Edward Witten: Symmetry and Emergence I discuss gauge and global symmetries in particle physics, condensed matter physics, and quantum gravity. In a modern understanding of particle physics, global symmetries are approximate and gauge symmetries may be emergent. (Based on a lecture at the April, 2016 meeting of the American Physical Society in Salt Lake City, Utah.)
On conservation laws for the supersymmetric sigma model Abstract. We derive conservation laws for Dirac-harmonic maps and their extensions to manifolds that have isometries, where we mostly focus on the spherical case. In addition, we discuss several geometric and analytic applications of the latter.
"Even the Harmonic Oscillator — the Most Basic of All Quantum Systems — Exhibits Supersymmetry!" Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. We find interesting new realizations of supersymmetry (SUSY) in quantum mechanics where Grassmann parity equals wavefunction parity.
Phenomenology with F-theory SU(5) We explore the low energy phenomenology of an F-theory based SU(5) model which, in addition to the known quarks and leptons, contains Standard Model (SM) singlets, and vector-like color triplets and SU(2) doublets. Depending on their masses and couplings, some of these new particles may be observed at the LHC and future colliders. We discuss the restrictions by CKM constraints on their mixing with the ordinary down quarks of the three chiral familes. The model is consistent with gauge coupling unification at the usual supersymmetric GUT scale, dimension five proton decay is adequately suppressed, while dimension-six decay mediated by the superheavy gauge bosons is enhanced by a factor of 5-7. The third generation charged fermion Yukawa couplings yield the corresponding lowenergy masses in reasonable agreement with observations. The hierarchical nature of the masses of lighter generations is accounted for via non-renormalisable interactions, with the perturbative vacuum expectation values (vevs) of the SM singlet fields playing an essential role.
Metastring Theory and Modular Space-time String theory is canonically accompanied with a space-time interpretation which determines S-matrix-like observables, and connects to the standard physics at low energies in the guise of local effective field theory. Recently, we have introduced a reformulation of string theory which does not rely on an a priori space-time interpretation or a pre-assumption of locality. This metastring theory is formulated in such a way that stringy symmetries (such as T-duality) are realized linearly. In this paper, we study metastring theory on a flat background and develop a variety of technical and interpretational ideas. These include a formulation of the moduli space of Lorentzian worldsheets, a careful study of the symplectic structure and consequently consistent closed and open boundary conditions, and the string spectrum and operator algebra. What emerges from these studies is a new quantum notion of space-time that we refer to as a quantum Lagrangian or equivalently a modular space-time. This concept embodies the standard tenets of quantum theory and implements in a precise way a notion of relative locality. The usual string backgrounds (non-compact space-time along with some toroidally compactified spatial directions) are obtained from modular space-time by a limiting procedure that can be thought of as a correspondence limit.
Quantum geometry of elliptic Calabi-Yau manifolds We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic fibrations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in the genus expansion as well as in the curve classes in the base. T-duality on the fibre implies that the topological string free energy also captures the BPSinvariants of D4-branes wrapping the elliptic fibre and a class in the base. We verify this proposal by explicit computation of the BPS invariants of 3 D4-branes on the rational elliptic surface.
Topological Strings on Elliptic Fibrations We study topological string theory on elliptically fibered Calabi-Yau manifolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomials of functions constructed from the special geometry of the deformation spaces. The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected behavior at special loci in moduli space. We further expand the amplitudes in the base moduli of the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial structure in terms of the modular forms of the modular group of the elliptic curve. We further find a recursive equation which captures this finer structure and which can be related to the anomaly equations for correlation functions.
Seiberg-Witten-Nekrasov Theory and Modular Properties of 6d Double Elliptic Systems If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge τ = θ2π +4πı g2 −→ − 1τ. The low-energy SeibergWitten prepotential F(a), however, is not explicitly invariant, because the flat moduli also change a −→ aD = ∂F/∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E2. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N) theory with two independent modular parameters τ and τˆ, the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N-dependent!) shift of τˆ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.
Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the E11 KacMoody algebra that was shown recently to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside E11, so that, by constructing the current algebra of E11, I obtain both internal gauge theory and gravity theory. The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved super-current. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein-Hilbert action.
Instantons on Calabi-Yau and hyper-Kähler cones The instanton equations on vector bundles over Calabi-Yau and hyper-K¨ahler cones can be reduced to matrix equations resembling Nahm’s equations. We complement the discussion of Hermitian Yang-Mills (HYM) equations on Calabi-Yau cones, based on regular semi-simple elements, by a new set of (singular) boundary conditions which have a known instanton solution in one direction. This approach extends the classic results of Kronheimer by probing a relation between generalised Nahm’s equations and nilpotent pairs/tuples. Moreover, we consider quaternionic instantons on hyper-K¨ahler cones over generic 3-Sasakian manifolds and study the HYM moduli spaces arising in this set-up, using the fact that their analysis can be traced back to the intersection of three Hermitian Yang-Mills conditions.
Quaternion-Kähler N = 4 Supersymmetric Mechanics Using the N = 4, 1D harmonic superspace approach, we construct a new type of N = 4 supersymmetric mechanics involving 4n-dimensional Quaternion-Kähler (QK) 1D sigma models as the bosonic core. The basic ingredients of our construction are local N = 4, 1D supersymmetry realized by the appropriate transformations in 1D harmonic superspace, the general N = 4, 1D superfield vielbein and a set of 2(n + 1) analytic “matter” superfields representing (n + 1) off-shell supermultiplets (4, 4, 0). Both superfield and component actions are given for the simplest QK models with the manifolds HHn = Sp(1, n)/[Sp(1)×Sp(n)] and HP n = Sp(1+n)/[Sp(1)×Sp(n)] as the bosonic targets. For the general case the relevant superfield action and constraints on the (4, 4, 0) “matter” superfields are presented. Further generalizations are briefly discussed.
K3 Elliptic Genus and an Umbral Moonshine Module Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of K3 string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the K3 elliptic genus. Inspired by the above two relations between moonshine and K3 string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the case of umbral moonshine corresponding to the Niemeier lattice with root system given by 6 copies of D4 root system. This CFT then leads to the construction of an infinite-dimensional graded module for the umbral group GD⊕6 4 whose graded characters coincide with the umbral moonshine functions. We also comment on how one can recover all umbral moonshine functions corresponding to the Niemeier root systems A⊕4 5 D4, A⊕2 7 D⊕2 5, A11D7E6, A17E7, and D10E⊕27.
Topological vertex formalism with O5-plane We propose new topological vertex formalism for Type IIB (p, q) 5-brane web with an O5-plane. We apply our proposal to 5d N = 1 Sp(1) gauge theory with Nf = 0, 1, 8 flavors to compute the topological string partition functions and check the agreement with the known results. Especially for the Nf = 8 case, which corresponds to E-string theory on a circle, we obtain a new, yet simple, expression of the partition function with two Young diagram sum.
Brane world models with bulk perfect fluid and broken 4D Poincaré invariance We consider 5D brane world models with broken global 4D Poincar´e invariance (4D part of the spacetime metric is not conformal to the Minkowski spacetime). The bulk is filled with the negative cosmological constant and may contain a perfect fluid. In the case of empty bulk (the perfect fluid is absent), it is shown that one brane solution always has a physical singularity in the bulk. The Kretschmann invariant goes to infinity in this point. We cut off this singularity in the case of compact two brane model and obtain regular exact solutions for both 4D Poincar´e broken and restored invariance. When the perfect fluid is present in the bulk, we get the master equation for the metric coefficients in the case of arbitrary bulk perfect fluid equation of state (EoS) parameters. In two particular cases of EoS, we obtain the analytic solutions for thin and thick branes. First one generalizes the well known Randall-Sundrum model with one brane to the case of the bulk anisotropic perfect fluid. In the second solution, the 4D Poincar´e invariance is restored. Here, the spacetime goes asymptotically to the anti-de Sitter one far from the thick brane.
Heterotic Hyper-Kähler flux backgrounds We study Heterotic supergravity on Hyper-K¨ahler manifolds in the presence of non-trivial warping and three form flux with Abelian bundles in the large charge limit. We find exact, regular solutions for multi-centered Gibbons-Hawking spaces and AtiyahHitchin manifolds. In the case of Atiyah-Hitchin, regularity requires that the circle at infinity is of the same order as the instanton number, which is taken to be large. Alternatively there may be a non-trivial density of smeared five branes at the bolt.
Highly symmetric D-brane-anti-D-brane effective actions The entire S-matrix elements of four, five and six point functions of D-braneanti D-brane system are explored. To deal with symmetries of string amplitudes as well as their all order α 0 corrections we first address a four point function of one closed string Ramond-Ramond (RR) and two real tachyons on the world volume of brane-anti brane system. We then focus on symmetries of string theory as well as universal tachyon expansion to achieve both string and effective field theory of an RR and three tachyons where the complete algebraic analysis for the whole S-matrix < VC−1 VT −1 VT0 VT0 > was also revealed. Lastly, we employ all the conformal field theory techniques to < VC−1 VT −1 VT0 VT0 VT0 >, working out with symmetries of theory and find out the expansion for the amplitude to be able to precisely discover all order singularity structures of D-brane-anti-D-brane effective actions of string theory. Various remarks about the so called generalized Veneziano amplitude and new string couplings are elaborated as well.
Emergent spacetime and quantum entanglement in matrix theory In the context of the Bank-Fishler-Shenker-Susskind Matrix theory, we analyze a spherical membrane in light-cone M theory along with two asymptotically distant probes. In the appropriate energy regime, we find that the membrane behaves like a smeared Matrix black hole; and the spacetime geometry seen by the probes can become non-commutative even far away from regions of Planckian curvature. This arises from nonlinear Matrix interactions where fast matrix modes lift a flat direction in the potential — akin to the Paul trap phenomenon in atomic physics. In the regime where we do have a notion of emergent spacetime, we show that there is non-zero entanglement entropy between supergravity modes on the membrane and the probes. The computation can easily be generalized to other settings, and this can help develop a dictionary between entanglement entropy and local geometry — similar to Ryu-Takayanagi but instead for asymptotically flat backgrounds.
Gromov–Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations We conjecture that the relative Gromov–Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice E8. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov–Witten potentials of the Schoen Calabi–Yau threefold (relative to P1) are E8 ×E8 quasi-biJacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi–Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed. In the Appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.
The ABC Conjecture and N = 4 Super-Yang-Mills Theory We establish a precise correspondence between the ABC Conjecture and N = 4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies’ method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d’enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N = 4 SYM.
Three-forms in Supergravity and Flux Compactifications We present a duality procedure that relates conventional four-dimensional mattercoupled N = 1 supergravities to dual formulations in which auxiliary fields are replaced by field-strengths of gauge three-forms. The duality promotes specific coupling constants appearing in the superpotential to vacuum expectation values of the fieldstrengths. We then apply this general duality to type IIA string compactifications on Calabi-Yau orientifolds with RR fluxes. This gives a new supersymmetric formulation of the corresponding effective four-dimensional theories which includes gauge three-forms.