Daniel Berwick-Evans
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View article: The Freed--Quinn line bundle from higher geometry
The Freed--Quinn line bundle from higher geometry Open
For a finite group $G$, and level $α\in Z^3(BG;{\rm U}(1))$, Freed and Quinn construct a line bundle over the moduli space of $G$-bundles on surfaces. Global sections determine the values of Chern--Simons theory at level $α$ on surfaces. I…
View article: Classifying spaces of infinity-sheaves
Classifying spaces of infinity-sheaves Open
We prove that the set of concordance classes of sections of an infinity-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss. This is reminiscent of an h-principle in which the role of isotopy is played by concorda…
View article: Elliptic cohomology and quantum field theory
Elliptic cohomology and quantum field theory Open
This survey provides an introduction to the Stolz-Teichner program on elliptic cohomology and quantum field theory.
View article: Power operations preserve Thom classes in twisted equivariant Real K-theory
Power operations preserve Thom classes in twisted equivariant Real K-theory Open
We construct power operations for twisted KR-theory of topological stacks. Standard algebraic properties of Clifford algebras imply that these power operations preserve universal Thom classes. As a consequence, we show that the twisted Ati…
View article: Smooth one-dimensional topological field theories are vector bundles with connection
Smooth one-dimensional topological field theories are vector bundles with connection Open
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rat…
View article: Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus
Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus Open
In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial…
View article: Chern characters for supersymmetric field theories
Chern characters for supersymmetric field theories Open
We construct a map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identify…
View article: The families analytic index for $1|1$-dimensional Euclidean field theories
The families analytic index for $1|1$-dimensional Euclidean field theories Open
We construct a ${\rm KO}$-valued families index for a class of $1|1$-dimensional Euclidean field theories. This realizes a conjectured cocycle map in the Stolz--Teichner program. We further show that a bundle of spin manifolds leads to a f…
View article: The families Clifford index and differential KO-theory
The families Clifford index and differential KO-theory Open
Extending ideas of Atiyah--Bott--Shapiro and Quillen, we construct a model for differential $\rm KO$-theory whose cocycles are families of Clifford modules with superconnection. The model is built to accommodate an analytic pushforward for…
View article: How do field theories detect the torsion in topological modular forms?
How do field theories detect the torsion in topological modular forms? Open
We construct deformation invariants of $2|1$-dimensional Euclidean field theories valued in a cohomology theory approximating topological modular forms. This implies several results anticipated by Stolz and Teichner and gives the first tor…
View article: Power operations in the Stolz–Teichnerprogram
Power operations in the Stolz–Teichnerprogram Open
The Stolz--Teichner program proposes a deep connection between geometric\nfield theories and certain cohomology theories. In this paper, we extend this\nconnection by developing a theory of geometric power operations for geometric\nfield t…
View article: Flat principal 2-group bundles and flat string structures
Flat principal 2-group bundles and flat string structures Open
For a weak 2-group, we construct a bicategory of flat 2-group bundles over differentiable stacks as a localization of a functor bicategory. This description is amenable to explicit geometric constructions. For example, we show that flat 2-…
View article: Supersymmetric field theories and the elliptic index theorem with complex coefficients
Supersymmetric field theories and the elliptic index theorem with complex coefficients Open
We present a cocycle model for elliptic cohomology with complex coefficients\nin which methods from 2-dimensional quantum field theory can be used to\nrigorously construct cocycles. For example, quantizing a theory of vector\nbundle-valued…
View article: Equivariant elliptic cohomology, gauged sigma models, and discrete torsion
Equivariant elliptic cohomology, gauged sigma models, and discrete torsion Open
For $G$ a finite group, we show that functions on fields for the 2-dimensional supersymmetric sigma model with background $G$-symmetry determine cocycles for complex analytic $G$-equivariant elliptic cohomology. Similar structures in super…
View article: Discrete Vector Bundles with Connection and the Bianchi Identity
Discrete Vector Bundles with Connection and the Bianchi Identity Open
We develop a combinatorial theory of vector bundles with connection that is natural with respect to appropriate mappings of the base space. The base space is a simplicial complex, the main objects defined are discrete vector bundle valued …
View article: Lie 2-algebras of vector fields
Lie 2-algebras of vector fields Open
We show that the category of vector fields on a geometric stack has the\nstructure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The\nconstruction uses a Lie groupoid that presents the geometric stack. We show\nthat the cate…
View article: Chern characters for supersymmetric field theories
Chern characters for supersymmetric field theories Open
We construct a map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identify…
View article: Chern characters and supersymmetric field theories
Chern characters and supersymmetric field theories Open
We construct a map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identify…
View article: Power operations in the Stolz--Teichner program
Power operations in the Stolz--Teichner program Open
The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theo…
View article: Classifying spaces of infinity-sheaves
Classifying spaces of infinity-sheaves Open
We prove that the set of concordance classes of sections of an infinity-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss. This is reminiscent of an h-principle in which the role of isotopy is played by concorda…
View article: Supersymmetric localization, modularity and the Witten genus
Supersymmetric localization, modularity and the Witten genus Open
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the…
View article: A model for complex analytic equivariant elliptic cohomology from quantum field theory
A model for complex analytic equivariant elliptic cohomology from quantum field theory Open
We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fiel…
View article: The equivariant Chern character as super holonomy on loop stacks
The equivariant Chern character as super holonomy on loop stacks Open
We study super parallel transport around super loops in a quotient stack, and show that this geometry constructs a global version of the equivariant Chern character.
View article: Twisted equivariant differential K-theory from gauged supersymmetric mechanics
Twisted equivariant differential K-theory from gauged supersymmetric mechanics Open
We use the geometry of the space of fields for gauged supersymmetric mechanics to construct the twisted differential equivariant K-theory of a manifold with an action by a finite group.
View article: Topological $q$-expansion and the supersymmetric sigma model
Topological $q$-expansion and the supersymmetric sigma model Open
The Hamiltonian and Lagrangian formalisms offer two perspectives on quantum field theory. This paper sets up a framework to compare these approaches for the supersymmetric sigma model. The goal is to use techniques from physics to construc…
View article: An effective field theory model for differential elliptic cohomology at the Tate curve
An effective field theory model for differential elliptic cohomology at the Tate curve Open
We construct a model for differential elliptic cohomology at the Tate curve whose cocycles are families of 2-dimensional effective supersymmetric field theories. A geometrically-motivated modularity condition requires partition functions t…
View article: Smooth one-dimensional topological field theories are vector bundles with connection
Smooth one-dimensional topological field theories are vector bundles with connection Open
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rat…