Wilson-’t Hooft Line Operators as Transfer Matrices

: We review the relation between half-BPS Wilson-’t Hooft line operators in N = 2 supersymmetric gauge theories on the twisted space-time S 1 × (cid:15) R 2 × R and the transfer matrices constructed from the trigonometric L-operators of an integrable system.


Introduction
The Wilson and 't Hooft line operators are important objects in quantum field theory to elucidate its vacuum structure [1], e.g., the Wilson line obeys an area law in the confining phase, while the Higgs phase can be characterized by a 't Hooft line obeying an area law. These have been studied extensively from various viewpoints. We discuss here a recent development of studies of Wilson-'t Hooft line operators in supersymmetric quantum field theory in 4d in relation with mathematical physics.
In supersymmetric quantum field theory, it is often the case that some quantities are related to the integrable systems. A classic example is the realization of the Seiberg-Witten theory [2], which specifies the low energy effective theory of N = 2 supersymmetric theory in 4d, as the classical Hitchin integrable system [3]. (See also [4][5][6][7].) This relation is generalized in [8] to the quantum level such that N = 2 theories on the certain gravitational background corresponds to quantum Hitchin systems. A recent example is the relation [9,10] between the superconformal index of N = 1 and N = 2 supersymmetric quiver gauge theories and the partition function of the integrable lattice model found in [11,12]. This latter relation with the lattice model was explained by using the so-called 4d Chern-Simons theory in [13,14].
In this paper we see the Wilson-'t Hooft line operators play a role in this context, focusing on N = 2 supersymmetric gauge theory. Specifically we consider certain Wilson-'t Hooft lines in N = 2 gauge theory which preserve a half of the supersymmetries on the twisted space-time S 1 × R 2 × R. The line operators wrap on S 1 . The vacuum expectation values (vevs) of the Wilson-'t Hooft lines are shown to be identified [15] with the transfer matrices of trigonometric L-operators which are obtained by a limit of the elliptic ones [16] satisfying the RLL relation with the elliptic R matrix [17][18][19].
This relation is a variant of the one found in [20,21] where certain half-BPS surface (codimension-2) defects in N = 1, 2 supersymmetric gauge theories have been identified with the transfer matrices of the L-operators of the elliptic integrable lattice model [16,22]. These results imply that an insertion of a class of line and surface defects, corresponds to an insertion of L-operators in the corresponding integrable system.
The reason why it is the case can be seen from the string theory [15]. Both the half-BPS line and surface defects in N = 2 supersymmetric gauge theory are realized on the worldvolume of intersecting D-branes, and these are related by duality in string theory with a minor difference that one direction is compactified on S 1 or not, which induces the difference between trigonometric and elliptic integrable models.
The organization of this paper is as follows. In section 2, we introduce the half-BPS Wilson and 't Hooft line operators in N = 2 supersymmetric gauge theory. The insertion of these operators is considered on the space-time geometry S 1 × R 2 × R and their vevs are computed [23] by using the localization method [24][25][26]. (See also [27][28][29][30].) We apply this to the N = 2 circular quiver gauge theory where SU(N ) gauge groups located on a circle and connected by hypermultiplets in the bi-fundamental representation of adjacent gauge groups.
In section 3, L-operators are discussed. We define an L-operator in an abstract manner, satisfying the RLL relation with the corresponding R-matrix. Their transfer matrix constructs an integrable system. An important example of the L-operators is the one introduced in [16] associated with the elliptic dynamical R-matrix in [17][18][19]. We then argue a trigonometric limit of the elliptic L-operator. Their transfer matrix reproduces the vevs of the Wilson-'t Hooft lines obtained in section 2.

Wilson-'t Hooft lines in supersymmetric gauge theories
Let us consider a gauge theory in 4d with gauge group G whose Lie algebra is g. Let a maximal torus of G be T with Lie algebra t. Also, let Λ r (g) ⊂ t * and Λ cr (g) ⊂ t be the root lattice and the coroot lattice of g, respectively. Their inner product is denoted by ·, · . Their duals are the coweight lattice Λ cw (g) = Λ r (g) ∨ ⊂ t and the weight lattice Λ w (g) = Λ cr (g) ∨ ⊂ t * .
Let L be a line in the space-time. We consider the following two types of operators defined on a support L and their mixtures: • A Wilson line operator is the worldline of a very heavy electrically charged particle, which is not dynamical. Let R be a representation of G, then the Wilson line on L is defined by where A is the gauge field.
• An 't Hooft line operator is the worldline of a very heavy magnetically charged monopole [1]. This is a disorder operator in the sense that the operator is defined by integrating over the fluctuation of the fields around the singular configuration at the location of the monopole in the path integral: where θ and φ are the polar angle and the azimuthal angle of the spherical coordinate in the direction orthogonal to the worldline and · · · represents less singular terms. For simplicity we set the gauge theory theta-angles to zero. The coefficient m is the magnetic charge of the monopole. Different singular gauge field configurations of the above form describe the same monopole if their magnetic charges are related by gauge transformation. It follows that m can be chosen from t, and the choice is meaningful only up to the action of the Weyl group W (G) of G.
The above expression of A is valid in a trivialization over a coordinate patch that contains the point θ = 0 of a two-sphere surrounding the monopole. Along θ = π, there is a Dirac string singularity which supports an unphysical magnetic flux. For the Dirac string to be invisible (or more precisely, for the gauge transformation by exp(imφ) which allows us to go to the coordinate patch containing θ = π to be well defined), we must have m, w ∈ Z (2.3) for every weight w ∈ t * of the representation of every field in the theory. The theory always contains fields in the adjoint representation, so m belongs to the coweight lattice: 1 m ∈ Λ cw (g)/W (G). Equivalently, m is specified by an irreducible representation of the Langlands dual L g of g. In general, m lies in a sublattice of Λ cw (g)/W (G) determined by the matter content.
• A Wilson-'t Hooft line operator is a worldline of a heavy particle that carries both magnetic and electric charges. In the path integral formalism, a Wilson-'t Hooft line is realized by an insertion of the Wilson line (2.1) and a singular boundary condition on the support L of the line a specified by the magnetic charge. The prescribed singularity (2.2) breaks the gauge symmetry to the stabilizer G m of m, so R is an irreducible representation of G m 2 . The data specifying such a pair (m, R) is actually the same as a pair (m, e) of coweight m and weight e modulo the Weyl group action: As emphasized in [31], this data has more information than a pair of irreducible representations of g and L g.
In what follows, we will label line operators by (m, e). In this notation, (0, e) and (m, 0) are purely a Wilson line and a 't Hooft line respectively.

Wilson-'t Hooft lines in
These Wilson-'t Hooft line operators were generalized to the ones preserving a half of supersymmetries in N = 2 gauge theories in [31]. We here consider these lines on the space-time geometry S 1 × R 2 × R. Here, S 1 × R 2 is the twisted product defined by the identification (2πβ, z) ∼ (0, e 2πi z), where z is the complex coordinate of R 2 C and β is the radius of S 1 . We always choose that the Wilson-'t Hooft line winds around S 1 and are located at the origin of R 3 .
To preserve a half of the supersymmetries, the gauge field in (2.1) must be paired up with the scalar field φ in the vector multiplet where R e is a representation specified by weight e and dτ is the line element along L. To define the 't Hooft loop L (m,0) , we do path-integral with the singular configuration of A (2.2) and the scalar field φ as well. By regarding S 1 as a time direction, the line operator defines a Hilbert space H L . Therefore the vev of a line operator is basically given by a trace over H L . The twisted space S 1 × R 2 leads to the insertion of e 2πi J , where J is the generator of the rotation of R 2 . To preserve a supersymmetry, this should be paired up with e 2πi I 3 where I 3 is the Cartan of SU(2) R . Then we have where m f are the mass parameters of the matter hypermultiplets associated with the Cartans of the flavor symmetry F f . These vevs depend holomorphically on the parameters [23] a ∈ t C , b ∈ t * C . (2.7) These have a semi-classical expansion in terms of the electric and magnetic holonomies, e iθe and e iθm , and the scalar field as [23,28] where ϑ is the theta angle of the gauge theory. The vev of a Wilson line L (0,e) is simply given by the classical value of the holonomy: For an 't Hooft line L (m,0) with magnetic charge m, the vev takes the form where m collectively denotes complex mass parameters. The summation over the coweights v in the shifted coroot lattice Λ cr (g) + m accounts for the so-called monopole bubbling, a phenomenon in which smooth monopoles are absorbed by the 't Hooft line and screen the magnetic charge. The norm v with respect to a Killing form is bounded by m , so this is a finite sum. The first two factors in the summand are the classical action and the one-loop determinant in the screened monopole background, respectively. The last factor is the nonperturbative contributions coming from degrees of freedom trapped on the 't Hooft line due to monopole bubbling. The one-loop contribution Z 1−loop has been computed in [23]. The one-loop contribution of the vector multiplet with gauge group G and the hypermultiplet transforming in the representation R with complex mass m are respectively given by where Φ(g) is the set of roots of g and P (R) is the set of weights of R. In the following, we will discuss gauge theories with multiple gauge groups and hypermultiplets transforming in the bi-fundamental representation where v, w = v 1 , w 1 + v 2 , w 2 and a, w = a 1 , w 1 + a 2 , w 2 . Here a 1 and a 2 are parameters in (2.8) for two gauge groups, G 1 and G 2 . The factor Z mono is subtle. The original computation in [23] did not give an answer that completely matches predictions from the AGT correspondence. The subtleties have been addressed in subsequent works [27,[32][33][34] but not resolved in full generality. For the Wilson-'t Hooft lines that we will discuss later, the screened magnetic charges are in the same W (G)-orbit as m. The corresponding contributions are therefore obtained by the W (G)-action from the perturbative term, for which v = m and Z mono = 1.

N = 2 circular quiver theory
We now consider the N = 2 gauge theory described by the quiver diagram (2.14) Each node represents a vector multiplet for an SU(N ) gauge group, 3 and each edge a hypermultiplet that transforms in the bi-fundamental representation under the gauge groups of the nodes it connects. Let the total number of nodes be n, and the mass parameters of the bi-fundamental hypermultiplets be m r , r = 1, . . . , n. We will refer to the gauge theory described by the above quiver diagram as circular quiver theory. Let us set up the notation of g = su N here. We denote by E ij the matrix that has 1 in the (i, j)th entry and 0 elsewhere, and by E * ij such that E ij , E * kl = δ ik δ jl . The roots of The positive roots are α ij , i < j, and the simple roots are The fundamental weights are dual of the coroots: The various lattices are given by For the circular quiver theory with G = SU(N ) n , the lattices are simply the direct product, e.g., the coweight lattice is Λ cw (g) = Λ cw (su N ) ⊕n . We now consider the 't Hooft line with the magnetic charge The classical and the one-loop contributions, for which i 1 = · · · = i n = 1, is given by The superscript r refers to the rth SU(N ) factor of G, with a n+1 = a 1 . By collecting the contributions from the other coweights, we get (2.20) We add the following electric charge This electric charge is in a sense a minimal one that is compatible with the choice of the magnetic charge This will be later compared with the transfer matrix of the L-operators.

Definition of L-operators
Let h be a finite-dimensional commutative complex Lie algebra and V a finite-dimensional diagonalizable h-module. Choosing a basis {v i } of V that is homogeneous with respect to weight decomposition, we denote the weight of v i by h i and the (i, j)th entry of a matrix M ∈ End(V ) by M i j . We write M h * for the field of meromorphic functions on the dual space h * of h.
Let R : C × h * → End(V ⊗ V ) be an End(V ⊗ V )-valued meromorphic function on C × h * that is invertible at a generic point (z, a) ∈ C × h * . The coordinate z is called the spectral parameter and a is called the dynamical parameter. Associated with R, let L : C → End(V ⊗ M h * ⊗ M h * ), which we think of as a matrix whose entries are linear operators on meromorphic functions on h * ×h * . We refer to this L as L-operator for R. This definition of L-operators depend on two independent dynamical parameters, thus is more general than the one given in [19]. This generalization appeared in [16] in the formalism of dynamical R-matrix. The L-operator satisfies the following two conditions: where L(z; a 1 , a 2 ) j i is a meromorphic function on C×h * ×h * and Here is a fixed complex parameter.
• The L-operator satisfies the RLL relation Equivalently, the operator relation holds on any meromorphic function f (a 1 , a 2 ).
We see that for R to satisfy the RLL relation with some L-operators, generally it must commute with h⊗1+1⊗h for all h ∈ h; in other words, R(z, a) kl ij = 0 unless h i +h j = h k +h l .

L-operator and quantum integrable system
Associated with an L-operator, there is an integrable quantum mechanical system consisting of particles moving in the space h * . 4 The Hilbert space of each particle is M h * , and that of the system is M ⊗n h * if n is the number of particles. To construct this system, we define the monodromy matrix M : Since T is an End(M ⊗n h * )-valued meromorphic function, each coefficient T m in the Laurent expansion T (z) = m∈Z T m z m is an operator acting on the Hilbert space M ⊗n h * . Then, one may pick a particular linear combination of these coefficients and declare that it is the Hamiltonian of the quantum mechanical system. The Hamiltonian thus obtained is a difference operator, which is typical of relativistic systems.
The integrability of the system is a consequence of the RLL relation. By repeated use of the RLL relation, one deduces that the monodromy matrix satisfies a similar relation: By multiplying both sides by R −1 (z − z , a 1 ) ij mn , setting a n+1 = a 1 and summing over i, j, m, n, one finds T (z)T (z ) = T (z )T (z). (3.7) In other words, transfer matrices at different values of the spectral parameter commute. It follows that the Laurent coefficients {T m } mutually commute and, in particular, commute with the Hamiltonian. Therefore the system has a series of commuting conserved charges.

Elliptic L-operator
An important example of an L-operator is one for the elliptic dynamical R-matrix [35][36][37], which is a representation of the elliptic quantum group for sl N . In this example, h is the Cartan subalgebra of sl N and V = C N is the vector representation of sl N . The Lie algebra sl N consists of the traceless complex N × N matrices. Let us define E ij ∈ gl N as the same matrix defined in section 2, and their dual E * ij the element of Since h is isomorphic to the quotient of the subspace of f N consisting of the diagonal matrices by the subspace spanned by the identity matrix Thus, h * may also be identified with the space of traceless diagonal matrices.
The natural action of sl N on C N defines the vector representation of sl N . In terms of the standard basis be Jacobi's first theta function. The elliptic dynamical R-matrix R ell is defined by [17][18][19] where a ij = a i − a j and The elliptic L-operator L ell , which satisfies the RLL relation with R ell , has the matrix elements given by [16] L ell w,y (z; a 1 , a 2 ) j i = . (3.11) The complex numbers w, y may be thought of as spectral parameters. The presence of the two parameters is due to the fact that R ell (z, a) is invariant under shift of a by a multiple of the identity matrix I and in the RLL relation (3.2) the spectral parameters z, z enter the R-matrix only through the difference z −z ; note also that the L-operator can be multiplied by any function of the spectral parameter. The elliptic dynamical R-matrix and the elliptic L-operator have many more properties than just that they satisfy the RLL relation. Most importantly, the R-matrix is a solution of the dynamical Yang-Baxter equation [17,18,38] and encodes the Boltzmann weights for a two-dimensional integrable lattice model [35][36][37]. This model is equivalent to the eightvertex model [39,40] and the Belavin model [41], an sl N generalization of the eight-vertex model, in the sense that the transfer matrices of the two models are related by a similarity transformation. The elliptic L-operator, on the other hand, satisfies the RLL relation with another R-matrix which describes an integrable lattice model called the Bazhanov-Sergeev model [11,12], whose spins variables take values in h * .

Trigonometric L-operators
The L-operators that appear in the correspondence with Wilson-'t Hooft lines are obtained from the elliptic L-operator L ell via the trigonometric limit τ → i∞. For comparison with gauge theory results, we actually need to express these L-operators in somewhat different forms.
We firstly describe L-operators in a quantum mechanical language. Consider quantum mechanics of a particle living in h * × h * , with Planck constant If (a 1 , a 2 ) ∈ h * × h * is the position of the particle, we write a r = N −1 i=1 q r i ω i , r = 1, 2. Similarly, we write the momenta (b 1 , b 2 ) ∈ h × h of the particle as b r = N −1 i=1 p r i α ∨ i . The corresponding position and momentum operatorsq r i ,p s i satisfy the canonical commutation relations: [q r i ,p s j ] = i δ rs δ ij , i, j = 1, . . . , N − 1.
(3.13) (As before, we are treating q r i , p r i as analytically continued variables.) To rewrite the commutation relations in a form that is invariant under the action of the Weyl group, we make a change of basis  By using these observables we can identify the matrix elements of an L-operator L with an operator in the Hilbert space of this quantum mechanical system: In quantum mechanics, there is an invertible map from functions on the classical phase space to operators in the Hilbert space, known as the Weyl transform: if q and p are canonically conjugate variables, it maps dx dy dp dq f (q, p)e i(x(q−q)+y(p−p)) . (3.17) The inverse map is the Wigner transform, which we denote by − : In the situation at hand (3.16), we have Next, we apply a similarity transformation to the elliptic L-operator. Assume Im > 0 and let 1 − e 2πi(mτ +n +z) (3.20) be the elliptic gamma function. Then, Γ(z) = e πiz 2 /2 Γ(z, τ, ) has the property that Γ(z + , τ, ) = g(τ, )θ 1 (z)Γ(z, τ, ) for some function g(τ, ). We define the conjugated L-operator L ell w,m (z) by It has the Wigner transform With these preparations, let us finally take the trigonometric limit to define the trigonometric L-operator: L w,m = lim τ →i∞ L ell w,m . (3.24) The trigonometric L-operator satisfies the RLL relation with the trigonometric limit R trig of the elliptic R-matrix R ell . Concretely, L w,m and R trig are obtained from L ell w,m and R ell by the replacement θ 1 (z) → sin(πz).
Once we are in the trigonometric setup, the quasi-periodicity in z → z + τ is lost and we can further take the limits w → ±i∞. This allows us to introduce more fundamental L-operators: