Workshop on Loops, Strings and Moduli Spaces

Chern Institute of Math, Tianjin, China

August 03-07, 2009

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Schedule:

LOCATION:All talks take place in Shiing-Shen building 214

Aug 3, Monday:

Chair: Ralph Cohen

9:20-9:30: Openning Address

9:30-10:30: Ulrike Tillmann

Title: Cobordism categories, some subcategories and their classifying spaces

Abstract: Cobordism categories have been successfully studied by associating classifying spaces to them. When their homotopy can be determined they tend to be well behaved infinite loop spaces with accessible cohomology. The homotopy type of the classifying space of the $d$-dimensional cobordism category had been determined for all $d$ in joint work with Galatius, Madsen, and Weiss and, for $d >1$ shown to be the same as that of the non-compact subcategory in which each connected component of a cobordism has a non-trivial outgoing boundary component. In my talk I will present some recent work of George Raptis concerning the homotopy type of certain cobordism subcategories. He shows that in dimension 1 the unoriented and the oriented non-compact categories have classifying spaces $\Omega ^{\infty -1} RP^\infty _{-1}$ and $\Omega ^\infty S^\infty$ respectively, thus derterminig the two cases that had been left out in the joint work mentioned above. Motivated by some work of Costella, also related categories of annuli (and their moduli spaces) are considered and their classifying spaces determined. The methods immediately also extend to the cobordism subcategories where the only morphisms that are not isomorphisms are spheres with $n$ disks removed (considered as cobordisms from $n$ to 0). The proof uses extensions of Quillen's Theorem B and the group completion method via homology fibration.

Group Photo and Tea Break

11:00-12:00: Tarje Bargheer

Title: String Topology Vividified

Abstract: We will describe work in preparation on coloured operads, acting via correspondances on Map(N,M) for N,M manifolds exhibiting suitable topological conditions. Monoids are to categories what operads are to coloured operads, and what we propose to put forth in this talk could be considered a 'colourification' of the spineless cacti operad. The main immediate advantage of this approach is that the situation easily generalizes to give spectral actions on higher free loop spaces; Map(S^n,M). For n=1, this is -- up to homotopy -- the Cohen-Jones construction of the Chas-Sullivan loop product. We will describe why N=S^n provides the best statements and how we -- in this case -- have an acting coloured operad, intimately related to E_{n+1}-operads; in the sense that they are disjoint union of coloured colour-contractible E_{n+1}-operads.

12:30: Lunch at Jiayuan Hotel

3:30-4:30: Andrew Stacey

Comparative Smootheology

Abstract: Smooth manifolds are extremely nice spaces. The fact that they have charts means that a vast amount of the theory of Euclidean spaces can be easily transferred to manifolds. This makes for a very useful subject. However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts. In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Fr?licher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.

Tea Break

5:00-6:00: Takuya Sakasai

Title: Lagrangian mapping class groups from group homological point of view

Abstract: We consider two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. Since these subgroups, called the Lagrangian preserving subgroup and the Lagrangian fixing subgroup, contain the Torelli group, we can expect that they might play important roles when we attack the (non-)triviality problem of even Morita-Miller-Mumford classes on the Torelli group. In this talk, we discuss group (co)homology of these subgroups.

6:30: Dinner at Jiayuan Hotel

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Aug 4, Tuesday:

Chairman: Nathalie Wahl

9:30-10:30: Johannes Ebert

Title: Characteristic classes of manifold bundles: (non-)vanishing theorems

Let f : E \to B be an oriented bundle of smooth closed n-manifolds. For any element c of H^* (BSO(n)), the associated generalzed Morita-Miller-Mumford (MMM-) class is defined to be f_{!} (c (T_v E)) \in H^{*-n} (B). This gives a linear map H^* (BSO(n)) \to H^{*-n} (B) that can be identified with the map induced by the Madsen-Tillmann map B \to MTSO(n). The central result is that the generalized MMM-class which is defined by the component of the Hirzebruch L-class is trivial if n is odd. The proof goes via index theory. On the bundle E \to B there is the odd signature operator; it is a self-adjoint elliptic operator that has an index in K^1(B). Using spectral theory, we show that this index is trivial as an element in K^1(B). The Atiyah-Singer theorem yields a specific class in K^1(MTSO(n)) which pulls back to the index of the signature operator by the Madsen-Tillmann map . A standard characteristic class computation gives the result for the L-class.

Tea Break

11:00-12:00: Alexander Berglund

Title: Koszul models and loop space homology algebras

Abstract: I will introduce the notion of a Koszul model for a dg-algebra, and I will talk about how Koszul models for the singular cochain algebra $C^*(M)$ of a simply connected closed oriented manifold $M$ can be used to compute the Pontrjagin algebra $H_*(\Omega M)$ and the Chas-Sullivan loop homology algebra $\mathbb{H}_*(LM)$.

12:30: Lunch at Jiayuan Hotel

3:30-4:30: Kathryn Hess

Title: How to model homotopy coincidence spaces

Abstract: Given two continuous maps $g,h :X\to Y$, their \emph{coincidence space}, $E_{g,h}$, is the equalizer of $g$ and $h$. For exemple, if $Y=X$, then $E_{g,Id_{X}}=\operatorname{Fix}(g)$, the space of fixed points of $g$. In particular, $E_{Id_{X},Id_{X}}=X$. A homotopy-invariant version of $E_{g,h}$ is the \emph{homotopy coincidence space}, $\mathcal E_{g,h}$, which we can choose to fit into the pullback diagram $$\xymatrix{ \mathcal E_{g,h}\ar[d] \ar [r]&Y^{I}\ar [d]^{(ev_{0},ev_{1})}\\ X\ar [r]^(0.4){(g,h)}&Y\times Y,}$$ where $Y^{I}$ is the space of unbased paths on $Y$ and $ev_{t}$ is the map evaluating a path at $t$. In other words, $$\mathcal E_{g,h}=\big\{(x,\ell)\in X\times Y^{I}\mid \ell (0)=g(x), \ell (1)=h(x)\big\}.$$ If $Y=X$, then $\mathcal E_{g,Id_{X}}=\operatorname{Fix}^{ho}(g)$, the space of homotopy fixed points of the self-map $g$. In particular, $E_{Id_{X},Id_{X}}$ is the space of free loops on $X$. More generally, if $X=U\times V$, where $U$ and $V$ are submanifolds of a manifold $Y$ and $g, h:U\times V\to Y$ are given by projection onto the first and second coordinates, respectively, then $\mathcal E_{g,h}$ is the space of open strings in $Y$ starting in $U$ and ending in $V$. In this talk I will present several models of $\mathcal E_{g,h}$, in the categories of both simplicial sets and chain complexes. The chain models admit rich algebraic structure, which I will explain in detail. I will also outline a number of interesting applications of the various models of $\mathcal E_{g,h}$.

Tea Break

5:00-6:00: Pokman Cheung

Vertex algebras and the Witten genus

This is the first report of an ongoing project aimed at finding a geometric interpretation of the Witten genus and other tmf classes in terms of vertex algebras and related notions. On each complex manifold satisfying certain conditions, Gorbounov, Malikov and Schechtman (GMS) have constructed a family of sheaves of vertex algebras whose sheaf cohomology provides a geometric interpretation of the Witten genus of the complex manifold. This is analogous to Hirzebruch-Riemann-Roch: Td(M)=\chi(O_M). In order to obtain an interpretation of the Witten genus for all string manifolds, I have constructed sheaves of dg vertex algebras that provide fine resolutions of the GMS sheaves; they should play the same role for the Witten genus as the Dolbeault resolution does for the Todd genus.

6:30: Dinner at Jiayuan Hotel

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Aug 5, Wednesday:

Morning: Huang-Ya-Guan Great Wall Excursion, Departure from Jiayuan hotel at 7:00am

Evening: 7:30pm@Shiing-Shen building 214, Film: The beauty of mathematics-about the life of Prof. Shiing-Shen Chern

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Aug 6, Thursday:

Chairman: Soren Galatius

9:30-10:30: Donald Stanley

Title: Rigid Poincare Duality

Tea Break

11:00-12:00: Natalia Dobrinskaya

Title: Loops on Toric Spaces

12:30: Lunch at Jiayuan Hotel

3:30-4:30: Carl-Friedrich Bodigheimer

Title: Symmetric Groups and Moduli Spaces

Tea Break

5:00-6:00: Fei Han

Title: Brief Introduction to SUSY QFTs and Algebraic Topology-Some Progresses of Stolz-Teichner Program

6:30: Banquet at Jiayuan Hotel

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Aug 7, Friday:

Chairman: Fei Han

9:30-10:30: Siye Wu

Title: Electric-magnetic duality and modular invariance

Abstract: Starting from the electric-magnetic duality of the classical Maxwell theory, we explain the role of Langlands dual in non-Abelian gauge theory. We then describe the quantum duality conjecture and its manifestations in various supersymmetric gauge theories. An example of such is the Vafa-Witten theory, whose partition functions are the generating functions of the Euler numbers of the instanton moduli spaces. When the gauge group is simply laced, their transformation under the modular group is consistent with the known topological calculations. It is also related to theta functions and quadratic reciprocity. In this talk, we propose the transformation law of the partition functions under the Hecke group when the gauge group is non-simply laced.

Tea Break

11:00-12:00: Eric Malm

Title: String Topology via the Chains of the Based Loop Space

12:30: Lunch at Jiayuan Hotel

3:30-4:30: Andres Angel

Title: Decompositions of cobordism groups of orbifolds

Abstract: Orbifolds are useful generalizations of manifolds that appear naturally in the study of moduli spaces, particularly those arising in Gromov- Witten theory, where in good cases, these moduli spaces are orbifolds and Gromov-Witten invariants are suitably deTned characteristic numbers. In this talk I will present a framework to study cobordism of orbifolds. As application, I will show calculations of several cobordism groups of orbifolds in terms of usual smooth bordism theory, these decompositions involve information around the singular sets and provide a way to deTne invariants for orbifolds.

Tea Break

5:00-6:00: John Jones

Title: Calculating String Homology

6:30: Dinner at Jiayuan Hotel

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