We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on R3 ⇥ S1, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory T[Σ,G] on L(k,1)×S1, the other is the LG "equivariant Verlinde formula", or equivalently partition function of LGℂ complex Chern-Simons theory on Σ×S1. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual LG. When G is not simply-connected, we provide a recipe of computing the index of T[Σ,G] as summation over indices of T[Σ,G̃ ] with non-trivial background 't Hooft fluxes, where G̃ is the simply-connected group with the same Lie algebra. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for G=SU(2) or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU(N) or PSU(N) and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

In this paper we study the support recovery problem for single index models Y=f(X⊺β,ε), where f is an unknown link function, X∼Np(0,𝕀p) and β is an s-sparse unit vector such that βi∈{±1s√,0}. In particular, we look into the performance of two computationally inexpensive algorithms: (a) the diagonal thresholding sliced inverse regression (DT-SIR) introduced by Lin et al. (2015); and (b) a semi-definite programming (SDP) approach inspired by Amini & Wainwright (2008). When s=O(p1−δ) for some δ>0, we demonstrate that both procedures can succeed in recovering the support of β as long as the rescaled sample size κ=nslog(p−s) is larger than a certain critical threshold. On the other hand, when κ is smaller than a critical value, any algorithm fails to recover the support with probability at least 12 asymptotically. In other words, we demonstrate that both DT-SIR and the SDP approach are optimal (up to a scalar) for recovering the support of β in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.

The operators with large scaling dimensions can be labeled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large N but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu– Wigner contraction of the u(2) algebra inside of the u(p) algebra of p giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large N limit.