Lectures on mathematical aspects of (twisted) supersymmetric gauge theories

Supersymmetric gauge theories have played a central role in applications of quantum field theory to mathematics. Topologically twisted supersymmetric gauge theories often admit a rigorous mathematical description: for example, the Donaldson invariants of a 4-manifold can be interpreted as the correlation functions of a topologically twisted N=2 gauge theory. The aim of these lectures is to describe a mathematical formulation of partially-twisted supersymmetric gauge theories (in perturbation theory). These partially twisted theories are intermediate in complexity between the physical theory and the topologically twisted theories. Moreover, we will sketch how the operators of such a theory form a two complex dimensional analog of a vertex algebra. Finally, we will consider a deformation of the N=1 theory and discuss its relation to the Yangian, as explained in arXiv:1308.0370 and arXiv:1303.2632.Comment: Notes from a lecture series by the first author at the Les Houches Winter School on Mathematical Physics in 2012. To appear in the proceedings of this conference. Related to papers arXiv:1308.0370, arXiv:1303.2632, and arXiv:1111.423

Charge Lattices and Consistency of 6D Supergravity

We extend the known consistency conditions on the low-energy theory of six-dimensional N = 1 supergravity. We review some facts about the theory of two-form gauge fields and conclude that the charge lattice Gamma for such a theory has to be self-dual. The Green-Schwarz anomaly cancellation conditions in the supergravity theory determine a sublattice of Gamma. The condition that this sublattice can be extended to a self-dual lattice Gamma leads to a strong constraint on theories that otherwise appear to be self-consistent.Comment: 15 pages. v2: minor changes; references, additional example added; v3: minor corrections and clarifications added, JHEP versio

Entropy of the self-dual string soliton

We compute the entropy and the corresponding central charge of the self-dual string soliton in the supergravity regime using the blackfold description of the fully localized M2-M5 intersection.Comment: 15 pages, 1 figure, harvma

On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups

We study the quantization of the M-theory G-flux on elliptically fibered Calabi-Yau fourfolds with singularities giving rise to unitary and symplectic gauge groups. We seek and find its relation to the Freed-Witten quantization of worldvolume fluxes on 7-branes in type IIB orientifold compactifications on Calabi-Yau threefolds. By explicitly constructing the appropriate four-cycles on which to calculate the periods of the second Chern class of the fourfolds, we find that there is a half-integral shift in the quantization of G-flux whenever the corresponding dual 7-brane is wrapped on a non-spin submanifold. This correspondence of quantizations holds for all unitary and symplectic gauge groups, except for SU(3), which behaves mysteriously. We also perform our analysis in the case where, in addition to the aforementioned gauge groups, there is also a 'flavor' U(1)-gauge group.Comment: 33 pages, 4 figure

D-brane anomaly inflow revisited

Axial and gravitational anomaly of field theories, when embedded in string theory, must be accompanied by canceling inflow. We give a self-contained overview for various world-volume theories, and clarify the role of smeared magnetic sources in I-brane/D-brane cases. The proper anomaly descent of the source, as demanded by regularity of RR field strengths H's, turns out to be an essential ingredient. We show how this allows correct inflow to be generated for all such theories, including self-dual cases, and also that the mechanism is now insensitive to the choice between the two related but inequivalent forms of D-brane Chern-Simons couplings. In particular, SO(6)_R axial anomaly of d=4 maximal SYM is canceled by the inflow onto D3-branes via the standard minimal coupling to C_4. We also propose how, for the anomaly cancelation, the four types of Orientifold planes should be coupled to the spacetime curvatures, of which conflicting claims existed previously.Comment: 41 pages, references updated; version to appear in JHE

BPS black holes, the Hesse potential, and the topological string

The Hesse potential is constructed for a class of four-dimensional N=2 supersymmetric effective actions with S- and T-duality by performing the relevant Legendre transform by iteration. It is a function of fields that transform under duality according to an arithmetic subgroup of the classical dualities reflecting the monodromies of the underlying string compactification. These transformations are not subject to corrections, unlike the transformations of the fields that appear in the effective action which are affected by the presence of higher-derivative couplings. The class of actions that are considered includes those of the FHSV and the STU model. We also consider heterotic N=4 supersymmetric compactifications. The Hesse potential, which is equal to the free energy function for BPS black holes, is manifestly duality invariant. Generically it can be expanded in terms of powers of the modulus that represents the inverse topological string coupling constant, $g_s$, and its complex conjugate. The terms depending holomorphically on $g_s$ are expected to correspond to the topological string partition function and this expectation is explicitly verified in two cases. Terms proportional to mixed powers of $g_s$ and $\bar g_s$ are in principle present.Comment: 28 pages, LaTeX, added comment

Mycobacterium avium-intracellulare cellulitis occurring with septic arthritis after joint injection: a case report

BACKGROUND: Cellulitis caused by Mycobacterium avium-intracellulare has rarely been described. Mycobacterium avium-intracellulare is a rare cause of septic arthritis after intra-articular injection, though the causative role of injection is difficult to ascertain in such cases. CASE PRESENTATION: A 57-year-old with rheumatoid arthritis treated with prednisone and azathioprine developed bilateral painful degenerative shoulder arthritis. After corticosteroid injections into both acromioclavicular joints, he developed bilateral cellulitis centered over the injection sites. Skin biopsy showed non-caseating granulomas, and culture grew Mycobacterium avium-intracellulare. Joint aspiration also revealed Mycobacterium avium-intracellulare infection. CONCLUSION: Although rare, skin and joint infections caused by Mycobacterium avium-intracellulare should be considered in any immunocompromised host, particularly after intra-articular injection. Stains for acid-fast bacilli may be negative in pathologic samples even in the presence of infection; cultures of tissue specimens should always be obtained

Gauge Fluxes in F-theory and Type IIB Orientifolds

We provide a detailed correspondence between G_4 gauge fluxes in F-theory compactifications with SU(n) and SU(n)x(1) gauge symmetry and their Type IIB orientifold limit. Based on the resolution of the relevant F-theory Tate models we classify the factorisable G_4-fluxes and match them with the set of universal D5-tadpole free U(1)-fluxes in Type IIB. Where available, the global version of the universal spectral cover flux corresponds to Type IIB gauge flux associated with a massive diagonal U(1). In U(1)-restricted Tate models extra massless abelian fluxes exist which are associated with specific linear combinations of Type IIB fluxes. Key to a quantitative match between F-theory and Type IIB is a proper treatment of the conifold singularity encountered in the Sen limit of generic F-theory models. We also shed further light on the brane recombination process relating generic and U(1)-restricted Tate models.Comment: 53 pages, 3 figures; v2: Refs added; v3: minor corrections to match version published in JHE

On Flux Quantization in F-Theory

We study the problem of four-form flux quantization in F-theory compactifications. We prove that for smooth, elliptically fibered Calabi-Yau fourfolds with a Weierstrass representation, the flux is always integrally quantized. This implies that any possible half-integral quantization effects must come from 7-branes, i.e. from singularities of the fourfold. We subsequently analyze the quantization rule on explicit fourfolds with Sp(N) singularities, and connect our findings via Sen's limit to IIB string theory. Via direct computations we find that the four-form is half-integrally quantized whenever the corresponding 7-brane stacks wrap non-spin complex surfaces, in accordance with the perturbative Freed-Witten anomaly. Our calculations on the fourfolds are done via toric techniques, whereas in IIB we rely on Sen's tachyon condensation picture to treat bound states of branes. Finally, we give general formulae for the curvature- and flux-induced D3 tadpoles for general fourfolds with Sp(N) singularities.Comment: 46 page

BPS dyons and Hesse flow

We revisit BPS solutions to classical N=2 low energy effective gauge theories. It is shown that the BPS equations can be solved in full generality by the introduction of a Hesse potential, a symplectic analog of the holomorphic prepotential. We explain how for non-spherically symmetric, non-mutually local solutions, the notion of attractor flow generalizes to gradient flow with respect to the Hesse potential. Furthermore we show that in general there is a non-trivial magnetic complement to this flow equation that is sourced by the momentum current in the solution.Comment: 25 pages, references adde