Hans Jockers

Nonabelian 2D gauge theories for determinantal Calabi-Yau varieties

A bstractThe two-dimensional supersymmetric gauged linear sigma model (GLSM) with abelian gauge groups and matter fields has provided many insights into string theory on Calabi-Yau manifolds of a certain type: complete intersections in toric varieties. In this paper, we consider two GLSM constructions with nonabelian gauge groups and charged matter whose infrared CFTs correspond to string propagation on determinantal Calabi-Yau varieties, furnishing another broad class of Calabi-Yau geometries in addition to complete intersections. We show that these two models — which we refer to as the PAX and the PAXY model — are dual descriptions of the same low-energy physics. Using GLSM techniques, we determine the quantum Kähler moduli space of these varieties and find no disagreement with existing results in the literature.

Conifold transitions in

We consider topology changing transitions for M-theory compactifications on Calabi– Yau fourfolds with background G-flux. The local geometry of the transition is generically a genus g curve of conifold singularities, which engineers a 3d gauge theory with four supercharges, near the intersection of Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta which solve the local quantization condition on G for a given geometry, including new solutions in which the flux is neither of horizontal nor vertical type. A local analysis of the flux superpotential shows that the potential has flat directions for a subset of these fluxes and the topologically different phases can be dynamically connected. For special geometries and background configurations, the local transitions extend to extremal transitions between global fourfold compactifications with flux. By a circle decompactification the M-theory analysis identifies consistent flux configurations in four-dimensional F-theory compactifications and flat directions in the deformation space of branes with bundles.

M

Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models, we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

{\mathcal{U}_{q}su(N)}

Knot Invariants from Topological Recursion on Augmentation Varieties

Uqsu(N). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation {2, 1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or—in the limit to classical groups—to determine color factors for Yang Mills amplitudes.

Torus Knots and the Topological Vertex

We propose a general formula for perturbative-in-α′ corrections to the Kähler potential on the quantum Kähler moduli space of Calabi–Yau n-folds, for any n, in their asymptotic large volume regime. The knowledge of such perturbative corrections provides an important ingredient needed to analyze the full structure of this Kähler potential, including nonperturbative corrections such as the Gromov–Witten invariants of the Calabi–Yau n-folds. We argue that the perturbative corrections take a universal form, and we find that this form is encapsulated in a specific additive characteristic class of the Calabi–Yau n-fold which we call the log Gamma class, and which arises naturally in a generalization of Mukai’s modified Chern character map. Our proposal is inspired heavily by the recent observation of an equality between the partition function of certain supersymmetric, two-dimensional gauge theories on a two-sphere, and the aforementioned Kähler potential. We further strengthen our proposal by comparing our findings on the quantum Kähler moduli space to the complex structure moduli space of the corresponding mirror Calabi–Yau geometry.

Flat connections in open string mirror symmetry

Using the duality between Wilson loop expectation values of SU(N) Chern–Simons theory on S3 and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on S3 and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the local mirror Calabi–Yau threefold of the resolved conifold, we generalize the topological recursion of the remodelled B-model in order to study branes beyond the class of toric Harvey–Lawson special Lagrangians—as required for analyzing non-trivial knots on S3. The basic ingredients for the proposed recursion are the spectral curve, given by the augmentation variety of the knot, and the calibrated annulus kernel, encoding the topological annulus amplitudes associated to the knot. We present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion. We further argue that—if an explicit form of the calibrated annulus kernel is provided for any other knot—the proposed topological recursion should still be applicable. We study the implications of our proposal for knot theory, which exhibit interesting consequences for colored HOMFLY polynomials of mutant knots.

Dual pairs of gauged linear sigma models and derived equivalences of Calabi–Yau threefolds

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S3. The key role is played by the