The String Coffee Table

The participating projects are the following:

A - String Theory

A1 - Particle Physics from String Compactification

A2 - Time-dependent Backgrounds in String Theory

A3 - Strings and QCD

A4 - Mathematical Foundations of String Theory

A5 - Algebraic Aspects of D-branes and Field Theories with Boundaries

A6 - Mathematical Aspects of String Compactifications

A7 - Pseudo-Riemannian Geometry and Supersymmetry

B - Particle Physics

B1 - Physics Beyond the Standard Model at ILC

B2 - Supersymmetry at LHC

B3 - Neutrinos in the Standard Model

B4 - Field Theoretic Aspects of New Physics

B5 - Simulations for Physics Beyond the Standard Model

B6 - Strong Interactions and New Physics at LHC

B7 - Physics of Heavy Quarks and Squarks

C - Cosmology

C1 - Extremely Energetic Cosmic Neutrinos

C2 - Search for Dark Matter

C3 - Leptogenesis and Dark Matter

C4 - Variations of Fundamental Constants

C5 - Type Ia Supernovae and Dark Energy

C6 - Scalar Fields in Cosmology: Inflation, Dark Matter, Dark Energy

C7 - Thermodynamics of Quantum Fields in nonstationary Spacetimes

Evidently, my personal interest (and supposed participation) is in project area A, especially in sections A4 and A5.

Here are some slides sketching the strategy behind these mathematically oriented projects sections $\to$.

Some general information about these projects is listed in the following:

A4 - Mathematical Foundations of String Theory

Summary: The aim of the project is to study mathematical structures - in particular representation categories of vertex algebras, module categories over these categories and gerbes - that appear in worldsheet theories for open and closed strings. They are not only to be studied as mathematical objects in themselves; in close contact with other groups, the question will be continuously raised of whether they appropriately formalize physical concepts.

Present state of knowledge in the field: The basic idea of string theory is to replace point particles by one-dimensional objects: in the case of open strings, by intervals, in the case of closed strings, by circles. As a consequence, the classical description of closed strings leads quite naturally to loop spaces, i.e. spaces of maps from the circle $S^1$ to a target space $M$. Worldsheet theories for closed strings can be thought of as quantizations of such loop spaces. For their study, infinite-dimensional symmetry structures, like the diffeomorphisms of the circle or the (semi-)group of changes of local coordinates, turn out to be crucial. Two different frameworks have, by now, been established as formalizations of the chiral symmetries of a conformal Field theory:

$•$ Conformal nets of operator algebras on the circle. While this framework presents advantages for the study of general structural questions, it is notoriously difficult to construct and investigate in this language explicit examples.

$•$ Vertex algebras, the framework in which more concrete work of this project will be done. The theory of vertex algebras has recently made spectacular progress [7] (for a recent monograph see [1]). Still, important questions in the theory of vertex algebras and their representations remain open.

Given a chiral conformal field theory whose sectors behave sufficiently “nicely” (which is in particular the case when they form a modular tensor category in the sense of Turaev and Reshetikhin), one can construct several local conformal field theories. This process is now fairly well-understood for rational conformal field theories. In contrast, for more general classes of conformal field theories (non-unitary, extended worldsheet structures, non-rational or even non-compact theories), a comparable level of understanding has not been achieved.

A different aspect of loop spaces will be important in the present project as well: Sometimes, it is also possible to capture certain aspects of (line bundles over) a loop space $LM$ through additional finite-dimensional structures on the target space $M$. This leads to the theory of (hermitian bundle) gerbes (as introduced in [6]); their holonomy describes Wess-Zumino terms in the worldsheet action.

In all these theories, fundamental questions are unanswered. Moreover, there are frequently reasons to doubt whether in all cases the correct mathematical axiomatization of physical notions has already been found. As a consequence, the importance of the interaction with theoretical physicists cannot be underrated; for its realization, the present collaborative research center will provide excellent conditions. On the other hand, there are also numerous relations to mathematical questions of current interest, ranging from differential geometry of gerbes, twisted K-theory, elliptic cohomology, generalized Galois theory to representation theory.

The long term goal underlying the whole project is to deepen our understanding of rational conformal field theories in such a way that it can be ultimately extended to non-rational compact and finally even non-compact conformal field theories.

Preliminary work by the participants: An important ingredient for the present program is the construction of correlation functions of rational conformal field theories [11,12] that combines algebraic structures in the representation category of a vertex algebra with three-dimensional topological field theory. The algebraic structure that has been identified in this work is the one of a (modular) tensor category $C$ and a module category over $C$. This situation is characteristic for Galois theory; additional expertise will therefore come from the algebraic topology group in the mathematics department [14].

The work [11,12] is the basic input for the problems (iv), (v) and (vi). It enters already in the very formulation of the goal of (iv), it is to be generalized in (v) to conformal field theories with additional structure, and its generalization in (vi) beyond the rational case is one of the driving motivations for the whole proposal.

The structural analogy of the Wess-Zumino term in local coordinates and the triangulation ansatz of [12] can be expected to give crucial insights in subprojects (ii) and (iii). Independent insight comes from earlier work on flux quantization [17] which has been by now reformulated in the appropriate language of gerbe modules [5]. The experience of the project leader will be complemented in a first phase of the SFB by the one of U. Schreiber on two-bundles [13].

A crucial input for the subproblems (i) and (ii) is [15] in which symmetries and dualities of rational conformal field theories are related to properties of the fusion algebra of (topological) conformal defects.

Project section plan (objectives, methods, work programme): It has been known for a long time that models of statistical mechanics, e.g. the Ising model, can have defect lines and disorder fields that create them. The corresponding structures in the conformal field theory that arises in the continuum limit have only been explored recently. While defects and disorder fields have not been of direct use in string theory so far, they still contain information about string compactifications: There is a natural notion of fusion of such defects, and it has been recently discovered [15] that the corresponding fusion rules contain information the internal symmetries and (Kramers-Wannier like) dualities of the conformal field theory and, thus, of the string background. The first two subprojects deal with defects in different approaches.

i) Determination of dualities and symmetries in concrete models: The results of [15] relate dualities and symmetries to the fusion rules of defects. The aim of this subproject is to apply this to various classes of models, including (super-)Virasoro minimal series, WZW theories and coset theories. Both the charge conjugation modular invariant and general invariants of simple current type will be investigated. The ultimate goal of the project is to understand in this way symmetries and (perturbative) dualities of Gepner and Kazama-Suzuki models in string theory.

ii) Defects and Wess-Zumino terms: The aim of this subproject is obtain a Lagrangian description of defects in WZW theories. In particular, we aim at gaining a better understanding of the Wess-Zumino term in the presence of defects. The Wess-Zumino term can be understood as the holonomy of a (hermitian bundle) gerbe. One is thus looking for geometric objects associated to a gerbe that admit a notion of fusion that reproducing the fusion of defects. Such objects are of independent mathematical interest: in the case of WZW theories on simply connected Lie groups, the fusion rules of defects just give the Verlinde algebra. A class of geometric objects related to gerbes that have these fusion rules have been proposed recently in [3]. The main motivation of this work was mathematical: to understand the product structure on twisted K-theory on a compact Lie group, as due to [4]. A possible relation of the structures proposed in [3] to defects, however, remains to be uncovered. (This part should take 12 post-doc months.)

The next project is also concerned of aspects of the Wess-Zumino term.

iii) Wess-Zumino terms for unorientable surfaces: ($\to$) It is by now well established by algebraic methods (see [12:II] for the approach through topological field theory) that WZW theories can be considered on an unorientable surface $X$. A formulation of the Wess-Zumino term, in this situation, is however, not known: while the worldsheet $X$ can still be seen as the boundary of a three-manifold, $X=\partial M$, the latter is, in general unoriented so that the integral of the usual three-form $H$ field strength on $M$ is not well-defined. More geometric input is needed in this case, and its precise form should be determined in the present subproject. A guiding principle is in this case is the structural analogy between the holonomy of a gerbe in local coordinates and the algebraic construction of correlators of (rational) conformal field theories through triangulations in [12].

The latter construction of correlators [12] uses algebraic structure in the representation category C of the (rational) vertex algebra; it is also at the basis of the following two projects:

iv) Partition functions and Brauer groups: It has recently become clear that the classification of conformal field theories amounts to the classification of (Morita classes of) certain Frobenius algebras in $C$. A special case of such Frobenius algebras are Azumaya algebras, their equivalence classes give rise to the Brauer group of a category [8,10]. As a consequence, the classification of (rational) conformal field theories for given chiral data is a far reaching generalization of the classification of division algebras over fields.

The aim of the present subproject is twofold: First to understand general properties of these Brauer groups, and second to gain a better understanding of them for specific classes of models. Since the calculation of a Brauer group can be reformulated as a representationtheoretic problem for a (weak) Hopf algebra, the determination of these Hopf algebras, and thus results of subproject (vi), for specific classes of models might be necessary.

v) More general worldsheet geometries: The results of [12] apply to rational conformal field theories. For many applications, in particular in string theory, more symmetry than just the conformal symmetry is present and requires additional geometric structure on the worldsheet. Prominent examples are superconformal theories that require a spin structure and orbifold theories that require the choice of a principal bundle on the worldsheet. The results of [11,12] should be extended to such theories with extended worldsheet geometry as well.

Another direction is to extend the results are non-unitary theories. In this case, the properties of the representation categories are not well-understood. A particularly useful case should be WZW theories at fractional level. The form of the Virasoro element of such theories - the usual affine Segal-Sugawara element has to be complemented by zero modes of the non-abelian currents since already the horizontal representation is infinite-dimensional - indicates that for these models, the worldsheet geometry should be extended as well.

Finally, we present a rather ambitious project to which all members of the group are supposed to contribute. We list it to indicate the long term perspective our our research programme.

vi) Representation theory of chiral symmetries The representation theory of vertex algebras provides many challenges. It seems to be interwoven with the representation theory of an associative algebra, called Zhu’s algebra. Its definition - in contrast to most other developments in this field - is not motivated by physical considerations. Its properties, however, seem to be intimately related to the behavior of the theory under the action of mapping class groups.

For general reasons, we know that the representation theory of a vertex algebra is can be described by the representation category of a weak Hopf algebra. This algebra is not unique; still, its knowledge allows to relate question in representation categories of vertex algebras to calculations with simpler mathematical objects. For example, the deformation theory of module categories is controlled by Davydov-Yetter cohomology; the latter can be computed as the Hochschild cohomology of a corresponding weak Hopf algebra. It is thus desirable to have explicit models of such weak Hopf algebras available for concrete classes of theories, and to clarify their relation to Zhu’s algebra.

An important breakthrough would be a concrete understanding of the representation categories of non-rational conformal field theories. It would bring within reach both a deformation theory of conformal field theories (since any deformation of a rational conformal field theory involves non-rational ones) and would open the way to the application of the algebraic results of [12] to phenomenologically interesting string compactifications and cosmologically interesting string backgrounds.

The full text can be found here.