Papers by: S.H. Abed, O. Amici, A. Asada, V.V. Astrhancev, S. Bacso, J.K. Beem, A. Bejancu, K. Belteky, I.M. Benn, L. Biesk, T.Q. Binh, N. Blazic, N. Bokan, K. Buchner, B. Casciaro, A.V. Chakmazjan, I. Comic, A. Crumeyrolle, I. Dimitric, C.T.J. Dodson, P. Dragila, S. Dragomir, J.J. Duistermaat, P.E. Ehrlich, J. Eichhorn, M. Falcitelli, M. Ferraris, S. Formella, M. Francaviglia, S.I. Goldberg, A. Gray, T. Hangan, M. Husty, G. Ivan, A. Jakubowicz, S. Janeczko, Z. Jankovic, J. Klein, F.C. Klepp, I. Kolar, Z. Kovacs, L. Kozma, B. Lukacs, V.S. Malakhovsky, K.B. Marathe, S. Markvorsen, I. Mihai, J. Mikesh, R. Miron, E. Molnar, A.M. Naveira, J. Nikic, J. Novotny, V. Oliker, T. Otsuki, A.M. Pastore, Z. Perjes, J.F. Pommaret, M. Puta, H. Reckziegel, A.H. Rocamora, O. Roschel, H. Sachs, A. Sebestyen, U. Simon, H. Singh, G.A.J. Sparling, P. Stavre, S. Steiner, R. Sulanke, J. Szenthe, J. Szilasi, L. Tamassy, E. Teufel, F. Tricerri, R.W. Tucker, C. Udriste, I. Vaisman, L. Vanhecke, L. Verstraelen, V.A. Vujicic, P.G. Walczak, H. Yasuda, A. Zajtz, V. Zoller

Topics in Differential Geometry

We address the construction of smooth bundles of fermionic Fock spaces, a problem that appears frequently in fermionic gauge theories. Our main motivation is the spinor bundle on the free loop space of a string manifold, a structure anticipated by Killingback, with a construction outlined by Stolz-Teichner. We develop a general framework for constructing smooth Fock bundles, and obtain as an application a complete and well-founded construction of spinor bundles on loop space.

Smooth Fock bundles, and spinor bundles on loop space

Given a manifold with a string structure, we construct a spinor bundle on its loop space. Our construction is in analogy with the usual construction of a spinor bundle on a spin manifold, but necessarily makes use of tools from infinite dimensional geometry. We equip this spinor bundle on loop space with an action of a bundle of Clifford algebras. Given two smooth loops in our string manifold that share a segment, we can construct a third loop by deleting this segment. If this third loop is smooth, then we say that the original pair of loops is a pair of compatible loops. It is well-known that this operation of fusing compatible loops is important if one wants to understand the geometry of a manifold through its loop space. In this work, we explain in detail how the spinor bundle on loop space behaves with respect to fusion of compatible loops. To wit, we construct a family of fusion isomorphisms indexed by pairs of compatible loops in our string manifold. Each of these fusion isomorphisms is an isomorphism from the relative tensor product of the fibres of the spinor bundle over its index pair of compatible loops to the fibre over the loop that is the result of fusing the index pair. The construction of a spinor bundle on loop space equipped with a fusion product as above was proposed by Stolz and Teichner with the goal of studying the Dirac operator on loop space". Our construction combines facets of the theory of bimodules for von Neumann algebras, infinite dimensional manifolds, and Lie groups and their representations. We moreover place our spinor bundle on loop space in the context of bundle gerbes and bundle gerbe modules.

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The Spinor Bundle on Loop Space and its Fusion product

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of 2-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle covering the fusion of loops, i.e., the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish a novel relation between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as a part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

Connes fusion of spinors on loop space

We construct a Cliﬀord algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the ﬁrst (fractional) Pontrjagin class of the manifold.

The Clifford algebra bundle on loop space

A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth free loop space of the manifold. 
First, a Clifford algebra bundle over the loop space is constructed from the vector bundle. A polarization class bundle is constructed, choosing continuously over each point of the loop space a polarization class of Lagrangian subspaces of the complexification of the real vector space from which the Clifford algebra is made. Being unable to choose a Lagrangian subspace continuously from the polarization class over each point, the thesis constructs a bundle gerbe over the loop space of the base manifold to encode over each loop all such subspaces, along with the isomorphisms between the Fock spaces made from them, resulting from their being in the same polarization class. 
The vanishing of the Dixmier-Douady class of the bundle gerbe implies that the latter has a trivialization, from which is constructed a spinor bundle.

A bundle gerbe construction of a spinor bundle from the smooth free loop of a vector bundle

As companies adopt increasingly experimentation-driven cultures, it is crucial to develop methods for understanding any potential unintended consequences of those experiments. We might have speciﬁc questions about those consequences (did a change increase or decrease gender representation equality among content creators?); we might also wonder whether if we have not yet considered the right question (that is, we don’t know what we don’t know). Hence we address the problem of unintended consequences in experimentation from two perspectives: namely, pre-speciﬁed vs. data-driven selection, of dimensions of interest. For a speciﬁed dimen-sion, we introduce a statistic to measure deviation from equal representation (DER statistic), give its asymptotic distribution, and evaluate ﬁnite-sample performance. We explain how to use this statistic to search across large-scale experimentation systems to alert us to any extreme unintended consequences on group representation. We complement this methodology by discussing a search for heterogeneous treatment eﬀects along a set of dimensions with causal trees, modiﬁed slightly for practicalities in our ecosystem, and used here as a way to dive deeper into experiments ﬂagged by the DER statistic alerts. We introduce a method for simulating data that closely mimics observed data at LinkedIn, and evaluate the performance of DER statistics in simulations. Last, we give a case study from LinkedIn, and show how these methodologies empowered us to discover surprising and important insights about group representation. Code for replication is available in an appendix.

Stuart Ambler

Papers

A bundle gerbe construction of a spinor bundle from the smooth free loop of a vector bundle

Representation-Aware Experimentation: Group Inequality Analysis for A/B Testing and Alerting