Effective Lagrangian for Non-Abelian Two-Dimensional Topological Field Theory
aa r X i v : . [ h e p - t h ] J a n Prepared for submission to JHEP
Effective Lagrangian for Non-AbelianTwo-Dimensional Topological Field Theory
Pongwit Srisangyingcharoen and Paul Mansfield
Centre for Particle Theory, University of Durham, Durham DH1 3LE, U.K.
E-mail: [email protected], [email protected]
Abstract:
An effective theory for 2 D non-Abelian topological BF theory is investigated.We develop a systematic approach by integrating out the non-Abelian gauge fields to obtainan effective theory containing solely scalar fields. Expressions for the SU (2) and SU (3)effective actions are explicitly stated. In the case of SU (2), we show that the effectiveaction can be interpreted as a winding number. By using the SU (2) effective action, thepartition function on a sphere for SU (2) Yang-Mills theory is calculated. Moreover, wegeneralise the theory to include a source term for the gauge field as well as calculate thevacuum expectation value of the Wilson loop based on the effective theory. The resultobeys the area law agreeing with known results. Keywords:
Topological field theory ontents SU (2) Effective BF Theory 43 Partition Function for SU (2) Yang-Mills Theory on Sphere 64 Generalization to an Arbitrary Lie Algebra 115 Diagrammatic Representation of the Inverse Matrix f M
146 Explicit Expressions for Effective SU (2) and SU (3) Lagrangians 177 The Topological Field Action with a Source Term and the ExpectationValue of the Wilson Loop 208 Conclusions 25A Expressions for Matrix Multiplications of the Matrix F Over the past few decades, the study of topological field theories has been important forboth mathematics and physics. The key feature of the theories is that observables dependonly on the global structure of the space where the theories are defined. Topological fieldtheories can be broadly categorized into two classes, namely Schwarz-type and Witten-type. Well-known examples of the Schwarz-type theories are the three-dimensional Chern-Simons model [1] as well as BF theories [2, 3]. A representative for the latter class istopological Yang-Mills theory [4, 5]. In this paper, we are interested in the BF theory intwo-dimensional Euclidean space.BF theories are the only known topological Schwarz-type theories which can be ex-tended to any arbitrary dimension of spacetime. They can be considered as a generalizationof Chern-Simons theory. Much research has been focused on this model [6–13].By using perturbative renormalization, the BF model exhibits finiteness for particulargauge fixings in two [14], three[15], four [16] and higher dimensions [17]. The proof involvesthe existence of the supersymmetric structure found in [18]. Moreover, the model plays animportant role as an alternative theory of gravity [19–25] and quite recently, it has becomeof interest in condensed matter physics [26–32].– 1 – BF theory is a diffeomorphism-invariant gauge theory. On a D -dimensional manifold M ( D ≥
2) with structure group, a Lie group G , the classical action of the non-AbelianBF theory takes the form S = 2 Z M tr( B ∧ F ) (1.1)where B is a ( D − G . F is a curvature 2-formof a connection 1-form A defined by F = dA + g [ A, A ]. The trace implies a scalar productin the algebra. Notice that the action is topologically invariant because it is independentof the metric. The equations of motion with respect to B and A are F = 0 and d A B = 0 (1.2)where d A is a covariant derivative defined as d A = d + g [ A, ]. The action is invariantunder local gauge transformation with gauge parameter ω as δA = d A ω and δB = [ B, ω ] + d A η. (1.3)The field η is a ( D − B field,namely B symmetry which only appears when D ≥ D = 2, one can express (1.1) as S = 2 Z M d ξ ǫ ij tr( φ F ij ) (1.4)or equivalently, S [ φ, A ] = Z M d ξ ig A iA A jB f ABC φ C − ∂ i φ A A Aj ! ǫ ij (1.5)where F ij = ∂ i A j − ∂ j A i + g [ A i , A j ] is the field-strength of the gauge field A . Both fields φ and A are elements of a non-Abelian group and so can be written in terms of a set ofgenerators { T R } as φ = φ R T R and A = A R T R . To obtain (1.5), the boundary term, i.e. R d ξ∂ i ( φ · A j ) ǫ ij , is assumed to vanish. Notice that in two dimensions, the B field is a0-form, thus, it is natural to replace it with the scalar field φ .The action (1.4) has a close connection to the Yang-Mills action in two dimensions asthey are equivalent in the zero coupling constant limit [5, 33]. This can be seen by addinga quadratic term with coupling constant e to the action and then integrating out the field φ in the path integral below using Gaussian integration Z D A Dφ exp (cid:18) Z M d ξ (cid:18) ǫ ij tr( φ F ij ) + e √ g tr( φφ ) (cid:19)(cid:19) = Z D A exp (cid:18) e Z M d ξ √ g tr( F ij F ij ) (cid:19) . (1.6)In this paper, we will obtain the effective Lagrangian purely in terms of the scalarfield φ for the BF theory resulting in a theory that is both gauge and Weyl invariant. tr( T A T B )= η AB and [ T A , T B ] = if ABC T C – 2 –ur motivation for constructing this model is to be able to generalise the results of [34],[35], and [36] in which it was shown that the Wilson loop for D-dimensional Abeliangauge theories could be obtained from a spinning string theory in which the physicaldegrees of freedom described by the string are electric lines of force. The interactionin this model is not the usual splitting/joining interaction of fundamental string theorybut rather (the supersymmetrisation of) a contact interaction that is supported on self-intersections of the string. If the string target space co-ordinates are denoted by X µ ( ξ , ξ )where ξ i are world-sheet parameters for an open string with fixed boundary curve C then d Σ µν ( ξ ) = d ξ ′ ǫ ij ∂ i X µ ( ξ ) ∂ j X µ ( ξ ) is an element of area in target space and the contactinteraction takes the form Z d Σ µν ( ξ ) δ D (cid:0) X ( ξ ) − X ( ξ ′ ) (cid:1) d Σ µν ( ξ ′ ) (1.7)with a coefficient proportional to the square of the electric charge. Averaging this over fluc-tuating world-sheets results in the gauge-field propagator connecting points X ( ξ ) and X ( ξ ′ )when they are on the boundary C . We would like to generalise this to non-Abelian gaugetheory so at the very least we need to introduce Lie algebra-valued world-sheet degrees offreedom φ ( ξ ) to try to reproduce the Lie algebra structure of Yang-Mills propagators Z d Σ µν ( ξ ) δ D (cid:0) X ( ξ ) − X ( ξ ′ ) (cid:1) d Σ µν ( ξ ′ ) tr (cid:0) φ ( ξ ) φ ( ξ ′ ) (cid:1) . (1.8)As a consequence of the δ -function this interaction is invariant under gauge transformationswhich are functions on target-space if φ transforms as φ ( ξ ) → g ( X ( ξ )) φ ( ξ ) g − ( X ( ξ )). Toconstruct a string theory describing non-Abelian lines of force we need a Lagrangian todescribe the dynamics of φ . This has to be gauge-invariant to preserve the space-time gaugeinvariance of the contact interaction and Weyl invariant to satisfy the usual organisingprinciple of string theories. It also has to generate the extra interactions of non-Abeliangauge theories which are absent from Abelian ones. A candidate for the dynamics of φ isobtained by integrating out the gauge fields in (1.4) leaving a Weyl invariant and gauge-invariant Lagrangian expressed in terms of φ only. We shall not address here the issueof whether the resulting theory does indeed generate the self-interactions of Yang-Millstheory but simply concentrate on deriving the model and addressing the examples of gaugegroups SU (2) and SU (3) relevant to the Standard Model.The outline of this paper is as follows. In section two, an explicit calculation for the SU (2) effective BF theory is shown. In section three we use the result to give a newderivation of the known expression for the partition function on a sphere of SU (2) Yang-Mills theory. The calculation for obtaining the effective theory is generalised for arbitraryLie algebras in section four. In section five, we construct a set of diagrams to represent theingredients appearing in the effective Lagrangian in order to aid our calculation. Using theresult found in section five we present the explicit form for the SU (3) effective Lagrangianin section six. Finally, in section seven, we investigate the BF model with a source termfor a gauge field A as well as calculate the expectation value of the Wilson loop in theeffective theory. – 3 – SU (2) Effective BF Theory
We begin our calculation with the simplest model for the non-Abelian two-dimensionaltopological field theory, i.e. the BF theory for SU (2). The partition function for thistheory is defined as Z = 1Vol Z DφD A e − S [ φ, A ] (2.1)where S [ φ, A ] is expressed in (1.5). The functional integral is divided by the volume of thegauge symmetry which is denoted by Vol.To obtain an effective theory for the scalar field φ , the gauge field A needs to beintegrated out. For that purpose, we express all fields in terms of a set of orthonormalbases, i.e. ˆ φ, ˆ E + , and ˆ E − , as φ A = ϕ ˆ φ A and A Ai = χ i ˆ φ A + a + i ˆ E A + + a − i ˆ E A − . (2.2)Note that these bases are ξ -dependent. They are defined throughout the manifold point bypoint. Obviously, we have chosen a unit vector ˆ φ to align in the direction of φ at each point.In terms of the usual cross products, the ξ -dependent bases give the following relations:ˆ φ × ˆ E + = ˆ E + , ˆ E + × ˆ E − = ˆ φ, ˆ E − × ˆ φ = ˆ E − . (2.3)Substituting (2.2) into (1.5), the action takes the form S [ φ, A ] = Z M d ξ igϕ a + i a − j − ∂ i φ A χ j ˆ φ A − ∂ i φ A a + j ˆ E A + − ∂ i φ A a − j ˆ E A − ! ǫ ij . (2.4)To obtain the first term, the relations (2.3) were utilised. Note that the structure constant f ABC is equal to ǫ ABC for SU (2).Rewriting all the fields using (2.2), the measure D A now turns into DχDa + Da − .Integrating out χ would generate a constraint via the Dirac delta function as Z Dχ i exp (cid:18) Z M d ξ φ A ∂ i φ A χ j ǫ ij (cid:19) = N Y ∀ ξ ∈M ϕ δ (2) ( ∂ϕ ) = N Y ∀ ξ ∈M δ (2) ( ∂ϕ ) . (2.5)This means ϕ (equivalently | φ | ) is constant throughout the space M .To proceed with the path integration with respect to the field a αi with α = ± , it isbetter to change the spacetime coordinates ξ and ξ into complex coordinates which aredefined by z = ξ + iξ and ¯ z = ξ − iξ . (2.6)In these new coordinates, the field a αi becomes complex fields b α where b α = 12 ( a α − ia α ) and ¯ b α = 12 ( a α + ia α ) . (2.7)Therefore, the path integral (2.1) takes the form Z = 1Vol Z DφDbD ¯ b Y ∀ ξ ∈M δ (2) ( ∂ϕ ) e − S [ φ,b, ¯ b ] (2.8)– 4 –here S [ φ, b, ¯ b ] = Z M d z − ¯ b α igϕǫ αβ b β + 2 ¯ ∂φ A ˆ E Aα b α − ∂φ A ˆ E Aα ¯ b α ! . (2.9)We can then use the Gaussian integration formula to integrate out the complex field b , Z DbD ¯ b e − R d z ( − ¯ b α M αβ b β + ¯ J α b α + J α ¯ b α ) = N e − R d z ( ¯ J α ( M − ) αβ J β Q ∀ ξ det( M ) . (2.10)According to (2.9), it is not hard to see that M αβ = 2 igϕǫ αβ , J α = − ∂φ A ˆ E Aα , and ¯ J α = 2 ¯ ∂φ A ˆ E Aα . (2.11)Using the fact that ǫ ij ǫ ij = 2, the inverse and the determinant of the matrix M are( M − ) αβ = − i gϕ ǫ αβ , and det( M ) = − g ϕ . (2.12)Consequently, we can express the path integral as Z ∼ Z Dφ Y ∀ ξ ∈M − i ( gϕ ) δ (2) ( ∂ϕ ) exp (cid:20) − Z M d z igϕ ¯ ∂φ A ∂φ B ( ˆ E Aα ǫ αβ ˆ E Bβ ) (cid:21) . (2.13)We can rewrite the term ˆ E Aα ǫ αβ ˆ E Bβ asˆ E Aα ǫ αβ ˆ E Bβ = ˆ E A + ˆ E B − − ˆ E B + ˆ E A − = ( ˆ E + × ˆ E − ) C ǫ ABC (2.14)which can be evaluated using (2.3). As a result, the cross product on the right-hand side issimply the unit vector ˆ φ . Thus, the effective action for two-dimensional SU(2) BF theorycan be written as Z M d z ig | φ | ¯ ∂φ A ∂φ B φ C ǫ ABC (2.15)or equivalently in the ( ξ , ξ ) coordinates as Z M d ξ i g | φ | ∂ i φ A ∂ j φ B ǫ ij φ C ǫ ABC . (2.16)Now, let us give an interpretation of the effective action (2.16). The effective actioncan be seen as a winding number (up to a constant). To see this, it needs to be noted thatthe unit vector ˆ φ ( ξ ) maps a point on the manifold M into a point on S , i.e. ˆ φ : M → S .Furthermore, the integrand of the action (2.16),12 ∂ i ˆ φ A ∂ j ˆ φ B ǫ ij ˆ φ C ǫ ABC , (2.17)is the area element on the target space S . This can be seen as follows: the variations of themanifold coordinates δξ and δξ correspond to two infinitesimal tangent vectors δξ ∂ ˆ φ and δξ ∂ ˆ φ on S . The cross product of these two vectors has direction ˆ φ and magnitude– 5 – A . Consequently, the triple product, δξ δξ ( ∂ ˆ φ × ∂ ˆ φ ) · ˆ φ , is basically an infinitesimalarea on the target space S as claimed.The integration over all manifold coordinates ξ of the integrand (2.17) yields the totalarea of the unit sphere times an integer corresponding to the winding number n as12 Z S d ξ∂ i ˆ φ A ∂ j ˆ φ B ǫ ij ˆ φ C ǫ ABC = 4 πn. (2.18)Note that the above term is proportional to the effective action (2.16) as the magnitude ofthe field φ , | φ | , is constant due to the constraint (2.5). SU (2) Yang-Mills Theory on Sphere
It is well known that a general expression for partition function for SU ( N ) Yang-Millstheory on a sphere is Z YM ( A ) = X R ( d R ) exp (cid:0) − e AC ( R ) (cid:1) (3.1)where A is an area of the sphere and R is an irreducible representation of SU ( N ). d R and C ( R ) are the dimension and the quadratic Casimir of the representation R respectively[37]. For SU (2), the representation R is characterized by a positive half-integer l . Thisyields d R = 2 l + 1 and C ( R ) = l ( l + 1) . (3.2)Therefore, the partition function takes the form Z YM ( A ) = ∞ X m =0 ( m + 1) exp (cid:0) − e A (( m + 1) − (cid:1) (3.3)where l = m/ SU (2) BF theory found in the previous section. To do this, we need to be morecareful in integrating out the complex b field in (2.8) as one may notice that ϕ in thedeterminant (2.12) will apparently get cancelled out by the Jacobian of the measure Dφ = ϕ dϕd Ω with Ω denoting a direction of the scalar field. If the previous statement were true,we would not get the prefactor in the formula (3.3). This implies that the cancellation needsto be partial. It is due to the difference in the degrees of freedom between the scalar fieldand the vector field.To put it into clearer perspective, let us evaluate the SU (2) partition function, i.e. Z = 1Vol Z DφDχD ¯ χDbD ¯ b exp (cid:0) − S [ φ, χ, ¯ χ ] − S [ φ, b, ¯ b ] (cid:1) (3.4)where S [ φ, χ, ¯ χ ] = 2 Z M d z ˆ φ A ( ∂φ A ¯ χ − ¯ ∂φ A χ ) (3.5)– 6 –nd S [ φ, b, ¯ b ] is expressed as (2.9). We then expand all the fields in terms of eigenfunctionsof the scalar Laplacian, ∇ u λ = λu λ . (3.6)Therefore, the expression for the real scalar field ϕ is ϕ = X λ =0 c λ u λ + ϕ (3.7)where the zero mode term ϕ = c u and those for the complex vector fields are b α = X λ =0 e αλ ∂u λ , ¯ b α = X λ =0 ¯ e αλ ¯ ∂u λ (3.8) χ = X λ =0 f λ ∂u λ , ¯ χ = X λ =0 ¯ f λ ¯ ∂u λ . (3.9)Note that there is no zero mode expansion for the vector fields as ∂u = 0 and u λ forms acomplete set of orthonormal basis satisfying Z d ξ √ gu λ ( ξ ) u λ ′ ( ξ ) = δ λλ ′ and √ g X λ u λ ( ξ ) u λ ( ξ ′ ) = δ (2) ( ξ − ξ ′ ) . (3.10)Now, let first take a look at the integral Z DχD ¯ χ exp (cid:0) − S [ φ, χ, ¯ χ ] (cid:1) . (3.11)By using the basis expansions, the integral (3.11) takes the form Z | J | Y λ df λ d ¯ f λ exp (cid:18) Z d z X λ,λ ′ c λ ( ∂u λ ¯ ∂u λ ′ ¯ f λ ′ − ¯ ∂u λ ∂u λ ′ f λ ′ ) (cid:19) (3.12)where J is the Jacobian determinant when changing variables from χ and ¯ χ to f λ and ¯ f λ .Therefore, it can be computed by J = det (cid:18) δ ( χ, ¯ χ ) δ ( f λ , ¯ f λ ) (cid:19) ≡ det( M ) (3.13)The determinant of the matrix can be evaluated from the relationdet( M ) = q det( M † M ) . (3.14)According to (3.9), δχ ( z ) δf λ = ∂u λ ( z ) and δ ¯ χ ( z ) δf λ = ¯ ∂u λ ( z ). Therefore, J = vuut det R d z ¯ ∂u λ ∂u λ ′ R d z ¯ ∂u λ ∂u λ ′ ! = Y λ λ, (3.15)where (3.10) was utilised to obtain the last expression and the product is over the non-zeroeigenvalues. – 7 –hen applying the completeness relation (3.10) to the exponent of (3.12), it is nothard to see that the integral becomes Z Y λ ( − iλ ) d (Re( f λ )) d (Im( f λ )) exp (cid:0) ic λ λ Im( f λ ) (cid:1) = Z Y λ d (Re( f λ ))( − πiλ ) δ (4 c λ λ )= Y λ Vol(Re( f λ ))( − πiδ ( c λ )) . (3.16)Similar to the expression (2.5), the above term provides a constraint on the theory via theDirac delta function δ ( c λ ) requiring the modulus of the scalar field ϕ to be constant, i.e. ϕ = ϕ , throughout the space.The volume of the real number, Vol(Re( f λ )), can be cancelled with the volume of thegauge symmetry in (3.4). To see this, let us apply a particular choice of gauge-fixing to ourcalculation. We consider a gauge condition that makes the direction of the scalar field, ˆ φ ,constant everywhere except for an infinitesimal region. After this gauge has been applied,there is left the residual gauge symmetry which does not alter the direction ˆ φ .Expanding an infinitesimal gauge transformation parameter ω as ω = ω φ ˆ φ + ω + ˆ E + + ω − ˆ E − (3.17)where all components are real, the gauge transformation of the scalar field (1.3) impliesthat the residual symmetry has ω ± = 0. Now, let us investigate the effect of this residualsymmetry on the gauge field A where A takes the form A = χ ˆ φ + b + ˆ E + + b − ˆ E − . (3.18)According to (1.3), a variation of the gauge field with respect to the residual gauge trans-formation is δ ω A = ∂ω + g [ A , ω ]= − i∂ω φ ˆ R − iω φ ∂ ˆ R + gω φ (cid:18) √ b + − b − ) ˆ B + i √ b + + b − ) ˆ B (cid:19) (3.19)where we have re-defined the bases to beˆ R = i ˆ φ, ˆ B = 1 √ E + + ˆ E − ) , ˆ B = − i √ E + − ˆ E − ) . (3.20)Note that these bases resemble a set of ordinary unit vectors in three-dimensional spherein which they obey the following algebras;[ ˆ R, ˆ B ] = ˆ B , [ ˆ B , ˆ R ] = ˆ B , [ ˆ B , ˆ B ] = ˆ R. (3.21)As a result, if the sphere is characterised by the usual polar angle α and azimuthal angle θ , the variation (3.19) becomes δ ω A = − i∂ω φ ˆ R + ω φ (cid:18) i ∂α∂z sin θ + g √ b + − b − ) (cid:19) ˆ B − iω φ (cid:18) ∂θ∂z − g √ b + + b − ) (cid:19) ˆ B . (3.22)– 8 –omparing the result to the actual variation of the gauge field (3.18), it implies that avariation of the field χ is in the residual gauge orbit when it is real. Remember thatthe variation of the field χ is equivalent to that of the function f λ according to (3.9)).Consequently, Vol(Re( f λ )) is the residual gauge volume as claimed.Moving on to the next integral to consider, the Gaussian functional integral of thevector fields b α in the partition function (3.4) can be written in terms of scalar functions e λ and ¯ e λ according to the Laplacian eigenfunction expansion (3.8) as | J | Z Y λ de λ d ¯ e λ e − S [ e, ¯ e ] (3.23)where J is the Jacobian determinant resulted from the change of variables from b and ¯ b into e and ¯ e . The action S [ e, ¯ e ] is defined as S [ e, ¯ e ] = Z M d z − igϕ X λ,λ ′ ¯ e αλ ǫ αβ e βλ ′ ¯ ∂u λ ∂u λ ′ + 2 ϕ ¯ ∂ ˆ φ A ˆ E Aα X λ e αλ ∂u λ − ϕ ∂ ˆ φ A ˆ E Aα X λ ¯ e αλ ¯ ∂u λ ! . (3.24)To obtain the above action, the constraint (3.16) is applied making the length of φ constant.The first term of the action (3.24) vanishes when λ = λ ′ due to the completeness relation(3.10) which yields S [ e, ¯ e ] =2 igϕ X λ λ ¯ e αλ ǫ αβ e βλ + 2 ϕ Z d z (cid:16) ¯ ∂ ˆ φ A ˆ E Aα X λ e αλ ∂u λ − ∂ ˆ φ A ˆ E Aα X λ ¯ e αλ ¯ ∂u λ (cid:17) . (3.25)The Jacobian determinant J is J = det (cid:18) δ ( b ( z ) , ¯ b ( z )) δ ( e λ , ¯ e λ ) (cid:19) . (3.26)By using the relation (3.14) and fact that δb α ( z ) δe βλ = δ αβ ∂u λ ( z ) and δ ¯ b α ( z ) δ ¯ e βλ = δ αβ ¯ ∂u λ ( z ), onecan obtain J = vuut det R d z ¯ ∂u λ ∂u λ ′ δ αβ R d z ¯ ∂u λ ∂u λ ′ δ αβ ! = Y λ λ . (3.27)where (3.10) was utilised and again the product is over non-zero eigenvalues.We can then calculate the Gaussian integral (3.23) over the complex field e λ and ¯ e λ using (2.10). It becomes Z Y λ λ de λ d ¯ e λ e − S [ e, ¯ e ] = exp( − S eff [ ϕ .n ]) Q λ − (2 gϕ ) (3.28)– 9 –here S eff [ ϕ , n ] = i ϕ g Z d z ¯ ∂ ˆ φ A ∂ ˆ φ B ˆ φ C ǫ ABC = 4 πni ϕ g . (3.29)Note that the effective action is related to the winding number n as shown in (2.18).The last element to consider is the decomposition of the measure Dφ . This can beobtained by considering a small variation of the field φ as δφ = δϕ ˆ φ + ϕδ ˆ φ (3.30)with δ ˆ φ = δω + ˆ E + + δω − ˆ E − (3.31)where δω ± are small variations in the tangent directions. The variations δϕ and δω ± canbe expanded in terms of the eigenfunction u λ as δϕ ( ξ ) = X m δc m u m ( ξ ) (3.32) δω ± ( ξ ) = X m δµ ± m u m ( ξ ) (3.33)Consequently, we can rewrite the measure as Dφ = | J | Y m dc m dµ + m dµ − m ≡ | J | Y m dc m d Ω (3.34)where the Jacobian determinant can be computed by J = det (cid:18) δφ A ( ξ ) δ ( c m , µ + p , µ − q ) (cid:19) ≡ det( M IJ ) . (3.35) M IJ is the Jacobian matrix where the row index I ≡ A, ξ and the column index J ≡ m, p, q .Again, the relation (3.14) is used to determine the Jacobian determinant.As δφ A ( ξ ) δc m = ˆ φ A ( ξ ) u m ( ξ ) and δφ A ( ξ ) δµ ± m = ˆ E A ± ( ξ ) ϕu m ( ξ ), It is not hard to see that M † M is (cid:0) R ξ u m ( ξ ) u m ′ ( ξ ) (cid:1) mm ′ (cid:0) R ξ ϕ u m ( ξ ) u m ′ ( ξ ) (cid:1) mm ′ (cid:0) R ξ ϕ u m ( ξ ) u m ′ ( ξ ) (cid:1) mm ′ (3.36)where R ξ is a shorthand for R √ gd ξ . Note that the objects in the parentheses are thematrix elements in row m and column m ′ . We can then utilise the fact that the value of ϕ is the constant ϕ throughout the space due to the constraint (3.16). This allows us toobtain the absolute value of the Jacobian determinant as | J | = Y m ( ϕ ) . (3.37)It is clear that the product of ( ϕ ) in (3.37) cannot be completely cancelled by theone in (3.28) as mentioned. The cancellation leaves a single factor of ( ϕ ) behind. Thisremaining factor accounts for the pre-factor of the partition function as we shall see later.– 10 –n consequence, when substituting (3.16), (3.28), (3.34), and (3.37) into (2.8), thegauge-fixed partition function takes the form Z = N Z dc (cid:18) Y λ dc λ δ ( c λ ) (cid:19) ϕ ∞ X n = −∞ exp( − S eff [ ϕ , n ]) (3.38)where S eff [ ϕ , n ] is expressed in (3.29).According to (1.6), we can relate the BF theory to two-dimensional Yang-Mills theoryby adding a quadratic term in the scalar field. Consequently, the partition function for 2DYang-Mills is Z = e N Z ∞ ϕ dϕ ∞ X n = −∞ exp( − S eff [ ϕ , n ] − e Z S d ξ √ gϕ )= e N Z ∞ ϕ dϕ ∞ X n = −∞ exp (cid:0) − πig nϕ − e ϕ A (cid:1) . (3.39)where A is the area of the sphere. The infinite sum of the Euler exponential provides aDirac delta function. This discretises the possible values of ϕ in the theory as ∞ X n = −∞ exp (cid:0) − πig nϕ (cid:1) = g δ (cid:0) ϕ mod g (cid:1) . (3.40)Therefore, it is not hard to see that the expression (3.39) turns into Z ∼ ∞ X m =1 m exp (cid:0) − ( eg ) Am (cid:1) . (3.41)The result (3.41) is in agreement with the expression (3.3). They differ by the factor − In this section, we would like to generalise the approach we used in section two to an arbi-trary Lie algebra. As seen in the earlier section, one of the key elements in our calculationis to expand the fields in terms of a set of suitable Lie bases. For a general Lie algebra, wewill work in the Cartan-Weyl basis.We will denote the Cartan generator H a and Weyl generator E α where a = 1 , . . . , N − α is a root of eigenvalue equation, ad H a ( E α ) = α a E α . The roots α forms a vectorspace Φ. The generators H a and E α satisfy the following algebra:[ H a , H b ] = 0 , [ H a , E α ] = α ( a ) E α , and [ E α , E β ] = ( N αβ E α + β if α + β ∈ Φ H α if α + β = 0 (4.1)where H α is defined as H α = α a H a . The Cartan generators H a are diagonal tracelessmatrices in the adjoint representation. – 11 –gain, we start the calculation with the action (1.5) with the path integral defined by(2.1). The calculation proceeds by expanding the fields φ and A i in the Cartan-Weyl basisas φ = φ a H a and A i = χ ia H a + a iα E α . (4.2)Similar to the SU(2) case, these bases are ξ -dependent. The Cartan generators were chosensuch that the field φ lies within their subalgebra.To relate Lie indices A with the Cartan and Weyl indices a and α , we introduce unitvectors ˆ H aA and ˆ E αA in Lie vector space which are defined as δ aA and δ αA respectively. As aresult, the inner products among the vectors areˆ H aA ˆ H Ab = η ab , ˆ E αA ˆ E Aβ = η αβ , ˆ H aA ˆ E Aα = 0 (4.3)and the completeness relation is ˆ H Aa ˆ H aB + ˆ E Aα ˆ E αB = δ AB . (4.4)It is not hard to write the field φ and A i in terms of the unit vectors as φ A = φ a ˆ H Aa and A Ai = χ ai ˆ H Aa + a αi ˆ E Aα . (4.5)Using the relations (4.5), one can find the topological field theory action (1.5) as S [ φ, χ, a ] = Z M d ξ igf ABC φ C a αi a βj ˆ E αA ˆ E βB − ∂ i φ A ) a αj ˆ E Aα − ∂ i φ A ) χ aj ˆ H Aa ! ǫ ij . (4.6)Notice that there is no contribution from diagonal components of A Ai to the first term asthe Cartan subalgebra is commutative.To obtain the effective Lagrangian of the field φ , we need to integrate out the variables χ ai and a αi . According to the action (4.6), integrating out χ ai would provide a constraintvia the Dirac-delta function as Z Dχ ja exp(2 Z d ξ ( ∂ i φ A ) χ ja ˆ H aA ǫ ij ) = N N − Y a =1 δ (2) (( ∂φ A ) ˆ H aA )= N N − Y a =1 δ (2) (2tr(( ∂φ ) H a )) . (4.7)This implies that the derivative of the field φ , i.e. ∂ i φ , has no H a component. This providesa constraint on the theory as tr( φ∂ i φ ) = 0.This constraint (4.7) also implies that the square of the field φ , i.e. φ A φ A ≡ | φ | ,is constant throughout the space which is similar to what we found earlier in the SU(2)theory. Apart from that, it also implies the existence of the new invariant quantity, d ABC φ A φ B φ C (4.8)where d ABC is a totally symmetric third rank tensor defined by d ABC = 2tr( { T A , T B } T C ) . (4.9)– 12 –p to this point we have ignored a boundary in (1.5). We will now consider the effect ofincluding this term 2 R d ξ∂ i ( φ A A Aj ) ǫ ij . It affects the constraints. To see this, let considerthe case when the manifold M has the topology of a disk. This manifold can be mappedto the upper-half plane parameterised by Cartesian coordinates. Therefore, the boundaryterm takes the form − Z d x δ ( y ) φ A A Ax . (4.10)By expanding the gauge field A as (4.5), this turns the theory constraints (4.7) into Y a δ (2tr( ∂ x φ ) H a ) δ (2tr( ∂ y φ − δ ( y ) φ ) H a ) . (4.11)This implies that the squared of the field φ is no longer constant throughout the manifold M . There is a discontinuity of | φ | at the boundary in the y direction as | φ | ( x, ǫ ) = 3 | φ | ( x, . (4.12)To perform the path integration with respect to the field a αi , we apply the same trickwe used in the previous section. We change the spacetime coordinates ξ and ξ intothe complex coordinates z and ¯ z which were previously defined in (2.6). Of course, thiscoordinate transformation modifies the field a αi into the complex field b α as stated in (2.7).As a result, the partition function now takes form Z = 1Vol Z Dφ A Db α D ¯ b α N − Y a =1 δ (2) (2tr(( ∂φ ) H a ))exp( − S [ φ, b, ¯ b ]) (4.13)where the action is expressed in the complex coordinates as S [ φ, b, ¯ b ] = 2 Z D d z igf ABC φ C b α ¯ b β ˆ E αA ˆ E βB − ( ∂φ A ¯ b α − ¯ ∂φ A b α ) ˆ E Aα ! . (4.14)The path integral of the complex fields b α and ¯ b α resembles a Gaussian integral whichcan be performed using (2.10). By comparing (4.14) with (2.10), one obtains M αβ = 2 gif ABC φ B ˆ E αA ˆ E βC , J α = − E Aα ∂φ A , ¯ J α = 2 ˆ E Aα ¯ ∂φ A . (4.15)Consequently, it is not hard to find that the effective Lagrangian with respect to the scalarfield φ is L eff ( φ ) = − i g ¯ J α ( f M − ) αβ J β = 2 ig ∂φ A ¯ ∂φ B (cid:16) ˆ E Aα ( f M − ) αβ ˆ E Bβ (cid:17) (4.16)where we used M αβ = 2 gi f M αβ .A general expression for an inverse matrix f M αβ is( f M − ) αβ = adj( f M ) αβ det( f M ) (4.17)– 13 –here adj( f M ) αβ = δ αj ...j n βi ...i n f M i j f M i j . . . f M i n j n , det( f M ) = δ j j ...j n i i ...i n f M i j f M i j . . . f M i n j n . (4.18) δ j j ...j n i i ...i n is a generalised Kronecker delta which is related to an anti-symmetrization ofordinary Kronecker deltas as δ j j ...j n i i ...i n = n ! δ j [ i δ j i . . . δ j n i n ] . (4.19)The integer n is the number of Weyl generators. In the case of SU(N), n is equal to N − N .To obtain the adjugate matrix and the matrix determinant expressed in (4.18), thematrices f M ij are contracted with each other depending on the permutations implicit by(4.19). For the adjugate matrix adj( f M ) αβ , the contractions lead to two types of terms.First, the matrices f M ij are contracted in such a way that they form a new matrix withindices α and β . This contraction generates a chain of matrix multiplications, for instance, f M αj f M j j f M j j f M j β . In this example, the matrices f M i j f M i j f M i j f M i j are contractedwith δ αi δ j i δ j i δ j i δ j β . Second, the contraction forms a trace of matrix products, i.e. tr( f M · f M . . . f M ). For example, when the same matrices f M i j f M i j f M i j f M i j are contractedwith δ j i δ j i δ j i δ j i . However, only the latter case contributes to the matrix determinantdet( f M ).In addition, the trace term vanishes when the number of matrices f M inside is odd.This can be seen explicitly by consideringtr( f M · f M . . . f M ) = f M αi f M i i . . . f M i k − i k − f M i k − α = f A B C φ B η C A f A B C φ B η C A · . . . · f A k B k C k φ B k η C k A . (4.20)We used the completeness relation (4.4) to obtain the last line. When we swap the firstand the third indices of each structure constant f ABC , it gives an extra ( −
1) to the lastline so the whole expression vanishes.The calculation of the inverse matrix (4.17) involves a lot of contractions correspondingto chains of matrix multiplications. To facilitate the calculation, it is sensible to develop aset of diagrams to represent them. These diagrams are presented in the next section. f M ji ≡ f M ij = ˆ E iA ( f ABC φ B ) ˆ E jC i j ≡ δ ij . Figure 1 . Diagrammatic representation for matrix element f M and Kronecker delta – 14 – α = f M αi f M ij f M jβ = ˆ E αA ( f ABC φ B η CD f DEF φ E η F G f GHI φ H ) ˆ E βI = f M ij f M jk f M kl f M li = f ABC φ B η CD f DEF φ E η F G × f GHI φ H η IJ f JKL φ K η LA Figure 2 . Examples for a strand and loop diagram representing certain matrix multiplications
According to the previous section, the inverse of the matrix f M is an essential ingredientof the SU ( N ) effective Lagrangian (4.16). To compute this object, the relation (4.17) isused. However, this is complicated by the large number of terms.For this reason, we would like to develop a set of diagrams to capture the contractionsbetween matrix elements f M αβ and Kronecker deltas δ ij . We represent these two objects asthe vertices and lines shown in figure 1.Based on this diagrammatic representation, matrix multiplication is represented byvertices connecting by a line. Note that no more than two lines are allowed to be connectedto each vertex. This fact implies that a diagram involved in the calculation is either a strandor a loop which corresponds to a chain of matrix multiplications and its trace respectively.Just for clarification, we show some examples for a loop diagram and a strand diagram aswell as their corresponding matrix representations in figure 2.According to (4.18), the adjugate matrix, adj( f M ) αβ , can be expressed diagrammati-cally as a summation of all possible products between a strand diagram and loop diagrams.The diagram includes n − n = N − N for SU ( N ) ( n is always evenfor N ≥ n −
1. Then, for each strand,loop diagrams can be created using the remaining vertices. Therefore, we can expand the– 15 –djugate matrix asadj( f M ) αβ = ( − n − ( ( n − α . . . β ( n −
1) terms − ( n − α . . . β ( n −
3) terms × (cid:18) n − (cid:19) − ( n − α . . . β ( n −
5) terms × (cid:18) n − (cid:19)" − (cid:0) (cid:1)(cid:0) (cid:1) − . . . − α β × (cid:18) n − n − (cid:19)" ( n − . . .. . . ( n −
2) terms + . . . + ( − n − − . . . ( n − / . (5.1)There is no contribution from loops with odd vertices as they are zero as discussed previ-ously. The minus sign factor comes from an antisymmetric permutation of the generalisedKronecker delta. Each time the diagram collapses to form smaller loops, an extra (-1)appears which corresponds to an odd permutation of the lower indices of the Kroneckerdelta in (4.19). The numbers in front of the diagrams count the multiplicities.One can also see that the indices α and β from the adj( f M ) αβ are embedded at theends of the strands corresponding to the basis ˆ E Aα . Consequently, we can always factorout these bases to write the adjugate matrix asadj( f M ) αβ = ˆ E αA Θ AB ˆ E βB (5.2)or equivalently Θ AB = ˆ E Aα (adj( f M ) αβ ) ˆ E βB . Due to the above relation, it is not hard to seethat the effective Lagrangian takes the form L eff ( φ ) = 2 ig f M ) ∂φ A Θ AB ¯ ∂φ B . (5.3)Unlike the adjugate matrix, only loop diagrams contribute to the matrix determinantdet( f M ). There are n vertices involve in the expression of the matrix determinant. Det( f M )is expressed as the sum over all product of loops. To obtain these, we can start with thebiggest loop of n vertices and then cut it down to form smaller loops. The expression fordet( f M ) is shown in the equation (5.4). To avoid overcounting, all diagrams in the squared– 16 –rackets contain the same number for fewer vertices than the loop in front of the bracket.Therefore, the general expression for the determinant isdet( f M ) = ( − n − ( ( n − . . .. . . n terms − (cid:18) n (cid:19) ( n − . . .. . . ( n −
2) terms × " − (cid:18) n (cid:19) ( n − . . .. . . ( n −
5) terms × " − (cid:0) (cid:1)(cid:0) (cid:1) − . . . − (cid:18) nn − (cid:19) × " ( − n − n − (cid:0) n − n (cid:1)(cid:0) n − n (cid:1) · . . . · (cid:0) (cid:1) ( n )! . . . ( n − / . (5.4) SU (2) and SU (3) Lagrangians
In this section, we show the explicit calculation to obtain the effective Lagrangians for2D topological field theory for SU (2) and SU (3) using the expression (5.3) together withthe diagrammatic representation for adjugate matrix and matrix determinant expressed in(5.1) and (5.4) respectively.For SU (2), the adjugate matrix isadj( f M ) αβ = ( − α β = − ˆ E αA ǫ ACB φ C ˆ E βB (6.1)where f ABC = ǫ ABC for SU(2). Therefore, Θ AB = − ǫ ACB φ C . The matrix determinant isdet( f M ) = ( −
1) = − ǫ ABC φ B η CD ǫ DEF φ E η AF = 2 η BD φ B φ D = 2 | φ | . (6.2)Thus, when substituting the above relations into (5.3), we obtain L eff ( φ ) = − ig | φ | ∂φ A ǫ ABC φ B ¯ ∂φ C (6.3)– 17 –hich is identical to what we found earlier in the equation (2.15).For SU(3), the diagrammatic expressions for the adjugate matrix and matrix determi-nant are adj( f M ) αβ =( − ( α β − α β × (cid:18) (cid:19) − α β × " − (cid:0) (cid:1)(cid:0) (cid:1) (6.4)and det( f M ) = ( − ( − × (cid:18) (cid:19) − (cid:0) (cid:1)(cid:0) (cid:1) ) (6.5)According to the above expressions, one can write the effective Lagrangian in the form(5.3) withΘ AB = − F AC F CD F DE F EF F F B ) + 3! · · ( F AC F CD F DB )( F EF F FE )+ F AB h F CD F DE F EF F EC ) − F CD F DC )( F EF F FE ) i (6.6)anddet( f M ) = − F AB F BC F CD F EF F FG F GA ) + 3! · · ( F AB F BC F CD F DA )( F EF F FE )+ 15( F AB F BA )( F CD F DC )( F EF F FE ) (6.7)where we used the notation F AB = f ACB φ C .We can further simplify the above terms by expanding them explicitly in the Cartan-Weyl basis for SU(3). The generators are I + = , I − = , I = 12 − ,U + = , U − = ,V + = 12 , V − = , Y = 13 − . (6.8)– 18 –e can determine the structure constants by considering all matrix commutators betweenthe elements. The generators I and Y are the Cartan subalgebra elements satisfying[ I , Y ] = 0 . (6.9)The plus and minus superscripts of the generators denote the raising and lowering operatorswithin the three su(2) subalgebras given by[ I + , I − ] = 2 I , [ U + , U − ] = 32 Y − I , [ V + , V − ] = 32 Y + I . (6.10)Note that the Hermitian conjugation of generators switches the plus and minus superscriptsof the generators within each SU(2) subgroup, i.e. ( I ± ) † = I ∓ , ( U ± ) † = U ∓ , ( V ± ) † = V ∓ .Apart from (6.10), the remaining non-zero commutators are[ I , I ± ] = ± I ± , [ I , U ± ] = ∓ U ± , [ I , V ± ] = ± U ± , [ Y, U ± ] = ± U ± , [ Y, V ± ] = ± V ± , [ I ± , U ± ] = ± V ± , [ I ± , V ∓ ] = ∓ U ∓ , [ U ± , V ∓ ] = ± I ∓ . (6.11)For convenience, we denote { I , Y, I + , I − , U + , U − , V + , V − } by { T , T , . . . , T } re-spectively. In this notation, the metric tensor η AB can be written as η AB = (6.12)which is directly from η AB = 2 tr( T A T B ).The field φ is an element of the Cartan subalgebra, i.e. φ = φ T + φ T . Consequently,one can find the adjoint representation of the field φ asad ( φ ) = − φ φ φ − φ − φ + 2 φ − φ − φ φ + 2 φ (6.13)– 19 –here ad( φ ) = if ABC φ B = i F AC . With this matrix (6.13), one can compute all loop andstrand diagrams appearing in the equations (6.6) and (6.7). Chains of matrix multiplica-tions of the matrix F are shown in the Appendix.From the calculation, we can further simplify the loop terms. The loop diagram withtwo vertices F AB F BC can be replaced by the absolute square of the field φ as F AB F BA = − φ ) −
12 ( φ − φ ) −
12 ( φ + 2 φ ) = − φ ) + 43 ( φ ) ) = − | φ | . (6.14)A similar pattern appears in the four-vertex loop as it is proportional to | φ | : F AB F BC F CD F DA = 2( φ ) + 18 ( φ − φ ) + 18 ( φ + 2 φ ) = 94 (( φ ) + 43 ( φ ) ) = 94 | φ | . (6.15)The six-vertex loop can be expressed in terms of two invariant objects, | φ | and d ABC φ A φ B φ C as F AB F BC F CD F DE F EF F FA = − φ ) −
132 ( φ − φ ) −
132 ( φ + 2 φ ) = − | φ | + 98 (cid:18) φ ) ( φ ) −
89 ( φ ) (cid:19) (6.16)where the quantity inside the parenthesis is d ABC φ A φ B φ C where d ABC is the totally sym-metric third rank tensor defined in (4.9).In consequence, we can rewrite the terms (6.6) and (6.7) asΘ AB = − F AC F CD F DE F EF F F B ) − F AC F CD F DB ) | φ | − F AB | φ | (6.17)and det( f M ) = − | φ | − d ABC φ A φ B φ C ) . (6.18) In this section we would like to generalise the action (1.4) further by adding a source termfor the gauge field A . Consider the action given by S [ J ] = 2 Z M d ξ ǫ ij tr( φ F ij + J i A j ) . (7.1)To obtain the effective Lagrangian for the field φ , we will integrate out the gauge field A as before. By doing so, we expand the field A in terms of the unit basis defined by (4.5).This gives the partition function as Z [ J ] = 1Vol Z Dφ A Da αi Dχ ja exp( − ˜ S [ J ]) . (7.2)– 20 –ith ˜ S [ J ] = Z M d ξ (cid:16) igf ABC φ C a αi a βj ˆ E αA ˆ E βB − (2 ∂ i φ A − J iA ) a αj ˆ E Aα − (2 ∂ i φ A − J iA ) χ ja ˆ H Aa (cid:17) ǫ ij . (7.3)It is not hard to see that the path integration of the last line leads to a constraint on thetheory. This appears in the form of a Dirac delta function N − Y a =1 2 Y i =1 δ (tr(2 ∂ i φ − J i ) H a ) . (7.4)The constraint implies that the difference between 2 ∂φ and J does not lie in the Cartansubalgebra.We can proceed with the calculation as in previous sections by changing from spacetimecoordinates ( ξ , ξ ) to the complex coordinates ( z, ¯ z ). The partition function now resemblesa Gaussian path integral with respect to the complex fields b and ¯ b expressed in (2.7) whichis Z [ J ] = N Vol Z Dφ A Db α D ¯ b α Y i =1 δ ( N − (tr((2 ∂ i φ − J i ) ˆ φ ))exp( − S [ φ, b, ¯ b, J ]) . (7.5)where S [ φ, b, ¯ b, J ] = Z M d z (cid:16) igf ABC φ C b α ¯ b β ˆ E αA ˆ E βB − ((2 ∂φ A − J A )¯ b α − (2 ¯ ∂φ A − ¯ J A ) b α ) ˆ E Aα (cid:17) . (7.6)Integrating out the b and ¯ b using (2.10) yields Z [ J ] = N Vol Z Dφ A Y i =1 δ ( N − (tr((2 ∂ i φ − J i ) ˆ φ ))exp (cid:16) − S eff ( φ, J ) (cid:17) (7.7)with S eff ( φ, J ) = Z d z i g f M ) (2 ∂φ A − J A )Θ AB (2 ¯ ∂φ B − ¯ J B ) . (7.8)Turning back to the ( ξ , ξ ) coordinates, the effective action takes the form S eff ( φ, J ) = Z d ξ i g f M ) (2 ∂ i φ A − J iA )Θ AB (2 ∂ j φ B − J jB ) ǫ ij . (7.9)It is known that one can relate a BF theory to 2 D Yang-Mills theory by introducingthe quadratic term for the field φ which is S qd = e Z d ξ √ g | φ | . (7.10)As a result, the partition function for the 2D gauge theory with the gauge field source J can be expressed as Z [ J ] = N Vol Z Dφ A Y i =1 δ ( N − (tr((2 ∂ i φ − J i ) ˆ φ ))exp (cid:16) − ( S eff + S qd ) (cid:17) . (7.11)– 21 – C M Figure 3 . Two-dimensional manifold M with a region D and a closed loop C With a suitable choice of the source term J , in principle we are able to compute theexpectation value of a Wilson loop in 2D Yang-Mills theory based on our effective BFtheory. However, we have to deal with the issue of path-ordering.The non-Abelian Wilson loop can be expressed as the trace of the path-ordered expo-nential of a line integral of the gauge field A along a closed loop C , W [ C ] = tr (cid:16) P (cid:16) e − g H C A· dξ (cid:17)(cid:17) . (7.12)The trace together with the path-ordering operator can be replaced by a functional integralover a complex anti-commuting field ψ [38, 39] as W [ C ] = Z Dψ † Dψ exp (cid:16) Z dτ ψ † ˙ ψ − g A iR ˙ ξ i ψ † T R ψ (cid:17) (7.13)where the loop C is now parameterized by τ . Therefore, the expectation value of theWilson loop takes the form h W [ C ] i = 1 Z ′ Z DφDψ † Dψ Y i =1 δ ( N − (tr((2 ∂ i φ − J i ) ˆ φ )) exp (cid:0) − ( S eff + S qd ) + Z dτ ψ † ˙ ψ (cid:1) (7.14)with J Ai ( ξ ) = − g I C ψ † ( ˜ ξ ) T A ψ ( ˜ ξ ) δ (2) ( ξ − ˜ ξ ) ǫ ij d ˜ ξ j (7.15)and the action S eff and S qd are expressed in (7.9) and (7.10) respectively. The term Z ′ inthe denominator is a normalization factor such that h i = 1.However, it turns out that the solution for the equation 2 ∂ i φ − J i = 0 with the sourceterm expressed above is not consistent as the line integral of J i is not path independentwhich contradicts to the equation itself. To deal with this, we will exploit gauge symmetry.For simplicity, we will proceed with the calculation in the context of SU (2) theory. Inthis setting, we choose the gauge fixing such that the unit vector ˆ φ is constant everywhereoutside a region D . Therefore, the manifold M now consists of the region D where thevalue of ˆ φ varies and the rest of the manifold where the ˆ φ is constant. We can furtherchoose that the region D does not intercept the loop C as depicted in the figure 3.– 22 –ccording to this gauge choice, the effective action term (7.9) becomes S eff ( φ, J ) = Z D d ξ i | φ | g ∂ i ˆ φ A ∂ j ˆ φ B ˆ φ C ǫ ABC ǫ ij + Z M /D d ξ i g | φ | (2 ∂ i φ A − J iA )(2 ∂ j φ B − J jB ) φ C ǫ ABC ǫ ij . (7.16)If we consider the case when the manifold M has a topology of unit sphere S , the firstterm can be related to a winding number as discussed in the earlier section. Note that | φ | is constant due to the absence of the source in D . Moreover, since the ˆ φ is constant in M /D , the non-vanishing contribution to the second line is Z M /D d ξ i g | φ | J iA J jB φ C ǫ ABC ǫ ij = ig I C I C d ˜ ξ i dξ ′ j (cid:18) ψ † T A ψ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ξ (cid:19)(cid:18) ψ † T B ψ (cid:12)(cid:12)(cid:12)(cid:12) ξ ′ (cid:19) δ (2) ( ˜ ξ − ξ ′ ) ǫ ij φ C | φ | ǫ ABC . (7.17)The term H C H C d ˜ ξ i dξ ′ j δ (2) ( ˜ ξ − ξ ′ ) ǫ ij counts the number of times the loop C intersects itself.Therefore, the above term can be set to zero provided that the loop C does not have aself-intersection. Subsequently, the effective action (7.16) turns into S eff ( φ ) = ig | φ | (4 πn ) (7.18)where n is the winding number of the map ˆ φ .At this point, the appearance of the fermionic field ψ in the effective action S eff hasbeen removed due to the gauge choice. Therefore, according to (7.14), the only term that issubject to the path-ordering operation is the source term J in the constraint. This allowsus to rewrite (7.14) as h W [ C ] i = 1 Z Z DφDχ tr (cid:20) P (cid:18) exp (cid:18) g I C ˆ φχ i dξ i (cid:19)(cid:19)(cid:21) × exp (cid:18) − ( S eff + S qd ) + Z d ξ ( ∂ i φ A ) ˆ φ A χ j ǫ ij (cid:19) (7.19)where the Dirac delta function is replaced by the functional integral over the field χ . Sincethe field ˆ φ is constant and commutes with itself throughout the loop, the path-orderingoperator P can be dropped. Denoting the eigenvalue of ˆ φ by λ , the trace of the exponentialin the first line takes the form X λ exp (cid:18) gλ Z d ξ I C δ (2) ( ξ − ˜ ξ ) χ i ( ξ ) d ˜ ξ i (cid:19) . (7.20)We then proceed with the calculation by integrating out the field χ . This generates aconstraint via a Dirac delta function as h W [ C ] i = 1 Z Z Dφ X λ Y i =1 δ (cid:18) ∂ i | φ | + gλ I C δ (2) ( ξ − ˜ ξ ) ǫ ij d ˜ ξ j (cid:19) e − ( S eff + S qd ) . (7.21)– 23 –t is not hard to see that the solution for the constraint, ∂ i | φ | + gλ I C δ (2) ( ξ − ˜ ξ ) ǫ ij d ˜ ξ j = 0 . (7.22)takes the form ϕ λ − ϕ = − gλ (cid:18) Z ξO I C δ (2) ( ξ ′ − ˜ ξ ) ǫ ij d ˜ ξ i dξ ′ j (cid:19) . (7.23)where ϕ λ and ϕ are the scalar fields at arbitrary point ξ and a reference point O respec-tively. The object in the parenthesis counts the number of oriented intersections betweentwo curves [40]. The solution above is independent of path, hence, it depends only on thereference point O . If we set the point O to be outside the loop C , ϕ λ − ϕ = ( − gλ , if ξ is inside the loop C , otherwise . (7.24)This allow us to compute the expectation value of the Wilson loop in 2D Yang-Millstheory (7.21) as h W [ C ] i = 1 Z X λ Z ∞ dϕ ∞ X n = −∞ exp (cid:20) − ig (4 πn ) ϕ − e Z M d ξ √ gϕ λ (cid:21) (7.25)The infinite m limit of the Dirichlet kernel, D m ( x ), represents the Dirac delta function aslim m →∞ D m ( x ) = lim m →∞ m X k = − m e imx = 2 πδ ( x ) (7.26)where x ∈ [0 , π ]. Therefore, (7.25) becomes h W [ C ] i = 1 Z X λ Z ∞ dϕ g δ ( ϕ mod g (cid:20) − e (cid:18) Z Γ d ξ √ gϕ λ + Z M / Γ d ξ √ gϕ λ (cid:19)(cid:21) (7.27)In the above expression, we separate the region M into Γ and M / Γ where Γ is all theregion inside the loop C with the boundary ∂ Γ = C . Denoting the surface area of theregion Γ and M / Γ by A and A subsequently together with (7.24), the relation (7.27)takes the form h W [ C ] i = g Z X λ ∞ X N =0 exp (cid:20) − (cid:18) eg (cid:19) (cid:18) A ( N − λ ) + A N (cid:19)(cid:21) . (7.28)In the case of SU(2), if we consider the eigenvalues of ˆ φ in the fundamental represen-tation, λ = ± /
2. This turns the expression (7.28) into h W [ C ] i = g Z (cid:18) ∞ X N = −∞ exp (cid:20) − e (cid:18) A ( N + 1 / + A N (cid:19)(cid:21) + exp (cid:20) − e A (cid:21)(cid:19) = g Z (cid:18) ϑ (cid:18) ie A π ; ie Aπ (cid:19) + 1 (cid:19) exp (cid:20) − e A (cid:21) (7.29)– 24 –here we re-define the Yang-Mills coupling constant e YM as eg and ϑ ( z ; τ ) is the Jacobi’sthird theta function defined as ϑ ( z ; τ ) = ∞ X N = −∞ exp(2 πiN z + πiN τ ) . (7.30)In the case that M is an infinitely large sphere, i.e. A → ∞ , the vacuum expectationvalue of the Wilson loop (7.28) turns into h W [ C ] i = gZ exp (cid:20) − e A (cid:21) (7.31)as the theta function becomes unity at this limit.The result (7.31) shows that the expectation value of the Wilson loop for 2D Yang-Millstheory obtained by the effective topological BF theory satisfies the area law. This agreeswith known results [41–43] as far as the exponent is concerned, which is the dominant piece.To compute the prefactor would require the computing the determinants arising from theGuassian integrals generalising the argument given above for the SU (2) partition function. To conclude, we constructed a gauge and Weyl invariant theory of a two-dimensional scalarfield by integrating out the gauge fields in BF theory. This model is a candidate forgeneralising a string theory contact interaction that describes Abelian gauge theory tothe non-Abelian case. The calculation was implemented by expanding the fields in theCartan-Weyl basis. By performing a Gaussian functional integration, we obtained theeffective theory with the Lagrangian (5.3) together with the constraint addressed in (4.7).The constraint implies that the magnitude of a scalar field, | φ | , as well as the quantity d ABC φ A φ B φ C are constant throughout the space.The adjugate and the determinant of the matrix f M play an important part in (5.3)where f M is defined as (4.15). We developed a diagrammatic approach to represent theseobjects. The diagrams are constructed from vertices connected to each other by lines. Nomore than two line are allowed to connect with one vertex. There are two type of diagrams,i.e. a strand and a loop. The adjugate matrix and the matrix determinant were expressedas summations over products of these diagrams as in (5.1) and (5.4) respectively.For the case of SU (2) and the manifold having the topology of a unit sphere, theeffective action (2.16) contains the winding number of the field ˆ φ which maps a point onthe manifold into a point on S . By using the SU (2) effective action and summing over thiswinding number, we re-formulated the partition function on a sphere of SU (2) Yang-Millstheory.Finally, we investigated the BF theory coupled to a source term for the gauge field.We exploited the gauge symmetry to deal with the path-ordering of the Wilson loop. Theresult showed that the vacuum expectation value of the Wilson loop exhibits the area lawagreeing with the well-known results [41–43].– 25 – eferences [1] Edward Witten. Quantum field theory and the jones polynomial. Comm. Math. Phys. ,121(3):351–399, 1989.[2] Gary T. Horowitz. Exactly soluble diffeomorphism invariant theories.
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Prog. Theor. Phys. , 104:1189–1265, 2000. – 28 – ppendix A Expressions for Matrix Multiplications of the Matrix F The expressions for matrix multiplication of the matrix F are given as follows: F AB F BC = − φ ) − φ ) − ( φ − φ ) − ( φ − φ ) − ( φ + 2 φ ) − ( φ + 2 φ ) (A.1) F AB F BC F CD = − i ( φ ) i ( φ ) i ( φ − φ ) − i ( φ − φ ) − i ( φ + 2 φ ) i ( φ + 2 φ ) (A.2) F AB F BC F CD F DE = φ ) φ ) ( φ − φ ) ( φ − φ ) ( φ + 2 φ ) ( φ + 2 φ ) (A.3)– 29 – AB F BC F CD F DE F EF = i ( φ ) − i ( φ ) − i ( φ − φ ) i ( φ − φ ) i ( φ + 2 φ ) − i ( φ + 2 φ ) (A.4) F AB F BC F CD F DE F EF F F G = − φ ) − φ ) − ( φ − φ ) − ( φ − φ ) − ( φ + 2 φ ) − ( φ + 2 φ )