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Indeed, given a point x on M, there always exists U, such that x ¢ U,. Choose a point u in 1*(x) and define M. Daniel and C. M. Viallet: Geometrical setting of Yang-Mills gauge theories This definition makes sense since u* Yu) is inde- pendent of the choice of the point wu in the fiber 1"(x). To see this, we need only use the property of 9,:

Figure 4 Indeed, given a point x on M, there always exists U, such that x ¢ U,. Choose a point u in 1*(x) and define M. Daniel and C. M. Viallet: Geometrical setting of Yang-Mills gauge theories This definition makes sense since u* Yu) is inde- pendent of the choice of the point wu in the fiber 1"(x). To see this, we need only use the property of 9,:

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_Any such change is represented by a smooth assign- ment of an element of the gauge group to any point of space-time, that is, a map: R*-~G. The graphs of these maps live in a space P=R*XG (Fig. 1).
_Any such change is represented by a smooth assign- ment of an element of the gauge group to any point of space-time, that is, a map: R*-~G. The graphs of these maps live in a space P=R*XG (Fig. 1).
A differentiable principal fiber bundle over M with group G consists of a manifold P and an action of G on P satisfying the following conditions: Suppose we have two strips of paper, (1) and (2) (see Fig. 2). We can glue (1) and (2) together in such a way
A differentiable principal fiber bundle over M with group G consists of a manifold P and an action of G on P satisfying the following conditions: Suppose we have two strips of paper, (1) and (2) (see Fig. 2). We can glue (1) and (2) together in such a way
2. Consider a circle S' and the group G=(1, -1)=2Z, (i.e., group with two elements) with the following action on S':+1= identity map, —1= antipodal map. The ac- tion of G is free. M =S1/G =P' =one-dimensional pro- jective space (i.e., space of diameters). .7 (point of S) =diameter through that point. We have to check the local triviality. Given an element in P’, that is, a di- rection in R? through the origin 0, there always exists an open cone U (in R?) containing that direction, and a3(U) =U x{+1, —1} (see Fig. 7). Hence S’ is the total space of a principal fiber bundle with group G and base space P'. We shall denote this bundle by S'(P", Z,). Actually, we can identify this circle (together with the action of Z,) with B’ (with the action defined above), since they are homotopically equivalent. C. More definitions
2. Consider a circle S' and the group G=(1, -1)=2Z, (i.e., group with two elements) with the following action on S':+1= identity map, —1= antipodal map. The ac- tion of G is free. M =S1/G =P' =one-dimensional pro- jective space (i.e., space of diameters). .7 (point of S) =diameter through that point. We have to check the local triviality. Given an element in P’, that is, a di- rection in R? through the origin 0, there always exists an open cone U (in R?) containing that direction, and a3(U) =U x{+1, —1} (see Fig. 7). Hence S’ is the total space of a principal fiber bundle with group G and base space P'. We shall denote this bundle by S'(P", Z,). Actually, we can identify this circle (together with the action of Z,) with B’ (with the action defined above), since they are homotopically equivalent. C. More definitions
U1, (respectively, U,) are the points of R°-{origin} that are not contained in the half-cone C, (respectively, C,) (see Fig. 11).
U1, (respectively, U,) are the points of R°-{origin} that are not contained in the half-cone C, (respectively, C,) (see Fig. 11).
The topology of the bundle spaces we introduce will turn out to be relevant to the study of the topologically stable gauge field configurations of the theories under consideration (see Sec. III). However, the bundle space is not given a priovi. What is usually provided is space- time and the gauge symmetry group. These entities are blended together by certain transition functions.
The topology of the bundle spaces we introduce will turn out to be relevant to the study of the topologically stable gauge field configurations of the theories under consideration (see Sec. III). However, the bundle space is not given a priovi. What is usually provided is space- time and the gauge symmetry group. These entities are blended together by certain transition functions.
The bundle £ is the Cartesian product R& X {closed curve C}. The bundle £’ is not such a Cartesian prod- uct.
The bundle £ is the Cartesian product R& X {closed curve C}. The bundle £’ is not such a Cartesian prod- uct.
By construction, w, acts on vectors tangent to the base space in U,. If VET,(M) and x ©U,U,, then both Wy and w, acton V. The above formula relates w,(V) and w,(V) for any such V. Note that #j4dy., is a 1 form on U, ‘\U, and takes values in the space tangent to the group at e, identified with @(G). _ ee
By construction, w, acts on vectors tangent to the base space in U,. If VET,(M) and x ©U,U,, then both Wy and w, acton V. The above formula relates w,(V) and w,(V) for any such V. Note that #j4dy., is a 1 form on U, ‘\U, and takes values in the space tangent to the group at e, identified with @(G). _ ee
Space. Consequently, we can identify the tangent vec- tors 8, with o,,9,. The giving of a local section o, is equivalent to the trivialization of P over U,. 1‘(U,) can be viewed as a product U,XG. Points on o, have coordinates (x,e), as shown in Fig. 16. o, reproduces U, in the bundle
Space. Consequently, we can identify the tangent vec- tors 8, with o,,9,. The giving of a local section o, is equivalent to the trivialization of P over U,. 1‘(U,) can be viewed as a product U,XG. Points on o, have coordinates (x,e), as shown in Fig. 16. o, reproduces U, in the bundle
This group is called the holonomy group (of the connec- tion) with reference point uw: }(u). M. Daniel and C, M. Viallet: Geometrical setting of Yang-Mills gauge theories It happens that if @(u) = G for some point and if M is connected, then @ is independent of uw. The connection is then said to be irreducible, and any two points of P can be joined by a horizontal curve in P.
This group is called the holonomy group (of the connec- tion) with reference point uw: }(u). M. Daniel and C, M. Viallet: Geometrical setting of Yang-Mills gauge theories It happens that if @(u) = G for some point and if M is connected, then @ is independent of uw. The connection is then said to be irreducible, and any two points of P can be joined by a horizontal curve in P.
The bundles P(S*, SU(n)) are characterized by the in- teger n= y Y2° dM (= Chern number = instanton number). n can be expressed as an integral over R?:
The bundles P(S*, SU(n)) are characterized by the in- teger n= y Y2° dM (= Chern number = instanton number). n can be expressed as an integral over R?:
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