This entry collects material on the June 2009 workshop on strings, fields, and topology at Oberwolfach.
Writeups of most of the talks have appeared as
Thomas Schick, Smooth cohomology theories.
Ullrich Bunke, Smooth K-theory.
Christoph Schweigert and Ingo Runkel , CFT and algebra in braided tensor categories I and II.
Kevin Costello, Factorization algebras in perturbative QFT.
Gabriel C. Drummond-Cole, $\infty$-Operads, $BU_\infty$ and $Hypercomm_\infty$.
Alex Kahle, Superconnections and index theory.
Chris Schommer-Pries, Topological defects, $D$-branes, and the classifications of TFTs in low dimensions.
Dan Freed, Geometry and topology of orientifolds.
Dan Freed, Evening informal session on RR fields and all that.
Scott Wilson, Some algebra related to mapping spaces and applications.
Urs Schreiber, Background fields in twisted differential nonabelian cohomology.
André Henriques, Invertible conformal nets.
André Henriques, Evening informal session more details about conformal nets.
Konrad Waldorf, String connections and Chern-Simons 2-gerbes.
Kevin Walker, Blob homology
Ralph Cohen, String topology, field theories and Fukaya categories.
Corbett Redden, String structures, 3-forms, and tmf classes.
Mike Hopkins, The Kervaire invariant.
Mike Hopkins, Evening informal session on technical details of the solution of the Kervaire invariant problem.
David Chataur, Evening informal session explaining recent results in opeards
Kishore Marathe, Evening informal session on the gauge theory to string theory correspondence
The basic idea of differential cohomology (also called smooth cohomology ) is to combine generalized cohomology (e.g. ordinary integral cohomology, or K-theory, or tmf, or …) and differential forms. Hopkins and Singer showed that a smooth refinement exists for any generalized cohomology theory, but didn’t provide an (easy for mere mortals to understand) explicit construction. In this talk, Thomas Schick advertised geometric models for multiplicative smooth cohomology with $S^1$-integration for $K$-theory (joint with Ulrich Bunke) and MU-bordism (joint with Ulrich Bunke, Schröder? and Wethamp?). They have proved a uniqueness theorem, stating that under certain assumptions any two smooth refinements of a generalized cohomology theory are naturally isomorphic.
In this talk, which led on from the previous one by Thomas Schick, Ulrich Bunke gave an explicit description of their smooth $K$-theory model in terms of geometric cocycles. Basically a cocycle consists of a pair $(\mathcal{E}, \rho)$ where $\mathcal{E}$ is a ‘geometric family’, namely a smooth proper submersion $E \rightarrow B$ with a fiberwise metric, a Clifford bundle $W$ and some connection of some sort. The object $\rho$ is an element of $\Omega(B, K) = C^\infty(B, \Lambda^*T^* B \otimes K^*)$. He also described smooth $K$-orientatiosn and the push-forward map, with an application to the $e$-invariant of Adams. There is also a Riemann-Roch theorem.
This talk explains how correlators for a rational two-dimensional conformal field theory can be constructed in the functorial TFT formalism. Based on a modular tensor category $C$, decoration data have been introduced in terms of special symmetric Frobenius algebras in $C$ and the correlator, as an element of $tft_C (\hat{X})$ with $\hat{X}$ a double cover of the surface $X$, has been expressed in terms ofthe invariant of a decorated 3-manifold $M_X$ with $\partial M_X = \hat{X}$. The correlators are invariant under the mapping class groups and obey the sewing constraint.
Morita equivalent special symemtric Frobenius algebras lead to an equivalent description of the correlators. A Morita invariant formulation is provided by the notion of a module category $M$ over the module tensor category $C$. The world sheet is now decorated by categories, functors between module categories and natural transformations.
A factorization algebra on a manifold $M$ is an object which associates to every ball $B \subseteq M$ a vector space (or cochain complex) $F(B)$; and to every collection $B_1 \coprod \cdots \coprod B_n \subset B_{n+1}$ of disjoint balls in a larger ball, a map $F(B_1) \otimes \cdots \otimes F(B_n) \rightarrow F(B_{n+1})$. These structures are the $C^\infty$ analogues of chiral algebras, as introduced by Beilinson and Drinfeld; they are closely related to algebras for the little n-disk operad $E_n$.
In this series of lectures, Kevin Costello set up a framework to construct factorization algebras from perturbative quantum field theory. The set up is analagous to the deformation quantization picture of quantum mechanics. Just as the observables of quantum mechanics are encoded in an associative algebra, Costello argues that the observables of a QFT on $M$ are encoded in a factorization algebra on $M$, similar to but slightly different from the waqy it works in (euclidean) AQFT. This factorization algebra arises by quantizing a commutative factorization algebra associated to classical field theory.
This series of lectures finished with the statement of a theorem allowing one to quantize the commutative factorization algebra associated to a classical field theory in a range of situations, including situations of physical interest. This is joint work with Owen Gwilliam.
Gabriel Drummond-Cole applies the machinery of the model category of operads to extend and explain the Barannikov-Kontsevich passage from differential BV algebras satisfying the $\partial-\overbar{\partial}$ lemma to hypercommutative algebras (Frobenius manifolds). This is joint work with Bruno Valette. The following theorem was proved:
Let $V$ be a differential BV-algebra over a field of characteristic zero. Let $H$ be its homology. Then: 1. If $V$ satisfies the noncommutative Hodge to de Rham degeneration condition, then there exists a hypercommutative $\infty$-structure on $H$. 2. If $V$ satisfies Park’s semiclassical condition then this structure is unique up to $Hypercomm_\infty$ quasi-isomorphism.
In this talk, Alex Kahle described results from his thesis work (his supervisor was Dan Freed) about superconnections and index theory. The idea is to prove a local index theorem using superconnections via direct geometric-analytical methods (i.e. not stochastic) for use with family index problems, determinant line bundles, etc. Just like ordinary connections on spinor buundles, a superconnection gives rise to a Dirac operator on a spinor bundle. The usual local index theorem (that the trace of the heat kernel converges as $t \rightarrow 0$ to a certain differential form) needs to be modified though, because forms of different degrees have different scaling behaviour (recall that a superconnection involves forms of different degrees). Alex worked out how to scale everything correctly so that one indeed gets a local index theorem for superconnections, leading to many potential applications.
Chris explained his classification result mentioned at (infinity,n)-category of cobordisms from his thesis of extended (he suggested calling them local) 2d TQFT’s via the explicit generators and relations he obtained on $2Cob$. He also showed how the higher-categorical viewpoint unites the following two ideas: the ‘open-closed’ theories and the ‘field theories with defects’ from Ingo Runkel and Christoph Schweigert’s talk. He showed how both these concepts are particular examples of a single notion, namely that of a collection of natural transformations between a restriction of FTQFT? $n$-functors (called “unnatural” or “supernatural” transformations to indicate that it is a transformation not of the entire functor, but just of its restriction to the 2-category of 2-manifolds with only inverttible 2-morphisms).
This is precisely the kind of viewpoint that Urs Schreiber and Jens Fjelstad suggested at the n-category cafe under the slogal D-branes from tin-cans (a D-brane is a boundary condition ordefect, a “tin-can diagram” is the naturality diagram for a higher natural transformation).
By the way, there’s much more to come from Chris, he is working on the 3d theory and much besides!
Freed described recent work with Jacques Distler and Greg Moore on a certain background structure in superstring theory called an [[orientifold], which is a bundle gerbe on a $\mathbb{Z}_2$ orbifold with a peculiar “twisted” equivariance condition. There is apparantly a beautiful string theory formula to calculate the ‘RR charge?’, namely something like
involving a certain normal bundle and the ‘Bott element in K-theory’, which looks similar to the Hirzebruch formula for the L-genus. This uses some kind of twisted differential cohomology version of KR theory. The whole thing can be viewed as a new sort of anomaly, having to do with an exotic orientation of some kind.
Summary appearing soon…
Wilson defined a partial algebra as a lax monoidal functor from the category of finite sets to chain complexes, such that the map $A(j \coprod k) \rightarrow A(j) \otimes A(k)$ is a quasi-isomorphism. He showed how these pop up in homotopy theory all the time.
(Remark: this formula is the basis for Jacob Lurie’s discussion of commutative algebra in an (infinity,1)-category in general and of symmetric monoidal (infinity,1)-category in particular).
For instance, if $Y$ is any space and $A$ a partial algebra, the resulting total complex can be seen as a generalization of Hochschild cohomology. One really interesting example (for Bruce) was where $Y$ is the interval, and $A$ is the partial algebra of forms on a Riemannian manifold $M$. It turns out that a certain canonical equation which pops out is precisely the Navier-Stokes equation. Gulp!
What can I (Bruce) say about Urs’s talk? Many mere mortals might think it impressive to pass from ordinary cohomology to twisted cohomology, or to differential cohomology, or to nonabelian cohomology. Urs does this all in one step! Yes, we’re talking about twisted differential nonabelian cohomology. As we all know and love, Urs has developed some $\infty$-machinery based on all sorts of work which enables him to ‘turn the crank’ and output all the cohomology theories which mathematicians and physicists currently are bumping into. A great application of his machinery is an understanding of the ‘Green-Schwarz anomaly cancellation’ mechanism in terms of twisted nonabelian String-gerbes with connection and the Chern-Simons 2-gerbe. Unfortunately Urs didn’t have time to quite get to that point, but he did at least mention other examples like how his machinery naturally produces the ‘twisted Bianchi identities’ in twisted flat differential cohomology, as well as nonabelian gerbes. Urs ended his talk by explaining how nonabelian cohomology on $X$ is related to twisted abelian cohomology on $X$. This ties in with Konrad’s talk, where the abelian viewpoint is taken, but the nonabelian viewpoint is essential for some applications.
Urs’s personal nLab write-up on Background fields in twisted differential nonabelian cohomology
In this talk, Henriques described joint work with Chris Douglas and Arthur Bartels on the program of establishing an equivalence between full conformal field theories and ‘conformal nets’. Preliminary write-ups of this work is available on his webpage. A conformal net is something like a factorization algebra (in the sense of Costello’s talk) on 1-dimensional manifolds taking values in the bicategory of Von Neumann algebras with bimodules as 1-morphisms. The $\mu$index $\mu(A)$ of a conformal net $A$ is defined to be the statistical dimension of the vacuum sector $H_0$, viewed as some kind of bimodule involving things which $A$ assigns to arcs on the unit circle. He stated the following result.
A conformal net is invertible if and only if $\mu(A)=1$, and it is fully dualizable (in the sense of Lurie) if and only if $\mu(A) \lt \infty$.
Notes by Dmitry Pavlov?
André responded to questions from the audience about further details regarding the ‘conformal nets’ programme.
Write-up about conformal nets on André Henriques’ webpage.
Konrad described the Chern-Simons 2-gerbe $CS_P$ which is a geometric object living on any manifold $M$ which is equipped with a Spin-bundle $P$. Its definition involves some kind of pullback of the basic bundle gerbe over the Spin group. He demonstrates that a string structure $S$ on $P$ is a trivialization of this 2-gerbe, in the sense of a morphism from the trivial 2-gerbe living over $M$ to $CS_P$. Such a trivialization only exists precisely if the first fractional Pontryagin class of the bundle vanishes, $\frac{1}{2} p_1(P) = 0$, otherwise the Chern-Simons 2-gerbe is globally nontrivial and a string structure doesn’t exist. He defined a string connection on a string structure $S$ (a connection on $P$ is needed here) on $P$ as a connection on this trivialization $S$ which is compatible with various things.
Basically, the data of a string structure together with a string connection on $P$ is a class in the $[CS_P]$-twisted differential cohomology of $M$ in Urs’s sense, for vanishing twist (the twist is the nontriviality of the String structure). Konrad described various results about string connections, such as the fact that they form a contractible space, confirming a conjecture by Stolz and Teichner. See his recent arXiv article.
Kevin Walker described a new way to think about extended TQFTs. Namely to an n-manifold $M$ and an n-category $C$ he defined a ‘blob complex’ $B_*(M, C)$. The construction produces a kind of ‘derived version’ of an extended TQFT. The idea is that the usual quantum invariants $Z(M)$ to manifolds of dimension $\leq n$ are produced out of this chain complex, with the Atiyah-Segal axioms following as corollaries of the setup. He listed a number of examples of the TQFT’s he has in mind in the beginning, such as the finite group model, one based on a -algebra, one based on a pivotal category, one based on a braided ribbon category, and even one based on contact structures.
This talk was about relating the subject of string topology, which is now 10 years old, with other field theories, especially in the light of recent work by Costello, Hopkins and Lurie. String theory was described as a ‘homological conformal field theory’. The slogan was that string topology simplifies when one applies Poincaré duality. A relation was sketched between string topology and Gromov-Witten symplectic field theory. Ultimately it was conjectured that the symplectic field theory of $T^*M$ is equivalent to string topology of $M$.
Corbett began my defining a string structure on a manifold $M$ as a certain lift of a classifying map; string structures up to homotopy are canonically isomorphic with things called ‘string classes’ on $M$ - namely, elements in the third integral cohomology of the total space of the bundle which behave on the fibers as the generator of the third cohomology of the Spin group. The point about string structures is that they transgress to give a spin structure on the loop space of the manifold, which is what one wants to study things like the Witten genus. He showed a beautiful way to obtain string structures from a metric on $M$, using Hodge representatives for the forms, and then passing to the ‘adiabatic limit’, to obtain a certain 3-form $H$. A theorem of Mazzeo-Melrose, Dai and Forman then says that the kernel of the Hodge laplacian extends smoothly in the adiabatic limit, and comes from a filtration in the Serre spectral sequence (something like that). He gave a hypothesis that if a manifold $M^n$ admits a spin and a string structure and a Riemannian metric $g$ such that the Ricci curvature of $g$ is positive and that $H=0$, then a certain invariant $\sigma(M, S)=0$ inside $tmf^{-n}(pt)$. He showed that all these conditions are necessary!
Mike Hopkins gave a nice story of the history and origins of the Kervaire invariant in the classification of manifolds. He spoke about Pontryagin’s work about the cobordism groups of stably framed manifolds in the 1930’s, and the mistake he made in dimension $n=2$! Basically he thought a certain function was linear, when in fact it was quadratic. This led to the Kervaire invariant being introduced. The problem about when the Kervaire invariant vanishes has been solved in dramatic fashion by Hopkins, Hill and Ravenel very recently… if a manifold has Kervaire invariant $1$, then its dimension must be either 2, 6, 14, 30, 62 or 126, with the final of these remaining open! See the notes for the great story. Hopkins ended with a nice geometric description of the Kervaire invariant on spheres in dimension 2 and 6, relating it to complex structures and exceptional Lie groups. He wondered if these things feature in dimension 126?
Notes needed… help…
Notes needed… help…
See also talk summaries.