On Goldman bracket for G 2 gauge group

: In this paper, we obtain an in(cid:12)nite dimensional Lie algebra of exotic gauge invariant observables that is closed under Goldman-type bracket associated with monodromy matrices of (cid:13)at connections on a compact Riemann surface for G 2 gauge group. As a by-product, we give an alternative derivation of known Goldman bracket for classical gauge groups GL( n; R ), SL( n; R ), U( n ), SU( n ), Sp(2 n; R ) and SO( n ).


JHEP02(2016)001
1 Introduction Traces of monodromy matrices of flat connections computed along closed loops on a Riemann surface Σ are known to satisfy the Goldman's Poisson bracket [9] for the following list of gauge groups: GL(n, R), GL(n, C), GL(n, H), SL(n, R), SL(n, C), SL(n, H), O(p, q), O(n, C), Sp(2n, H), U(p, q), Sp(2n, R), Sp(p, q) Sp(2n, C) and SU(p, q). The problem of computation of the bracket between traces of monodromy matrices for exceptional gauge groups so far remains open. The goal of this paper is to fill this gap for G 2 , the simplest exceptional Lie group. In the process, we learn that traces of G 2 monodromy matrices are not sufficient to close an algebra under such Poisson bracket and hence demands for the introduction of an infinite set of exotic G 2 -gauge invariant observables that together with the canonical observables of traces of G 2 -monodromy matrices form a closed infinite dimensional Lie algebra under such Goldman-type bracket. Given the fundamental group π of a closed oriented surface S and a Lie group G, the space Hom(π, G)/G is defined as the quotient of the analytic variety Hom(π, G) by the action of G by conjugation. In his study (see, for example, [22]) of Weil-Petersson symplectic geometry of Teichmüller space (with G = PSL(2, R)), Scott A. Wolpert discovered an expression of the Poisson bracket between geodesic length functions (Hamiltonian potential of the flow generated by Fenchel-Nielsen vector fields) in terms of the geometry of the underlying surface. William Goldman, in [9], went steps further to investigate the geometry of the symplectic structure of Hom(π, G)/G for Lie group G, satisfying fairly general conditions with the help of family of invariant functions on Hom(π, G)/G. He obtained the Poisson bracket between such invariant functions due to transversally intersecting homotopically inequivalent loops on S. An instructive example of invariant functions in the JHEP02(2016)001 sense of Goldman [9] comprises of the traces of monodromy matrices of flat connections on the principal G-bundle over S, also known as Wilson loop in QCD jargon. M. Chas and D. Sullivan, in the late 90's, generalized Goldman's Poisson bracket by looking at H * (LM ), the singular homology of the loop space in a compact oriented manifold M of dimension d (see [7] for detail). This generalized Poisson bracket, known as the string bracket, reproduces Goldman's bracket when d = 2 and * = 0. All that we have discussed so far is intimately tied with the Q-vector space generated by free homotopy classes of loops on S. If this infinite dimensional vector space is quotiented out by null-homotopic loops, then the resulting vector space V , equipped with a well-defined bracket [ , ] : V ⊗ V → V induced from the underlying string bracket and a well-defined cobracket δ : V → V ⊗ V , has a Lie bialgebra structure [18]. The cobracket of the Lie bialgebra (V, [ , ], δ) is known as Turaev cobracket in modern literature. Also, of considerable importance in theoretical Physics is the notion of Batalin-Vilkovisky (BV) operator that arises in the context of string topology as ∆ : H * (LM ) → H * +1 (LM ). Roughly speaking, it produces the (d + 1)-dimensional family of loops by rotating the d-dimensional ones. The string operations make H * (LM ) a BV-algebra. Construction of BV-algebra on moduli space of Riemann surfaces and its application in closed string field theory was elucidated by Sen and Zwiebach in [17]. An excellent treatment of the passage from string topology to topological field theory by Cattaneo et al. can be found in [6] where the authors used BV (Lagrangian) or BRST (Hamiltonian) formalism depending on whether the dimension of the base manifold is odd or even, respectively, the relation between the two formalisms being explained in its appendix. Turning ourselves to the physical side, an interesting variant of the usual supersymmetric nonlinear sigma model exhibiting BRST-like fermionic symmetry was studied by Witten in [21].
Consider a G-valued (G is the Lie algebra of G) flat connection A = A z (z,z)dz + Az(z,z)dz on a compact Riemann surface Σ of genus g. The Atiyah-Bott bracket on the space of flat connections can be derived from the Chern-Simons action on the 3-dimensional manifold Σ × R. Let us represent the connection 1-forms as A i = The space of flat connections modulo gauge transformation is finite dimensional and traces of the monodromy matrices of flat connections can be chosen to be the underlying gauge invariant observables. Goldman in [9] derived the Poisson bracket between traces of the monodromy matrices for classical groups already listed at the start of the introduction. For example, for any two transversally intersecting oriented closed curves γ 1 and γ 2 on Σ, the Poisson bracket between tarces of GL(n, R) monodromy matrices reads (1.3)
The extensions of Goldman's results to exceptional Lie groups were not known before. Of all five exceptional Lie groups, G 2 is simultaneously the smallest and one of the most important ones. Recently, it played pivotal roles in exceptional geometry (see [4]) and in Lattice QCD (see [11,16,19] and [10], for example). Manifolds admitting G 2 holonomy are also of special interest in M-theory (see [3] and [1] for a brief review). Quantization of Goldman bracket for loops on surfaces and its relation to (2+1)-quantum gravity are also investigated in ( [14,15]). In ( [14]), it has been shown how signed area phases appear in the quantized version of the classical brackets due to Goldman.
The goal of this paper is to generalize Goldman's bracket to the case of G 2 gauge group. Consider the 7 × 7 monodromy matrices M γ 1 and M γ 2 to be in the fundamental representation of G 2 . The existence of a new type of gauge invariant observables emerges by noting that Here O 1 , . . . , O 7 are skew symmetric 7 × 7 matrices representing the right action of purely imaginary octonions lying in 6-dimensional sphere S 6 (see [13] for a detailed discussion on left and right octonionic operators).
φ i e i be a purely imaginary octonion and the matrices {O i } in (1.5) represent the octonionic imaginary units {e i } in the sense of [13]. Then the action of O i on φ is defined as We proceed further to show that there is an infinite set of such exotic gauge invariant observables for the case of G 2 gauge group, the Poisson bracket between two such observables being again a linear combination of exotic G 2 gauge invariant observables. The Poisson bracket between the trace of a G 2 monodromy matrix and an exotic observable of this type can also be found to be a linear combination of exotic G 2 -gauge invariant observables, hence proving the closedness of the algebra of G 2 -gauge invariant observables (canonical+exotic) under such Goldman-type Poisson bracket. The organization of the paper is as follows. In section 2, we recall how the Atiyah-Bott bracket originates from the Hamiltonian Chern-Simons theory and derive an auxiliary JHEP02(2016)001 expression for Poisson bracket of traces of monodromy matrices along intersecting loops. In section 3, we show how this general expression can be used to derive this bracket for a few cases from Goldman's list. In section 4, we derive the Poisson bracket between traces of G 2 monodromy matrices using the formalism developed in section 2 and show how it leads to an infinite set of exotic G 2 -gauge invariant observables.

Poisson brackets for traces of monodromy matrices from the Atiyah-Bott bracket
This section is devoted to the review of basic facts of Hamiltonian formulation of Chern-Simons theory which will enable us to understand the symplectic structure of the infinite dimensional phase space of flat gauge connections and eventually lead us to construct the moduli space of flat connections, the dimension of which is given by (2g − 2) dimG where g is the genus of the underlying Riemann surface Σ and G is the gauge group as described in the Introduction 1. An elegant treatment of the Hamiltonian formulation of Chern-Simons theory and the related aspects of Goldman's Poisson bracket can be found in [2]. In this paper, space-time is modelled as a 3-manifold Σ × R where Σ representing "space" is a compact Riemann surface and R represents "time". For an arbitrary real gauge group G, the Chern-Simons action functional on Σ × R reads The connection 1-forms on the principal G-bundle, taking their values in the Lie algebra G of the gauge group G, are given by Denote the generators of the group G are given by {t a } and write where the space-time label i = z,z, 0. Here, {t a } are chosen such that the following holds

JHEP02(2016)001
The space of flat connections modulo gauge transformation turns out to be a finite dimensional space; traces of monodromy matrices of flat connections computed along intersecting loops on Σ can be used as gauge invariant observables. In this section, we compute the Poisson bracket between traces of the monodromy matrices along two homotopically inequivalent loops that intersect transversally at a single point using the formalism originating from the Hamiltonian theory of Solitons. The generalization to many intersection points is straight forward.
Denote two loops on Σ by γ 1 and γ 2 that intersect transversally at a single point on Σ. Without loss of generality, let us assume that the paths γ 1 and γ 2 intersect orthogonally at O. These two loops are illustrated schematically by x 1 x 2 x 1 and y 1 y 2 y 1 in figure 1. The parts x 1 Ox 2 and y 1 Oy 2 are taken to lie along X and Y axes, respectively. The relevant transition matrices are denoted by T (x 1 , x 2 ) and T (y 1 , y 2 ). Let us denote the monodromy matrices computed along γ 1 and γ 2 by M γ 1 and M γ 2 , respectively. They are given by where M γ 1 and M γ 2 are the remaining contributions of monodromy matrices M γ 1 and M γ 2 due to the paths x 2 x 1 and y 2 y 1 , respectively (see figure 1). The matrices M γ 1 and M γ 2 Poisson commute with each other and with other transition matrices in question since they are due to parts of the loops far away from the intersection point O and hence have nothing to do with each other. There are two distinct ways of resolving the point of intersection O. One of them is shown in figure 1 to obtain the loop γ 1 • γ 2 . Monodromy matrix around the loop γ 1 • γ 2 is denoted by M γ 1 •γ 2 .
Here, the matrices M γ 1 and M γ 2 take their values in the gauge group G. Let us represent the connection 1-form A on Σ as A = A z (z,z)dz + Az(z,z)dz = A 1 (x, y)dx + A 2 (x, y)dy. (2.7) In view of (2.7), the 1-forms, restricted to the real and imaginary axes, read A(x, 0) = A 1 (x, 0)dx, and A(0, y) = A 2 (0, y)dy. (2.8) In terms of the real and imaginary parts of the connection 1-forms, i.e. A 1 and A 2 , the Atiyah-Bott bracket (2.5) reduces to Lemma 2.1. The fundamental Poisson brackets between G valued 1-forms are given by where Γ is the Casimir tensor for G given by
Remark 2.1. We should emphasize in the context of lemma 2.1 that the basis of the underlying Lie algebra is chosen in such a way that the trace form between the group generators is diagonalised in order to comply with what was used in the derivation of the Atiyah-Bott bracket (2.5). The statement of lemma 2.1 is independent of the representation of the Lie algebra, though. All it means is that the same representation has to be chosen during both the derivations of the Atiyah-Bott brackets and the fundamental Poisson brackets.
Using lemma 2.1, one obtains the Poisson bracket between transition matrices along two small paths of the given loops around the intersection point O as illustrated in figure 1.
and T (y 1 , y 2 ) be the transition matrices corresponding to paths x 1 Ox 2 and y 1 Oy 2 as indicated in figure 1. The Poisson brackets between them is given by where Γ is the Casimir tensor given by (2.11). Here, T (x 1 , y 2 ) and T (y 1 , x 2 ), appearing in the right side of (2.12), are computed along the loop γ 1 • γ 2 of figure 1.
Proofs. The Poisson brackets between transition matrices in the context of Hamiltonian theory of Solitons are given in ( [8], page 192). In our setting, this formula gives The Poisson bracket between traces of monodromy matrices is as follows where M γ 1 and M γ 2 are given by (2.6). In (2.14), Tr and Tr 12 denote trace in the vector spaces R n and R n ⊗ R n , respectively.

JHEP02(2016)001
Proofs. Using (2.6), one obtains where we have exploited the fact that M γ 1 and M γ 2 both Poisson commute with T (x 1 , x 2 ) and T (y 1 , y 2 ), and among themselves. Using lemma 2.2, one obtains Taking trace on both sides of equation (2.16) and subsequently making use of the cyclic property of trace, one finally obtains

Examples of Poisson brackets between traces of monodromy matrices for some known real Lie groups
In the previous section, we obtained an auxiliary formula (2.14) for Poisson brackets between traces of monodromy matrices computed along free homotopy classes of loops on Σ. In this section, we shall use it to reproduce Goldman's brackets for GL(n, R), U(n), SL(n, R), SU(n), Sp(2n, R) and SO(n) gauge groups. We, first, note that the generalized Gell-Mann matrices in n dimensions read Here, e jk is an n × n matrix with 1 in the (j, k) entry and 0 elsewhere. A couple of preparatory lemmas are required in order to derive the Poisson bracket between traces of monodromy matrices for some known real Lie groups from (2.14).
Consider the right side of (3.3) and compute which leads to Lemma 3.2. Given f n k,j for k < j and k > j as in (3.1), the following holds Proofs. We also have,

JHEP02(2016)001
Let us now compute the Casimir tensor Γ appearing in (2.14) for the specific cases of GL(n, R), U(n), SL(n, R), SU(n) and SO(n). In what follows, the n 2 × n 2 permutation matrix is denoted by P . Given two n × n matrices A and B, P enjoys the following properties: (3.10) Proposition 1. For GL(n, R) and U(n) gauge groups, the Casimir tensor in the auxiliary expression (2.14) of Poisson bracket between traces of the monodromy matrices reads e jk ⊗ e kj is the Permutation matrix. Proofs.
Case 1: GL(n, R). The Lie algebra associated with GL(n, R) is gl(n, R), the vector space of all real n × n matrices. The dimension of this vector space is n 2 . We choose the matrix h n 1 , (n − 1) matrices h n k with 1 < k ≤ n, (n 2 −n) 2 matrices f n k,j with k < j, and another (n 2 −n) 2 matrices if n k,j with k > j from (3.1) to form a basis of gl(n, R). Here, in (2.4), associated with the preceding choice of generators for GL(n, R), f (a) = −1 for the (n 2 −n) 2 basis elements if n k,j with k > j. For the rest of the n 2 basis elements, we have f (a) = 1.
With the above choice of the basis of gl(n, R), the Casimir tensor Γ reads, Using lemma 3.1 together with lemma 3.2 in (3.12), one obtains the Casimir tensor for the case of GL(n, R) gauge group: Case 2: U(n). An appropriate choice of basis for the Lie algebra u(n), in the context of (2.4), will be the n 2 skew-Hermitian matrices (see (3.1)) ih n 1 , ih n k for 1 < k ≤ n and if n k,j for k = j. In accordance with the choice of these generators of unitary group U(n), f (a) = −1 in (2.4) for a = 1, 2 . . . , n 2 . The corresponding Casimir tensor Γ reads off JHEP02(2016)001 (3.14) Here, again, we use lemma 3.1 and lemma 3.2 to arrive at (3.14). Direct application of proposition 1 in (2.14) and subsequent use of the properties of P , enumerated in (3.10), yield the formula of Poisson bracket for traces of GL(n, R) and U(n) monodromy matrices as given by the following theorem: Proposition 3. The Casimir tensor in (2.14) for SL(n, R) or SU(n) gauge group reads e jk ⊗ e kj being the Permutation matrix and I being the n 2 × n 2 identity matrix. Proofs.
Case 1: SL(n, R). The Lie algebra sl(n, R) consists of traceless n × n real matrices. We, therefore, choose (n − 1) matrices h n k with 1 < k ≤ n, (n 2 −n) 2 matrices f n kj for k ≤ j and another (n 2 −n) 2 real matrices if n kj with k > j from the ones enumerated in (3.1). As was in the case of gl(n, R), f (a) = −1 in (2.4) holds only for the SL(n, R) group generators given by if n kj . Therefore, the associated Casimir tensor reads

JHEP02(2016)001
Case 2: SU(n). The real Lie algebra su(n) consists of n × n traceless skew-Hermitian matrices. As a basis of su(n), we choose (n − 1) traceless skew-Hermitian matrices ih n k with 1 < k ≤ n and another (n 2 − n) such matrices if n kj for k = j from the matrices enumerated in (3.1). Here, we only have f (a) = −1 in (2.4) for all such (n 2 − 1) group generators of SU(n). The corresponding Casimir tensor then reads We have repeatedly used lemma 3.1 and lemma 3.2 in establishing (3.17) and (3.18).
Following the use of proposition 3 in (2.14) and subsequent use of the properties of P as given by (3.10), one obtains the Poisson bracket for SL(n, R) and SU(n) monodromy matrices.
Theorem 4. The Poisson bracket between traces of monodromy matrices for SL(n, R) and SU(n) gauge groups is given by In course of proving theorem 4, one also makes use of the identity Tr 12 (A ⊗ B) = Tr A Tr B for any two n × n matrices A and B.
We shall now consider the case when the gauge group is Sp(2n, R). It is being dealt separately since an appropriate choice of basis for the associated Lie algebra sp(2n, R), in view of (2.4), is unrelated with the generalized Gell-Mann matrices enumerated in (3.1).
The Lie algebra sp(2n, R) is an n(2n+1) dimensional real vector space. An appropriate choice of basis, along with respective f (a) = ±1 for a = 1, 2, . . . , n(2n + 1) in (2.4), is outlined in table 1. Now, the Casimir tensor for the structure Lie group Sp(2n, R) is provided by the following proposition Proposition 5. The Casimir tensor Γ in (2.14), for Sp(2n, R) gauge group, reads 2 (e i,j+n + e j,i+n + e j+n,i + e i+n,j ) 1 2 (e i,j+n + e j,i+n − e j+n,i − e i+n,j ) -1 2 (e ij + e ji − e i+n,j+n − e j+n,i+n ) 1 2 (e ij − e ji + e i+n,j+n − e j+n,i+n ) -1 n 2 −n 2 1 ≤ k ≤ n e kk − e k+n,k+n 1 n We shall be calling χ as the defect matrix henceforth.
Proofs. In order to prove proposition 5, we first note that, for any two n × n matrices a and b, the following holds Using the above fact, we have the following for 1 ≤ i < j ≤ n: = e i,j+n ⊗ e j+n,i + e i,j+n ⊗ e i+n,j + e j,i+n ⊗ e j+n,i + e j,i+n ⊗ e i+n,j + e j+n,i ⊗ e i,j+n + e j+n,i ⊗ e j,i+n + e i+n,j ⊗ e i,j+n + e i+n,j ⊗ e j,i+n . (3.23) We also compute for 1 ≤ k ≤ n, (e k,n+k + e n+k,k ) ⊗ (e k,n+k + e n+k,k ) − (e k,n+k − e n+k,k ) ⊗ (e k,n+k − e n+k,k ) = 2(e k,n+k ⊗ e n+k,k + e n+k,k ⊗ e k,n+k ). (3.24) Again, considering another set of n 2 − n generators and applying (3.22), one obtains for 1 ≤ i < j ≤ n,

JHEP02(2016)001
Adding (3.23) to (3.25) and (3.24) to (3.26) followed by summing over 1 ≤ i < j ≤ n and 1 ≤ k ≤ n, respectively and finally adding up the two summands, we obtain, (e i,j+n ⊗ e j+n,i + e j,i+n ⊗ e i+n,j + e j+n,i ⊗ e i,j+n + e i+n,j ⊗ e j,i+n + e ij ⊗ e ji + e j+n,i+n ⊗ e i+n,j+n + e ji ⊗ e ij + e i+n,j+n ⊗ e j+n,i+n ) + 1≤k≤n (e k,n+k ⊗ e n+k,k + e n+k,k ⊗ e k,n+k + e kk ⊗ e kk + e k+n,k+n ⊗ e k+n,k+n ) (e i,j+n ⊗ e i+n,j + e j,i+n ⊗ e j+n,i + e j+n,i ⊗ e j,i+n + e i+n,j ⊗ e i,j+n − e ij ⊗ e i+n,j+n − e j+n,i+n ⊗ e ji − e ji ⊗ e j+n,i+n − e i+n,j+n ⊗ e ij ) + 1≤k≤n (e k,n+k ⊗ e n+k,k + e n+k,k ⊗ e k,n+k − e kk ⊗ e k+n,k+n − e k+n,k+n ⊗ e kk )   = P + χ. (3.27) We require the following lemma to prove the main result regarding the Poisson bracket for Sp(2n, R) monodromy matrices. Proofs. Given the 2n × 2n symplectic matrix B, its inverse is given by the following sets of equations: For the matrix entries with 1 ≤ i < j ≤ n, (3.29) Whereas, for the matrix entries with 1 ≤ k ≤ n, one obtains We now prove the main theorem concerning the Poisson bracket between traces of Sp(2n, R) monodromy matrices. Proofs. Plugging the Casimir tensor Γ (see (3.20)) back in (2.14) and using the identity from lemma 3.3, one obtains (3.33)

JHEP02(2016)001
We now proceed to compute the Poisson bracket between traces of SO(n) monodromy matrices. Let e ij denote an n × n matrix with 1 in (i, j) entry and 0 elsewhere. There are (n 2 −n) 2 basis elements for the corresponding Lie algebra so(n) given by t a = e ij − e ji with 1 ≤ i < j ≤ n. It can immediately be seen that the index a runs from 1 to n(n−1)
In this case, f (a), appearing in (2.4) is −1 for all a. The Casimir tensor Γ for the Lie algebra so(n) now reads where P is the so-called Permutation matrix and χ, which we refer to as the defect matrix for the Lie algebra so(n), is given by We state the following lemma before deriving the Poisson bracket between traces of SO(n) monodromy matrices. Tr(Ae ij ) Tr(Be ij ) Using the expression (3.35) for the Casimir tensor Γ of the Lie algebra so(n) in the general formula (2.14) and repeating the same computations as in the proof of theorem (6), one obtains the Poisson brackets between the traces of SO(n) monodromy matrices. We state this main result for the case of rotation group SO(n) by means of the following theorem.
Theorem 7. The Poisson bracket between traces of SO(n) monodromy matrices M γ 1 and M γ 2 is given by

Poisson bracket between G 2 -gauge invariant observables
This section is dedicated to the computation of the Poisson bracket between G 2 -gauge invariant observables which has not been considered in the literature so far. The starting point here is to compute the Poisson bracket between traces of G 2 -monodromy matrices corresponding to two transversally intersecting loops on the underlying Riemann surface Σ.
The exceptional real Lie group G 2 is 14-dimensional. Below is a list of the appropriately normalized (in view of (2.4)) 14 basis elements of the corresponding exceptional real simple Lie algebra g 2 as given in [5]. The Casimir tensor Γ corresponding to the fundamental representation of the Lie algebra g 2 is provided by the following proposition.
where the matrices {O i } are given by  The proof is given in the appendix A. The skew-symmetric 7 × 7 matrices {O i } are reminiscent of the imaginary units of the normed division algebra of octonions, the multiplication table of which can be constructed out of the following relations (see [13]): where ijk is totally antisymmetric and is unity for the following set of combinations: {123, 145, 176, 246, 257, 347, 365}.

JHEP02(2016)001
Note that {O i } can be obtained from the 8 × 8 matrix representations of {e i } (see [13]) by deleting the first row and the first column of the respective matrices. Use of proposition 8 in (2.14) yields the Poisson bracket between traces of G 2 monodromy matrices computed along homotopically inequivalent loops intersecting transversally at a ponit on Σ.The Poisson bracket between such canonical gauge invariant observables is provided by the following proposition.
Proposition 9. The Poisson bracket between traces of G 2 monodromy matrices M γ 1 and M γ 2 , computed along homotopically inequivalent and transversally intersecting loops γ 1 and γ 2 , respectively, is given by Proofs.
Remark 4.1. We remark here that the term We can obtain an infinite set of such gauge invariant observables using the following Lemma ( [12]): Proofs. The proof is based on the fact that G 2 is the automorphism group of octonions. Let e 1 , e 2 , .., e 7 denote the 7 imaginary octonions that obey the multiplication rule given by (4.4) and 1 denote the identity. Using where the orthogonality of g ∈ G 2 is exploited. Similarly, application of g on the right hand side of (4.8) yields where we have used the fact that g(1) = 1. Now comparing (4.10) with (4.11), one finds Multiplying both sides of (4.12) by g −1 jl and summing over j, we obtain the following for any m and l: (4.13) . Now we proceed to state the main result of the paper by means of a theorem that concerns an infinite set of G 2 -gauge invariant observables. The theorem is as follows Theorem 4.1. Given 4 non-negative integers r, s, n 1 and n 2 satisfying r ≤ n 1 , s ≤ n 2 and two (0, 1)-matrices K and Q of order t × n 1 and t × (2n 2 − s), respectively and t being another positive integer satisfying t ≤ n 1 + 2n 2 , the following expression involving monodromy matrices M γ j corresponding to pairwise transversally intersecting n 1 + t loops γ j 's with j = 1, 2, . . . , n 1 + t, is a G 2 gauge invariant observable lj ∈{1,2,...,7} . Also, here, each column of the (0, 1)-matrix K has exactly one entry equal to 1. The (0, 1)-matrix Q of order t × (2n 2 − s), on the other hand, has exactly two entries equal to 1 in each column of the t × s-block and one entry equal to 1 in each column of the adjacent t × (2n 2 − 2s)-block.
The proof is given in the appendix A.
Remark 4.2. It is worth remarking at this point that for a fixed tuple of non-negative integers (r, n 1 , s, n 2 , t), with r ≤ n 1 , s ≤ n 2 and t ≤ n 1 + 2n 2 , there are only finitely many choices for the (0, 1)-matrices K and Q to obtain various G 2 -gauge invariant observables. By varying r, n 1 , s, n 2 and t subject to the above mentioned constraints, one thus obtains an infinite set of such gauge invariant observables.
We will refer to the observables obtained in Theorem 4.1 as exotic gauge invariant observables. The Poisson bracket between two such exotic G 2 gauge invariant observables can be seen to be a linear combination of exotic G 2 gauge invariant observables by using formula (2.16) and Lemma 4.1 repeatedly. On the other hand, the Poisson bracket between a canonical G 2 gauge invariant observable (trace of a G 2 monodromy matrix) and an exotic G 2 gauge invariant observable can again be shown to be a linear combination of exotic G 2 gauge invariant observables. We will only prove the latter fact for the fact that the Poisson bracket between two exotic G 2 -gauge invariant observables being a linear combination of gauge invariant observables of the same type can be proven using exactly similar techniques.
Let us, give a few concrete examples of such exotic gauge invariant observables and elucidate what roles the non-negative parameters r, n 1 , s, n 2 , t and the (0, 1)-matrices K and Q, enumerated in theorem 4.1, have got to play in these contexts.

Examples of exotic gauge invariant observables
We encountered the first instance of exotic G 2 -gauge invariant observables in proposition 9, namely, the term Tr(M γ 1 O i ) Tr(M γ 2 O i ) while computing the Poisson bracket between traces of G 2 monodromy matrices. In the preceding example, r = n 1 = 1, s = n 2 = 0 and t = 1 so that we have the (0, 1)-matrix K to be just 1 as implicated by theorem 4.1.

JHEP02(2016)001 5 Conclusion and outlook
In this paper, we have generalized the Goldman bracket to the case of G 2 gauge group. The expression for the Poisson bracket between traces of G 2 -valued monodromy matrices reveals the existence of a gauge invariant term of new type which were not present in the cases of classical gauge groups obtained by Goldman in [9]. An infinite set of such exotic G 2 -gauge inavriant observables is obtained that is closed under Poisson bracket. As a byproduct, we present an alternative derivation of the well-known Goldman's bracket for the following gauge groups: GL(n, R), U(n), SL(n, R), SU(n), Sp(2n, R) and SO(n). In future, we plan to extend our formalism to find Goldman-type brackets for the other exceptional gauge groups: F 4 , E 6 , E 7 and E 8 . It would also be interesting to see how the quantum Goldman bracket turns out to be for the G 2 gauge group and for the other exceptional groups in comparison with the one computed in (p 428, [15]) for SL(2, R). We plan to investigate the underlying quantum geometry for the exceptional gauge groups by looking at the relevant quantum Goldman bracket in a future publication.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.