Positroid Stratification of Orthogonal Grassmannian and ABJM Amplitudes

A novel understanding of scattering amplitudes in terms of on-shell diagrams and positive Grassmannian has been recently established for four dimensional Yang-Mills theories and three dimensional Chern-Simons theories of ABJM type. We give a detailed construction of the positroid stratification of orthogonal Grassmannian relevant for ABJM amplitudes. On-shell diagrams are classified by pairing of external particles. We introduce a combinatorial aid called `OG tableaux' and map each equivalence class of on-shell diagrams to a unique tableau. The on-shell diagrams related to each other through BCFW bridging are naturally grouped by the OG tableaux. Introducing suitably ordered BCFW bridges and positive coordinates, we construct the complete coordinate charts to cover the entire positive orthogonal Grassmannian for arbitrary number of external particles. The graded counting of OG tableaux suggests that the positive orthogonal Grassmannian constitutes a combinatorial polytope.Comment: 32 pages, 23 figures; v2. minor corrections; v3. several clarifications and minor improvement

Little strings on DnD_n orbifolds

We explore two classes of 6d $\mathcal{N}=(1,0)$ little string theories obtained from type IIA/IIB NS5-branes probing $D_n$ singularities. Their tensor branches are described by effective gauge theories whose instanton solitons are macroscopic little strings. We specifically study two families of 2d $\mathcal{N}=(0,4)$ gauge theories which describe at low energy the worldsheet dynamics of the type IIA/IIB little strings. These gauge theories are useful to calculate the supersymmetric partition functions of the little string theories on $\mathbf{R}^4 \times T^2$. We establish the T-duality of the little string theories by utilizing their BPS spectra as a probe.Comment: 29 pages, 9 figures; v2: minor changes, published versio

Two Single-Reference Approaches to Singlet Biradicaloid Problems: Complex, Restricted Orbitals and Approximate Spin-Projection Combined With Regularized Orbital-Optimized M{\o}ller-Plesset Perturbation Theory

We present a comprehensive study of two single-reference approaches to singlet biradicaloids. These two approaches are based on the recently developed regularized orbital-optimized M{\o}ller-Plesset method ($\kappa$-OOMP2). The first approach is to combine the Yamaguchi's approximate projection (AP) scheme and $\kappa$-OOMP2 with unrestricted (U) orbitals ($\kappa$-UOOMP2). By capturing only essential symmetry breaking, $\kappa$-UOOMP2 can serve as a suitable basis for AP. The second approach is $\kappa$-OOMP2 with complex, restricted (cR) orbitals ($\kappa$-cROOMP2). Though its applicability is more limited due to the comparative rarity of cR solutions, $\kappa$-cROOMP2 offers a simple framework for describing singlet biradicaloids with complex polarization while removing artificial spatial symmetry breaking. We compare the scope of these two methods with numerical studies. We show that AP+$\kappa$-UOOMP2 and $\kappa$-cROOMP2 can perform similarly well in the TS12 set, a data set that includes 12 data points for triplet-singlet gaps of several atoms and diatomic molecules with a triplet ground state. This was also found to be true for the barrier height of a reaction involving attack on a cysteine ion by a singlet oxygen molecule. However, we also demonstrate that in highly symmetric systems like $\text{C}_{30}$ ($\text{D}_{5h}$) $\kappa$-cROOMP2 is more suitable as it conserves spatial symmetry. Lastly, we present an organic biradicaloid that does not have a $\kappa$-cROOMP2 solution in which case only AP+$\kappa$-UOOMP2 is applicable. We recommend $\kappa$-cROOMP2 whenever complex polarization is essential and AP+$\kappa$-UOOMP2 for biradicaloids without essential complex polarization but with essential spin-polarization

Elliptic Genus of E-strings

We study a family of 2d N=(0,4) gauge theories which describes at low energy the dynamics of E-strings, the M2-branes suspended between a pair of M5 and M9 branes. The gauge theory is engineered using a duality with type IIA theory, leading to the D2-branes suspended between an NS5-brane and 8 D8-branes on an O8-plane. We compute the elliptic genus of this family of theories, and find agreement with the known results for single and two E-strings. The partition function can in principle be computed for arbitrary number of E-strings, and we compute them explicitly for low numbers. We test our predictions against the partially known results from topological strings, as well as from the instanton calculus of 5d Sp(1) gauge theory. Given the relation to topological strings, our computation provides the all genus partition function of the refined topological strings on the canonical bundle over 1/2 K3.Comment: 49 pages, 2 figure

Regularized Orbital-Optimized Second-Order M{\o}ller-Plesset Perturbation Theory: A Reliable Fifth-Order Scaling Electron Correlation Model with Orbital Energy Dependent Regularizers

We derive and assess two new classes of regularizers that cope with offending denominators in the single-reference second-order M{\o}ller-Plesset perturbation theory (MP2). In particular, we discuss the use of two types of orbital energy dependent regularizers, $\kappa$ and $\sigma$, in conjunction with orbital-optimized MP2 (OOMP2). The resulting fifth-order scaling methods, $\kappa$-OOMP2 and $\sigma$-OOMP2, have been examined for bond-breaking, thermochemistry, and biradical problems. Both methods with strong enough regularization restore restricted to unrestricted instability (i.e. Coulson-Fischer points) that unregularized OOMP2 lacks when breaking bonds in $\text{H}_{2}$, $\text{C}_2\text{H}_6$, $\text{C}_2\text{H}_4$, and $\text{C}_2\text{H}_2$. The training of the $\kappa$ and $\sigma$ regularization parameters was performed with the W4-11 set. We further developed scaled correlation energy variants, $\kappa$-S-OOMP2 and $\sigma$-S-OOMP2, by training on the TAE140 subset of the W4-11 set. Those new OOMP2 methods were tested on the RSE43 set and the TA14 set where unmodified OOMP2 itself performs very well. The modifications we made were found insignificant in these data sets. Furthermore, we tested the new OOMP2 methods on singlet biradicaloids using Yamaguchi's approximate spin-projection. Unlike the unregularized OOMP2, which fails to converge these systems due to the singularity, we show that regularized OOMP2 methods successfully capture strong biradicaloid characters. While further assessment on larger datasets is desirable, $\kappa$-OOMP2 with $\kappa$ = 1.45 $E_{h}^{-1}$ appears to combine favorable recovery of Coulson-Fischer points with good numerical performance