Surface operators in 𝒩 = 2 abelian gauge theory

Abstract

We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly prove that for any embedding of surface operators in a general, twisted N = 2 pure abelian theory on an arbitrary spin (or non-spin) four-manifold, the parameters transform naturally under the SL(2, Z) (or Γ 0 (2)) duality of the theory. However, for nontrivially-embedded surface operators, exact S-duality holds if and only if the "quantum" parameter effectively vanishes, while the overall SL(2, Z) (or Γ 0 (2)) duality holds up to a c-number at most, regardless. Nevertheless, this observation sets the stage for a physical proof of a remarkable mathematical result by Kronheimer and Mrowka [1]-that expresses a "ramified" analog of the Donaldson invariants solely in terms of the ordinary Donaldson invariants-which, will appear, among other things, in forthcoming work [2]. As a prelude to that, the effective interaction on the corresponding u-plane will be computed. In addition, the dependence on second Stiefel-Whitney classes and the appearance of a Spin c structure in the associated low-energy Seiberg-Witten theory with surface operators, will also be demonstrated. In the process, we will stumble upon an interesting phase factor that is otherwise absent in the "unramified" case. * On leave of absence from the National University of Singapore.

Supersymmetric surface operators, four-manifold theory and invariants in various dimensions

“…This means that the various fields of the theory are necessarily coupled to the gauge field A with field strength F . This important fact was first pointed out in [17], and further exploited in [16] to prove an S-duality in a general, abelian N = 2 theory without matter in the presence of surface operators.…”

“…In this case, X = T 3 × S 1 is symplectic Kähler with b + 2 (X) = b 1 (M ) = 3, and according to [30], χ(HF * (T 3 )) = +1. 16 What about Gr(c 1 (E))? Well, although b + 2 (X) > 1, because b 1 (X) > 0, one cannot read off from our result in (5.9) (which is defined for b 1 (X) = 0 and b + 2 (X) > 1).…”

“…16), the x's are just the points which span the space H of dimension zero; in (4.17), E is some nontrivial complex line bundle with a self-dual u(1)-valued connection A E and curvature c 1 (E): recall from our discussion in §2.2 that a choice of an extension of E over Σ results in α being t-valued, and since the maximal torus T of SU (2) is actually U (1), α and therefore F + (i.e., 2πc 1 (E)) are actually u(1)-valued in(2.22). Such a complex line bundle E…”

Surface Operators

“…The construction described in the end of the previous section can be easily generalized to define half-BPS surface operators in N = 2 gauge theories, see e.g. [7,8,9,10,11] and subsequent work. As a result, one finds a fairly large class of surface operators labeled by the Levi subgroup L ⊆ G and continuous parameters (α, η) ∈ T × T ∨ /W, which in N = 2 theories conveniently unify into holomorphic combinations…”

S-duality in Abelian gauge theory revisited

“…We also note that if the Gaussian integral in (23) i.e., the restricted Feynman measure would turn out to be the singular measure cot(πc) i dc then modular properties of Z(M V , g V , τ ) are also recovered.…”