Abstract
We introduce the description of a Wilson surface as a 2-dimensional topological quantum field theory with a 1-dimensional Hilbert space. On a closed surface, the Wilson 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 theories modifying their partition functions. This gives a new interpretation of the results obtained in [1]. We compute explicitly the partition function of the 2-dimensional Yang-Mills theory interacting with a Wilson surface for the cases G = SU(N)/ℤm, G = Spin(4l)/(ℤ2 ⊕ ℤ2) and obtain a general formula for any compact connected Lie group.
Highlights
We introduce the description of a Wilson surface as a 2-dimensional topological quantum field theory with a 1-dimensional Hilbert space
The Wilson surface theory defines a topological invariant of the principal G-bundle P → Σ
We compute explicitly the partition function of the 2-dimensional Yang-Mills theory interacting with a Wilson surface for the cases G = SU(N )/Zm, G = Spin(4l)/(Z2 ⊕ Z2) and obtain a general formula for any compact connected Lie group
Summary
Recall the construction of Wilson surface observables from [1]. Let G be the gauge group, g its Lie algebra, (x, y) → Tr(xy) an invariant scalar product on g and P a principal Gbundle over a surface Σ. A Wilson surface observable is described by an auxiliary 2-dimensional gauge theory on the surface Σ The fields in this theory are a g∗-valued scalar field b and a g-valued 1-form a. Acting by the contraction we obtain: ıξ Tr(bFA+a) = ıξ Tr (b(FA+a2+dgg−1a+adgg−1+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 ıξ (dg) = −ξg, ıξ A = ξ (by definition of connection), ıξ FA = 0, ıξ a = 0 (FA and a are horizontal) This computation proves that G-invariant form Tr(bFA+a) is horizontal, and basic. A representation of Gcan be considered as a projective representation of G, and λ is allowed to take values in Λ∗G, and, as we will see later, these are exactly the values which describe the presence of nontrivial Wilson surfaces
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
