Seiberg-Witten prepotential for E-string theory and random partitions

We find a Nekrasov-type expression for the Seiberg-Witten prepotential for the six-dimensional non-critical E_8 string theory toroidally compactified down to four dimensions. The prepotential represents the BPS partition function of the E_8 strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta. We show that our expression exhibits expected modular properties. In particular, we prove that it obeys the modular anomaly equation known to be satisfied by the prepotential.Comment: 14 page

Seiberg-Witten prepotential for E-string theory and global symmetries

We obtain Nekrasov-type expressions for the Seiberg-Witten prepotential for the six-dimensional (1,0) supersymmetric E-string theory compactified on T^2 with nontrivial Wilson lines. We consider compactification with four general Wilson line parameters, which partially break the E_8 global symmetry. In particular, we investigate in detail the cases where the Lie algebra of the unbroken global symmetry is E_n + A_{8-n} with n=8,7,6,5 or D_8. All our Nekrasov-type expressions can be viewed as special cases of the elliptic analogue of the Nekrasov partition function for the SU(N) gauge theory with N_f=2N flavors. We also present a new expression for the Seiberg-Witten curve for the E-string theory with four Wilson line parameters, clarifying the connection between the E-string theory and the SU(2) Seiberg-Witten theory with N_f=4 flavors.Comment: 22 pages. v2: comments and a reference added, version to appear in JHE

Towards Deconstruction of the Type D (2,0) Theory

We propose a four-dimensional supersymmetric theory that deconstructs, in a particular limit, the six-dimensional $(2,0)$ theory of type $D_k$. This 4d theory is defined by a necklace quiver with alternating gauge nodes $\mathrm{O}(2k)$ and $\mathrm{Sp}(k)$. We test this proposal by comparing the 6d half-BPS index to the Higgs branch Hilbert series of the 4d theory. In the process, we overcome several technical difficulties, such as Hilbert series calculations for non-complete intersections, and the choice of $\mathrm{O}$ versus $\mathrm{SO}$ gauge groups. Consistently, the result matches the Coulomb branch formula for the mirror theory upon reduction to 3d

Mapping 6D N = 1 supergravities to F-theory

We develop a systematic framework for realizing general anomaly-free chiral 6D supergravity theories in F-theory. We focus on 6D (1, 0) models with one tensor multiplet whose gauge group is a product of simple factors (modulo a finite abelian group) with matter in arbitrary representations. Such theories can be decomposed into blocks associated with the simple factors in the gauge group; each block depends only on the group factor and the matter charged under it. All 6D chiral supergravity models can be constructed by gluing such blocks together in accordance with constraints from anomalies. Associating a geometric structure to each block gives a dictionary for translating a supergravity model into a set of topological data for an F-theory construction. We construct the dictionary of F-theory divisors explicitly for some simple gauge group factors and associated matter representations. Using these building blocks we analyze a variety of models. We identify some 6D supergravity models which do not map to integral F-theory divisors, possibly indicating quantum inconsistency of these 6D theories.Comment: 37 pages, no figures; v2: references added, minor typos corrected; v3: minor corrections to DOF counting in section

5-dim Superconformal Index with Enhanced En Global Symmetry

The five-dimensional $\mathcal{N}=1$ supersymmetric gauge theory with Sp(N) gauge group and SO(2N_f) flavor symmetry describes the physics on N D4-branes with $N_f$ D8-branes on top of a single O8 orientifold plane in Type I' theory. This theory is known to be superconformal at the strong coupling limit with the enhanced global symmetry $E_{N_f+1}$ for $N_f\le 7$. In this work we calculate the superconformal index on $S^1\times S^4$ for the Sp(1) gauge theory by the localization method and confirm such enhancement of the global symmetry at the superconformal limit for $N_f\le 5$ to a few leading orders in the chemical potential. Both perturbative and (anti)instanton contributions are present in this calculation. For $N_f=6,7$ cases some issues related the pole structure of the instanton calculation could not be resolved and here we could provide only some suggestive answer for the leading contributions to the index. For the Sp(N) case, similar issues related to the pole structure appear.Comment: 70 pages, references added, published versio

Static Charges in the Low-Energy Theory of the S-Duality Twist

We continue the study of the low-energy limit of N=4 super Yang-Mills theory compactified on a circle with S-duality and R-symmetry twists that preserve N=6 supersymmetry in 2+1D. We introduce external static supersymmetric quark and anti-quark sources into the theory and calculate the Witten Index of the resulting Hilbert space of ground states on a torus. Using these results we compute the action of simple Wilson loops on the Hilbert space of ground states without sources. In some cases we find disagreement between our results for the Wilson loop eigenvalues and previous conjectures about a connection with Chern-Simons theory.Comment: 73 pages, two paragraphs added, one to the introduction and one to the discussio

An Instanton Toolbox for F-Theory Model Building

Several dimensionful parameters needed for model building can be engineered in a certain class of SU(5) F-theory GUTs by adding extra singlet fields which are localized along pairwise intersections of D7-branes. The values of these parameters, however, depend on dynamics external to the GUT which causes the singlets to acquire suitable masses or expectation values. In this note, we demonstrate that D3-instantons which wrap the same 4-cycle as one of the intersecting D7's can provide precisely the needed dynamics to generate several important scales, including the supersymmetry-breaking scale and the right-handed neutrino mass. Furthermore, these instantons seem unable to directly generate the \mu term suggesting that, at least in this class of models, it should perhaps be tied to one of the other scales in the problem. More specifically, we study the simple system consisting of a pair of D7-branes wrapping del Pezzo surfaces which intersect along a curve $\Sigma$ of genus 0 or 1 and classify all instanton configurations which can potentially contribute to the superpotential. This allows one to formulate topological conditions which must be imposed on \Sigma for various model-building applications. Along the way, we also observe that the construction of arXiv:0808.1286 which engineers a linear superpotential in fact realizes an O'Raifeartaigh model at the KK scale whose 1-loop Coleman-Weinberg potential generically leads to a metastable, long-lived SUSY-breaking vacuum.Comment: 18 pages, 2 figures; v2: updated to reflect corrections in v2 of 0808.128