Duality Covariant Formulation of Extended Supergravity: Central Charges,BPS States and Black-Hole Entropy

In these lectures we give a geometrical formulation of N-extrended supergravities which generalizes N = 2 special geometry of N = 2 theories. In all these theories duality symmetries are related to the notion of “flat symplectic bundles” and central charges may be defined as “sections” over these bundles. Attractor points giving rise to “fixed scalars” of the horizon geometry and Bekenstein-Hawking entropy formula for extremal black-holes are discussed in some details.     

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL _k Heisenberg chain is dual to the special reduced k + 2-points gl _N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.     

Spectral duality in integrable systems from AGT conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl _N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the Alday-Gaiotto-Tachikawa relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous Adams-Harnad-Hurtubise duality.     

Duality for Z(N) gauge theories

The duality properties of simple Z(N) gauge theories are discussed. For N ? 4 we find self duality in four dimensions and we give the transition points. For N > 4 these systems are not self dual. Also, the order parameter is discussed. The general Z(N) gauge theory is found to be self dual for all N.     

Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials

Original proofs of the AGT relations with the help of the Hubbard–Stratanovich duality of the modified Dotsenko–Fateev matrix model did not work for β ≠ 1, because Nekrasov functions were not properly reproduced by Selberg–Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper, we describe a complete proof of AGT in the simple case of GL(2) (N = 2) Yang–Mills theory, i.e., the 4-point spherical conformal block of the Virasoro algebra.     

Black holes and groups of type E 7

We report some results on the relation between extremal black holes in locally supersymmetric theories of gravity and groups of type E ₇, appearing as generalized electric-magnetic duality symmetries in such theories. Some basics on the covariant approach to the stratification of the relevant symplectic representation are reviewed, along with a connection between special K?hler geometry and a ??generalization?? of groups of type E ₇.     

Duality relations and equivalences for models with O(N) and cubic symmetry

Formal expression for high-temperature series are derived for models with O(N) and cubic symmetry, with a special form of nearest neighbor interactions on the honeycomb lattice. By deriving low-temperature series for a class of generalized solid-on-solid and cubic models, a duality relation is established. Equivalences between cubic and SOS type models are also found. In the large-N limit, the series reduce to those of the hard hexagon model.     

Strings on a cone and black hole entropy

String propagation on a cone with deficit angle 2π(1 − 1/N) is described by constructing a non-compact orbifold of a plane by a Z_N subgroup of rotations. It is modular invariant and has tachyons in the twisted sectors that are localized at the tip of the cone. A possible connection with the quantum corrections to the black hole entropy is outlined. The entropy computed by analytically continuing in N would receive contribution only from the twisted sectors and be naturally proportional to the area of the event horizon. Evidence is presented for a new duality for these orbifolds similar to the R → 1/R duality.     

Low-energy analysis of M- and F-theories on Calabi-Yau threefolds

We elucidate the interplay between gauge and supersymmetry anomalies in six-dimensional N = 1 supergravity with generalized couplings between tensor and vector multiplets. We derive the structure of the five-dimensional supergravity resulting from the S₁ reduction of these models and give the constraints on Chem-Simons couplings that follow from duality to M-theory compactified on a Calabi-Yau threefold. The duality is supported only on a restricted class of Calabi-Yau threefolds and requires a special type of intersection form. We derive five-dimensional central-charge formulas and briefly discuss the associated phase transitions. Finally, we exhibit connections with F-theory compactifications on Calabi-Yau manifolds that admit elliptic fibrations. This analysis suggests that F-theory unifies type-IIb three-branes and M-theory five-branes.     

Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d ${\mathcal{N} = 1}$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N + 1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of knot theory, generalized AGT duality for (3 + 3)d theories, and a 2d vortex partition function are described.     

First order flows for N=2 extremal black holes and duality invariants

We derive explicitly the superpotential W for the non-BPS branch of $N=2$ extremal black holes in terms of duality invariants of special geometry. Although this is done for a one-modulus case (the $t^3$ model), the example gives $Z\ne 0$ black holes and captures the basic distinction from previous attempts on the quadratic series (vanishing C tensor) and from the other $Z=0$ cases. The superpotential W turns out to be a non-polynomial expression (containing radicals) of the basic duality invariant quantities. These are the same which enter in the quartic invariant $I_4$ for $N=2$ theories based on symmetric spaces. Using the flow equations generated by W, we also provide the analytic general solution for the warp factor and for the scalar field supporting the non-BPS black holes.     

F-theory,T-duality on K3 surfaces,and N = 2 supersymmetric gauge theories in four dimensions

We construct T-duality on K3 surfaces. The T-duality exchanges a 4-brane RR charge and a 0-brane RR charge. We study the action of the T-duality on the moduli space of 0-branes located at points of K3 and 4-branes wrapping it. We apply the construction to F-theory compactified on a Calabi-Yau 4-fold and study the duality of N = 2 SU(N_c) gauge theories in four dimensions. We discuss the generalization to the N = 1 duality scenario.     

Electric dipole moment for supersymmetric monopoles

The electric dipole moment for the monopoles that can be present in N = 2 and N = 4 supersymmetric SU(2) gauge theories, spontaneously broken by imposing a non-zero expectation value of a scalar field at infinity, is determined by considering the response to a weak external electric field. The magnetic g factor g_M = 2 which is in accord with the duality conjecture of Montonen and Olive.     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Triplet Z(N) model in a triangular lattice

The phase structure of a class of two-dimensional spin models with three-body interactions defined on a triangular lattice is studied. This class of models, containing the Baxter-Wu model as a special case, is shown to share the duality properties of a wide class of spin theories in two and three dimensions and the Z(N) gauge theory in four dimensions. Like these models, our theory is shown to possess a massless, Kosterlitz-Thouless-like phase when the number of available spin states exceeds a critical value.     

Two-dimensional models as testing ground for principles and concepts of local quantum physics

In the past two-dimensional models of QFT have served as theoretical laboratories for testing new concepts under mathematically controllable condition. In more recent times low-dimensional models (e.g., chiral models, factorizing models) often have been treated by special recipes in a way which sometimes led to a loss of unity of QFT. In the present work, I try to counteract this apartheid tendency by reviewing past results within the setting of the general principles of QFT. To this I add two new ideas: (1) a modular interpretation of the chiral model Diff(S)-covariance with a close connection to the recently formulated local covariance principle for QFT in curved spacetime and (2) a derivation of the chiral model temperature duality from a suitable operator formulation of the angular Wick rotation (in analogy to the Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational chiral theories. The SL (2, Z) modular Verlinde relation is a special case of this thermal duality and (within the family of rational models) the matrix S appearing in the thermal duality relation becomes identified with the statistics character matrix S. The relevant angular “Euclideanization” is done in the setting of the Tomita-Takesaki modular formalism of operator algebras. I find it appropriate to dedicate this work to the memory of J.A. Swieca with whom I shared the interest in two-dimensional models as a testing ground for QFT for more than one decade. This is a significantly extended version of an “Encyclopedia of Mathematical Physics” contribution hep-th/0502125.     

Conservation laws in the quantum Hall Liouvillian theory and its generalizations

It is known that the localization length scaling of noninteracting electrons near the quantum Hall plateau transition can be described in a theory of the bosonic density operators, with no reference to the underlying fermions. The resulting “Liouvillian” theory has a U(1|1) global supersymmetry as well as a hierarchy of geometric conservation laws related to the noncommutative geometry of the lowest Landau level (LLL). Approximations to the Liouvillian theory contain quite different physics from standard approximations to the underlying fermionic theory. Mean-field and large-N generalizations of the Liouvillian are shown to describe problems of noninteracting bosons that enlarge the U(1|1) supersymmetry to U(1|1)×SO(N) or U(1|1)×SU(N).These noninteracting bosonic problems are studied numerically for 2⩽N⩽8 by Monte Carlo simulation and compared to the original N=1 Liouvillian theory. The N>1 generalizations preserve the first two of the hierarchy of geometric conservation laws, leading to logarithmic corrections at order 1/N to the diffusive large-N limit, but do not preserve the remaining conservation laws. The emergence of nontrivial scaling at the plateau transition, in the Liouvillian approach, is shown to depend sensitively on the unusual geometry of Landau levels.