Modular and duality properties of surface operators in N=2* gauge theories

We calculate the instanton partition function of the four-dimensional N=2* SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure N=2 or to N=2* gauge theories.Comment: 51 pages, v3: references added, typos fixed, footnote added, some small changes in the text, appendix B streamlined. Matches the published versio

Surface operators in 5d gauge theories and duality relations

We study half-BPS surface operators in 5d N=1 gauge theories compactified on a circle. Using localization methods and the twisted chiral ring relations of coupled 3d/5d quiver gauge theories, we calculate the twisted chiral superpotential that governs the infrared properties of these surface operators. We make a detailed analysis of the localization integrand, and by comparing with the results from the twisted chiral ring equations obtain constraints on the 3d and 5d Chern-Simons levels so that the instanton partition function does not depend on the choice of integration contour. For these values of the Chern-Simons couplings, we comment on how the distinct quiver theories that realize the same surface operator are related to each other by Aharony-Seiberg dualities.Comment: 39 pages. v2: A few sentences rephrased, references added, and typos corrected. Matches version published in JHE

Notes on unoriented D-brane instantons

In the first lecture, we discuss basic aspects of worldsheet and penta-brane instantons as well as (unoriented) D-brane instantons, which is our main focus here, and threshold corrections to BPS-saturated couplings. The second lecture is devoted to non-perturbative superpotentials generated by `gauge' and `exotic' instantons living on D3-branes at orientifold singularities. In the third lecture we discuss the interplay between worldsheet and D-string instantons on $T^4/Z_2$. We focus on a 4-fermi amplitude, give Heterotic and perturbative Type I descriptions, and offer a multi D-string instanton interpretation. We conclude with possible interesting developments.Comment: 31 pages. Based on lectures delivered by M. Bianchi at the Fourth Young Researchers Workshop of the European Superstring Theory Network in Kounnas Bay, Cyprus, September 200

N=1/2 quiver gauge theories from open strings with R-R fluxes

We consider a four dimensional N=1 gauge theory with bifundamental matter and a superpotential, defined on stacks of fractional branes. By turning on a flux for the R-R graviphoton field strength and computing open string amplitudes with insertions of R-R closed string vertices, we introduce a non-anticommutative deformation and obtain the N=1/2 version of the theory. We also comment on the appearance of a new structure in the effective Lagrangian.Comment: 30 pages, 5 figures, JHEP class (included); some comments and a reference adde

Surface operators, chiral rings and localization in N = 2 gauge theories

We study half-BPS surface operators in supersymmetric gauge theories in four and five dimensions following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and five-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and five-dimensional gauge theories, and find a perfect match with the results obtained from the chiral ring equations. In the five-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for ramified instantons. We also comment on the necessity of including Chern-Simons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator.Comment: 41 pages. v3: typos corrected in the text and in some formulae. Some sentences rephrased according to the suggestions of the referee. Matches the version published on JHE

Surface operators, dual quivers and contours

We study half-BPS surface operators in four dimensional N=2 SU(N) gauge theories, and analyze their low-energy effective action on the four dimensional Coulomb branch using equivariant localization. We also study surface operators as coupled 2d/4d quiver gauge theories with an SU(N) flavour symmetry. In this description, the same surface operator can be described by different quivers that are related to each other by two dimensional Seiberg duality. We argue that these dual quivers correspond, on the localization side, to distinct integration contours that can be determined by the Fayet-Iliopoulos parameters of the two dimensional gauge nodes. We verify the proposal by mapping the solutions of the twisted chiral ring equations of the 2d/4d quivers onto individual residues of the localization integrand.Comment: 42 pages. v2: Sections 4 and 5 partially restructured in order to describe in a more compact ad unified way the association of contours to quiver

D3 instantons in Calabi-Yau orientifolds with(out) fluxes

We investigate the instanton effect due to D3 branes wrapping a four-cycle in a Calabi-Yau orientifold with D7 branes. We study the condition for the nonzero superpotentials from the D3 instantons. For that matter we work out the zero mode structures of D3 branes wrapping a four-cycle both in the presence of the fluxes and in the absence of the fluxes. In the presence of the fluxes, the condition for the nonzero superpotential could be different from that without the fluxes. We explicitly work out a simple example of the orientifold of $K3 \times T^2/Z_2$ with a suitable flux to show such behavior. The effects of D3-D7 sectors are interesting and give further constraints for the nonzero superpotential. In a special configuration where D3 branes and D7 branes wrap the same four-cycle, multi-instanton calculus of D3 branes could be reduced to that of a suitable field theory. The structure of D5 instantons in Type I theory is briefly discussed.Comment: 17 pages; Typos corrected, arguments improved and references adde

N=4 superconformal mechanics as a Non linear Realization

An action for a superconformal particle is constructed using the non linear realization method for the group PSU(1,1|2), without introducing superfields. The connection between PSU(1,1|2) and black hole physics is discussed. The lagrangian contains six arbitrary constants and describes a non-BPS superconformal particle. The BPS case is obtained if a precise relation between the constants in the lagrangian is verified, which implies that the action becomes kappa-symmetric.Comment: new subection, references added and new acknowledgment

Instantons in N=1/2 Super Yang-Mills Theory via Deformed Super ADHM Construction

We study an extension of the ADHM construction to give deformed anti-self-dual (ASD) instantons in N=1/2 super Yang-Mills theory with U(n) gauge group. First we extend the exterior algebra on superspace to non(anti)commutative superspace and show that the N=1/2 super Yang-Mills theory can be reformulated in a geometrical way. By using this exterior algebra, we formulate a non(anti)commutative version of the super ADHM construction and show that the curvature two-form superfields obtained by our construction do satisfy the deformed ASD equations and thus we establish the deformed super ADHM construction. We also show that the known deformed U(2) one instanton solution is obtained by this construction.Comment: 32 pages, LaTeX, v2: typos corrected, references adde

Multi-Instanton Corrections to Superpotentials in Type II Compactifications

We consider two simple examples of multi-instanton configurations in type II 4d N=1 superstring compactifications. The first one involves O(1) and U(1) D2-instantons embedded in T^6/(Z_2 x Z_2') geometry with SU(4) gauge symmetry coming from D6-branes and the second is related to quiver gauge theory of orientifolded orbifold of conifold containing fractional D(-1)-instantons and D3-branes. The additional zero modes of instantons which appear at the intersection points are lifted through interactions stemming from dimensionally reduced F- and D-terms. It is shown that multi-instanton configurations effectively generate corrections to the usual non-perturbative superpotential for chiral matter.Comment: LaTeX, 17 pages, 3 Postscript figure