let us denote the number of edges, or equivalently the number of vertices, of of polygons by f . The total number of vertices, edges and faces of the network is then V = 1 X pi, E = 1 X pi, F = f. Furthermore, denote the number of boundary edges by B which vanish if the connected component has the topology of torus. From Euler’s theorem we should have 1 X pi ≥ 0 1 X pi ≤ 0. From this we see that the connected component has the topology of the torus, and every polygon is a hexagon. Therefore, the possible A2 networks on T 2 are mapped into the bipartite hexagon tilings with three corner condition at every vertex. It is interesting to note at this point that bipartite hexagon tilings of the torus appeared in the string theory literature in the context of brane tilings [41–43]. In this case the bipartite hexagon tilings corresponded to Abelian orbifolds of C3. Now let us make the dictionary between the bipartite hexagon tilings and the dyonic charges. Instead of thinking of filling a torus by hexagons, we can take the quotient of the bipartite hexagon tiling filling the entire plane, and then we define the vectors e1 and e2 hexagonal tilings and the dyonic charges that already appeared in our analysis so far. From these examples, we can find the general map. We first naturally identify the A2 this condition. Let us end this section by exhibiting some more examples of the mapping between the dyonic charges and the hexagonal tilings, see figure 9. In this section we only discussed the A2 case. It would be interesting to find a general map from the charge of the dyonic loops to the networks for or general Ak>2 cases. Some examples in the superconformal index So far we have been studying the properties of the networks of the class S theories. In this section, we would like to study a few concrete manifestations of our analysis when the four-dimensional side is S1 ×S3, or equivalently, in the setting of the superconformal index. As has been well established, the superconformal index of a class S theory of type SU(N ), defined by a Riemann surface C of genus g with n full punctures, is given by the (zero-area limit of the) (p, q, t)-deformed Yang-Mills theory on the same Riemann surout, and the superconformal index becomes the standard q-deformed Yang-Mills theory, defined in [19, 20, 44]. In this case, the supercharge defining the superconformal index is compatible with the presence of a half-BPS line wrapping the S1 direction [25, 45]. Therefore, in this setup, a defect of a class S theory, specified by a network with labels on the Riemann surface C should be realized as a concrete object in the q-deformed YangMills theory, defined by the same network with labels on C. Below, we show that they are just given by the Wilson lines8 and the Wilson junctions in the q-deformed Yang-Mills 8This statement was independently obtained by W. Peelaers and L. Rastelli. Ig(a1, . . . , an) = Tr(−1)F qΔ−I3 a1 · · · an where the trace is over the Hilbert space of the system on S3 or equivalently on the space of to the i-th full puncture. This is known to be given by the following explicit expression: Ig(a1, . . . , an) = qρ = diag(q(N−1)/2, q(N−3)/2, . . . , q(1−N)/2), theory. In particular, we compute the superconformal index of the nontrivial line operator with three full punctures. Reassuringly, we will find that the result shows an enhancement of the symmetry from SU(3)3 to E6, as it should be, since the T3 theory is the E6 theory of Minahan and Nemeshcansky. Below, we will first very briefly review the relation between the superconformal index and the q-deformed Yang-Mills in section 5.1. Then we check that the superconformal index with a Wilson line in the 4d gauge theory is given by the Wilson line in the 2d gauge theory in section 5.2. Finally, we will compute the superconformal index of a network in the T3 theory in section 5.3. Note that in this section we use the 2d Yang-Mills normalization of the junctions as discussed in section 3.3.2. The superconformal indices without lines Consider the class S theory Sg,n for a genus g surface with n full punctures. Its superconformal index is defined as K(a) = PE 1 − q N − 1 + X ai/aj i6=j K0 = PE 1 − q n≥1 and K(a), K0 are given by Here we took a to be diagonal, and PE is the plethystic exponential, defined by n≥1 a = diag(a1, . . . , aN ) ∈ SU(N ), X antn = Y (1 − tn)−an . Take two class S theories Sg,n and Sg0,n0, pick one full puncture from each, say the last one from Sg,n and the first one from Sg0,n0. Let us then couple SU(N ) gauge multiplets to the diagonal combination of the SU(N ) symmetries carried by the two full punctures thus chosen. The combined system has the superconformal index Ig,g0(a1, . . . , an−1; b2, . . . , bn0) where [dz]Haar is the natural measure on the Cartan of SU(N ) given by [dz]Haar = 1 N−1 N ! i=1 2π√−1zi i6=j Y(1 − zi/zj). Plugging in the expression (5.2) to (5.7) and using the orthogonality of characters full punctures. This is as it should be. SU(N )3 enhances to E6. The decomposition is 4d Wilson lines Let us move on to the superconformal index in the presence of a loop operator wrapping S1 [45, 46]. In this subsection, for simplicity, we only study a loop operator that is just a Wilson line with respect to a 4d gauge group. For class S theories, this covers every operator that is a genuine loop L labeled by a representation R on the Riemann surface (i.e. all networks without junctions), since we can always cut the Riemann surface along L to go to a duality frame where that particular loop wraps a tube once. To compute the superconformal index with a Wilson loop, let us first consider a more general situation. Take a theory X with G flavor symmetry whose superconformal index is IX (a). Suppose we can couple it with the G vector multiplet, such that the gauge coupling is exactly marginal. Let us insert a BPS Wilson line in the representation R of G. The resulting index is simply IR(q) = [dz]HaarK(z)−2χR(z)IX (z). This reduces to the formula (5.7) when R is a trivial representation, as it should be. Now consider the case when X consists of two copies of the TN theory, such that we gauge a diagonal combination of G symmetries. We have IR(q) = × K(z)−2 K(z−1)K(a3)K(a4) X χλ0(z−1)χλ0(a3)χλ0(a4) K(a1)K(a2)K(a3)K(a4) The result (5.13) is, up to the prefactor involving K, the unnormalized correlator of the q-deformed Yang-Mills theory with the Wilson line with the representation R, around the tube associated to the gauge group, see figure 10. There, we displayed full punctures as boundaries, as would be more common in the 2d Yang-Mills viewpoint. For non-deformed 2d Yang-Mills theory, this is an immediate consequence from the fact that the Hilbert space of the system on S1 is the space of class functions on G, that fact that the structure of the tensor product decomposition of the representations of the quantum group Gq is not deformed as long as q is generic. A network in the T3 theory As an example of the network that is not just a loop, let us consider the T3 theory and the 2d network shown in figure 11. The superconformal index is given by IT3,network = K(a1)K(a2)K(a3) In the undeformed 2d Yang-Mills theory, cR1,R2,R3 is given by the integral cR1,R2,R3 = × ijk ¯i¯jk¯(U1)¯ii(U2)¯jj(U3)k¯, (5.16) k where U1,2,3 are holonomies of the 2d gauge field from one junction point to the other junction point, along three different segments. of equations In the q-deformed Yang-Mills theory, we need to perform the integral above in the sense of the quantum group [19, 20]. The U3 integral gives a nonzero result only when SU(3)q. Since the rules of irreducible decompositions of tensor products are unchanged under the q deformation, we see that cR1,R2,R3 is nonzero only when the highest weight of R1 is given by adding to the highest weight of R2 one of the three weights of V . This means that there is an arrow connecting R1 → R2 in the diagram of irreducible representations as shown in figure 12. We immediately see that there we should have arrows similarly for R2 → R3 and R3 → R1. Therefore cR1,R2,R3 is nonzero only when R1 → R2 → R3 → R1 forms a triangle in figure 12. Next, consider what happens when the holonomy at a puncture a3 is set to the special value a3 = q have the situation in figure 13. Using the skein relation, we see that the left hand side and the right hand side should be proportional by a factor of [2]. The right hand side is R2→R3→R1→R2 cR1,R2,R3 χR3 (qρ) = [2]qδR1→R2 where the sum on the right hand side is over R3 that fit in the triangle. This equation can be recursively solved starting from the triangle closest to the origin. We can check that cR1,R2,R3 is nonzero only when the arrows form a triangle R1 → R2 → R3 → R1. the following gives the general solution:9 dimq(nω1 + mω2) = χnω1+mω2 (qρ) = [n + 1][m + 1][n + m + 2] [n + 1][m + 1][n + m + 1] [n + 1][m + 1][n + m + 3] Plugging them into (5.15), we find it nicely becomes a sum of representations of E6: IT3,Line(q, α) = q1/2χ 0 Acknowledgments The authors would like to thank Dongmin Gang, Takuya Okuda and Masahito Yamazaki for helpful discussions. YT’s work is supported in part by JSPS Grant-in-Aid for Scientific Research No. 25870159. NW is supported by the Advanced Leading Graduate Course for Photon Science, one of the Program for Leading Graduate Schools lead by Japan Society for the Promotion of Science, MEXT, and YT are also supported in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. 9The authors thank the twitter user @LT shu for kindly providing this general solution in https://twitter.com/LT shu/status/349136595632390144. Examples of OPE and charge/network dictionary Here we present some more examples of the OPE and the correspondence of the dyonic be removed and those with 2 can be replaced by the reversed ones with 1. We use the electric one is unrestricted. Pure dyonic loops Dyonic loops can be roughly classified into two, which we call pure and complex. The pure ones are those that can be mapped to a Wilson loop in a duality frame, and the complex ones are those without any such duality frame. Let us first discuss the representation of the pure ones as loops on the torus. arrow without any label: sr−1 sr q ≥ p : p ≥ q : This is essentially the same as the discussion in [8]. Let us compute the skein relation of W and T . Comparing with what we expect from the gauge theory, we can then identify various networks with complex dyonic loops. From + q−1 + q−2 Let us next consider W(2,1) × T : = q2/N  The first term is a new one, the second and third ones are elementary and the final one is pure. Then we identify: = q3/N  + q−1 + q−2 The first and the third terms are new, the second one is elementary and the fourth one is appear and indeed this case is the same as Example A.5. We can therefore identify: D[(ω1,2hi(6=1)+hj(>i))] ∼ , D[(ω1,hi(6=1)+hj(>i)+hk(>j))] ∼ Our next example is W = q3/N  − 2 + q−1  + q−2 The first term is new, the fifth one is elementary, the seventh one is pure and others have + q−3 As a further example, let us consider WAdj × T : D[(ω1,h1+hi(6=1)+hj(>i))] ∼ + q−1 The second term is new, and the first and the third ones are obtained by some duality actions of some elementary one. Our identifications are: D[(ω1,hi(6=1)−hj(6=i)+hj(>i))] ∼ We now move on to the example WAdj × TAdj: + (2[N − 2] + [N − 4]) + (3 + q2 + q−2) + q−2 + q−1  D[(λAdj,hi(6=1,N)−h1)] ∼ D[(λAdj,−h1+hi(6=1,N))] ∼ Then the charge/network dictionary for the new ones is: D[(λAdj,h1−hi(6=1,N))] ∼ D[(λAdj,hN −hi(6=1,N))] ∼ , (A.18) = qk−`/N  + q−1 An OPE of complex dyonic loops Finally we give an example of the OPE of two elementary dyonic loops D(10,1) and D(`0,k) for `, k ≤ N/2 in N =4 SYM: Here we require js differ for each s. It would be interesting to apply the diagrammatic approach to more complicated OPEs and read off the charge information from networks in Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. approximation, arXiv:0907.3987 [INSPIRE]. four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE]. JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE]. [INSPIRE]. polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE]. Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [INSPIRE]. Phys. B 300 (1988) 360 [INSPIRE]. [arXiv:0909.0945] [INSPIRE]. theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE]. CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE]. (2011) 114 [arXiv:1008.4139] [INSPIRE]. [arXiv:1312.5001] [INSPIRE]. plane graphs, Enseign. Math. 44 (1998) 325. U.K. (1996). Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge Chern-Simons theory, Commun. Math. Phys. 172 (1995) 317 [hep-th/9403066] [INSPIRE]. Commun. Math. Phys. 170 (1995) 669 [hep-th/9405126] [INSPIRE]. Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE]. symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE]. [27] T. Okuda, Line operators in supersymmetric gauge theories and the 2D-4D relation, [28] S. Gukov, Surface operators, arXiv:1412.7127 [INSPIRE]. Korean Math. Soc. 49 (2012) 993, [math/0506403]. [31] M.J. Jeong and D. Kim, The quantum sl(n, C) representation theory and its applications, J. [INSPIRE]. [33] A. Chirvasitu and M. Tucker-Simmons, Remarks on quantum symmetric algebras, J. Algebra [34] Y. Tachikawa, On the 6D origin of discrete additional data of 4D gauge theories, JHEP 05 [INSPIRE]. [38] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE]. [1] D. Gaiotto , N = 2 dualities, JHEP 08 ( 2012 ) 034 [arXiv:0904.2715] [INSPIRE]. [2] D. Gaiotto , G.W. Moore and A. Neitzke , Wall-crossing, Hitchin systems and the WKB [3] L.F. Alday , D. Gaiotto and Y. Tachikawa , Liouville correlation functions from [5] A. Gadde , E. Pomoni , L. Rastelli and S.S. Razamat , S-duality and 2D topological QFT, [6] A. Gadde , L. Rastelli , S.S. Razamat and W. Yan , The 4D superconformal index from q-deformed 2D Yang-Mills , Phys. Rev. Lett . 106 ( 2011 ) 241602 [arXiv:1104.3850] [7] A. Gadde , L. Rastelli , S.S. Razamat and W. Yan , Gauge theories and Macdonald [8] N. Drukker , D.R. Morrison and T. Okuda , Loop operators and S-duality from curves on [9] E.P. Verlinde , Fusion rules and modular transformations in 2D conformal field theory , Nucl. [10] L.F. Alday , D. Gaiotto , S. Gukov , Y. Tachikawa and H. Verlinde , Loop and surface operators [11] N. Drukker , J. Gomis , T. Okuda and J. Teschner , Gauge theory loop operators and Liouville [12] N. Drukker , D. Gaiotto and J. Gomis , The virtue of defects in 4D gauge theories and 2D [13] J. Gomis and B. Le Floch , 't Hooft operators in gauge theory from Toda CFT , JHEP 11 [14] D. Xie , Higher laminations, webs and N = 2 line operators , arXiv:1304 .2390 [INSPIRE]. [15] M. Bullimore , Defect networks and supersymmetric loop operators , JHEP 02 ( 2015 ) 066 [16] G. Kuperberg , Spiders for rank 2 Lie algebras , Comm. Math. Phys . 180 ( 1996 ) 109 [17] H. Murakami , T. Ohtsuki and S. Yamada , Homfly polynomial via an invariant of colored [18] C. G´omez, M. Ruiz-Altaba and G. Sierra , Quantum groups in two-dimensional physics , [19] A.Y . Alekseev , H. Grosse and V. Schomerus , Combinatorial quantization of the hamiltonian [20] E. Buffenoir and P. Roche , Two-dimensional lattice gauge theory based on a quantum group , [21] A. Kapustin , Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality, [22] J.A. Minahan and D. Nemeschansky , An N = 2 superconformal fixed point with E6 global [23] S. Gukov and E. Witten , Gauge theory, ramification, and the geometric Langlands program , [24] D. Gaiotto , G.W. Moore and A. Neitzke , Framed BPS states, Adv. Theor. Math. Phys. 17 [25] T. Dimofte , D. Gaiotto and S. Gukov , 3 -manifolds and 3D indices, Adv. Theor. Math. Phys. [26] Y. Ito , T. Okuda and M. Taki , Line operators on S1 × R3 and quantization of the Hitchin [29] N. Hama and K. Hosomichi , Seiberg-Witten theories on ellipsoids, JHEP 09 ( 2012 ) 033 [30] J. Gomis , T. Okuda and V. Pestun , Exact results for 't Hooft loops in gauge theories on S4 , [32] S. Cautis , J. Kamnitzer and S. Morrison , Webs and quantum skew Howe duality , Math. Ann. [35] L.H. Kauffman , Knots and physics, Series on Knots and Everything volume 1 , World [36] E. Witten , Gauge theories and integrable lattice models, Nucl . Phys . B 322 ( 1989 ) 629 [37] V. Pestun , Localization of gauge theory on a four-sphere and supersymmetric Wilson loops , [39] N. Saulina , A note on Wilson-'t Hooft operators , Nucl. Phys . B 857 ( 2012 ) 153 [40] R. Moraru and N. Saulina , OPE of Wilson-'t Hooft operators in N = 4 and N = 2 SYM with [41] A. Hanany and K.D. Kennaway , Dimer models and toric diagrams , hep-th/ 0503149 [42] S. Franco , A. Hanany , K.D. Kennaway , D. Vegh and B. Wecht , Brane dimers and quiver [43] A. Hanany , D. Orlando and S. Reffert , Sublattice counting and orbifolds , JHEP 06 ( 2010 ) [44] M. Aganagic , H. Ooguri , N. Saulina and C. Vafa , Black holes, q-deformed 2D Yang-Mills and non-perturbative topological strings, Nucl . Phys . B 715 ( 2005 ) 304 [hep-th /0411280] [45] D. Gang , E. Koh and K. Lee , Line operator index on S1 × S3 , JHEP 05 ( 2012 ) 007 [46] D. Gang , E. Koh and K. Lee , Superconformal index with duality domain wall , JHEP 10

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282015%29186.pdf

Yuji Tachikawa, Noriaki Watanabe. On skein relations in class S theories, Journal of High Energy Physics, 2015, 1-43, DOI: 10.1007/JHEP06(2015)186