Non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation and its implication to quark confinement
We give a gauge-independent definition of magnetic monopoles in the Yang-Mills theory through the Wilson loop operator. For this purpose, we give an explicit proof of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation of the gauge group to derive a new form for the non-Abelian Stokes theorem. The new form is used to extract the magnetic-monopole contribution to the Wilson loop operator in a gauge-invariant way, which enables us to discuss confinement of quarks in any representation from the viewpoint of the dual superconductor vacuum.
The Wilson loop operator Wilson74 is the physical quantity of fundamental importance in gauge theories due to its gauge invariance. Indeed, quark confinement is judged by the area law of the vacuum expectation value of the Wilson loop operator, which is the so-called Wilson criterion for quark confinement. Recently, it has been shown KKSS15 that the non-Abelian Stokes theorem (NAST) for the Wilson loop operator is quite useful to understand quark confinement based on the dual superconductor picture dualsuper . Here the NAST for the Wilson loop operator refers to the alternative expression in which the line integral defining the original Wilson loop operator is replaced by the surface integral. In particular, we want to obtain the NAST which eliminates the path ordering. Such a version of the NAST was indeed derived for the first time by Diakonov and Petrov for the Wilson loop operator based on a specific method in DP89 (See also DP96 ). Later, it was recognized that the Diakonov-Petrov version of the NAST can be derived as a path-integral representation using the coherent state of the Lie group in a unified way KondoIV ; KT00b ; KT00 ; Kondo08 . The NAST is rederived based on the coherent state in KondoIV . In a similar way, the NAST has been extended into the gauge group in KT00b and in KT00 ; Kondo08 to discuss the quarks in the fundamental representation Kondo99Lattice99 ; Kondo00 ; Kondo08b . See KKSS15 for a review. There exist other versions of the NAST, see HU99 ; Halpern79 ; Bralic80 ; Arefeva80 ; Simonov89 ; Lunev97 ; HM97 .
Let be the Lie algebra valued connection one-form for the gauge group :
where is the generator of the Lie algebra of the group and is the dimension of the group , i.e., for . In what follows, the summation over the repeated indices should be understood unless otherwise stated. For a given loop, i.e., a closed path , the Wilson loop operator in the representation is defined by
where denotes the path ordering and the normalization factor is equal to the dimension of the representation , to which the probe of the Wilson loop belongs, ensuring . We introduce the Yang-Mills coupling constant for later convenience, although this can be absorbed by scaling the field .
For the gauge group , for instance, any representation is characterized by a single index . In fact, the Wilson loop operator in the representation of is rewritten into the surface-integral form DP89 ; KondoIV :
where () are the Pauli matrices with being the diagonal matrix, is an group element and is the product of an invariant measure on over :
The purpose of this paper is to extend the Diakonov-Petrov version of the NAST for the Wilson loop operator to an arbitrary representation of the group () to derive a new form for the NAST, which enables one to define a gauge-invariant magnetic monopole in the Yang-Mills theory and to extract the magnetic-monopole contribution to the Wilson loop operator in a gauge-invariant way. The new form of the NAST has been obtained already for the fundamental representation of () in Kondo08 . The new form is useful to discuss quark confinement in an arbitrary representation from the viewpoint of the dual superconductor picture. The relevance of the Wilson loop to quark confinement can be observed by calculating the magnetic monopole current , whose definition is proposed in this paper. In fact, one of the authors and his collaborators have used the new form of the NAST to calculate the average of the Wilson loop operator for and in the fundamental representation using the numerical simulations on a lattice. Through the simulations, they have examined the dual superconductivity picture for quark confinement. See chapter 9 of KKSS15 . The new form of the NAST will be used to extend the preceding works to any representation of in subsequent works.
Last but not least we must mention the facts that an original form of the NAST for arbitrary representation of the group () was already announced in the second paper of Ref.DP96 and that the same form for the NAST has been derived in an independent way specifically for the fundamental representation of the gauge group in Lunev97 . However, the formula given there is not appropriate for our purpose stated above. Although the formula given originally in the second paper of DP96 is correct, indeed, nontrivial (mathematical) works are required to derive the new form from it. Moreover, to the best of our knowledge, there are no available proofs of the NAST for any representation in the published literature. Therefore, we give an explicit proof of the NAST as a preliminary step toward our purpose.
Ii non-Abelian Stokes theorem for the Wilson loop operator
Let be the gauge transformation of the Yang-Mills gauge field by the group element :
Using a reference state , we define the one-form from the Lie algebra valued one-form by
where is the product of the invariant integration measure at each point on the loop :
Now the argument of the exponential is an Abelian quantity, since is no longer a matrix, just a number. Therefore, we can apply the (usual) Stokes theorem,
to replace the line integral along the closed loop to the surface integral over the surface bounded by . See Fig. 1. Thus we obtain a NAST:
where the is the curvature two-form defined by
and the integration measure on the loop is replaced by the integration measure on the surface ,
by inserting additional integral measures, for .
The field strength is calculated as
where we have introduced
We define the Lie algebra valued field which we call the precolor (direction) field by
For a Lie algebra valued operator , we obtain the relation:
where we adopted the normalization for the generator:
Therefore, the field strength is written as
Notice that the final term is not gauge invariant and disappears finally after the integration with respect to the gauge-invariant measure . Therefore, it is omitted in what follows.
Iii Color direction field
As a reference state , we can choose the highest-weight state defined by the (normalized) common eigenvector of the generators in the Cartan subalgebra with the eigenvalues :
where is the rank of , i.e., . Then we have
by taking into account the normalization .
Let be a subsystem of positive (negative) roots.111 The root vector is defined to be the weight vector of the adjoint representation. A weight is called positive if its last nonzero component is positive. With this definition, the weights satisfy . Then the highest-weight state satisfies the following properties:
is annihilated by all the (off-diagonal) shift-up operators with :
is annihilated by some shift-down operators with , not by other with :
The adjoint rotation of a generator can be written as a linear combination of the generators :
since is written by using the commutator repeatedly:
and the commutator is closed with the structure constant . Hence, the precolor field (15) is written as
Multiplying from the left and from the right, on the other hand, (23) yields
which is cast after multiplying into the form:
where we have used the fact that the matrix is a real-valued and unitary , in other words, is an orthogonal matrix satisfying for the transposed matrix of , because the structure constant is real-valued.
where we have used in the second equality the fact that the generators other than the Cartan generators , i.e., the shift-up and shift-down generators in the Cartan basis have the property:
since or is the eigenvector with the eigenvalue and obeys
because the eigenvectors with the different eigenvalues are orthogonal for . We have used (20) in the last equality.
We introduce Lie algebra valued fields defined by
Then we arrived at the important relation:
Notice that (23) is determined by the commutation relation alone and, hence, does not depend on the representation adopted. Therefore, does not depend on the representation
and we can use the fundamental representation to calculate and to calculate the precolor field .
We define by
In what follows, the summation over should be understood. Then it satisfies the relation:
On the other hand, we find
This relation follows from
where we have used in the second equality and following from in the third equality. Therefore, we obtain another relation for the precolor field :
The relation (41) is used to write the third term in as
Thus, we have arrived at the final form of the NAST for in arbitrary representation:
We can introduce also the normalized 222 This color field is normalized in the fundamental representation. In general, , which is equal to in the fundamental representation. and traceless field which we call the color (direction) field Kondo08 :
to rewrite the NAST into
In what follows, we work out the case for concreteness. For , we choose the highest-weight state as the reference state. Then the highest-weight vector of the representation with the Dynkin indices is given by
The fields and are independent of the representation and, hence, can be calculated in the fundamental representation:
with the components:
where and are the diagonal matrices of the Gell-Mann matrices () for the Lie algebra . The parametrization of a group element and the explicit form of the integration measure can be found in KT00 .
For the fundamental representation , the color field takes the value in the Lie algebra of (See Fig. 2):
This is also the case for the fundamental representation :
The fundamental representations have the same structure characterized by the degenerate matrix: the two of the three diagonal elements are equal, despite their different appearance.
For the adjoint representation , on the other hand, the color field takes the value in the Lie algebra of (see Fig. 3):
Here the matrix between and is not degenerate: the three diagonal elements take different values.
For the general representation with the Dynkin index , the color field reads
where is called the maximal stability subgroup.
Thus,we can show that every representation of specified by the Dynkin index belongs to (I) or (II):
Minimal case: If ( or ), the maximal stability group is given by
with generators . In the minimal case, is minimal. Such a degenerate case occurs when the highest-weight vector is orthogonal to some root vectors. In the minimal case, the coset is given by the complex projective space:
For example, the fundamental representation has the maximal stability subgroup with the generators , where
See Fig. 4.
Maximal case: If ( and ), is the maximal torus group:
with generators . In the maximal case, is maximal. This is a non-degenerate case. In the maximal case, the coset is given by the flag space:
For example, the adjoint representation has the maximal stability subgroup with the generators