Urs Schreiber - Academia.edu (original) (raw)

Papers by Urs Schreiber

Quantum field theory allows more general symmetries than groups and Lie algebras. For instance qu... more Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor -- categorical groups, groupoids, Lie algebroids and their higher analogues -- appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties themselves are replaced by suitable categories of sheaves; another in which the gauge groups are categorified to higher groupoids. Together with their gauge groups, also the fiber bundles themselves become categorified, and their gluing (or descent data) is given by nonabelian cocycles, generalizing group cohomology, where infinity-groupoids appear in the role both of the domain and the coefficient...

In the foundational logical framework of homotopy-type theory we discuss a natural formalization ... more In the foundational logical framework of homotopy-type theory we discuss a natural formalization of secondary integral transforms in stable geometric homotopy theory. We observe that this yields a process of non-perturbative cohomological quantization of local pre-quantum field theory; and show that quantum anomaly cancellation amounts to realizing this as the boundary of a field theory that is given by genuine (primary) integral transforms, hence by linear polynomial functors. Recalling that traditional linear logic has semantics in symmetric monoidal categories and serves to formalize quantum mechanics, what we consider is its refinement to linear homotopy-type theory with semantics in stable infinity-categories of bundles of stable homotopy types (generalized cohomology theories) formalizing Lagrangian quantum field theory, following Nuiten and closely related to recent work by Haugseng and Hopkins-Lurie. For the reader interested in technical problems of quantization we provide non-perturbative quantization of Poisson manifolds and of the superstring; and find insight into quantum anomaly cancellation, the holographic principle and motivic structures in quantization. For the reader inclined to the interpretation of quantum mechanics we exhibit quantum superposition and interference as existential quantification in linear homotopy-type theory. For the reader inclined to foundations we provide a refinement of the proposal by Lawvere for a formal foundation of physics, lifted from classical continuum mechanics to local Lagrangian quantum gauge field theory.

Trends in Mathematics, 2015

It is shown that the Pohlmeyer invariants of the classical bosonic string are a proper subset of ... more It is shown that the Pohlmeyer invariants of the classical bosonic string are a proper subset of the classical DDF invariants. This makes the quantization of the Pohlmeyer invariants particularly transparent and allows to generalize them to the superstring.

Advances in Theoretical and Mathematical Physics, 2012

Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged... more Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged under a fiber bundle thereby turns into a string charged under a 2-bundle, or gerbe. This stringification is nothing but categorification. We look at supersymmetric quantum mechanics on loop space and demonstrate how deformations here give rise to superstring background fields and boundary states, and, when generalized, to local nonabelian connections on loop space. In order to get a global description of these connections we introduce and study categorified global holonomy in the form of 2-bundles with 2-holonomy. We show how these relate to nonabelian gerbes and go beyond by obtaining global nonabelian surface holonomy, thus providing a class of action functionals for nonabelian strings. The examination of the differential formulation, which is adapted to the study of nonabelian p-form gauge theories, gives rise to generalized nonabelian Deligne hypercohomology. The (possible) relation of this to strings in Kalb-Ramond backgrounds, to M2/M5-brane systems, to spinning strings and to the derived category description of D-branes is discussed. In particular, there is a 2-group related to the String-group which should be the right structure 2-group for the global description of spinning strings.

Just as gauge theory describes the parallel transport of point particles using connections on bun... more Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge

We present a menagerie of examples for Lie n-algebras, study their morphisms and discuss applicat... more We present a menagerie of examples for Lie n-algebras, study their morphisms and discuss applications to higher order connections, in par- ticular String 2-connections and Chern-Simons 3-connections.

Journal of High Energy Physics, 2004

We show how the Pohlmeyer invariants of the bosonic string are expressible in terms of DDF invari... more We show how the Pohlmeyer invariants of the bosonic string are expressible in terms of DDF invariants. Quantization of the DDF observables in the usual way yields a consistent quantization of the algebra of Pohlmeyer invariants. Furthermore it becomes straightforward to generalize the Pohlmeyer invariants to the superstring as well as to all backgrounds which allow a free field realization of the worldsheet theory.

Journal of High Energy Physics, 2004

Journal of High Energy Physics, 2004

Motivated by the representation of the super Virasoro constraints as generalized Dirac-Kähler con... more Motivated by the representation of the super Virasoro constraints as generalized Dirac-Kähler constraints (d ± d † ) |ψ = 0 on loop space, examples of the most general continuous deformations d → e −W d e W are considered which preserve the superconformal algebra at the level of Poisson brackets. The deformations which induce the massless NS and NS-NS backgrounds are exhibited. Hints for a manifest realization of S-duality in terms of an algebra isomorphism are discussed. It is shown how the first order theory of 'canonical deformations' is reproduced and how the deformation operator W encodes vertex operators and gauge transformations.

It is shown that 1-morphisms between 2-functors from 2-paths to the category BiTor(H) define bibu... more It is shown that 1-morphisms between 2-functors from 2-paths to the category BiTor(H) define bibundles with bibundle connection the way they appear in nonabelian bundle gerbes. An arrow theoretic description of bibundle connections is given using synthetic dierential geometry. In a sequel to this paper this will be used to show that pre-trivializations of 2-functors to BiTor(H) are in bijection with fake flat nonabelian bundle gerbes. (Unfinished and unscrutinized private draft.)

Communications in Mathematical Physics, 2009

Advances in Theoretical and Mathematical Physics, 2012

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinar... more What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth ∞-groups, i.e., by smooth groupal A∞spaces. Namely, we realize differential characteristic classes as morphisms from ∞-groupoids of smooth principal ∞-bundles with connections to ∞-groupoids of higher U (1)-gerbes with connections. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.

We promote geometric prequantization to higher geometry (higher stacks), where a prequantization ... more We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields -- which is the higher Poisson bracket of local observables -- and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry.

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connecti... more Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. We observe that in the context of higher Chern-Weil theory in smooth infinity-groupoids this statement generalizes from Lie algebras to L-infinity-algebras and further to L-infinity-algebroids. It turns out that the symplectic

Homology, Homotopy and Applications, 2014

Electronic Proceedings in Theoretical Computer Science, 2014

International Journal of Geometric Methods in Modern Physics, 2013

Quantum field theory allows more general symmetries than groups and Lie algebras. For instance qu... more Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor -- categorical groups, groupoids, Lie algebroids and their higher analogues -- appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties themselves are replaced by suitable categories of sheaves; another in which the gauge groups are categorified to higher groupoids. Together with their gauge groups, also the fiber bundles themselves become categorified, and their gluing (or descent data) is given by nonabelian cocycles, generalizing group cohomology, where infinity-groupoids appear in the role both of the domain and the coefficient...

In the foundational logical framework of homotopy-type theory we discuss a natural formalization ... more In the foundational logical framework of homotopy-type theory we discuss a natural formalization of secondary integral transforms in stable geometric homotopy theory. We observe that this yields a process of non-perturbative cohomological quantization of local pre-quantum field theory; and show that quantum anomaly cancellation amounts to realizing this as the boundary of a field theory that is given by genuine (primary) integral transforms, hence by linear polynomial functors. Recalling that traditional linear logic has semantics in symmetric monoidal categories and serves to formalize quantum mechanics, what we consider is its refinement to linear homotopy-type theory with semantics in stable infinity-categories of bundles of stable homotopy types (generalized cohomology theories) formalizing Lagrangian quantum field theory, following Nuiten and closely related to recent work by Haugseng and Hopkins-Lurie. For the reader interested in technical problems of quantization we provide non-perturbative quantization of Poisson manifolds and of the superstring; and find insight into quantum anomaly cancellation, the holographic principle and motivic structures in quantization. For the reader inclined to the interpretation of quantum mechanics we exhibit quantum superposition and interference as existential quantification in linear homotopy-type theory. For the reader inclined to foundations we provide a refinement of the proposal by Lawvere for a formal foundation of physics, lifted from classical continuum mechanics to local Lagrangian quantum gauge field theory.

Trends in Mathematics, 2015

It is shown that the Pohlmeyer invariants of the classical bosonic string are a proper subset of ... more It is shown that the Pohlmeyer invariants of the classical bosonic string are a proper subset of the classical DDF invariants. This makes the quantization of the Pohlmeyer invariants particularly transparent and allows to generalize them to the superstring.

Advances in Theoretical and Mathematical Physics, 2012

Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged... more Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged under a fiber bundle thereby turns into a string charged under a 2-bundle, or gerbe. This stringification is nothing but categorification. We look at supersymmetric quantum mechanics on loop space and demonstrate how deformations here give rise to superstring background fields and boundary states, and, when generalized, to local nonabelian connections on loop space. In order to get a global description of these connections we introduce and study categorified global holonomy in the form of 2-bundles with 2-holonomy. We show how these relate to nonabelian gerbes and go beyond by obtaining global nonabelian surface holonomy, thus providing a class of action functionals for nonabelian strings. The examination of the differential formulation, which is adapted to the study of nonabelian p-form gauge theories, gives rise to generalized nonabelian Deligne hypercohomology. The (possible) relation of this to strings in Kalb-Ramond backgrounds, to M2/M5-brane systems, to spinning strings and to the derived category description of D-branes is discussed. In particular, there is a 2-group related to the String-group which should be the right structure 2-group for the global description of spinning strings.

Just as gauge theory describes the parallel transport of point particles using connections on bun... more Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge

We present a menagerie of examples for Lie n-algebras, study their morphisms and discuss applicat... more We present a menagerie of examples for Lie n-algebras, study their morphisms and discuss applications to higher order connections, in par- ticular String 2-connections and Chern-Simons 3-connections.

Journal of High Energy Physics, 2004

We show how the Pohlmeyer invariants of the bosonic string are expressible in terms of DDF invari... more We show how the Pohlmeyer invariants of the bosonic string are expressible in terms of DDF invariants. Quantization of the DDF observables in the usual way yields a consistent quantization of the algebra of Pohlmeyer invariants. Furthermore it becomes straightforward to generalize the Pohlmeyer invariants to the superstring as well as to all backgrounds which allow a free field realization of the worldsheet theory.

Journal of High Energy Physics, 2004

Journal of High Energy Physics, 2004

Motivated by the representation of the super Virasoro constraints as generalized Dirac-Kähler con... more Motivated by the representation of the super Virasoro constraints as generalized Dirac-Kähler constraints (d ± d † ) |ψ = 0 on loop space, examples of the most general continuous deformations d → e −W d e W are considered which preserve the superconformal algebra at the level of Poisson brackets. The deformations which induce the massless NS and NS-NS backgrounds are exhibited. Hints for a manifest realization of S-duality in terms of an algebra isomorphism are discussed. It is shown how the first order theory of 'canonical deformations' is reproduced and how the deformation operator W encodes vertex operators and gauge transformations.

It is shown that 1-morphisms between 2-functors from 2-paths to the category BiTor(H) define bibu... more It is shown that 1-morphisms between 2-functors from 2-paths to the category BiTor(H) define bibundles with bibundle connection the way they appear in nonabelian bundle gerbes. An arrow theoretic description of bibundle connections is given using synthetic dierential geometry. In a sequel to this paper this will be used to show that pre-trivializations of 2-functors to BiTor(H) are in bijection with fake flat nonabelian bundle gerbes. (Unfinished and unscrutinized private draft.)

Communications in Mathematical Physics, 2009

Advances in Theoretical and Mathematical Physics, 2012

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinar... more What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth ∞-groups, i.e., by smooth groupal A∞spaces. Namely, we realize differential characteristic classes as morphisms from ∞-groupoids of smooth principal ∞-bundles with connections to ∞-groupoids of higher U (1)-gerbes with connections. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.

We promote geometric prequantization to higher geometry (higher stacks), where a prequantization ... more We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields -- which is the higher Poisson bracket of local observables -- and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry.

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connecti... more Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. We observe that in the context of higher Chern-Weil theory in smooth infinity-groupoids this statement generalizes from Lie algebras to L-infinity-algebras and further to L-infinity-algebroids. It turns out that the symplectic

Homology, Homotopy and Applications, 2014

Electronic Proceedings in Theoretical Computer Science, 2014

International Journal of Geometric Methods in Modern Physics, 2013