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Given the recent geometrical classification of 6d $(1,0)$ SCFTs, a major question is how to compute for this large class their elliptic genera. The latter encode the refined BPS spectrum of the SCFTs, which determines geometric invariants of the associated elliptic non-compact Calabi-Yau threefolds. In this paper we establish for all 6d $(1,0)$ SCFTs in the atomic classification blowup equations that fix these elliptic genera to large extent. The latter fall into two types: the unity- and the vanishing blowup equations. For almost all rank one theories, we find unity blowup equations which determine the elliptic genera completely. We develop several techniques to compute elliptic genera and BPS invariants from the blowup equations, including a recursion formula with respect to the number of strings, a Weyl orbit expansion, a refined BPS expansion and an $\epsilon_1,\epsilon_2$ expansion. For higher-rank theories, we propose a gluing rule to obtain all their blowup equations based on those of rank one theories. For example, we explicitly give the elliptic blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of $-2$ curves and conformal matter theories. We also give the toric construction for many elliptic non-compact Calabi-Yau threefolds which engineer 6d $(1,0)$ SCFTs with various matter representations., Comment: 164 pages, 3 figures, 24 tables, typos corrected, refs added

In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate (GS) energy of the system is compared and, for small lattice sizes (e.g. $3 \times 3$ with $18$ spinors), shown to agree with the analytically derived value of the energy up to a deviation of $0.09\%$. Moreover, the wave-functions we find have $99.89\%$ overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2+1)d and 2d CFTs, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the $2$d Ising model., Comment: (15+5) pages, 11 figures, 5 tables: Minor journal revisions and additions. + Journal ref

In this work we present solutions to Knizhnik-Zamolodchikov (KZ) equations corresponding to conformal block wavefunctions of non-Abelian Ising- and Fibonacci-Anyons. We solve these equations around regular singular points in configuration space in terms of hypergeometric functions and derive explicit monodromy representations of the braid group action. This confirms the correct non-Abelian statistics of the solutions. One novelty of our approach is that we explicitly keep track of spin basis states and identify conformal blocks uniquely with such states at relevant points in moduli space., 39 pages, 3 figures

An alumina coating with a thickness of 20–50 μm was deposited by plasma electrolytic oxidation (PEO) on an AA 7075 Al alloy in an alkali-aluminate electrolyte using a 5 kW DC power supply. After the formation of the coating, the effects of current density, electrolyte concentration, and process time on the coating characteristics were investigated comprehensively. In order to achieve improved tribological and corrosion properties, the prepared samples were evaluated by means of various characterization techniques including scanning electron microscopy (SEM) and X-ray diffraction (XRD) analyses, micro-hardness, thickness and roughness measurements, corrosion potentio-dynamic, and pin-on-disk wear tests. Electrophoretic graphite absorption was subsequently carried out on the samples, which resulted in the desirable experimental results to boost the coating in terms of roughness, tribological, and corrosion properties. The positive effects of graphite absorption on the PEO coatings were also elaborated in detail.

The present study concentrated on the metallurgical, micro- and macroscopic observations, hardness evaluation, as well as X-ray diffraction (XRD) and energy-dispersive spectroscopy (EDS) analysis induced during the high-temperature operation of a pipeline that resulted in its relatively short-time failure. An ASTM A213-T12 superheater pipeline is used in a fossil fuel power-plant unit and is constructed by an array of steel tubes joined together by circumferential weldments. Macroscopic observations indicated a thinning and prolonged deformation of the superheater tube in the region close to the fire zone besides thick corrosion products on the other side of the pipe. Hardness measurements showed a remarkable decrease in hardness values of superheater metal near to the failure zone, indicating that grain coarsening or any other microstructure weakening must happen. Scanning electron microscopy (SEM) observations proved the ductile nature of cracking and justified the prolonged failure of the tube. The observed voids in the microstructure of the failed section demonstrated the occurrence of creep phenomenon. XRD and EDS analyses demonstrated the nature of sediments formed on the tube and described how the formation of these corrosion products contribute in the removal of any possible protective passive layer, exposing the tube to further corrosive deterioration and creating loop stresses which, in turn, will induce creep phenomenon and accelerate the occurrence of a failure in the tube.

In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve., Comment: v2 typos corrected, added explanations on brane constructions, added references

We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains., 54 pages, 10 tables, references added

Compactifications of 6d N $$ \mathcal{N} $$ = (1, 0) SCFTs give rise to new 4d N $$ \mathcal{N} $$ = 1 SCFTs and shed light on interesting dualities between such theories. In this paper we continue exploring this line of research by extending the class of compactified 6d theories to the D- type case. The simplest such 6d theory arises from D5 branes probing D-type singularities. Equivalently, this theory can be obtained from an F-theory compactification using −2- curves intersecting according to a D-type quiver. Our approach is two-fold. We start by compactifying the 6d SCFT on a Riemann surface and compute the central charges of the resulting 4d theory by integrating the 6d anomaly polynomial over the Riemann surface. As a second step, in order to find candidate 4d UV Lagrangians, there is an intermediate 5d theory that serves to construct 4d domain walls. These can be used as building blocks to obtain torus compactifications. In contrast to the A-type case, the vanishing of anomalies in the 4d theory turns out to be very restrictive and constraints the choices of gauge nodes and matter content severely. As a consequence, in this paper one has to resort to non- maximal boundary conditions for the 4d domain walls. However, the comparison to the 6d theory compactified on the Riemann surface becomes less tractable.

Quantum curves arise from Seiberg-Witten curves associated to 4d $\mathcal{N}=2$ gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $\mathcal{N}=(1,0)$ SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a $\mathbb{C}^2/\mathbb{Z}_k$ singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $\mathcal{N}=2$ theories are known as class $\mathcal{S}_k$. We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values., Comment: 65 pages, 3 figures

M5 branes probing D-type singularities give rise to 6d (1,0) SCFTs with $SO \times SO$ flavor symmetry known as D-type conformal matter theories. Gauging the diagonal $SO$-flavor symmetry leads to a little string theory with an intrinsic scale which can be engineered in F-theory by compactifying on a doubly-elliptic Calabi-Yau manifold. We derive Seiberg-Witten curves for these little string theories which can be interpreted as mirror curves for the corresponding Calabi-Yau manifolds. Under fiber-base duality these models are mapped to D-type quiver gauge theories and we check that their Seiberg-Witten curves match. By taking decompactification limits, we construct the curves for the related 6d SCFTs and connect to known results in the literature by further taking 5d and 4d limits., Comment: 34 pages, 5 figures

We propose quantum states for Little String Theories (LSTs) arising from M5 branes probing A- and D-type singularities. This extends Witten's picture of M5 brane partition functions as theta functions to this more general setup. Compactifying the world-volume of the five-branes on a two-torus, we find that the corresponding theta functions are sections of line bundles over complex 4-tori. This formalism allows us to derive Seiberg-Witten curves for the resulting four-dimensional theories. Along the way, we prove a duality for LSTs observed by Iqbal, Hohenegger and Rey., 27 pages, 1 Figure. v3: included further explanations and corrected typos. This the version published in JHEP

6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the $A$ case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the $D$ and $E$ cases as well., version 2: typos corrected, 30 pages, 8 figures

Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N=4 topological Yang-Mills theory on P^2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge., 45 p

We study strings associated with minimal 6d SCFTs, which by definition have only one string charge and no Higgs branch. These theories are labelled by a number n with 1 2 the strings interact with the bulk gauge symmetry. In this paper we find a quiver description for the n = 4 string using Sen's limit of F-theory and calculate its elliptic genus with localization techniques. This result is checked using the duality of F-theory with M-theory and topological string theory whose refined BPS partition function captures the elliptic genus of the SCFT strings. We use the topological string theory to gain insight into the elliptic genus for other values of n.

A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural network, thereby increasing the modelling capacity of the network. Both the Boltzmann machine and the derived feedforward neural network can be successfully trained via standard gradient- and non-gradient-based optimization techniques., Comment: 29 pages, 11 figures, final version published in Neurocomputing

M2 branes suspended between adjacent parallel M5 branes lead to light strings, the `M-strings'. In this paper we compute the elliptic genus of M-strings, twisted by maximally allowed symmetries that preserve 2d (2,0) supersymmetry. In a codimension one subspace of parameters this reduces to the elliptic genus of the (4,4) supersymmetric A_{n-1} quiver theory in 2d. We contrast the elliptic genus of N M-strings with the (4,4) sigma model on the N-fold symmetric product of R^4. For N=1 they are the same, but for N>1 they are close, but not identical. Instead the elliptic genus of (4,4) N M-strings is the same as the elliptic genus of (4,0) sigma models on the N-fold symmetric product of R^4, but where the right-moving fermions couple to a modification of the tangent bundle. This construction arises from a dual A_{n-1} quiver 6d gauge theory with U(1) gauge groups. Moreover we compute the elliptic genus of domain walls which separate different numbers of M2 branes on the two sides of the wall., Comment: 75 pages, 16 figures. Minor corrections to the paper, references added

We show how recent progress in computing elliptic genera of strings in six dimensions can be used to obtain expressions for elliptic genera of strings in five-dimensional field theories which have a six-dimensional parent. We further connect our results to recent mathematical results about sheaf counting on ruled surfaces., 17 pages, 4 figures. v3: Definitions clarified, Section "Concluding Thoughts" added

E-strings arise from M2 branes suspended between an M5 brane and an M9 plane. In this paper we obtain explicit expressions for the elliptic genus of two E-strings using a series of string dualities. Moreover we show how this can be used to recover the elliptic genus of two ${E}_{8}\ifmmode\times\else\texttimes\fi{}{E}_{8}$ heterotic strings using the Ho\ifmmode \check{r}\else \v{r}\fi{}ava--Witten realization of heterotic strings in M-theory. This involves highly nontrivial identities among Jacobi forms and is remarkable in light of the fact that E-strings are ``sticky'' and form bound states whereas heterotic strings do not form bound states.

We consider M-theory in the presence of M parallel M5-branes prob- ing a transverse AN−1 singularity. This leads to a superconformal theory with (1,0) supersymmetry in six dimensions. We compute the supersymmetric partition func- tion of this theory on a two-torus, with arbitrary supersymmetry preserving twists, using the topological vertex formalism. Alternatively, we show that this can also be obtained by computing the elliptic genus of an orbifold of recently studied M-strings. The resulting 2d theory is a (4,0) supersymmetric quiver gauge theory whose Higgs branch corresponds to strings propagating on the moduli space of SU(N) M−1 instan- tons on R 4 where the right-moving fermions are coupled to a particular bundle.

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest., Comment: 80 pages, 48 figures

Background Based on Acute Kidney Injury Network (AKIN) criteria, we considered acute kidney injury (AKI) as an absolute increase in the serum creatinine (sCr) level of more than or equal to 0.3 mg/dl or 50%. The introduction of Urinary neutrophil gelatinase-associated lipocalin (UNGAL) has conferred earlier diagnosis of AKI. Pentoxifylline (PTX), a non-specific phosphodiesterase inhibitor, can suppress the production of some factors of inflammatory response and presumably prevent AKI. We examined the PTX on the development of AKI in cardiac surgery patients by measuring the levels of UNGAL. Materials and methods We performed a double blind randomized clinical trial, enrolling 28 consecutive patients undergoing elective coronary artery bypass graft (CABG) surgery. Patients were divided into two groups, one to receive PTX 5 mg/kg intravenous bolus injection, followed by 1.5 mg/kg/h continuous intravenous infusion until 3 hours after cessation of CPB and the other group received placebo. UNGAL was measured before, 3 and 24 hours after surgery. In addition serum creatinine was measured before and 24, 48, 72 and 96 hours after surgery and C-reactive protein (CRP) only 24 hours postoperatively. Results Both groups did not differ in demographic and baseline characteristics. 12 patients developed AKI 48 hours after surgery; 5 of them were in the intervention group and 7 in the control group (p= 0.445). There was an increase of UNGAL in both groups postoperatively, although not significant. Mean sCr was significantly increased in the control group at 24 and 48 hours after surgery (24-h mean: 0.79 ± 0.18 mg/dl vs. 1.03 ± 0.43 mg/dl, P value = 0.02; 48-h mean: 1.17 ± 0.24 mg/dl vs. 0.98 ± 0.20 mg/dl, P value = 0.03, respectively). PTX had a positive effect in preventing AKI reflecting in changes in sCr, and the increase of UNGAL was consistent with the emergence of AKI (Pearson's correlation = 0.30). Conclusion Our study demonstrates a weak correlation between UNGAL and sCr after cardiac surgery. The rise of UNGAL in these patients may be reduced by administration of PTX although we did not show significance. PTX could reduce the occurrence of AKI as determined by attenuation of sCr rise without causing hemodynamic instability or increased bleeding. Overall, we suggest future studies with larger sample sizes to elucidate this effect and determine the different aspects of administrating PTX. Trial Registration ISRCTN: IRCT138807302622N1

Safety of and humoral immune response to the anti-influenza vaccine in coronary artery disease (CAD) patients were evaluated. The trivalent vaccine was administered to 137 eligible CAD patients and 67 age- and sex-matched healthy individuals. Antibody (Ab) titers were measured before and 1 month after vaccination. CAD and HC groups were not significantly different in serologic response and magnitude of change in antibody titers against each of the vaccine antigens. In multivariate analyses, regular exercise and using multivitamin supplements were independently associated with better antibody response among CAD patients. There were no major cardiac or general adverse effects. Influenza vaccine was found safe in CAD patients and antibody responses were similar to HCs.

We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the $\frac{1}{N^2}$ expansion of matrix models with more then one cut., 47 pages, 3 figures; v2: fixed typos, added journal-ref

We present solutions for the higher genus topological string amplitudes on Calabi-Yau-manifolds, which are realized as complete intersections in Grassmannians. We solve the B-model by direct integration of the holomorphic anomaly equations using a finite basis of modular invariant generators, the gap condition at the conifold and other local boundary conditions for the amplitudes. Regularity of the latter at certain points in the moduli space suggests a CFT description. The A-model amplitudes are evaluated using a mirror conjecture for Grassmannian Calabi-Yau by Batyrev, Ciocan-Fontanine, Kim and Van Straten. The integrality of the BPS states gives strong evidence for the conjecture., Comment: 36 pages, 1 eps figure

Postoperative arrhythmias are among the most common complications of cardiac surgery. Total serum magnesium concentration will change after coronary bypass surgery but compensatory prophylactic administration of magnesium has remained a controversial issue. We studied whether prophylactic administration of magnesium could prevent post-coronary artery bypass grafting (CABG) arrhythmias and evaluated the effects of diabetes mellitus on prophylactic magnesium administration.In a clinical trial, 345 consecutive CABG candidates were randomly assigned to study (n = 166, 48.1%) and control groups. Patients in study group received supplemental magnesium infusion as following: 2 g [corrected] after induction of anesthesia until cardio-pulmonary bypass and then 8 g upon arrival in Intensive Care Unit (ICU) until 24 hr. Total serum magnesium concentration was measured at four designated time points: onset of induction, and 0, 24 and 48 hr after ICU admission. Cardiac arrhythmias were sought with a 12-lead electrocardiogram (ECG) from the end of surgery up till discharge.Atrial Fibrillation (Af) occurred in 34 patients (9.9%). Total serum magnesium concentration was significantly higher in patients who received supplemental magnesium (P0.001) and significantly lower in Af patients (P= 0.02). Among non-diabetics, Af incidence was significantly lower in study group compared with control group.The occurrence of atrial fibrillation correlates with serum magnesium level. Diabetes mellitus probably hampers prophylactic effect of supplemental magnesium in preventing the occurrence of Af.