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Daniel Berwick-Evans

Publications and source records attributed to Daniel Berwick-Evans.

At least 19 recordsLinked to original sources
arXiv2026

Discrete Vector Bundles with Connection

We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete exterior covariant derivative, a forward-difference operator defined on bundle-valued cochains. Many standard objects in differential geometry (e.g., curvature, connection 1-forms, gauge transformations) can be understood via the discrete covariant derivative operator, with their defining formulas identical to the smooth setting. These discrete objects satisfy all of the expected algebraic identities, such as naturality with respect to simplicial maps, and a Bianchi identity for discrete curvature. We also show that flat discrete connections determine a cochain complex that computes twisted de Rham cohomology in a local coefficient system determined by the discrete vector bundle, with twisted Poincare duality (of densities) being one application. Finally, a coarsening operation applied to bundle-valued cochains provides a direct and concrete comparison with the recent framework for discrete bundles of Christiansen and Hu.

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arXiv2025

Flat principal 2-group bundles and flat string structures

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-group bundles can be described in terms of ordinary $G$-bundles together with a trivialization of a certain 2-gerbe. This specializes to a characterization of flat string structures on vector bundles over differentiable stacks.

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arXiv2025

The Freed--Quinn line bundle from higher geometry

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. In this paper, we provide an alternate construction using tools from higher geometry: the pair $(G,α)$ determines a 2-group group, and the Freed--Quinn line arises as a categorical truncation of the bicategory of 2-group bundles.

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arXiv2025

Classifying spaces of infinity-sheaves

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 concordance. As an application, we offer an answer to the question: what does the classifying space of a Segal space classify?

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arXiv2024

Power operations preserve Thom classes in twisted equivariant Real K-theory

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 Atiyah-Bott-Shapiro orientation commutes with power operations.

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arXiv2023

Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus

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 cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson's cochain product defined using Whitney and de Rham maps.

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arXiv2023

Smooth one-dimensional topological field theories are vector bundles with connection

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 rather than gluing laws. We make this idea precise through a smooth version of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential-geometric arguments.

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arXiv2023

The families Clifford index and differential KO-theory

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 bundles of spin manifolds, affording a differential refinement of Atiyah and Singer's families index.

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arXiv2023

How do field theories detect the torsion in topological modular forms?

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 torsion invariants of field theories valued in $π_*{\rm TMF}$. The framework leads to a version of the elliptic Euler class described in terms of field-theoretic data, as well as a partial construction of the supersymmetric sigma model with target a string manifold.

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arXiv2023

The families analytic index for $1|1$-dimensional Euclidean field theories

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 family of partially-defined $1|1$-Euclidean field theories, yielding a cocycle refinement of the families analytic index. The methods are chosen with the goal of generalizing to $2|1$-dimensional field theories, where analogous structures are expected to provide an analytic index valued in topological modular forms.

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arXiv2021

Chern characters for supersymmetric field theories

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 identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parameterized generalization of Fei Han's realization of the Chern character in K-theory as dimensional reduction for $1|1$-dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of $2|1$-dimensional tori and the derived geometry of complex analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of $\mathcal{N}=(0,1)$ supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.

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arXiv2020

A de Rham model for complex analytic equivariant elliptic cohomology

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins' "theorem of the cube" construction of the ${\rm MString}$-orientation of elliptic cohomology.

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arXiv2020

Equivariant elliptic cohomology, gauged sigma models, and discrete torsion

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 supersymmetric mechanics determine cocycles for equivariant K-theory with complex coefficients. The path integral for gauge theory with a finite group constructs wrong-way maps associated to group homomorphisms. When applied to an inclusion of groups, we obtain the induced character formula of Hopkins, Kuhn, and Ravenel. For the homomorphism $G\to *$ we obtain Vafa's formula for gauging with discrete torsion. The image of equivariant Euler classes under gauging constructs modular form-valued invariants of representations that depend on a choice of string structure. We illustrate nontrivial dependence on the string structure for a 16-dimensional representation of the Klein 4-group.

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arXiv2020

Supersymmetric field theories and the elliptic index theorem with complex coefficients

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushfoward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $σ$-model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and 2-dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai--Quillen Thom form in complexified ${\rm KO}$-theory and a cocycle representative of the $\hat{A}$-class for a family of oriented manifolds.

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arXiv2020

A model for complex analytic equivariant elliptic cohomology from quantum field theory

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 fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-gauge fields carries a multiplication compatible with the complex analytic group structure on the universal (dual) elliptic curve, with the Euler class providing a choice of coordinate. This provides a physical manifestation of the elliptic group law central to the homotopy-theoretic construction of elliptic cohomology.

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arXiv2020

Power operations in the Stolz--Teichner program

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 theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.

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arXiv2020

Lie 2-algebras of vector fields

We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise formulation of Morita invariance of the construction, we verify that the Lie 2-algebra structure defined in this way is well-defined on the underlying stack.

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