Correction to: Lett Math Phys https://doi.org/10.1007/s11005-016-0910-2

In Section 2 and Section 3 of the original version of the article, the author adapted several results from [12] and [16] to the Kapustin-Witten equations. Even though the Section 3 is not used anywhere in the proof of the main theorem 1.1 in original version of our article, the exposition of Theorem 2.4 and Section 3 followed Dr. Mares’s thesis [12] closely and the original version of the article failed to cite Mares’s thesis. The original version of the article also did not point out the similarity of Corollary 1.2 to [4,Corollary 2], Theorem 2.9 to [12, Theorem 4.2.1], did not give references to some Definitions in Pages 2,4,14,15,20. The author sincerely apologizes for these mistakes. The following list is the details.

\(\bullet \) The conclusion of Corollary 1.2 is obtained from the setting of Kapustin-Witten equations, but the conditions of Corollary 1.2 are the same as those in [4,Corollary 2].

\(\bullet \) The estimate in Theorem 2.4 is a standard application of the Weitzenböck formula. A similar estimate was proved for the Vafa-Witten equations in [12, Theorem 3.1.1]. In the setting of the Kapustin-Witten equations, it was proved in [16, Section 2(b)]. In original version of the article, we briefly review the proof of the statement for the readers’ convenience.

\(\bullet \) Following the strategy of [12, Theorem 4.2.1], we prove a unique continuation result (see Theorem 2.9) for the Kapustin-Witten equations.

\(\bullet \) In Section 3, we study the regularity theory for the Kapustin-Witten equations. The exposition is modeled on the corresponding theory for the PU(2) monopole equations [6]. The general pattern is that one typically has global \(L^{2}_{1}\) estimates for solutions to the Kapustin-Witten equations, but when we need estimates with respect to stronger norms, we must get by with only local estimates. We also present the construction of the Kuranishi complex associated with the Kapustin-Witten equations in Section 3.1. The computations are standard in mathematical gauge theories, and our exposition is adapted from [12, Section 3.2.1]. Using methods from [6], Mares [12] studied the regularity of solutions of the Vafa-Witten equations. The arguments of [12] work verbatim for the Kapustin-Witten equations. The rest of Section 3 reviews the relevant results.

The Proposition 3.2 taken from Corollary 3.4 of [6], as described in Theorem 3.3.1 of [12].

The Proposition 3.3 taken from Corollary 3.11 of [6], as described in Theorem 3.3.3 of [12].

The Theorem 3.4 taken from Theorem 3.13 of [6], as described in Theorem 3.3.4 of [12].

The Proposition 3.5 taken from Theorem 3.3.5 of [12] which follows analogue of Corollary 3.15 in [6].

The Theorems 3.6, 3.7 taken from Theorems 3.3.6, 3.3.7 of [12].

\(\bullet \) The statement of definition of the Kapustin-Witten equations in Page 2 taken from [16].

The statement of definition of Sololev space in Page 4 taken from Section 4.1 of [4].

The statement of definitions of \(L^{\sharp }\) and \(L^{\sharp ,2}\) in Page 15 taken from Page 28 of [3].

The Definition 2.7 of irreducible connection in Page 7 taken form Definition 2.1 of Y. Tanaka, Some boundedness properties of solutions to the Vafa-Witten equations on closed 4-manifolds, Q. J. Math. 68 (2017), 1203–1225.

The definitions of gauge-equivariant map VW(A, B, C) and VW-moduli space in Page 20 are first appear in [12]. The definitions of gauge-equivariant map \(KW(A,\phi )\) and KW-moduli space in Page 3 follow from the setting of Vafa-Witten equations.