S-duality and loop operators in canonical formalism

We study the gauge invariant t' Hooft operator in canonical formalism for Yang-Mills theory as well as the $\mathcal{N} =4 $ Super Yang-Mills theory. It is shown that the spectrum of the t' Hooft operator labeled by the dual representation of the gauge group is the same as the spectrum of the Wilson operator labeled by the same representation. So it is possible to construct a unitary operator $ S $ making the two kinds of loop operators transformed into each other. S-duality transformation could be realized by the operator $ S $. We compute the supersymmetry variations of the loop operators with the fermionic couplings turned off. The result is consistent with the expectation that the action of $ S $ should make the supercharges transform with a $ U(1)_{Y} $ phase.


INTRODUCTION
In U (1) gauge theory, S-duality [1][2][3] has a simple realization in canonical formalism. The canonical coordinates are A i with the conjugate momentum Π i , i = 1, 2, 3. In temporal gauge, gives the S-transformation of the theory. For the gauge potential eigenstate |A , is the eigenstate of Π i . In U (1) gauge theory, Wilson and t' Hooft operators for the spacial loop C are given by and T R * (τ ; λ I , λ a , C) with λ I and λ a the parameters characterizing the scalar and the fermionic couplings. We show that it is possible to construct S with S −1 T R * (τ ; λ I , λ a , C)S = W R * (− 1 τ ; λ I , e − iθ 2 λ a , C) , (1.8) where e iθ 2 = (τ /τ ) 1 4 . This is consistent with the expectation that the S-transformation will make the t' Hooft operator labeled by R * in theory with the coupling constant τ and the gauge group G mapped into the Wilson operator labeled by R * in theory with the coupling constant −1/τ and the gauge group G * [5].
With the S-duality transformation operator S given, we may study whether the theory is duality invariant. For N = 4 SYM theory, if the supercharges transform as the theory will be duality invariant [6]. We calculate the supersymmetry variations of the loop operators with λ a = 0 and show that the constructed S is consistent with (1.10). To prove (1.10), the situation with λ a = 0 should also be considered, which is left for future study.
t' Hooft operator is usually defined in path integral formalism [7]. We investigate the relation between these two kinds of definitions and show that it is possible to extract T R (τ ; λ I , λ a , C) in canonical formulation from the path integral.
The rest of the paper is organized as follows: section 2 is a review for the electric and the magnetic weight lattices of the gauge group; in section 3, we give a definition for the gauge invariant t' Hooft operator in canonical formalism and compute the generic commutation relations for t' Hooft and Wilson operators in the arbitrary representations; in section 4, we consider the T-transformation rule for loop operators; in section 5, we study the S-transformation of the loop operators; the discussion is in section 6.

THE ELECTRIC AND MAGNETIC WEIGHT LATTICES
For a semi-simple and simply connected group G with the rank r, simple roots and simple coroots are given by the r-dimensional vectors { α A | A = 1, 2, · · · , r} and { α * A = α A /| α A | 2 | A = 1, 2, · · · , r}. The fundamental roots are { λ A | A = 1, 2, · · · , r} satisfying The electric weight lattice Υ(G) and the magnetic weight lattice Υ * (G) are generated by the fundamental roots { λ A } and the simple coroots { α * A }: The magnetic weight lattice Υ * (G) can be identified with the electric weight lattice of the GNO dual group G * [4]. Υ * (G) = Υ(G * ), Υ * (G * ) = Υ(G). The center and the fundamental group of G * are isomorphic to the fundamental group and the center of G.
When G = U (N ) that is not semi-simple but could be locally decomposed as SU (N ) × U (1), with A = 1, 2, · · · , N . The fundamental weights and the simple coroots are . (2.14) The electric and the magnetic weight lattices are identical, Υ(U (N )) = Υ * (U (N )) and U (N ) = When R is the fundamental representation and m = (1, 0, · · · , 0), The physical Hilbert space H ph is composed by states invariant under the action of U . ∀ |ψ ∈ H ph , ∀ U ∈ G, U |ψ = |ψ . ∀ |A i , the corresponding |A i ph ∈ H ph is obtained as For the arbitrary spatial loop C, the Wilson loop in representation R labeled by m is given by where d R is the dimension of the representation and A i R is the gauge potential in representation R. For the fundamental representation, we will use A i and W with the omitted subscript.
Instead of (3.3), the Wilson loop in representation R also has an equivalent definition as a path integral over all gauge transformations periodic along the loop C [8,9]: In (3.4), all fields are in the fundamental representation with the information on R encoded in H m . The action of the Wilson operator W R (C) is give by Based on (3.4), another operator W R (C) can be introduced with W R (C) can be written in terms of W R (C): The integration is taken over all of the local gauge transformation operators U ∈ G. Obviously, In [10], t' Hooft operator T (C) is introduced satisfying the commutation relation with the Wilson operator W(C) in fundamental representation. When G = SU (N ), the center element Z = exp{2πi/N }. l(C 1 , C 2 ) is the linking number of the two spatial loops C 1 and C 2 .
In [11], an operator T R (C) satisfying (3.9) was explicitly constructed. The action of T R (C) on |A i is given by where ω(Σ C , x) = 4π . Locally, T R (C)A i and A i are related by a gauge transformation. So the path-ordered integrations and 14) starting and ending at the point x ∈ Σ C , moving along a loop C intersecting Σ C once at x will differ by Z, i.e. I(T R (C)A i ; C ) = ZI(A i ; C ). As a result, W(T R (C)A i ; C ) = ZW(A i ; C ).
T R (C) is an operator satisfying When G = SU (N ) and R is the fundamental representation, T R is abbreviated as T , Z = exp{2πi/N }, is a Σ C -dependent function with the value jumping Z at Σ C . Depending on the group G, Z K = 1 for some integer K, so Z k Ω m (Σ C , x) with k = 0, 1, · · · , K − 1 glued together form a K-folded C-dependent continues functionΩ m (C) taking values {Z k Ω m , k = 0, 1, · · · , K − 1} at each point. iΩ −1 m (C)∂ iΩ m (C) = H m a i (C) holds in every branch.
When e −4πiH m = I, which is the situation for G = U (N ) and R the arbitrary representation everywhere. Even though, due to the singularity at C, H m a i (C) is not a pure gauge and T R (C) is not a gauge transformation. For example, . The magnetic field transforms as is a unit magnetic field loop located at C, i.e.
. The Wilson loop of A i and A i − 1 2 a i (C) are the same, but A i and A i − 1 2 a i (C) are not gauge equivalent. T (C) defined in this way is identical to (1.3). T R (C) is not a physical operator. For |ψ ∈ H ph , T R (C)|ψ may not be a state in H ph . Consider the action of T R (C) on a gauge transformation operator U giving by (3.1), is still a single-valued continuous function taking values in G and approaching I at infinity, which requires [u, H m ] = 0 at C, sinceΩ m (C) is singular at C.
In this case, ∀ |ψ ∈ H ph Otherwise, U T R (C)|ψ = T R (C)|ψ . T R (C)|ψ is a state that would only be affected by the gauge transformation at C. In this respect, T R (C) is quite similar with W R (C).
The gauge invariant t' Hooft operator could be constructed as From (3.15), T R (C) defined in (3.17) is labeled by H m related with the highest weight λ ∈ Υ(G).
the linking number l(C 1 , C 2 ), We have Instead of (3.8), the Wilson operator W R (C 2 ) can also be written as where U (C 2 ) are gauge transformations equal to I away from a torus surrounding C 2 . (3.8) and (3.25) are equivalent if the proper normalization are assumed for the integration over U (C 2 ).
where we have used the fact that is still a gauge transformation equal to I away from the torus surrounding C 2 . From (3.26), This is the generic commutation relation for loop operators in the arbitrary representation.
When H m and H m are in the electric and the magnetic weight lattices Υ(G) and Υ * (G), When G = SU (N ), according to (2.8), Especially, when R and R are fundamental representation characterized by H, When G = U (N ), for the arbitrary R and R , from (2.15), exp{iL(R, R ; C 1 , C 2 )} = 1,

T-TRANSFORMATION OF LOOP OPERATORS
S-duality transformation is generated by the T-transformation and the S-transformation.
In canonical quantization formalism, T-transformation could be realized by a unitary operator the Chern-Simons term.
Obviously, Wilson operator is invariant under the action of g, For the T-transformation of the t' Hooft operator, we can compute the action of T R (C) on X(A): 5) or equivalently, Under the T-transformation, T R (C) is multiplied by W R (C) in the same representation. On the other hand, T-transformation for the gauge invariant t' Hooft operator T R (C) is given by where [T W] R (C) could be taken as the Wilson-t Hooft operator originally proposed in path integral formulation [5]. Here, W and T in [T W] R are both labeled by the representation R.

S-TRANSFORMATION OF LOOP OPERATORS
S-transformation is expected to make the t' Hooft operator T R * (C) in theory with the gauge group G and the coupling −1/τ mapped into the Wilson operator W R * (C) in theory with the gauge group G * and the coupling τ [5]. For it to be possible, two kinds of operators should be equivalent in physical Hilbert space. They should have the same spectrum and degeneracy and could be related by a unitary transformation. In this section, we will study the spectrum and eigenstates of the t' Hooft operator in YM theory as well as the N = 4 SYM theory. We will show that it is possible to construct a unitary operator S relating the Wilson operator W R * (τ ; C) and the t' Hooft operator T R * (−1/τ ; C).
So S-transformation could be realized at the kinematical level. At the dynamical level, if S could also make the Hamiltonian with the coupling −1/τ transformed into the Hamiltonian with the coupling τ , the theory is S-duality invariant. For N = 4 SYM theory, the condition for the S-duality invariance is that the supercharges should transform with a U (1) Y phase. We will calculate the supersymmetry variations of the loop operators and provide the evidence for the U (1) Y transformation of the supercharges under the action of S.

Spectrum and eigenstates of t' Hooft operator in YM theory
In YM theory with the gauge group G, the complete orthogonal bases of the Hilbert space H can be selected as {|A | ∀ A}, eigenstates of the gauge potential. The action of the Wilson operator in representation R on |A is given by For the arbitrary spacial loop C and the arbitrary local gauge transformation operator U ∈ G, the action of T (±C) and U could make {|A | ∀ A} divided into the equivalent classes, where for some U i and C i , |A and |A belong to the same equivalent class. In other words, for the group L defined as Eigenstates of T R (C) can be constructed in the sub Hilbert space H[E(Â)] generated by where the firstÂ in the subscript (Â,Â) means it is a state in H[E(Â)] and the second indicates it is to be constructed as an eigenstate with the eigenvalue W R (Â; C).
with X(A) the Chern-Simons term.
where we use 1 in L(1, R; ±C k , C) to stand for T (±C k ) in fundamental representation. The action of T R (C) on |D (Â,Â) is given by When G = SU (N ), for R characterized by m, from (3.29), In this case, the highest weight of R is also in the magnetic lattice Υ * (SU (N )) = S-transformation is supposed to make T R * (C) in theory with the group G mapped into W R * (C) in theory with the group G * [5]. The necessity to replace With the reference state |Â chosen, |A ∈ E(Â) can be written as |A = L|Â , L ∈ L. If L|Â = |Â only when L = I, there is a one-to-one correspondence between |A and L, so E(Â) can be identified with the group manifold L. |D (Â,Â) can be rewritten as If L|Â = |Â also for some L = I, such L can only be a gauge transformation, under which, g is a wave function on L, so with the reference state modified to the arbitraryÂ , one may get is an eigenstate of T R * (C) with the eigenvalue W R * (Â; C) for the arbitrary gauge potentialÂ that does not necessarily belong to E(Â ). When the gauge group is U (1), T (C) = T (C), W(C) = W (C), where we have used Finally, In theory with the group G, let for some e ih(Â ,A) , will make The spectrum of T R * (C) and W R * (C) are highly degenerate, so (5.34) can only make S is the momentum eigenstate, and the operator U making is the duality operator in quantum mechanics. Since

Modified t' Hooft operator
The standard t' Hooft operator T R (C) satisfies the commutation relation does not need to be gauge invariant.
If there is a gauge invariant operator K(A) with  This is the T-transformation of the t' Hooft operator.

t' Hooft operator in path integral formalism and canonical formalism
In path integral formulation, t' Hooft operator is introduced by expanding the quantum fields around the singular configurations. For the globally defined gauge potential A µ , the Bianchi identity is automatically satisfied: also has the solution denoted as G µ , which is not globally defined. (5.55) can be reduced to To get the spatial t' Hooft operator T R * (C) located at the time t = t 0 , j µ is taken to be The corresponding G µ can be selected as and a i are given by (3.18), (3.12) and (3.11). ω(Σ C , x) jumps 4π when x crosses Σ C , but e iω is continues except for the singularity at C. To solve for (5.56), note that a i is not a pure gauge, but a i = −ie −iω ∂ i e iω , so locally, we have a i = ∂ i ω. (5.56) becomes with the non-globally defined ω.
In (5.56), G µ is determined up to the addition of an arbitrary globally defined gauge potential A µ . The generic solution of (5.55) is A µ = G µ +Ã µ .
Consider the YM theory with the gauge group G and the coupling constant τ , τ = τ 1 + iτ 2 .
According to the relation between the YM theory and D3 branes in type IIB string theory, τ 2 = 1/g s , where g s is the type IIB string coupling constant. The gauge coupling is g = √ 2πg s .
1/g 2 = τ 2 2π . The action of the YM theory is In temporal gauge,Π i = − τ 2 2πF i0 + τ 1 2πB i , so replacingS by S amounts to adding the operator is the standard t' Hooft operator in the dual representation R * [11], is an operator constructed fromÃ. T R * (C) and T R * (C) are equivalent if the suitable K(A) can be obtained as is in (5.46).
For A µ = G µ +Ã µ , under the gauge transformation U , To preserve the gauge invariance, the integration should cover the background U G µ U −1 , or more concretely, U (H * m ω)U −1 for the arbitrary U . The obtained gauge invariant operator is Under the T-transformation,Π i →Π i +B i 2π . According to (5.65), and then This is the manifestation of the T-transformation rule in path integral formalism.
In canonical formulation, t' Hooft operator, in parallel with the Wilson operator, is determined by the field content with no dynamical information involved. But in path integral formalism, t' Hooft operator is action-dependent.
where Y (C) depends on action. We may consider YM theories with the arbitrary higher order interactions, and for each theory, there is a corresponding Y (C). It is expected that T R * (C) and T R * (C) are equivalent.
The Lagrangian can be expanded as Canonical quantization ofL in temporal gauge gives So adding the background fields amounts to adding the operator into the path integral.
is the standard t' Hooft operator generating the singular gauge transformation in N = 4 SYM theory.
is an operator constructed fromÃ i ,Φ I ,Ψ a , whose form depends on action as well as the explicit solutions for φ I and ψ a .
Under the local gauge transformation, representation R * can be defined as where W R * (τ ; λ I , λ a , C) can also be written as where A R * , Φ R * , Ψ R * are fields in representation R * [13,14]. Suppose |Λ := |A i , Φ I , Ψ a is the common eigenstate of (A i , Φ I , Ψ a ), The corresponding supersymmetric t' Hooft operator is T R (C) can be defined via its action on |A i , Φ I , Ψ a : Loop operators satisfy the commutation relation and then Especially, is the equivalent class generated by the action of T and U . T stands for T R with R the fundamental representation of G.

Modified supersymmetric t' Hooft operator
There is no exact distinction between the spectrum of and the spectrum of Suppose y I (x) and z a (x) are functions specifying a three dimensional hypersurface in superspace and W R (τ ; λ I , λ a , C) could be taken as the Wilson loop of the gauge potential The corresponding supersymmetric t' Hooft operator T R (τ ; λ I , λ a , C) and T R (C) can also be related by a unitary transformation. For operator where we have used  In path integral formulation, the obtained t' Hooft operator is with Y (τ ; λ I , λ a , C) a unitary operator constructed from Λ. It is expected that T R * (τ ; λ I , λ a , C) and T R * (τ ; λ I , λ a , C) are equivalent.
In addition to S 1 , another unitary physical operator S 2 inducing a rescaling can also be introduced: S-transformation operator is taken to be S = S 1 S 2 with which is the expected transformation rule for supersymmetric loop operators.
When θ β aẋ αβ + ( τ 2 2π )  The supersymmetry variation of W Ψ R * (τ ; λ a , C) contains the conjugate momentum Π αβ and Π ab . For completeness, we should consider the Wilson loops constructed from both A i , Φ I , Ψ a and Π i , Π I , Π a . For such operators, |Λ is not the eigenstate, so it is not straightforward to determine their transformation under S 1 which is defined in (5.130) through the action on |Λ ph .

CONCLUSION AND DISCUSSION
In this paper, we studied the gauge invariant t' Hooft operator in canonical formalism.