aa r X i v : . [ h e p - t h ] M a y Quantum Wilson surfaces and topological interactions.
Olga Chekeres
Department of Mathematics, University of Geneva,2-4 rue du Li`evre, c.p. 64, 1211 Gen`eve 4, Switzerland
E-mail:
Abstract:
We introduce the description of a Wilson surface as a 2-dimensional topolog-ical quantum field theory with a 1-dimensional Hilbert space. On a closed surface, theWilson surface theory defines a topological invariant of the principal G -bundle P → Σ.Interestingly, it can interact topologically with 2-dimensional Yang-Mills and BF theoriesmodifying their partition functions. We compute explicitly the partition function of the2-dimensional Yang-Mills theory with a Wilson surface. The Wilson surface turns out to benontrivial for the gauge group G non-simply connected (and trivial for G simply connected).In particular we study in detail the cases G = SU ( N ) / Z m , G = Spin (4 l ) / ( Z ⊕ Z ) andobtain a general formula for any compact connected Lie group. Keywords:
Wilson surface, topological interactions, 2d Yang-Mills, gauge theories ontents G = Spin (4 l ) / ( Z ⊕ Z ) 12 The discussion of surface observables in gauge theories has been ongoing for quite sometime. Wilson surfaces, domain walls, surface defects etc appear in many domains of physicsand mathematics, from gauge theories to condensed matter. They have been studiedextensively in literature [1–7]. In most cases a 1-dimensional observable, namely a Wilsonline [8–16], is generalized to 2 dimensions by introducing higher gauge fields defined onsurfaces. Our approah is different, it is based as well on a definition of a Wilson line, butdoesn’t involve introducing higher gauge fields.The basis for our construction is a 1-form standard gauge field taking values in a Liealgebra. A Wilson surface is defined by an orientable surface Σ and a representation R λ of the gauge group G . In [17] we obtained its description as a 2-dimensional topological σ -model: S λ ( a, b, A ) = Z Σ Tr( b ( d ( A + a ) + ( A + a ) )) = Z Σ Tr( bF A + a ) , (1.1)where λ ∈ Λ ∗ is the highest weight of the representation R λ , b is a scalar field takingvalues in g ∗ and constrained to be a conjugate of λ ∈ Λ ∗ , A is a background gauge field,and a is an auxiliary gauge field. Interpreting A + a as a new gauge field allows us tointerpret a Wilson surface as an independent 2-dimensional BF-theory with a constrainton the B-field.In this article we start with the action functional (1.1) and canonically quantize it. Ourfirst result is the partition function formula for this Wilson surface theory. To describe theformula we first recall some topological facts. Principal G -bundles P → Σ over a closed– 1 –urface Σ are classified by the elements γ ∈ π ( G ) in the fundamental group of the gaugegroup G [18, 19]. The gauge group can be represented as G = ˜ G/ Γ, where ˜ G is the universalcover of G and Γ ⊂ Z ( ˜ G ) is a proper subgroup of the center of ˜ G . Since Γ ∼ = π ( G ), forevery element γ ∈ π ( G ) in the fundamental group there exists a corresponding element C γ ∈ Γ ⊂ Z ( ˜ G ) in the center of the covering group. Then the partition function for theparticular equivalence class of principal bundles P → Σ, defined by γ ∈ π ( G ), is given by: Z Σ W S ( C γ , λ ) = χ λ ( C γ ) d λ = e iϕ γ ∈ U (1) , (1.2)where χ λ ( C γ ) is a value of the character χ λ on the element C γ and d λ is the dimensionof the representation. This is a 2-dimentional topological quantum field theory with a1-dimensional Hilbert space.Our second result is the description of topological interactions of Wilson surfaces with2-dimensional topological gauge theories, namely with BF and Yang-Mills theories. Theirpartition functions on a surface Σ are obtained by summation over all the classes of principal G -bundles defined over the given surface [20–22]. When we insert a Wilson surface into 2-dimensional Yang-Mills or BF, it interacts topologically with the background gauge theory,as the gauge connections A and A + a are defined on the same principal G -bundle P → Σ.The presence of a Wilson surface modifies the partition function of the background theorymultiplying by a phase (1.2) the individual contributions for each class of principal bundles: Z interact = X γ ∈ π ( G ) Z backgr ( C γ ) · e iϕ γ . (1.3)Next, we study concrete examples of 2-dimensional Yang-Mills theory with a Wilsonsurface for the gauge groups G = SU ( N ) / Z m ( m divides N ) and G = Spin (4 N ) / ( Z ⊕ Z ). Since Yang-Mills in 2 dimensions is exactly solvable, we can obtain explicit formulas forthe partition function in the presence of a Wilson surface.In case of G = SU ( N ) / Z m the fundamental group is π ( G ) ∼ = Z m and the Wilsonsurface phase e iϕ k is defined by the angle: ϕ k = 2 πkm · [ λ ] , where k = 0 , , ..., m − π ( G ), and [ λ ] ∈ Z m is an integer mod m denoting the equivalence class of the highest weight λ characterising the Wilson surface.For G = Spin (4 N ) / ( Z ⊕ Z ) the fundamental group is π ( G ) ∼ = Z ⊕ Z and the Wilsonsurface is defined by the angle: ϕ k ,k = π ( k [ λ ] + k [ λ ]) , where a pair ( k , k ) labels the elements of π ( G ), with k , k ∈ { , } , and [ λ ] , [ λ ] ∈ Z are integers modulo 2 given by two linear combinations of the components of same highest The examples of G = U (1) , SU (2) , SO (3) were computed in [17]. – 2 –eight λ characterizing the Wilson surface.Eventually, we obtain the formula of the partition function for 2D-YM with a Wilsonsurface for any compact connected Lie group G = ˜ G/ Γ. In this case the Wilson surfacephase is defined by the angle: ϕ k ,...,k i = X i k i < λ, c i > = 2 π X i k i m i [ λ i ] . Here we account for the most general case, when the fundamental group of G is given by aproduct of i cyclic groups: π ( G ) = Z m × ...... × Z m i . Then m i is the number of elements in Z m i , the index k i = 0 , ..., m i − i -th factor, k i c i ∈ h ⊂ Lie ( ˜ G ) isan element of the Cartan subalgebra such that e i P i k i c i = C k ,...,k i ∈ Γ ∼ = π ( G ) is a centralelement of the covering group ˜ G , λ ∈ h ∗ is the highest weight of the representation of G characterizing the Wilson surface, <, > is the invariant scalar product defined on Lie ( ˜ G )and [ λ i ] ∈ Z m i are integers modulo m i given by i linear combinations of the componentsof same highest weight λ .For a closed surface the Wilson surface is nontrivial for G non-simply connected, andit is not visible ( e iϕ = 1) for G simply connected. Also the value of λ plays a role: for λ being the highest weight of a representation of the gauge group G itself the Wilson surfaceis trivial, and it is nontrivial if λ labels a representation of the universal cover ˜ G whichdoes not descend to G . On a closed surface, the partition function of the Wilson surface isa topological invariant of the principal G -bundle. Acknowledgements.
Our deepest gratitude is to A. Alekseev for inspiration through-out this work. We also thank D. Nedanovski, who participated in the early stages of theproject and independently confirmed our computation for G = SU ( N ) / Z N , F. Valachfor illuminating discussions and all the inhabitants of Villa Battelle Math Department inGeneva for inspiring atmosphere. Our research was supported in part by the grant 178794,the grant MODFLAT of the European Research Council (ERC) and the NCCR SwissMAPof the Swiss National Science Foundation. Recall the construction of Wilson surface observables from [17]. Let G be the gauge group, g its Lie algebra, ( x, y ) → Tr( xy ) an invariant scalar product on g and P a principal G -bundle over a surface Σ. We denote by h ⊂ g a Cartan subalgebra and by Λ ∗ ⊂ h ∗ theweight lattice.A Wilson surface observable is described by an auxiliary 2-dimensional gauge theoryon the surface Σ. The fields in this theory are a g ∗ -valued scalar field b and a g -valued1-form a . The action depends on the following data: the background gauge filed A and the– 3 –eight λ ∈ Λ ∗ . For a trivial G -bundle it is given by S λ ( a, b, A ) = R Σ Tr( b ( F A − ( dgg − + A ) + ( dgg − + A + a ) )= R Σ Tr( b ( d ( A + a ) + ( A + a ) )) , (2.1)where we identifed g ∗ with g using the scalar product and integrated by parts using theequality Tr b [ dgg − , a ] = Tr[ b, dgg − ] a = − Tr( db ) a . The field b = gλg − belongs to thesame conjugacy class as the fixed element λ , the combination A + a is a new gauge field.Note that integrating out a in (2.1) yields the Diakonov-Petrov action [17, 23] for a Wilsonsurface: S DP = Z Σ Tr( b ( F A − ( dgg − + A ) ) . (2.2)The construction (2.1) also works on nontrivial bundles. A ∈ Ω ( P, g ) is the connectionon P , its curvature F A ∈ Ω hor ( P, g ) G is a horizontal 2-form taking values in g . The auxiliarygauge field a is such that a ∈ Ω hor ( P, g ) G and the sum A + a defines a new connection on P with a curvature F A + a = d ( A + a ) + ( A + a ) , F A + a ∈ Ω hor ( P, g ) G . The field b takesvalues in Ω hor ( P, g ) G . The combination Tr( bF A + a ) is then a basic 2-form which decends toΣ. One can show this in the following way. The 2-form bF A + a is G -equivariant, i.e. withrespect to a gauge transformations by h : Σ → G : g hg , A hAh − − dhh − , b hbh − , a hah − , it transforms as b h F A h + a h = hbF A + a h − , yeilding Tr( b h F A h + a h ) = Tr( bF A + a ). Acting bythe contraction we obtain: ı ξ ♯ Tr( bF A + a ) = ı ξ ♯ Tr ( b ( F A + a + dgg − a + adgg − + Aa + aA )) = Tr ( b ( − ξa + aξ + ξa − aξ )) = 0 , where ξ ∈ g induces the fundamental vector field ξ ♯ ∈ X (Σ), and we have used that ı ξ ♯ ( dg ) = − ξg , ı ξ ♯ A = ξ (by definition of connection), ı ξ ♯ F A = 0, ı ξ ♯ a = 0 ( F A and a arehorizontal). This computation proves that G -invariant form Tr( bF A + a ) is horizontal, andhence basic.The meaning of λ ∈ Λ ∗ + is as follows. The integral weights of the representations of G form the weight lattice Λ ∗ ⊂ h ∗ . Dominant integral weights Λ ∗ + ⊂ Λ ∗ are in one to onecorrespondence with irreducible representations of G [24, 25]. The element λ ∈ Λ ∗ + is thehighest weight of some representation of G , it is a parameter characterising the Wilsonsurface. For example, for G = SU (2) or G = SO (3) we talk about a Wilson surface of spin λ . Note that in case when G is not simply connected but is a quotient G = ˜ G/ Γ, where ˜ G is its universal cover and Γ ⊂ Z ( ˜ G ) is a subgroup of the center of ˜ G , the weight lattices arerelated as Λ ∗ G ⊂ Λ ∗ ˜ G . A representation of ˜ G can be considered as projective representationof G , and λ is allowed to take values in Λ ∗ ˜ G , and, as we will see later, these are exactly thevalues which describe the presence of nontrivial Wilson surfaces.– 4 – .2 Quantum Wilson surfaces Rewriting the action for a Wilson surface (2.1) with respect to the new connection A + a gives us a 2-dimensional BF theory [26] described on Σ: S λ ( a, b, A ) = Z Σ Tr( bF A + a ) . (2.3)Recall canonical quantization of BF -theory on a surface [20, 21]. For simplicity letfirst Σ be a cylinder C , the G -bundle P will be necessarily trivial. We chose space and timecoordinates ( x, t ) in a way that the boundary of C is given by two closed curves γ and γ ,situated on equal time slices, and x is a periodic coordinate of period L . We associate to γ i a gauge invariant wave function ψ ( A ) which is a function of holonomy of A around γ i : ψ [ U i ] = ψ [ P e R L dxA ] . The Hilbert space H γ of such a theory is given by G -invariant L functions on G . H γ admits a natural basis in terms of characters of representations, and any wave function ψ ( A ) ∈ H γ has an expansion in characters χ R ( U ), where R is a representation. The bound-aries are oriented, the wave functions on the incoming and outgoing boundary componentsare denoted by χ R ( U i ) and χ R ( U j ) respectively.In BF-theory Hamiltonian vanishes (as expected for a topological field theory), so thepartition function reduces to: Z CBF ( U , U ) = X R χ R ( U ) χ R ( U ) . For a generic surface with genus g and r boundary components the BF partitionfunction reads: Z Σ BF ( U , ..., U r ) = X R d − g − rR χ R ( U ) ...χ R ( U r ) , where d R is the dimension of the representation and all the boundaries are chosen to beoutgoing.To obtain the formula for a closed surface we proceed as follows. The partition functionwill necessarily depend on the equivalence class of G -bundle over Σ. Recall the classificationof principal G -bundles over Σ by the elements of the fundamental group of G : π ( G ) ∼ = Γ ⊂ Z ( ˜ G ). Consider a surface with just one puncture, i.e. one boundary component. Gluingthis puncture to an infinitesimal disc yeilds a closed surface, and this operation is describedby identifying U = C i , where C i ∈ Γ is a central element of ˜ G .The contribution to the partition function of each class [ P ] of a principal G -bundleover the surface is given by: Z Σ BF ( C i ) = X R d − gR χ R ( C i ) . (2.4)The total partition function for BF-theory on a closed surface Σ is then a sum over– 5 –quivalence classes of principal G -bundles over Σ: Z Σ BF = 1 X C i ∈ Γ X R d − gR χ R ( C i ) , (2.5)where R in (2.5) converges only for surfaces Σ with genus g > g ∗ -valued field b = gλg − is now the conjugation of thesame fixed element λ ∈ Λ ∗ ˜ G . The Hilbert space becomes one-dimensional choosing onerepresentation R λ , and the partition function is just a phase. The normalization of thestates is such that || < χ λ ( U ) | χ λ ( U ) > || = 1.The state corresponding to a disc is given by Z discW S ( U, λ ) = χ λ ( U ) . (2.6)The orientation of the disc chosen in a way that U is a holonomy of the connection A + a around an outgoing boundary.The Wilson surface on a pair of pants has the formula: Z p − o − pW S ( U , U , U , λ ) = χ λ ( U ) χ λ ( U ) χ λ ( U ) , (2.7)where U , U , U are holonomies of A + a around one incoming and 2 outoing boundarycomponents.Any other orientable surface can be obtained by gluing those elementary componentstogether. The partition function for a surface of arbitrary genus with r boundary compo-nents is a product of r states living on the boundaries (here chosen to be outgoing): Z Σ W S ( U , ...., U r , λ ) = χ λ ( U ) ......χ λ ( U r ) . The expression for a closed surface for a particular class [ P ] of principal bundles P → Σis Z Σ W S ( C i , λ ) = χ λ ( C i ) d λ . (2.8)For a nontrivial central element C i this is an element of U (1): Z Σ W S ( C i , λ ) ∈ U (1). Incase when P is a trivial bundle, the Wilson surface is always trivial: Z Σ W S ( P triv , λ ) = 1. In general case our observable could be understood as a surface defect embedded into ahigher dimensional space-time. But in this paper we want to test it in the context of 2-dimensional gauge theories, which can be solved exactly. In this case the Wilson surface isa “global” observable, defined on the entire 2-dimensional space-time Σ.– 6 – .1 BF theory with a Wilson surface
The action functional for BF theory with a Wilson surface is: S λBF ( A, a, B, b ) = Z Σ Tr( BF A ) + Z Σ Tr( bF A + a ) , (3.1)where B ∈ Ω ( P, g ∗ ) G , A ∈ Ω ( P, g ) is the background gauge field, F A ∈ Ω hor ( P, g ) G itscurvature.These two BF theories are not completely independent, they interact topologically:the connections A and A + a are defined on the same principal G -bundle, so the characters χ R ( U ( A )) and χ λ ( U ( A + a )) are taken on the same central element.Then the partition function for a closed surface with a Wilson surface of weight λ isobtained by taking a product of partition functions defined in the previous section for eachclass [ P ] and then summing over all the equivalence classes: Z λBF = 1 X C i ∈ Γ χ λ ( C i ) d λ X R d − gR χ R ( C i ) . (3.2) Consider 2D-YM in the first order formalism. The action functional for the theory with aWilson surface is: S λY M ( A, a, B, b ) = Z Σ Tr( BF A + e B d σ ) + Z Σ Tr( bF A + a ) , (3.3)where B is an auxiliary field taking values in g ∗ and d σ is the area element on Σ. Again, wesee that the action splits into two theories interacting topologically through the connections A and A + a defined on the same principal bundle. The partition function for each class[ P ] will be a product of the partition function for 2D-YM and the partition function forthe Wilson surface.The Hamiltonian of the first theory is H = e Tr B . The Hamiltonian of the secondtheory vanishes. The basis for the Hilbert space is given by gauge invariant functions ψ R ( A, a ) = χ R ( U ( A )) χ λ ( U ( A + a )), where R runs through the irreps of G , λ choses oneirrep of G , and U ( A ), U ( A + a ) are holonomies of the connections A and A + a respectively.The eigenvalues of the Hamiltonian on ψ ( A, a ) are given by quadratic Casimir C ( R ) of therepresentation R , just like for the 2D YM without Wilson surface, as only the Hamiltonianof 2D YM contributes to the total theory.Then the time evolution operator takes value e − τC ( R ) on the functions ψ R ( A, a ), where τ = e σ absorbs the YM coupling constant e and the area of the surface σ .The partition function for a closed surface with a Wilson surface of weight λ is givenby the following formula: Z λY M ( τ ) = 1 X C i ∈ Γ X R d − gR e − τC ( R ) χ R ( C i ) χ λ ( C i ) d λ . (3.4)– 7 – Exact results for 2D-YM theory interacting with a Wilson surface
Yang-Mills theory in 2 dimensions is exactly solvable [27–36], this allows us to obtainexplicit formulas for partition function in the presence of a Wilson surface. In [17] wecomputed the partition functions for 2D Yang-Mills with a Wilson surface for the gaugegroups U (1), SU (2) and SO (3). Now we are going to generalize this result to G being anycompact connected Lie group. To visualize the result of topological interactions with a Wilson surface, we first performa detailed computation for the case of ˜ G = SU ( N ). The center of SU ( N ) is given by: Z ( SU ( N )) = { e πikN Id N | k = 0 , ..., N − } = Z N . And the subgroups of the centre areΓ = Z m where m devides N . We consider G = SU ( N ) / Z m .The rank of SU ( N ) is equal to N −
1, i.e. the basis of Cartan subalgebra h has N − h is given by: h n = diag(0 ......., , − , ...... h nn = 1, h n +1 ,n +1 = −
1. Then any element h ∈ h can be representedin terms of the basis as h = P N − n =1 a n h n , with a n ∈ R linear coefficients. Exponenti-ating elements h ∈ h we obtain the maximal torus of SU ( N ): H = e ih = e i P a n h n =diag(e i θ , e i θ , ...., e i θ N − , e − i P N − θ i ) ∈ T.The center Z ( SU ( N )), and hence its proper subgroup Γ ⊂ Z ( SU ( N )) ⊂ T , is asubgroup of the maximal torus: Γ ∋ C k = e iθ k Id N ∈ T , with θ k = 2 πk/m .Consider the elements of the Cartan subalgebra c k = diag( θ k , θ k , ......, − (N − θ k ) N × N ∈ h , such that C k = e ic k ∈ Z ( SU ( N )). In terms of the basis of h they are given as follows: c k = diag( θ k , θ k , ......, − (N − θ k ) = θ k · diag(1 , , ......., , − (N − θ k · P N − nh n .Then the central elements of SU ( N ) are given by C k = e iθ k P N − n =1 nh n ∈ Z ( SU ( N )).The irreducible representations of SU ( N ) are labeled by highest weights with N − µ = ( µ , ..., µ N − ).The central elements in the representation R µ of highest weight µ are obtained as: R µ ( e iθ k P N − n =1 nh n ) = e iθ k P N − n =1 nR µ ( h n ) .The natural choice for the basis of R µ is in terms of the weight vectors v i with v thehighest weight vector. In this basis R µ ( h n ) are diagonal and yield weights while acting onthe basis vectors. A central element C k is a multiple of identity, therefore R µ ( C k ) has tobe a multiple of identity as well, so it’s enough to compute it just on the highest weightvector: R µ ( C k ) = e i πkm P N − n =1 nµ n · Id d Rµ . (4.1)The linear combination P N − n =1 nµ n is an integer, but the expression (4.1) depends only onthe value of this sum modulo m , as e i πkm P N − n =1 nµ n = e i πkm ( P N − n =1 nµ n + m ) . This allows us todefine the equivalence classes of the highest weight µ :[ µ ] ≡ [ N − X n =1 nµ n ] ∈ Z m . (4.2)– 8 –ote that the irreps of ˜ G descend to the irreps of G if P N − n =1 nµ n = 0 mod m . Interms of weight lattices Λ ∗ G ⊂ Λ ∗ ˜ G ⊂ h ∗ , where Λ ∗ ˜ G is the weight lattice for SU ( N ), Λ ∗ G isthe weight lattice for G = SU ( N ) / Z m .The characters of the central elements in the representations R µ are as follows: χ R µ ( C k ) = T r ( e i πkm P N − n =1 nµ n · Id d Rµ ) = d R µ · ( e i πkm ) [ µ ] . (4.3)We keep the notation Z Y M ( τ ) for the partition function of the free 2D-YM theory and Z λY M ( τ ) for the theory with a Wilson surface of the highest weight λ .Without Wilson surface the partition function for SU ( N ) / Z m is given by Z Y M ( τ ) = 1 m m − X k =0 X R µ ( SU ( N )) d R µ − g e − τC ( R µ ) χ R µ ( C k ) , (4.4)where the sum is over the representations R µ of SU ( N ), and k labels central elements inthe subgroup Γ.The computation gives the following result: Z Y M ( τ ) = 1 m m − X k =0 X R µ ( SU ( N )) d − gR µ e − τC ( R µ ) ( e i πkm ) [ µ ] = X R µ ( SU ( N )) d − gR µ e − τC ( R µ ) m m − X k =0 ( e i πkm ) [ µ ] , (4.5)where the sum over k is equal to m for [ µ ] = 0 and zero otherwise. The condition cor-responds to those representations of SU ( N ) in which the central elements are all trivial,that is to the representations of SU ( N ) / Z m : Z Y M ( τ ) = X R µ ( G = SU ( N ) / Z m ) d − gR µ e − τC ( R µ ) . (4.6)Now let us introduce a Wilson surface of weight λ : Z λY M ( τ ) = P m − k =0 P R µ ( SU ( N )) d − gR µ e − τC ( R µ ) χ R µ ( C k ) χ λ ( C k ) d λ = m P m − k =0 P R µ ( SU ( N )) d − gR µ e − τC ( R µ ) d λ · ( e i πkm ) [ λ ] d λ d R · ( e i πkm ) [ µ ] = m P m − k =0 P R µ ( SU ( N )) d − gR µ e − τC ( R µ ) ( e i πkm ) [ µ ]+[ λ ] , (4.7)where [ λ ] = [ P N − n =1 nλ n ] ∈ Z m are equivalence classes of the Wilson surface weight λ .In more detail, we consider a quotient map Λ ∗ ˜ G ∋ λ ( P N − n =1 nλ n ) mod m ∈ Z m and thehighest weights for Wilson surfaces will belong to equivalence classes [ λ ] ∈ Λ ∗ ˜ G / Λ ∗ G ∼ = Z m .Note that in case when λ ∈ Λ ∗ G , i.e. P N − n =1 nλ n = 0 mod m , the Wilson surface is notvisible: Z λ ( τ ) = Z ( τ ).The sum over k in (4.7) is different from zero only for [ µ ] + [ λ ] = 0, and the partitionfunction formula for 2D-YM with a Wilson surface yields: Z λY M ( τ ) = X µ ∈ Λ ∗ ˜ G , [ µ + λ ]=0 dim − gR µ e − τC ( R µ ) = X µ ∈ [ − λ ] dim − gR µ e − τC ( R µ ) , – 9 –here the sum now goes over the representations R µ + λ of G = SU ( N ) / Z m of highestweights µ + λ , i.e. [ µ ] = [ − λ ] ∈ Λ ∗ ˜ G / Λ ∗ G .Note that in case when G = SU ( N ), i.e. the gauge group is simply connected, thepresence of the Wilson surface makes no impact on the partition function. Let us look atthis situation in more detail. There is just one class of principal SU ( N )-bundles over asurface Σ – trivial SU ( N )-bundle. The SU ( N ) partition function without Wilson surfaceis given by: Z SU ( N ) Y M ( τ ) = X R µ d − gR µ e − τC ( R µ ) χ R ( e ) = X R µ d − gR µ e − τC ( R µ ) , (4.8)where e is identity.And adding a Wilson surface of weight λ leaves the partition function unchanged: Z λ,SU ( N ) Y M ( τ ) = X R µ d − gR µ e − τC ( R µ ) χ R ( e ) χ λ ( e ) d λ = X R µ d − gR µ e − τC ( R µ ) . (4.9) The result explained in the explicit example of the previous section remains valid for allcompact connected Lie groups. All of them (with the exception of exceptional ones) haveas a universal cover one of the following groups: SU ( N ), Spin ( N ), Sp ( N ) and can beobtained by taking a quotient by a subgroup Γ of the center.The case of ˜ G = SU ( N ) has been discussed in the previous section. For Spin ( N ), N ≥ Z ( Spin ( N )) = Z if N = 2 l + 1, Γ = Z , Z if N = 4 l + 2, Γ = Z or Γ = Z , Z ⊕ Z if N = 4 l , Γ = Z or Γ = Z ⊕ Z . (4.10)The group Sp ( N ) has the center Z ( Sp ( N )) = Z .Among the exceptional groups only E and E are interesting for our purposes, therest of them ( G , F and E ) are simply-connected and have a trivial center. The realcompact forms of E and E are not simply-connected. The universal cover of E has thecenter Z ( ˜ E ) = Z , and the universal cover of E has the center Z ( ˜ E ) = Z .We consider the gauge group G = ˜ G/ Γ. The center of the cover Z ( ˜ G ) ⊂ T is asubgroup of the maximal torus. T is given by the elements H = e ih ∈ T , where h ∈ h is inthe Cartan subalgebra.The irreducible representations of ˜ G are labeled by highest weight with n independentelements, where n is the rank of ˜ G : ( µ , ..., µ n ).In most cases the center of ˜ G , or its proper subgroup Γ, is given by Z m for some m ∈ Z ,and the calculation looks similar to the ˜ G = SU ( N ) example. But in general Γ can berepresented by a product of i cyclic groups: Γ = Z m × ...... × Z m i . We take the elements P i k i c i ∈ h in the Cartan subalgebra h of ˜ G and exponentiate them to get the centralelements C k ....k i = e i P i k i c i ∈ Z ( ˜ G ). Here we account for the structure of Γ: the index i – 10 –efers to the i -th factor in the product and the coefficient k i labels the elements inside eachfactor Z m i .In terms of the basis of the Cartan subalgebra h j ∈ h we can express P i k i c i = P i π k i m i P nj =1 ( a i ) j h j , where n is the dimension of h , m i is the number of the elementsin Z m i and ( a i ) j are real linear coefficients describing c i and depending on the choice of abasis h i .The representation R µ of a central element C k ,...,k i is given by the formula: R µ ( C k ,...,k i ) = R µ ( e i P i k i c i ) = e i P i k i R µ ( c i ) . The characters of the central elements in the representations R µ are: χ R µ ( C k ,...,k i ) = T r ( e i P i k i R µ ( c i ) ) = d Rµ · e i P i k i <µ,c i > = d Rµ · e i π P i kimi [ µ i ] . (4.11)Here we have rewritten the pairing P i k i < µ, c i > in the following way: < µ, P i k i c i > = i π P i k i m i P nj =1 ( a i ) j µ j = i π P i k i m i [ µ i ], where ( a i ) j are linear coefficients producing dif-ferent linear combinations of µ j s for each k i -th element. The i different linear combi-nations P nj =1 ( a i ) j µ j ∈ Z define i types of equivalence classes of the highest weight µ :[ P nj =1 ( a i ) j µ j ] ≡ [ µ i ] ∈ Z m i , where [ µ i ] is an integer modulo m i .Without Wilson surface the partition function for G is given by: Z Y M ( τ ) = X i m i m i − X k i =0 X R µ ( G ) d − gR µ e − τC ( R µ ) χ R µ ( C k ,...,k i ) , (4.12)where the sum is over the representation R µ of G = ˜ G/ Γ and i coefficients k i label a centralelement in the subgroup Γ.Using (4.11) we compute: Z Y M ( τ ) = P i m i P m i − k i =0 P R µ ( G ) d − gR µ e − τC ( R µ ) · e ik i <µ,c i > = P R µ ( G ) d − gR µ e − τC ( R µ ) P i m i P m i k i =0 e i π kimi [ µ i ] . (4.13)Here each sum over k i in the second line is different from zero and is equal to m i only if µ i = 0 mod m i (i.e. [ µ i ] = 0) . This condition corresponds to choosing onlythose representations of ˜ G in which the elements C k ,....,k i ∈ Γ are all trivial, i.e. therepresentations of G = ˜ G/ Γ: Z Y M ( τ ) = X R µ ( G = ˜ G/ Γ) d − gR µ e − τC ( R µ ) . (4.14)Now let us introduce a Wilson surface of weight λ . Just like any highest weight, λ will belong to i types of equivalence classes defined by the pairing k i < λ, c i > =2 π k i m i P nj =1 ( a i ) j λ j , where ( a i ) j are real linear coefficients for the pairing with the k i -thelement and P nj =1 ( a i ) j λ j is an integer. Then the weight λ will be characterised by belong-– 11 –ng to i types of equivalence classes: [ λ i ] = [ P nj =1 ( a i ) j λ j ] ∈ Z m i . The partition functionwith a Wilson surface of weight λ is given by: Z λY M ( τ ) = P i m i P m i − k i =0 P R µ ( G ) d − gR µ e − τC ( R µ ) χ R µ ( C k ,...,k i ) χ λ ( C k ,...,ki ) d λ = P i m i P m i − k i =0 P R µ ( G ) d − gR µ e − τC ( R µ ) e ik i <µ,c i > e ik i <λ,c i > = P i m i P m i − k i =0 P R µ ( G ) d − gR e − τC ( R µ ) e i π kimi [ µ i ] e i π kimi [ λ i ] = P R µ ( G ) d − gR e − τC ( R µ ) P i m i P m i − k i =0 e i π kimi [ µ i + λ i ] . (4.15)Now each sum over k i is different from zero and is equal to m i only if [ µ i + λ i ] = 0 forall i. This results in: Z λY M ( τ ) = X R µ + λ ( G = ˜ G/ Γ) dim − gR µ e − τC ( R µ ) , (4.16)where the sum is over such representations R µ ( ˜ G ), that the representations of ˜ G with thehighest weight µ + λ would correspond to the representations of G = ˜ G/ Γ. G = Spin (4 l ) / ( Z ⊕ Z )Now let us illustrate the formulas (4.15), (4.16) with an example of a gauge group with π ( G ) ∼ = Γ = Z m × ...... × Z m i . The covering group ˜ G = Spin (4 l ) has the center givenby a product of two copies of Z : Z ( Spin (4 l )) = Z ⊕ Z . If we factorize by the entirecenter we get G = Spin (4 l ) / Z ⊕ Z . We start with the central elements of ˜ G = Spin (4 l ): C k k = e i ( k c + k c ) ∈ Z ⊕ Z , where k c + k c = πk P ni =1 ( a ) i h i + πk P ni =1 ( a ) i h i ∈ h are in the Cartan subalgebra of Spin (4 l ), the coefficients k j = 0 , Z , and ( a ) i , ( a ) i are real coefficients describing the elements c and c respectively and depending on the choice of a basis h i .The representation R µ of a central element C k k is given by the formula: R µ ( C k k ) = R µ ( e i ( k c + k c ) ) = e i ( k R µ ( c )+ k R µ ( c )) . The characters of the central elements in the rep-resentations R µ are: χ R µ ( C k k ) = T r ( e i ( k R µ ( c )+ k R µ ( c )) ) = d Rµ · e i ( k <µ,c > + k <µ,c > ) = d Rµ · e iπ ( k [ µ ]+ k [ µ ]) . (4.17)Here we’ve computed the pairing < µ, k c + k c > explicitly: iπ ( k P ni =1 ( a ) i µ i + k P ni =1 ( a ) i µ i ) ≡ iπ ( k [ µ ] + k [ µ ]). We denote by [ µ ] and [ µ ] two different linearcombinations ( P ni =1 ( a ) i µ i ∈ Z and P ni =1 ( a ) i µ i ∈ Z ) of the components of the samehighest weight µ modulo 2.Without Wilson surface the partition function for G = Spin (4 l ) / Z ⊕ Z is given by:– 12 – Y M ( τ ) = P k =0 P k =0 P R µ ( Spin (4 l )) d − gR µ e − τC ( R µ ) χ R µ ( C k k )= P k =0 P k =0 P R µ ( Spin (4 l )) d − gR µ e − τC ( R µ ) · e i ( k <µ,c > + k <µ,c > ) = P R µ ( Spin (4 l )) d − gR µ e − τC ( R µ ) · P k =0 P k =0 e iπ ( k [ µ ]+ k [ µ ]) = P R µ ( Spin (4 l ) / ( Z ⊕ Z )) d − gR µ e − τC ( R µ ) . (4.18)In more detail, the sum in the first line of (4.18) runs over the representations R µ of Spin (4 l ) and in the last line - over the representations R µ of Spin (4 l ) / ( Z ⊕ Z ). Thechange happens for the following reason. Each sum over k i in the second line is equal tozero, or to 2 if P ni =1 a i µ i is even, i.e. [ µ i ] = 0. This condition corresponds to chosing onlythose representations of Spin (4 l ) in which the elements C k k ∈ Z ⊕ Z are all trivial, i.e.the representations of Spin (4 l ) / ( Z ⊕ Z ).When we introduce a Wilson surface of weight λ , it will involve defining two equiv-alence classes for λ from the pairing < λ, k i c i > : [ λ ] = [ P nj =1 ( a ) j λ j ] ∈ Z and [ λ ] =[ P nj =1 ( a ) j λ j ] ∈ Z . The partition function in the presence of a Wilson surface is modifiedin the following way: Z λY M ( τ ) = P k =0 P k =0 P R µ ( Spin (4 l )) d − gR µ e − τC ( R µ ) χ R µ ( C k k ) χ λ ( C k k ) d λ = P k =0 P k =0 P R µ ( Spin (4 l )) d − gR µ e − τC ( R − µ ) e i<µ,k c + k c > e i<λ,k c + k c > = P k =0 P k =0 P R µ ( Spin (4 l )) d − gR e − τC ( R ) e i ( k [ µ ]+ k [ µ ]) e i ( k [ λ ]+ k [ λ ]) = P R µ ( Spin (4 l )) d − gR e − τC ( R ) 14 P k =0 P k =0 e iπ ( k [ µ + λ ]+ k [ µ + λ ]) = P µ ∈ [ − λ ] , µ ∈ [ − λ ] dim − gR µ e − τC ( R µ ) , (4.19)Here the sum over each k i is different from zero only for µ i + λ i even. This conditionreduces the sum in the last line to the sum over such representations R µ ( Spin (4 l )) thatthe representations of Spin (4 l ) with the highest weight µ + λ would correspond to therepresentations of Spin (4 l ) / ( Z ⊕ Z ). References [1] A.Cattaneo, C. Rossi,
Wilson surfaces and higher dimensional knot invariants , Commun.Math.Phys. (2005) 513 [math-ph/0210037].[2] B. Chen, W. He, J.-B. Wu and L. Zhang,
M5-branes and Wilson surfaces, JHEP (2007)067 [arXiv:0707.3978].[3] I. Chepelev, Non-Abelian Wilson Surfaces, JHEP (2002) 013 [hep-th/0111018]. – 13 –
4] O. Ganor,
Six-dimensional tensionless strings in the large N limit, Nucl. Phys.
B 489 (1997)95 [hep-th/9605201].[5] S.Gukov, A. Kapustin,
Topological Quantum Field Theory, Nonlocal Operators, and GappedPhases of Gauge Theories , arXiv:1307.4793[6] A. Kapustin,
Bosonic topologial insulators and paramagnets: a view from cobordisms ,arXiv:1404.6659.[7] A.J. Parzygnat,
Gauge invariant surface holonomy and monopoles , Theory and Applicationsof Categories , (2015)42:1319-1428, [arXiv:1410.6938].[8] K. Wilson, Confinement of quarks, Phys. Rev.
D 10 (1974) 2445.[9] R. Giles,
Reconstruction of gauge potentials from Wilson loops, Phys. Rev.
D 24 (1981) 2160.[10] E. Witten,
Quantum Field Theory and the Jones Polynomial, Comm. Math. Phys. (1989) 351.[11] A. Alekseev, L. Faddeev, S. Shatashvili,
Quantization of symplectic orbits of compact Liegroups by means of the functional integral, J. Geom. Phys (1988) 391.[12] A. P. Balachandran, S. Borchardt, A. Stern, Lagrangian And Hamiltonian Descriptions ofYang-Mills Particles, Phys. Rev.
D 17 (1978) 3247.[13] H. B. Nielsen, D. Rohrlich,
A Path integral to quantize Spin, Nuci. Phys.
B 299 (1988) 471.[14] D. Diakonov, V. Petrov,
Phys. Lett.
B 224 (1989) 131.[15] S. Elitzur, G. Moore, A. Schwimmer, N. Seiberg,
Remarks on the Canonical Quantization ofthe Chern-Simons-Witten Theory, Nucl. Phys.
B 326 (1995) 108.[16] C. Beasley,
Localization for Wilson Loops in Chern-Simons Theory , in J. Andersen, H.Boden, A. Hahn, and B. Himpel (eds.)
Chern-Simons Gauge Theory: 20 Years AfterAMS/IP Studies in Adv. Math. (2011), Adv. Theor. Math. Phys. (2013) 1[arXiv:0911.2687].[17] A. Alekseev, O. Chekeres, P. Mnev, Wilson surface observables from equivariant cohomology,JHEP (2015) 093 [arXiv:1507.06343].[18] N. Steenrod, The topology of Fiber Bundles , Princeton Mathematical Series , PrincetonUniversity Press (1951).[19] D. Husemoeller, Fiber bundles , 3rd ed., Springer Science+Business Media LLC, 3rd ed.(1994).[20] E. Witten,
On Quantum gauge theories in two dimensions, Commun. Math. Phys. (1991) 153.[21] E. Witten,
Two Dimensional Gauge Theories Revisited, J.Geom.Phys. (1992) 303[hep-th/9204083].[22] S. Cordes, G. Moore, S. Ramgoolam, Lectures on 2D Yang-Mills Theory, EquivariantCohomology and Topological Field Theories, Nucl. Phys. Proc. Suppl. (1995) 184[hep-th/9411210].[23] D. Diakonov, V. Petrov, Non-Abelian Stokes theorem and quark-monopole interaction [hep-th/9606104], Published version:
Nonperturbative approaches to QCD, Proceedings of theInternat. workshop at ECT* , Trento, July 10-29, 1995, D.Diakonov (ed.), PNPI (1995). – 14 –
24] D. P. Zhelobenko,
Compact Lie Groups And Their Representations , Translations ofMathematical Monographs , American Mathematical Society (1978).[25] R. Bott, The Geometry and Representation Theory of Compact Lie Groups , in
Representation Theory of Lie Groups , London Mathematical Society Lecture Note Series,Cambridge University Press (1979).[26] D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Topological Field Theories , Phys.Rep. (1991) 129.[27] A. Migdal,
Recursion Relations in Gauge Theories, Zh. Eksp. Teor. Fiz. (1975) 810 ( Sov.Phys. Jetp. Exact Computation of Loop Averages in Two-Dimensional Yang-Mills Theory,Phys. Rev.
D 22 (1980) 3090.[29] V. Kazakov, I. Kostov,
Non-linear Strings in Two-Dimensional U ( ∞ ) Gauge Theory, Nucl.Phys.
B 176 (1980) 199.[30] V. Kazakov, I. Kostov,
Computation of the Wilson Loop Functional in Two-Dimensional U ( ∞ ) Lattice Gauge Theory, Phys. Lett.
B 105 (1981) 453.[31] V. Kazakov,
Wilson Loop Average for an Arbitrary Contour in Two Dimensional U(N)Gauge Theory, Nuc. Phys.
B 179 (1981) 283.[32] L. Gross, C. King, A. Sengupta,
Two-Dimensional Yang-Mills via Stochastic DifferentialEquations, Ann. of Phys. (1989) 65.[33] B. Rusakov,
Loop Averages And Partition Functions in U(N) Gauge Theory OnTwo-Dimensional Manifolds, Mod. Phys. Lett.
A 5 (1990) 693.[34] D. Fine,
Quantum Yang-Mills On The Two-Sphere, Commun. Math. Phys. (1990) 273.[35] D. Fine,
Quantum Yang-Mills On A Riemann Surface, Commun. Math. Phys. (1991)321.[36] M. Blau, G. Thompson,
Quantum Yang-Mills Theory On Arbitrary Surfaces , Int. J. Mod.Phys.
A 7 (1992) 3781.(1992) 3781.