The isomorphism between two-dimensional conformal field theories with the affine-sl(2) and theN=2 superconformal symmetry algebras

We discuss the one-to-one correspondence between two-dimensional conformal field theories with the affine-sl(2) and the N=2 superconformal symmetry algebras. We obtain the formulas relating zero-norm states in the two theories and the formulas expressing the partition function of the affine-sl(2) theory through the partition function of the theory with the N=2 superconformal symmetry algebra, and vice versa. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 29–45, April. 1998.     

On mirror symmetry in Wess-Zumino-Novikov-Witten models

We examine the problem of constructing N=2 superconformal algebras out of N=1 non-semi-simple affine Lie algebras. These N=2 superconformal theories share the property that the super Virasoro central charge depends only on the dimension of the Lie algebra. We find, in particular, a construction having a central charge c=9. This provides a possible internal space for string compactification and where mirror symmetry might be explored.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 55–63, July, 1995.     

Spectral Networks and Snakes

We apply and illustrate the techniques of spectral networks in a large collection of A _K-1 theories of class S, which we call “lifted A ₁ theories.” Our construction makes contact with Fock and Goncharov’s work on higher Teichmüller theory. In particular, we show that the Darboux coordinates on moduli spaces of flat connections which come from certain special spectral networks coincide with the Fock–Goncharov coordinates. We show, moreover, how these techniques can be used to study the BPS spectra of lifted A ₁ theories. In particular, we determine the spectrum generators for all the lifts of a simple superconformal field theory.     

Quelques remarques sur les actions analytiques des réseaux des groupes de Lie de rang supérieur

All the actions considered here are (real) analytic. Let г be a subgroup of finite index of SL(n,). We prove, in particular, the (global) homotopical rigidity, for both its standard affine action on the torus T^a (n ≥ 3), and its standard projective action on the sphere Sⁿ⁻¹ (n, ≥ 4).     

On the Elliptic Genera of Manifolds of Spin(7) Holonomy

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.     

On a second conjecture of Stolarsky: the sum of digits of polynomial values

Let q, r ≥ 2 be integers, and denote by s _q the sum-of-digits function in base q. In 1978, K.B. Stolarsky conjectured that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N} \frac{s_2(n^r)}{s_2(n)} \leq r.$$ In this paper we prove this conjecture. We show that for polynomials ${P_1(X), P_2(X) \in \mathbb{Z}[X]}$ of degrees r ₁, r ₂ ≥ 1 and integers q ₁, q ₂ ≥ 2, we have $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N}\frac{s_{q_1}(P_1(n))}{s_{q_2}(P_2(n))} = \frac{r_1 (q_1 - 1) {\rm log}q_2}{r_2(q_2 - 1) {\rm log} q_1}.$$ We also present a variant of the problem to polynomial values of prime numbers.     

Combinatorial expansions of conformal blocks

A representation of Nekrasov partition functions in terms of a nontrivial two-dimensional conformal field theory was recently suggested. For a nonzero value of the deformation parameter ∈ = ∈ ₁ + ∈ ₂ , the instanton partition function is identified with a conformal block of the Liouville theory with the central charge c = 1 + 6∈ ² /∈ ₁ ∈ ₂ . The converse of this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with nondegenerate Virasoro representations, has a nontrivial decomposition into a sum over Young diagrams that differs from the natural decomposition studied in conformal field theory. We provide some details about this new nontrivial correspondence in the simplest case of the four-point correlation functions.     

A generalization of the Poisson integral formula for the functions harmonic and biharmonic in a ball

We construct an analytic solution to the problem of extension to the unit N-dimensional ball of the potential on its values on an interior sphere. The formula generalizes the conventional Poisson formula. Bavrin’s results obtained for the two-dimensional case by methods of function theory are transferred to the N-dimensional case (N ≥ 3). We also exhibit a solution to a similar extension problem for some operator expressions depending on a potential known on an interior sphere. A connection is established between solutions to the moment problem on a segment and on a semiaxis.     

The Ω Deformed B-model for Rigid N = 2 Theories

We give an interpretation of the Ω deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four-dimensional rigid N = 2 theories explicitly in general Ω-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N = 2 supersymmetric theories. The rigid N = 2 field theories we focus on are the conformal rank one N = 2 Seiberg–Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N = 2 theories arising from compactifications on local Calabi–Yau manifolds, we consider the theory of local ${\mathbb{P}^2}$ . We calculate motivic Donaldson–Thomas invariants for this geometry and make predictions for generalized Gromov–Witten invariants at the orbifold point.     

Three-dimensional superconformal field theories, indices, and monopoles

We explain the recent progress in three-dimensional superconformal field theories based on the index for magnetic monopole operators and discuss applications to M2-branes and the AdS/CFT duality.     

Spectral Networks

We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional ${\mathcal{N} = 2}$ theories coupled to surface defects, particularly the theories of class S. In these theories, spectral networks provide a useful tool for the computation of BPS degeneracies; the network directly determines the degeneracies of solitons living on the surface defect, which in turn determines the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat ${{\rm GL}(K, \mathbb{C})}$ connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover ${\Sigma}$ of C. This construction produces natural coordinate systems on moduli spaces of flat ${{\rm GL}(K, \mathbb{C})}$ connections on C, which we conjecture are cluster coordinate systems.     

Stanley Reisner rings of generalized truncation polytopes and their moment angle manifolds

We consider simple polytopes \(P = vc^k \left( {\Delta ^{n_1 } \times \ldots \times \Delta ^{n_r } } \right)\) for n ₁ ≥ … ≥ n _r ≥ 1, r ≥ 1, and k ≥ 0, that is, k-vertex cuts of a product of simplices, and call them generalized truncation polytopes. For these polytopes we describe the cohomology ring of the corresponding moment-angle manifold \(\mathcal{Z}_P\) and explore some topological consequences of this calculation. We also examine minimal non-Golodness for their Stanley-Reisner rings and relate it to the property of \(\mathcal{Z}_P\) being a connected sum of sphere products.     

Towards a classification of modular invariant partition functions for theories based onN=4 superconformal algebras

The representation theory of the doubly extended N=4 superconformal algebra is reviewed. The modular properties of the corresponding characters can be derived, using characters sumrules for coset realizations of these N=4 algebras. Some particular combinations of massless characters are shown to transform as affine SU(2) characters under S and T, a fact used to completely classify the massless sector of the partition function.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 385–394, May, 1993.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

The external constraint 4 nonempty part sandwich problem

List partitions generalize list colourings. Sandwich problems generalize recognition problems. The polynomial dichotomy (NP-complete versus polynomial) of list partition problems is solved for 4-dimensional partitions with the exception of one problem (the list stubborn problem) for which the complexity is known to be quasipolynomial. Every partition problem for 4 nonempty parts and only external constraints is known to be polynomial with the exception of one problem (the 2K₂-partition problem) for which the complexity of the corresponding list problem is known to be NP-complete. The present paper considers external constraint 4 nonempty part sandwich problems. We extend the tools developed for polynomial solutions of recognition problems obtaining polynomial solutions for most corresponding sandwich versions. We extend the tools developed for NP-complete reductions of sandwich partition problems obtaining the classification into NP-complete for some external constraint 4 nonempty part sandwich problems. On the other hand and additionally, we propose a general strategy for defining polynomial reductions from the 2K₂-partition problem to several external constraint 4 nonempty part sandwich problems, defining a class of 2K₂-hard problems. Finally, we discuss the complexity of the Skew Partition Sandwich Problem.     

The Collet–Eckmann condition for rational functions on the Riemann sphere

We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.     

Derivation algebra and automorphism group of generalized topological N = 2 superconformal algebra

We determine the derivation algebra and the automorphism group of the generalized topological N = 2 superconformal algebra.     

On the Representation of Band-Dominant Functions on the Sphere Using Finitely Many Bits

A band-dominant function on the Euclidean sphere embedded in R ^q+1 is the restriction to this sphere of an entire function of q+1 complex variables having a finite exponential type in each of its variables. We develop a method to represent such a function using finitely many bits, using the values of the function at scattered sites on the sphere. The number of bits required in our representation is asymptotically the same as the metric entropy of the class of such functions with respect to any of the L ^p norms on the sphere.