Upsilon Invariants from Cyclic Branched Covers

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

Gaps of saddle connection directions for some branched covers of tori

We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at zero and quadratic tail decay. We also construct examples of translation surfaces in any genus $d>1$ that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked tori and saddle connections between distinct marked points with a specific orientation. These results can be interpreted as the gap distribution of slopes of affine lattices. We obtain our results by translating the question of gap distributions to a dynamical question of return times to a transversal under the horocycle flow on an appropriate moduli space.

From symmetric product CFTs to AdS3

Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator thus has the form of an integral over the entire world-sheet moduli space weighted with a Nambu-Goto-like action. Quite strikingly, at leading order this action can also be written as the absolute value of the Schwarzian of the covering map.Given the equivalence of the symmetric product CFT to tensionless string theory on AdS3, this provides an explicit realisation of the underlying mechanism of gauge-string duality originally proposed in [1] and further refined in [2].

Free field world-sheet correlators for AdS3

We employ the free field realisation of the $$ \mathfrak{psu}{\left(1,1\left|2\right.\right)}_1 $$ psu 1 1 2 1 world-sheet theory to constrain the correlators of string theory on AdS3× S3× 𝕋4 with unit NS-NS flux. In particular, we directly obtain the unusual delta function localisation of these correlators onto branched covers of the boundary S2 by the (genus zero) world-sheet — this is the key property which makes the equivalence to the dual symmetric orbifold manifest. In our approach, this feature follows from a remarkable ‘incidence relation’ obeyed by the correlators, which is reminiscent of a twistorial string description. We also illustrate our results with explicit computations in various special cases.

Explicit computation of some families of Hurwitz numbers, II

We continue our computation, using a combinatorial method based on Gronthendieck’s dessins d’enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2, …, 2], [2h + 1, 3, 2, …, 2], $\begin{array}{} \displaystyle \pi=[d_i]_{i=1}^\ell. \end{array}$ We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the di’s.