Nuclides as a liquid phase of SU(2)L×SU(2)RSU(2)_L \times SU(2)_R chiral perturbation theory I: emergence of pion-less SU(2) χ\chi PT

The Standard Model of particle physics, augmented with neutrino mixing, is at least very nearly the complete theory of interactions of known particles at energies accessible to Nature on Earth. Candidate effective theories of nuclear structure must therefore reflect SM symmetries, especially the chiral global $SU(2)_L \times SU(2)_R$ symmetry of two-massless-quark QCD. For ground-state nuclei, SU(2) chiral perturbation theory (XPT) enables perturbation in inverse powers of $\Lambda_{XSB}\simeq 1 GeV$, with analytic operators renormalized to all loop orders. We show that pion-less "Static Chiral Nucleon Liquids" (SXNL) emerge as a liquid phase of SU(2) XPT of protons, neutrons and 3 Nambu-Goldstone boson pions. Far-IR pions decouple from SXNL, simplifying the derivation of saturated nuclear matter and microscopic liquid drops (ground-state nuclides). We trace to the global symmetries of two-massless-quark QCD the power of pion-less SU(2) XPT to capture experimental ground-state properties of certain nuclides with even parity, spin zero, even proton number Z, and neutron number N. We derive the SXNL effective SU(2) XPT Lagrangian, including all order $\Lambda_{XSB},\Lambda^0_{XSB}$ operators. These include: all 4-nucleon operators that survive Fierz rearrangement in the non-relativistic limit, and effective Lorentz-vector iso-vector neutral "$\rho$-exchange" operators. SXNL motivate nuclear matter as non-topological solitons at zero pressure: the Nuclear Liquid Drop Model and Bethe-Weizsacker Semi-Empirical Mass Formula emerge in an explicit Thomas-Fermi construction provided in the companion paper. For chosen nuclides, nuclear Density Functional and Skyrme models are justified to order $\Lambda_{\chi SB}^0$. We conjecture that inclusion of higher order operators will result in accurate "natural" Skyrme, No-Core-Shell, and neutron star models

The Quantum McKay Correspondence for polyhedral singularities

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the crepant resolution conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].Comment: Introduction rewritten. Issue regarding non-uniqueness of conifold resolution clarified. Version to appear in Inventione