Strings, Gauge Fields, and the Geometry Behind

The following sections are included:

• Foreword by the Editors

• Contents

We present the mathematical considerations which determine all gauge invariant actions and anomaly candidates in gauge theories of standard type such as ordinary or gravitational Yang–Mills theories. Starting from elementary concepts of field theory the discussion tries to be explicit and complete, only the cohomology of simple Lie algebras is quoted from the literature.

The application and extension of well-known BRST cohomological methods to supersymmetric field theories are discussed. The focus is on the emergence and particular features of supersymmetry algebra cohomology in this context. In particular it is discussed and demonstrated that supersymmetry algebra cohomology emerges within the cohomological analysis of standard supersymmetric field theories whether or not the commutator algebra of the symmetry transformations closes off-shell.

We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of the Racah symbols for the algebra SU_q. Then, the HOMFLY polynomials can be extended to the entire space of time-variables. The so extended HOMFLY polynomials are no longer knot invariants, they depend on the choice of the braid representation, but instead one can naturally discuss their explicit integrable properties. The generating functions of torus knot/link coefficients are turned to satisfy the Plücker relations and can be associated with τ-function of the KP hierarchy, while generic knots correspond to more involved systems. On the other hand, using the expansion into the Schur functions, one can immediately derive difference equations (A-polynomials) for knot polynomials which play a role of the string equation. This adds to the previously demonstrated use of these character decompositions for the study of β-deformations from HOMFLY to superpolynomials.

We explain some applications of bicategories in both classical and quantum field theory. This includes a modern perspective on some pioneering work of Max Kreuzer and Bert Schellekens on rational conformal field theory.

We give a detailed derivation of the spherical harmonics which correspond to the spectrum in the tables of the 1985 paper by Kim, Romans and the author on the Kaluza–Klein dimensional reduction of IIB supergravity on S₅ to 5-dimensional maximal anti de Sitter supergravity. We show how all methods that have been used in the literature to obtain spectra if the internal space is S_n can be derived from embedding S_n in ℝ^{n + 1}. A crucial observation for spin 1/2 spherical harmonics is that one also needs a local Lorentz rotation if one transforms to polar or stereographic coordinates. The relation to the vector spherical harmonics of electromagnetism is also worked out.

Any account of Max's career in physics must be bound up with the history of the study of Calabi–Yau manifolds, to which Max contributed at many levels. There were many currents in this study and work was not done in isolation. Work often advances through a series of challenges, and in reaction to other work. Insofar as I have myself been involved in some of these researches it is inevitable that I will have to recall some of these projects that were, at times, inextricably linked with Max's work. For this deficiency of the account let me make this single apology…

We study the Poincaré polynomials of all known Calabi–Yau three-folds as constrained polynomials of Littlewood type, thus generalising the wellknown investigation into the distribution of the Euler characteristic and Hodge numbers. We find interesting fractal behaviour in the roots of these polynomials, in relation to the existence of isometries, distribution versus typicality, and mirror symmetry.

There exist exactly 166 4-dimensional reflexive polytopes Δ such that the corresponding 4-dimensional Gorenstein toric Fano variety ℙ_Δ has at worst terminal singularities in codimension 3 and the anticanonical divisor of ℙ_Δ is divisible by 2 in its Picard group. For every such a polytope Δ, one naturally obtains a family of Fano hypersurfaces X ⊂ ℙ_Δ with at worst conifold singularities. A generic 3-dimensional Fano hypersurface can be interpreted as a flat conifold degeneration of some smooth Fano 3-folds Y whose classification up to deformation was obtained by Iskovskikh, Mori and Mukai. In this case, both Fano varieties X and Y have the same Picard number r. Using toric mirror symmetry, we define a r-dimensional generalized hypergeometric power series Φ associated to the dual reflexive polytope Δ*. We show that if r = 1 then Φ is a normalized regular solution of a modular D3-equation that appears in the Golyshev correspondence. We expect that the power series Φ can be used to compute the small quantum cohomology ring of all Fano 3-folds Y with the Picard number r ≥ 2 if Y admits a conifold degeneration .

I summarize some of the ideas and motivations behind a recently performed conformal field theory analysis of closed strings in three-form flux backgrounds. This suggests an underlying nonassociative structure for the coordinates.

The problem of counting points is revisited from the perspective of reflexive 4-dimensional polytopes. As an application, the Hilbert series of the 473, 800, 776 reflexive polytopes (equivalently, their Calabi–Yau hypersurfaces) are computed.

We review several approaches to the construction and classification of realistic string vacua, and emphasize recent successes in construction of Standard Model-like vacua in string theory, mostly in the heterotic string.

We briefly survey our present understanding of the string landscape, and use it to discuss the chances that we will see low energy supersymmetry at the LHC.

Using factorizable Hopf algebras, we construct modular invariant partition functions of charge conjugation, or Cardy, type as characters of coends in categories that share essential features with the ones appearing in logarithmic CFT. The coefficients of such a partition function are given by the Cartan matrix of the theory.

This article is based on a talk given at the Memorial Conference for Maximilian Kreuzer at the ESI in Vienna and contains a compact summary of a recent collaboration with P.A. Grassi. A nonlinear projection from the space of SO(10) Weyl spinors to the space of pure spinors is presented together with some of its particular properties. This projection can be used to remove the constraints from Berkovits' pure spinor superstring while introducing additional gauge symmetries. This should allow to make transitions to equivalent formulations which might shed light on the origin of the pure spinor ghosts. It might also be useful in the context of path integral measures for the pure spinor string.

We review evidence for an action of the largest Mathieu group on the space of BPS states contributing to the elliptic genus of K3 (‘Mathieu moonshine’). As a first step in trying to understand this action we discuss a theorem classifying symmetries of nonlinear sigma-models on K3 which commute with the superconformal algebra. We illustrate this theorem by discussing as a particular example the sigmamodel on .

We generalize the combinatorial description of the orbifold (Chen–Ruan) cohomology and of the Grothendieck ring of a Deligne–Mumford toric stack and its associated stacky fan in a lattice N in the presence of a deformation parameter β ∈ N ⊗ ℂ. As an application, we construct a topological mirror symmetry map that produces a complete system of Γ-series solutions to the better behaved version of the GKZ hypergeometric system for β ∈ N ⊗ ℂ.

Fano polytopes are the convex-geometric objects corresponding to toric Fano varieties. We give a brief survey of classification results for different classes of Fano polytopes.

We propose a class of AdS/CFT dual pairs which have small internal dimensions on the gravity side. Starting from known Freund–Rubin AdS/CFT dual pairs, we use 7-branes to nearly cancel the curvature energy of the internal dimensions while maintaining their stabilization. This leads to a new corner of the landscape — a class of AdS solutions with a hierarchically large AdS radius — with a dual field theory given (implicitly) by the infrared limit of a concrete brane construction involving D3-branes, 7-branes, and curvature. We first construct a class of hierarchical AdS5/CFT4 dual pairs with a simple formula for the number of degrees of freedom which we interpret in the dual QFT. We then generalize these to AdS4/CFT3 duals, and suggest extensions of the method to obtain de Sitter solutions.

In these notes we consider relations between strings, integrable models and gauge theories. The considerations are based on the two examples where one can identify the integrable structures on the string side and then proceeds with the relation to the dual field theory.

We extend the analysis of arXiv:1009.2624, which constructed the nonlinear realisation of the semi-direct product of E₁₁ and the l₁ representation at level zero, to level one. Thus we add to the previously considered NS ⊗ NS fields those of the R ⊗ R sector.

This is a brief review of simple current techniques and their application to string theory constructions, centered around Max Kreuzer's contribution to this field.

Over the past few years the arithmetic Langlands program has found applications in two different problems that arise in string physics. The first of these is concerned with the fundamental problem of deriving the geometry of spacetime from the worldsheet dynamics, leading to a realization of the notion of an emergent spacetime in string theory. The second problem is concerned with the idea of using automorphic black holes as probes of spacetime. In this article both of these physical applications of the Langlands program are described.

Two of my collaborations with Max Kreuzer involved classification problems related to string vacua. In 1992 we found all 10,839 classes of polynomials that lead to Landau–Ginzburg models with c=9 (Klemm and Schimmrigk also did this); 7,555 of them are related to Calabi–Yau hypersurfaces. Later we found all 473,800,776 reflexive polytopes in four dimensions; these give rise to Calabi–Yau hypersurfaces in toric varieties. The missing piece — toric constructions that need not be hypersurfaces — are the reflexive Gorenstein cones introduced by Batyrev and Borisov. I explain what they are, how they define the data for Witten's gauged linear sigma model, and how one can modify our classification ideas to apply to them. I also present results on the first and possibly most interesting step, the classification of certain basic weights systems, and discuss limitations to a complete classification.

This article provides a complete user's guide to version 2.1 of the toric geometry package PALP by Maximilian Kreuzer and others. In particular, previously undocumented applications such as the program Nef.X are discussed in detail. New features of PALP 2.1 include an extension of the program mori.x which can now compute Mori cones and intersection rings of arbitrary dimension and can also take specific triangulations of reflexive polytopes as input. Furthermore, the program nef.x is enhanced by an option that allows the user to enter reflexive Gorenstein cones as input. The present documentation is complemented by a Wiki which is available online.

The following sections are included:

• Author Index