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We discuss two physics-inspired approaches to derivation of the eigenfunctions and eigenvalues of $A_N$ Ruijsenaars-Schneider model. First approach which was recently proposed by the authors relies on the computations of superconformal indices of class $\mathcal{S}$ $4d$ ${\mathcal N}=2$ theories with the insertion of surface defects. Second approach uses computations of Nekrasov-Shatashvili limit of $5d$ ${\mathcal N} = 1^*$ instanton partition functions in the presence of co-dimension two defect. We compare results of these two approaches for the low-lying levels of Ruijsenaars-Schneider model. We also discuss different previously proposed exact quantization conditions for the Coulomb branch parameters of the instanton partition functions and their interpretations in terms of index calculations., Comment: 25 pages

Level density $\rho$ is derived for a finite system with strongly interacting nucleons at a given energy E, neutron N and proton Z particle numbers, projection of the angular momentum M, and other integrals of motion, within the semiclassical periodic-orbit theory (POT) beyond the standard Fermi-gas saddle-point method. For large particle numbers, one obtains an analytical expression for the level density which is extended to low excitation energies U in the statistical micro-macroscopic approach (MMA).The interparticle interaction averaged over particle numbers is taken into account in terms of the extended Thomas-Fermi component of the POT. The shell structure of spherical and deformed nuclei is taken into account in the level density. The MMA expressions for the level density $\rho$ reaches the well-known macroscopic Fermi-gas asymptote for large excitation energies U and the finite combinatoric power-expansion limit for low energies U. We compare our MMA results for the averaged level density with the experimental data obtained from the known excitation energy spectra by using the sample method under statistical and plateau conditions. Fitting the MMA $\rho$ to these experimental data on the averaged level density by using only one free physical parameter - inverse level density parameter K - for several nuclei and their long isotope chain at low excitation energies U, one obtains the results for K. These values of K might be much larger than those deduced from neutron resonances. The shell, isotopic asymmetry, and pairing effects are significant for low excitation energies., Comment: 31 pages, 7 figures, 1 table

Level density $\rho(E,N,Z)$ is calculated for the two-component close- and open-shell nuclei with a given energy $E$, and neutron $N$ and proton $Z$ numbers, taking into account pairing effects within the microscopic-macroscopic approach (MMA). These analytical calculations have been carried out by using the semiclassical statistical mean-field approximations beyond the saddle-point method of the Fermi gas model in a low excitation-energies range. The level density $\rho$, obtained as function of the system entropy $S$, depends essentially on the condensation energy $E_{\rm cond}$ through the excitation energy $U$ in super-fluid nuclei. The simplest super-fluid approach, based on the BCS theory, accounts for a smooth temperature dependence of the pairing gap $\Delta$ due to particle number fluctuations. Taking into account the pairing effects in magic or semi-magic nuclei, excited below neutron resonances, one finds a notable pairing phase transition.Pairing correlations sometimes improve significantly the comparison with experimental data., Comment: 8 pages, 2 figures, 2 tables

We derive ground state eigenfunctions and eigenvalues of various relativistic elliptic integrable models. The models we discuss appear in computations of superconformal indices of four-dimensional theories obtained by compactifying six-dimensional models on Riemann surfaces. These include, among others, the Ruijsenaars-Schneider model and the van Diejen model. The derivation of the eigenfunctions builds on physical inputs, such as conjectured Lagrangian across dimensions IR dualities and assumptions about the behavior of the indices in the limit of compactifications on surfaces with large genus/number of punctures/flux., Comment: 14 pages, 1 figure, supplementary Wolfram Mathematica file; v2: minor typos fixed, references added; v3: minor changes, published version

We discuss a 4d Lagrangian descriptions, across dimensions IR dual, of compactifications of the 6d $(\text{D},\text{D})$ minimal conformal matter theory on a sphere with arbitrary number of punctures and a particular value of flux as a gauge theory with a simple gauge group. The Lagrangian has the form of a ``star shaped quiver'' with the rank of the central node depending on the 6d theory and the number and type of punctures. Using this Lagrangian one can construct across dimensions duals for arbitrary compactifications (any, genus, any number and type of $\text{USp}$ punctures, and any flux) of the $(\text{D},\text{D})$ minimal conformal matter gauging only symmetries which are manifest in the UV., Comment: 7 pages, 4 figures

We identify a categorical structure of the set of all CFTs. In particular, we show that the set of all CFTs has a natural monoidal strict $2$-category structure with the $1$-morphisms being sequences of deformations and $2$-morphisms determined by $0$-form symmetries of the CFTs., Comment: 11 pages, 7 figures

What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

Level density $\rho(E,{\bf Q})$ is derived within the micro-macroscopic approximation (MMA) for a system of strongly interacting Fermi particles with the energy $E$ and additional integrals of motion ${\bf Q}$, in line with several topics of the universal and fruitful activity of A.S. Davydov. Within the extended Thomas Fermi and semiclassical periodic orbit theory beyond the Fermi-gas saddle-point method we obtain $\rho\propto I_\nu(S)/S^\nu$, where $I_\nu(S)$ is the modified Bessel function of the entropy $S$. For small shell-structure contribution one finds $\nu=\kappa/2+1$, where $\kappa$ is the number of additional integrals of motion. This integer number is a dimension of ${\bf Q}$, ${\bf Q}=\{N, Z, ...\}$ for the case of two-component atomic nuclei, where $N$ and $Z$ are the numbers of neutron and protons, respectively. For much larger shell structure contributions, one obtains, $\nu=\kappa/2+2$. The MMA level density $\rho$ reaches the well-known Fermi gas asymptote for large excitation energies, and the finite micro-canonical combinatoric limit for low excitation energies. The additional integrals of motion can be also the projection of the angular momentum of a nuclear system for nuclear rotations of deformed nuclei, number of excitons for collective dynamics, and so on. Fitting the MMA total level density, $\rho(E,{\bf Q})$, for a set of the integrals of motion ${\bf Q}=\{N, Z\}$, to experimental data on a long nuclear isotope chain for low excitation energies, one obtains the results for the inverse level-density parameter $K$, which differs significantly from those of neutron resonances, due to shell, isotopic asymmetry, and pairing effects., Comment: 24 pages, 4 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2109.01830

In this set of five lectures we present a basic toolbox to discuss the dynamics of four dimensional supersymmetric quantum field theories. In particular we overview the program of geometrically engineering the four dimensional supersymmetric models as compactifications of six dimensional SCFTs. We discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be naturally imbedded in the geometric constructions. The lectures mostly review results which previously appeared in the literature but also contain some unpublished derivations., Comment: 174 pages, 66 Figures; v2: references added

This is a Snowmass whitepaper on physical mathematics. It briefly summarizes and highlights some of the key questions drawn from a much more extensive essay, by the same authors, entitled, "A Panorama Of Physical Mathematics 2021.", Comment: 9 pp

Main properties of isoscalar giant multipole resonances (up to L=3), including L=0,2 overtones, in medium-mass closed-shell nuclei are described within the semimicroscopic particle-hole dispersive optical model. Main properties are characterized by the energy-averaged strength distribution, projected (one-body) transition density, and probabilities of direct one-nucleon decay. Calculation results obtained for the parameters of the mentioned resonances in the 48Ca, 90Zr, and 132Sn nuclei are compared with available experimental data., Comment: 35 pages, 9 figures, 7 tables. arXiv admin note: text overlap with arXiv:2011.03315

We investigate the unflavoured Schur indices of class $\mathcal S$ theories of modest rank, and in the case of $\mathcal{N}=4$ super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their modular properties. We find closed form expressions for many of these theories in terms of quasimodular forms of level one or two, with the curious feature that in general they are sums of quasimodular forms of different weights. For type $\mathfrak{a}_1$ theories, the index can be fixed by taking a simple Ansatz for the family of quasimodular forms appearing in the expansion of this type and demanding that the result be sufficiently regular as $q\to0$. For higher rank cases, an equally simple construction is lacking, but we nevertheless find that these indices can be expressed in terms of mixed-weight quasimodular forms., Comment: 13 pages plus references, two column format

Level density $\rho(E,N,Z)$ is derived for a nuclear system with a given energy $E$, neutron $N$, and proton $Z$ particle numbers, within the semiclassical extended Thomas-Fermi and periodic-orbit theory beyond the Fermi-gas saddle-point method. We obtain $~~\rho \propto I_\nu(S)/S^\nu$,~~ where $I_\nu(S)$ is the modified Bessel function of the entropy $S$, and $\nu$ is related to the number of integrals of motion, except for the energy $E$. For small shell structure contribution one obtains within the micro-macroscopic approximation (MMA) the value of $\nu=2$ for $\rho(E,N,Z)$. In the opposite case of much larger shell structure contributions one finds a larger value of $\nu=3$. The MMA level density $\rho$ reaches the well-known Fermi gas asymptote for large excitation energies, and the finite micro-canonical limit for low excitation energies. Fitting the MMA $\rho(E,N,Z)$ to experimental data on a long isotope chain for low excitation energies, due mainly to the shell effects, one obtains results for the inverse level density parameter $K$, which differs significantly from that of neutron resonances., Comment: 26 pages, 3 figures, 1 table. arXiv admin note: text overlap with arXiv:2103.16480

We discuss certain structural analogies between supersymmetric quiver gauge theories and lattice models leading to fracton phases of matter. In particular, classes of quiver models can be viewed as lattice models having sub-system symmetries, dimensions of moduli spaces growing linearly with the size of the lattice, and having excitations with limited mobility (with "excitations" and "mobility" properly defined)., Comment: 5 pages, 4 figures

We consider supersymmetric surface defects in compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. For the case of $N=1$ such defects are introduced into the supersymmetric index computations by an action of the $BC_1\,(\sim A_1\sim C_1)$ van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras $A_{N=1}$, $C_{N=1}$, and $(A_1)^{N=1}$. The indices of these $4d$ theories give rise to three different Kernel functions for the $BC_1$ van Diejen model. We then consider the generalizations with $N>1$. The operators introducing defects into the index computations are certain $A_{N}$, $C_N$, and $(A_1)^{N}$ generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the $A_N$ case, and derive various properties these operators have to satisfy as a consequence of $4d$ dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal $(D_5,D_5)$ conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces., Comment: 42 pages + Appendices, 23 Figures, minor typos corrected

Level density $\rho(E,A)$ is derived for a one-component nucleon system with a given energy $E$ and particle number $A$ within the mean-field semiclassical periodic-orbit theory beyond the saddle-point method of the Fermi gas model. We obtain $~~\rho \propto I_\nu(S)/S^\nu$, with $I_\nu(S)$ being the modified Bessel function of the entropy $S$. Within the micro-macro-canonical approximation (MMA), for a small thermal excitation energy, $U$, with respect to rotational excitations, $E_{\rm rot}$, one obtains $\nu=3/2$ for $\rho(E,A)$. In the case of excitation energy $U$ larger than $E_{\rm rot}$ but smaller than the neutron separation energy, one finds a larger value of $\nu=5/2$. A role of the fixed spin variables for rotating nuclei is discussed. The MMA level density $\rho$ reaches the well-known grand-canonical ensemble limit (Fermi gas asymptotic) for large $S$ related to large excitation energies, and also reaches the finite micro-canonical limit for small combinatorial entropy $S$ at low excitation energies (the constant "temperature" model). Fitting the $\rho(E,A)$ of the MMA to the experimental data for low excitation energies, taking into account shell and, qualitatively, pairing effects, one obtains for the inverse level density parameter $K$ a value which differs essentially from that parameter derived from data on neutron resonances., Comment: 21 pages, 4 figures, 1 Table

We consider compactifications of rank $Q$ E-string theory on a genus zero surface with no punctures but with flux for various subgroups of the $\text{E}_8\times \text{SU}(2)$ global symmetry group of the six dimensional theory. We first construct a simple Wess-Zumino model in four dimensions corresponding to the compactification on a sphere with one puncture and a particular value of flux, the cap model. Using this theory and theories corresponding to two punctured spheres with flux, one can obtain a large number of models corresponding to spheres with a variety of fluxes. These models exhibit interesting IR enhancements of global symmetry as well as duality properties. As an example we will show that constructing sphere models associated to specific fluxes related by an action of the Weyl group of $\text{E}_8$ leads to the S-confinement duality of the $\text{USp}(2Q)$ gauge theory with six fundamentals and a traceless antisymmetric field. Finally, we show that the theories we discuss possess an $\text{SU}(2)_{\text{ISO}}$ symmetry in four dimensions that can be naturally identified with the isometry of the two-sphere. We give evidence in favor of this identification by computing the `t Hooft anomalies of the $\text{SU}(2)_{\text{ISO}}$ in 4d and comparing them with the predicted anomalies from 6d., Comment: 57 pages, 6 figures; v2: a few minor changes, published in SciPost Phys

The semimicroscopic particle-hole dispersive optical model (PHDOM) is implemented to describe main properties of Isoscalar Giant Multipole Resonances (up to L=3) in medium-heavy closed-shell nuclei. The main properties are characterized by the strength distribution, transition density, partial and total probabilities of direct one-nucleon decay. Calculation results obtained for the 208Pb nucleus are compared with available experimental data., Comment: 33 pages, 6 tables, 8 figures

In principle, there is no obstacle to gapping fermions preserving any global symmetry that does not suffer a 't Hooft anomaly. In practice, preserving a symmetry that is realised on fermions in a chiral manner necessitates some dynamics beyond simple quadratic mass terms. We show how this can be achieved using familiar results about supersymmetric gauge theories and, in particular, the phenomenon of confinement without chiral symmetry breaking. We present simple models that gap fermions while preserving a symmetry group under which they transform in chiral representations. For example, we show how to gap a collection of 4d fermions that carry the quantum numbers of one generation of the Standard Model, but without breaking electroweak symmetry. We further show how to gap fermions in groups of 16 while preserving certain discrete symmetries that exhibit a mod 16 anomaly., Comment: 21 pages. v2: references added

We have studied the uncharacteristic behavior of the measured values of the isoscalar and isovector centroid energies, ECEN, of giant resonances with multipolarity L=0-3 in 44Ca, 54Fe, 64,68Zn and 56,58,60,68Ni. For this purpose, we carried out calculations of ECEN within the spherical Hartree-Fock (HF)-based random phase approximation (RPA) theory with 33 different Skyrme-type effective nucleon-nucleon interactions. We have also determined the Pearson linear correlation coefficients between the calculated centroid energies and the various nuclear matter (NM) properties associated with each interaction and determined the sensitivity of ECEN to NM properties. We compared the calculated centroid energies of the giant resonances with experimental data and discuss the results. We note in particular, that we obtain good agreement between the calculated ECEN of isovector giant dipole resonance and the available experimental data.

Statistical level density $\rho(E,A)$ is derived for nucleonic system with a given energy $E$, particle number $A$ and other integrals of motion in the micro-macroscopic approximation beyond the standard saddle-point method of the Fermi gas model. This level density reaches the two limits; the well-known Fermi gas grand-canonical ensemble limit for a large entropy $S$ related to large excitation energies, and the finite micro-canonical limit for a small combinatorical entropy $S$ at low excitation energies. The inverse level density parameter $K$ as function of the particle number $A$ in the semiclassical periodic orbit theory, taking into account the extended Thomas-Fermi and Strutinsky shell corrections, is calculated and compared with experimental data., Comment: 13 pages, 3 figures

We show that the $4d$ ${\cal N}=1$ $SU(3)$ $N_f=6$ SQCD is the model obtained when compactifying the rank one E-string theory on a three punctured sphere (a trinion) with a particular value of flux. The $SU(6)\times SU(6)\times U(1)$ global symmetry of the theory, when decomposed into the $SU(2)^3\times U(1)^3\times SU(6)$ subgroup, corresponds to the three $SU(2)$ symmetries associated to the three punctures and the $U(1)^3 \times SU(6)$ subgroup of the $E_8$ symmetry of the E-string theory. All the puncture symmetries are manifest in the UV and thus we can construct ordinary Lagrangians flowing in the IR to any compactification of the E-string theory. We generalize this claim and argue that the ${\cal N}=1$ $SU(N+2)$ SQCD in the middle of the conformal window, $N_f=2N+4$, is the theory obtained by compactifying the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter SCFT on a sphere with two maximal $SU(N+1)$ punctures, one minimal $SU(2)$ puncture, and with a particular value of flux. The $SU(2N+4)\times SU(2N+4)\times U(1)$ symmetry of the UV Lagrangian decomposes into $SU(N+1)^2\times SU(2)$ puncture symmetries and the $U(1)^3\times SU(2N+4)$ subgroup of the $SO(12+4N)$ symmetry group of the $6d$ SCFT. The models constructed from the trinions exhibit a variety of interesting strong coupling effects. For example, one of the dualities arising geometrically from different pair-of-pants decompositions of a four punctured sphere is an $SU(N+2)$ generalization of the Intriligator-Pouliot duality of $SU(2)$ SQCD with $N_f=4$, which is a degenerate, $N=0$, instance of our discussion., Comment: 38 pages, 17 figures; v2: minor clarification in sec 3.1, v3: typos corrected, JHEP version

We classify ${\cal N}=1$ gauge theories with simple gauge groups in four dimensions which possess a conformal manifold passing through weak coupling. A very rich variety of models is found once one allows for arbitrary representations under the gauge group. For each such model we detail the dimension of the conformal manifold, the conformal anomalies, and the global symmetry preserved on a generic locus of the manifold. We also identify, at least some, sub-loci of the conformal manifolds preserving more symmetry than the generic locus. Several examples of applications of the classification are discussed. In particular we consider a conformal triality such that one of the triality frames is a $USp(6)$ gauge theory with six fields in the two index traceless antisymmetric representation. We discuss an IR dual of a conformal $Spin(5)$ gauge theory with two chiral superfields in the vector representation and one in the fourteen dimensional representation. Finally, an extension of the conformal manifold of ${\cal N}=2$ class ${\cal S}$ theories by conformally gauging symmetries corresponding to maximal punctures with the addition of two adjoint chiral superfields is commented upon., Comment: 109 pages, 3 figures; v2: references added, corrections in subsection 9.2

We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index., Comment: 21 pages, 1 figure

We suggest three new ${\cal N}=1$ conformal dual pairs. First, we argue that the ${\cal N}=2$ $E_6$ Minahan-Nemeschansky (MN) theory with a $USp(4)$ subgroup of the $E_6$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of an $SU(2)^5$ quiver gauge theory. Second, we argue that the ${\cal N}=2$ $E_7$ MN theory with an $SU(2)$ subgroup of the $E_7$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of a conformal ${\cal N}=1$ $USp(4)$ gauge theory. Finally, we claim that the ${\cal N}=2$ $E_8$ MN theory with a $USp(4)$ subgroup of the $E_8$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of an ${\cal N}=1$ $Spin(7)$ conformal gauge theory. We argue for the dualities using a variety of non-perturbative techniques including anomaly and index computations. The dualities can be viewed as ${\cal N}=1$ analogues of ${\cal N}=2$ Argyres-Seiberg/Argyres-Wittig duals of the $E_n$ MN models. We also briefly comment on an ${\cal N}=1$ version of the Schur limit of the superconformal index., Comment: 23 pages, two figures

The development of a modern and more realistic nuclear energy density functional (EDF) for accurate predictions of properties of nuclei is the subject of enhanced activity, since it is very important for the study of properties of nuclear matter (NM), giant resonances and, in particular, of properties of rare nuclei with unusual neutron-to-proton ratios. Here, we provide a short review of the current status of the nuclear EDF and the theoretical results obtained for properties of nuclei and nuclear matter. We will first describe a method for determining the parameters of the EDF, associated with the Skyrme type effective interaction, by carrying out a Hartree-Fock based fit to extensive set of data of ground state properties and constraints. We will then describe the fully self-consistent Hartree-Fock based random-phase-approximation theory for calculating the strength functions S(E) and centroid energies E_CEN of giant resonances and provide results for E_CEN of isoscalar and isovector giant resonances of multipolarities L=0-3 for a wide range of spherical nuclei, using 33 EDFs associated with standard form of the Skyrme type interactions, commonly employed in the literature. We investigate the sensitivities E_CEN of the giant resonances to bulk properties of NM and determine constraints on NM properties, such as the incompressibility coefficient and effective mass, by comparing with experimental data on E_CEN of giant resonances., Comment: Based on the talk presented at annual conference of Institute for Nuclear Research held in Kyiv, Ukraine (April, 8-12, 2019)

We study the compactification of the 6d ${\cal N}=(2,0)$ SCFT on the product of a Riemann surface with flux and a circle. On the one hand, this can be understood by first reducing on the Riemann surface, giving rise to 4d ${\cal N}=1$ and ${\cal N}=2$ class ${\cal S}$ theories, which we then reduce on $S^1$ to get 3d ${\cal N}=2$ and ${\cal N}=4$ class ${\cal S}$ theories. On the other hand, we may first compactify on $S^1$ to get the 5d ${\cal N}=2$ Yang-Mills theory. By studying its reduction on a Riemann surface, we obtain a mirror dual description of 3d class ${\cal S}$ theories, generalizing the star-shaped quiver theories of Benini, Tachikawa, and Xie. We comment on some global properties of the gauge group in these reductions, and test the dualities by computing various supersymmetric partition functions., Comment: 24 pages, 10 figures

We consider compactifications of $6d$ minimal $(D_{N+3},D_{N+3})$ type conformal matter SCFTs on a generic Riemann surface. We derive the theories corresponding to three punctured spheres (trinions) with three maximal punctures, from which one can construct models corresponding to generic surfaces. The trinion models are simple quiver theories with $\mathcal{N}=1$ $SU(2)$ gauge nodes. One of the three puncture non abelian symmetries is emergent in the IR. The derivation of the trinions proceeds by analyzing RG flows between conformal matter SCFTs with different values of $N$ and relations between their subsequent reductions to $4d$. In particular, using the flows we first derive trinions with two maximal and one minimal punctures, and then we argue that collections of $N$ minimal punctures can be interpreted as a maximal one. This suggestion is checked by matching the properties of the $4d$ models such as `t Hooft anomalies, symmetries, and the structure of the conformal manifold to the expectations from $6d$. We then use the understanding that collections of minimal punctures might be equivalent to maximal ones to construct trinions with three maximal punctures, and then $4d$ theories corresponding to arbitrary surfaces, for $6d$ models described by two $M5$ branes probing a $\mathbb{Z}_k$ singularity. This entails the introduction of a novel type of maximal puncture. Again, the suggestion is checked by matching anomalies, symmetries and the conformal manifold to expectations from six dimensions. These constructions thus give us a detailed understanding of compactifications of two sequences of six dimensional SCFTs to four dimensions., Comment: 57 pages, 27 figures, v3: anomaly notations improved (see footnote 3)

We discuss compactifications of rank $Q$ E-string theory on a torus with fluxes for abelian subgroups of the $E_8$ global symmetry of the $6d$ SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as $E[USp(2Q)]$. This model has a variety of non trivial properties. In particular the global symmetry is $USp(2Q)\times USp(2Q)\times U(1)^2$ with one of the two $USp(2Q)$ symmetries emerging in the IR as an enhancement of an $SU(2)^Q$ symmetry of the UV Lagrangian. The $E[USp(2Q)]$ model after dimensional reduction to $3d$ and a subsequent Coulomb branch flow is closely related to the familiar $3d$ $T[SU(Q)]$ theory, the model residing on an S-duality domain wall of $4d$ $\mathcal{N}=4$ $SU(Q)$ SYM. Gluing the $E[USp(2Q)]$ models by gauging the $USp(2Q)$ symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of $4d$ theories with emergent IR symmetries. We support our claims by various checks involving computations of anomalies and supersymmetric partition functions. Many of the needed identities satisfied by the supersymmetric indices follow directly from recent mathematical results obtained by E. Rains., Comment: 54 pages, 21 figures

SCFTs in six dimensions are interrelated by networks of RG flows. Compactifying such models on a Riemann surface with flux for the $6d$ global symmetry, one can obtain a wide variety of theories in four dimensions. These four dimensional models are also related by a network of RG flows. In this paper we study some examples of four dimensional flows relating theories that can be obtained from six dimensions starting with different SCFTs connected by $6d$ RG flows. We compile a dictionary between different orders of such flows, $6d\to 6d\to 4d$ and $6d\to 4d\to 4d$, in the particular case when the six dimensional models are the ones residing on M5 branes probing different $A$-type singularities. The flows we study are triggered by vacuum expectation values (vevs) to certain operators charged under the six dimensional symmetry. We find that for generic choices of parameters the different orders of flows, $6d\to 6d\to 4d$ and $6d\to 4d\to 4d$, involve compactifications on different Riemann surfaces with the difference being in the number of punctures the surface has., Comment: Added some clarification in Section 2

We consider on one hand the possibility that a supersymmetric ${\cal N}=1$ conformal gauge theory has a strongly coupled locus on the conformal manifold at which a different, dual, conformal gauge theory becomes a good weakly coupled description. On the other hand we discuss the possibility that strongly coupled theories, e.g. SCFTs in class ${\cal S}$, having exactly marginal ${\cal N}=1$ deformations admit a weakly coupled gauge theory description on some locus of the conformal manifold. We present a simple algorithm to search for such dualities and discuss several concrete examples. In particular we find conformal duals for ${\cal N}=1$ SQCD models with $G_2$ gauge group and a model with $SU(4)$ gauge group in terms of simple quiver gauge theories. We also find conformal weakly coupled quiver theory duals for a variety of class ${\cal S}$ theories: $T_4$, $R_{0,4}$, $R_{2,5}$, and rank $2n$ Minahan-Nemeschansky $E_6$ theories. Finally we derive conformal Lagrangians for four dimensional theories obtained by compactifying the E-string on genus $g>1$ surface with zero flux. The pairs of dual Lagrangians at the weakly coupled loci have different symmetries which are broken on a general point of the conformal manifold. We match the dimensions of the conformal manifolds, symmetries on the generic locus of the conformal manifold, anomalies, and supersymmetric indices. The simplicity of the procedure suggests that such dualities are ubiquitous., Comment: 14 pages, 13 figures, revtex; v2: footnote 1 added

We study several cases of IR enhancements of global symmetry in four dimensions. In particular, we consider a sequence of $Spin(n+4)$ supersymmetric gauge theories ($8\geq n\geq 1$) with $n$ vectors and spinor matter with $32$ components. We show that the subgroup of the flavor symmetry of these theories rotating the matter in the spinor representations in the UV, when proper gauge singlet fields are added, enhances to the commutant of $SU(2)$ in $E_{9-n}$. We discuss several other interesting cases of enhanced symmetries and the interplay between symmetry enhancement and self-duality. We also make some observations about possible interconnections between chiral ring relations and symmetry enhancement. Finally, we conjecture relations of the discussed models to compactifications of certain conformal matter models in six dimensions on tori. The conjecture is based on deriving a relation between five dimensional models with $Spin$ gauge groups and conformal theories in six dimensions. As a by product of our considerations we discover a new instance of a simple self-duality of a theory with an $SU(6)$ gauge group., Comment: 37 pages + appendices, 5 figures

We discuss supersymmetric surface defects in compactifications of six dimensional minimal conformal matter of type SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are N=1 quiver gauge theories with SU(3) and SU(4) gauge groups respectively. The defects are engineered by giving space-time dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter the defects carry SU(2) flavor symmetry which is not a symmetry of the four dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six dimensional compactifications and four dimensional models., Comment: 19 pages, six figures, harvmac

Recently a very interesting three-dimensional $\mathcal{N}=2$ supersymmetric theory with $SU(3)$ global symmetry was discussed by several authors. We denote this model by $T_x$. This was conjectured to have two dual descriptions, one with explicit supersymmetry and emergent flavor symmetry and the other with explicit flavor symmetry and emergent supersymmetry. We discuss a third description of the model which has both flavor symmetry and supersymmetry manifest. We then investigate models which can be constructed by using $T_x$ as a building block gauging the global symmetry and paying special attention to the global structure of the gauge group. We conjecture several cases of $\mathcal{N}=2$ mirror dualities involving such constructions with the dual being either a simple $\mathcal{N}=2$ Wess-Zumino model or a discrete gauging thereof., Comment: 21 pages, 7 figures, harvmac; v2: references added, typos fixed

We study compactifications on Riemann surfaces with punctures of N=(1,0) 6d SCFTs with a one dimensional tensor branch and no continuous global symmetries. The effective description of such models on the tensor branch is in terms of pure gauge theories with decoupled tensor. For generic Riemann surfaces, the resulting theories in four dimensions are expected to have N=1 supersymmetry. We compute the anomalies expected from the resulting 4d theories by integrating the anomaly polynomial of the 6d theory on the Riemann surface. For the cases with 6d gauge models with gauge groups SU(3) and SO(8) we further propose a field theory construction for the resulting 4d theories. For the 6d SU(3) theory, we argue that the theories in four dimensions are quivers with SU(3) gauge nodes and free chiral fields. The theories one obtains from the 6d SO(8) gauge theory are quivers with SU(4) gauge groups and chiral fields with R charge a half. In the last case the theories constructed for general Riemann surfaces involve gauging of symmetries appearing at strong coupling. The conformal manifolds of the models are constructed from gauge couplings and baryonic superpotentials. We support our conjectures by matching the dimensions of the conformal manifolds with complex structure moduli of the Riemann surfaces, matching anomalies between six and four dimensions, and checking the dualities related to different pair of pants decompositions of the surfaces. As a simple application of the results we conjecture that SU(3) gauge theory with nine flavors in four dimensions has a duality group acting on the seven dimensional conformal manifold which is the mapping class group of sphere with ten marked points., Comment: 10 pages, 8 figures

In this paper we study compactifications of ADE type conformal matter, N M5 branes probing ADE singularity, on torus with flux for global symmetry. We systematically construct the four dimensional theories by first going to five dimensions and studying interfaces. We claim that certain interfaces can be associated with turning on flux in six dimensions. The interface models when compactified on a circle comprise building blocks for constructing four dimensional models associated to flux compactifications of six dimensional theories on a torus. The theories in four dimensions turn out to be quiver gauge theories and the construction implies many interesting cases of IR symmetry enhancements and dualities of such theories., Comment: 66 pages; 38 figures

We study 4d N=1 supersymmetric theories of class S_k, obtained from flux compactifications on a Riemann surface of 6d (1,0) conformal theories describing the low energy physics on a stack of M5 branes probing a Z_k singularity. We conjecture that the protected spectrum of class S_k theories contains a freely generated ring, generalizing the Coulomb branch of the N=2 theories. We derive this by examining a limit of the supersymmetric index of 4d N=1 class S_k theories. The limit generalizes the Coulomb limit of N=2 theories, which coincides with the case of k=1 for a particular choice of flux. We conjecture a general simple formula for the index in the aforementioned limit., Comment: 26 pages, 5 figures

We discuss the four dimensional models obtained by compactifying a single M5 brane probing $D_{N}$ singularity (minimal D-type $(1,0)$ conformal matter in six dimensions) on a torus with flux for abelian subgroups of the $SO(4N)$ flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel ${\cal N}=1$ dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter., Comment: 38 pages, 13 figures

We study the supersymmetric index of four dimensional theories obtained by compactifications of the six dimensional E string theory on a Riemann surface. In particular we derive the difference operator introducing certain class of surface defects to the index computation. The difference operator turns out to be, up to a constant shift, an analytic difference operator discussed by van Diejen., Comment: 20 pages, three figures; v.2 minor clarifications

We study two cases of interrelations between enhancement of symmetries in the infra red (IR) and duality properties of supersymmetric quantum field theories in four dimensions. First we discuss an SU(2) N=1 model with four flavors, singlet fields, and a superpotential. We show that this model flows to a conformal field theory with E_6\times U(1) global symmetry. The enhancement of the flavor symmetry follows from Seiberg duality. The second example is concerned with an SU(4) gauge theory with matter in the fundamental and antisymmetric representations. We argue that this model has enhanced SO(12) symmetry in the IR, and then guided by this enhancement, we deduce a new IR duality., Comment: four pages, four figures

For the recently developed particle-hole dispersive optical model, weak violations of unitarity due to a phenomenological description of the spreading effect are considered. Methods for unitarity restoration are proposed and implemented for the 208Pb nucleus in the description of the energy-averaged isoscalar monopole double transition density and strength functions in a wide excitation energy interval that includes the isoscalar giant monopole resonance and its overtone. To illustrate abilities of the model, direct neutron decay of the mentioned giant resonance is also considered., Comment: 11 pages, 2 tables, 5 figures

In this paper we discuss $3d$ ${\cal N}=2$ supersymmetric gauge theories and their IR dualities when they are compactified on a circle of radius $r$, and when we take the $2d$ limit in which $r\to 0$. The $2d$ limit depends on how the mass parameters are scaled as $r\to 0$, and often vacua become infinitely distant in the $2d$ limit, leading to a direct sum of different $2d$ theories. For generic mass parameters, when we take the same limit on both sides of a duality, we obtain $2d$ dualities (between gauge theories and/or Landau-Ginzburg theories) that pass all the usual tests. However, when there are non-compact branches the discussion is subtle because the metric on the moduli space, which is not controlled by supersymmetry, plays an important role in the low-energy dynamics after compactification. Generally speaking, for IR dualities of gauge theories, we conjecture that dualities involving non-compact Higgs branches survive. On the other hand when there is a non-compact Coulomb branch on at least one side of the duality, the duality fails already when the $3d$ theories are compactified on a circle. Using the valid reductions we reproduce many known $2d$ IR dualities, giving further evidence for their validity, and we also find new $2d$ dualities.

We discuss USp(2n) supersymmetric models with eight fundamental fields and a field in the antisymmetric representation. Turning on the most generic superpotentials, coupling pairs of fundamental fields to powers of the antisymmetric field while preserving an R symmetry, we give evidence for the statement that the models are connected by a large network of dualities which can be organized into orbits of the Weyl group of E_8. We make also several curious observations about such models. In particular, we argue that a USp(2m) model with the addition of singlet fields and even rank m flows in the IR to a CFT with E_7 U(1) symmetry. We also discuss an infinite number of duals for the USp(2) theory with eight fundamentals and no superpotential., Comment: 15 pages, harvmac. v2: added references

We study compactifications of the 6d E-string theory, the theory of a small E_8 instanton, to four dimensions. In particular we identify N=1 field theories in four dimensions corresponding to compactifications on arbitrary Riemann surfaces with punctures and with arbitrary non-abelian flat connections as well as fluxes for the abelian sub-groups of the E_8 flavor symmetry. This sheds light on emergent symmetries in a number of 4d N=1 SCFTs (including the `E7 surprise' theory) as well as leads to new predictions for a large number of 4-dimensional exceptional dualities and symmetries., Comment: 85 pages, 25 figures, harvmac