jim stasheff - Academia.edu (original) (raw)

Papers by jim stasheff

Abstract. We study the cohomological physics of fivebranes in type II and heterotic string theory... more Abstract. We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the “magnetic dual ” version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly

Lecture Notes in Mathematics, 1970

arXiv: Mathematical Physics, 2020

Leibniz algebras have been increasingly used in gauging procedures in supergravity. Their relatio... more Leibniz algebras have been increasingly used in gauging procedures in supergravity. Their relationship with LinftyL_\inftyLinfty-algebras and tensor hierarchies have been explored in the physics literature. This paper is devoted to showing that a Leibniz algebra VVV gives rise to a non-positively graded LinftyL_\inftyLinfty-algebra. We call such an LinftyL_\inftyLinfty-algebra an '$L_\infty$-extension of the Leibniz algebra VVV' and show that this construction is functorial. We will also use the opportunity of building this functor to provide a more clear and straightforward construction of the differential graded Lie algebra structure equipping the tensor hierarchy, previously presented in arXiv:1708.07068. We do not claim that the LinftyL_\inftyLinfty-algebra thus obtained from a Leibniz algebra should be the 'correct' one, that physicists should use in their models, though many of them do. However, we stress that a canonical and functorial construction exists, hence justifying that there is room for wel...


arXiv: Algebraic Topology, 2016

This is my attempt to rediscover how Dennis realized that he had discovered/invented LinftyL_\inftyLinfty-a... more This is my attempt to rediscover how Dennis realized that he had discovered/invented LinftyL_\inftyLinfty-algebras a.k.a. strongly homotopy Lie algebras before my colleagues and I defined them.

arXiv: Quantum Algebra, 2018

Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days ... more Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of AinftyA_\inftyAinfty-structures and later of LinftyL_\inftyLinfty -structures and their progeny, I hope to highlight some old results which seem not to have garnered the attention they deserve as well as some tantalizing new connections.

Bulletin of the American Mathematical Society, 1969

Pure and Applied Mathematics Quarterly, 2013

This is an expanded and updated version of a talk given at the Conference on Topics in Geometry a... more This is an expanded and updated version of a talk given at the Conference on Topics in Geometry and Physics at the University of Southern California, November 6, 1992. It is a survey talk, aimed at mathematicians AND physicists, which attempts to bring together the topics in the title without assuming much background in any of them. Closed string field theory leads to a (strong homotopy) generalization of Lie algebra, which is strongly related to the way the moduli spaces CalM0,N+1\Cal M_{0,N+1}CalM0,N+1 fit together as an ``operad''. The latter in turn plays an important role in the understanding of vertex operator algebras.

Communications in Mathematical Physics, 1984

Proceedings of a Symposium in Honor of John C. Moore. (AM-113)

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞al... more We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1)→U(H)→PU(H) to higher categorical central extensions, like the String-extension BU(1)→String(G)→G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe t...

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, ... more Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when ‘gauge parameters’ act in a field dependent way. Such symmetries appear in several field theories, most notably in a ‘Poisson induced’ class due to Schaller and Strobl [SS94] and to Ikeda[Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich’s deformation quantization [Kon97]. Consideration of ‘particles of spin > 2 led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to study ‘field dependent parameters’ in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-lie algebra (L∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries.

Abstract. Traditionally symmetries of field theories are encoded via Lie group actions, or more g... more Abstract. Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when ‘gauge parameters ’ act in a field dependent way. Such symmetries appear in several field theories, most notably in a ‘Poisson induced ’ class due to Schaller and Strobl [SS94] and to Ikeda[Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich’s deformation quanti-zation [Kon97]. Consideration of ‘particles of spin> 2 led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to study ‘field dependent parameters ’ in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-lie algebra (L∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more gen-eral class of theories with field dependent symmetries. 1.

Cohomological physics is a phrase I introduced sometime ago in the context of anomalies in gauge ... more Cohomological physics is a phrase I introduced sometime ago in the context of anomalies in gauge theory, but it all began with Gauss in 1833, if not sooner (cf. Kirchoff's laws, as Nash's "Topology AND Physics - an historical survey" [?] reminded me). The cohomology referred to in Gauss was that of differential forms, div, grad, curl and especially Stokes Theorem (the de Rham complex).

For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H... more For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that gαβgβγ is homotopic to gαγ as a map of Uα ∩ Uβ ∩ Uγ into H. Moreover, on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis [17]. Here we present it using language that has been developed in the interim. We also show how this points a direction ‘on beyond gerbes’.

Abstract. We study the cohomological physics of fivebranes in type II and heterotic string theory... more Abstract. We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the “magnetic dual ” version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly

Lecture Notes in Mathematics, 1970

arXiv: Mathematical Physics, 2020

Leibniz algebras have been increasingly used in gauging procedures in supergravity. Their relatio... more Leibniz algebras have been increasingly used in gauging procedures in supergravity. Their relationship with LinftyL_\inftyLinfty-algebras and tensor hierarchies have been explored in the physics literature. This paper is devoted to showing that a Leibniz algebra VVV gives rise to a non-positively graded LinftyL_\inftyLinfty-algebra. We call such an LinftyL_\inftyLinfty-algebra an '$L_\infty$-extension of the Leibniz algebra VVV' and show that this construction is functorial. We will also use the opportunity of building this functor to provide a more clear and straightforward construction of the differential graded Lie algebra structure equipping the tensor hierarchy, previously presented in arXiv:1708.07068. We do not claim that the LinftyL_\inftyLinfty-algebra thus obtained from a Leibniz algebra should be the 'correct' one, that physicists should use in their models, though many of them do. However, we stress that a canonical and functorial construction exists, hence justifying that there is room for wel...


arXiv: Algebraic Topology, 2016

This is my attempt to rediscover how Dennis realized that he had discovered/invented LinftyL_\inftyLinfty-a... more This is my attempt to rediscover how Dennis realized that he had discovered/invented LinftyL_\inftyLinfty-algebras a.k.a. strongly homotopy Lie algebras before my colleagues and I defined them.

arXiv: Quantum Algebra, 2018

Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days ... more Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of AinftyA_\inftyAinfty-structures and later of LinftyL_\inftyLinfty -structures and their progeny, I hope to highlight some old results which seem not to have garnered the attention they deserve as well as some tantalizing new connections.

Bulletin of the American Mathematical Society, 1969

Pure and Applied Mathematics Quarterly, 2013

This is an expanded and updated version of a talk given at the Conference on Topics in Geometry a... more This is an expanded and updated version of a talk given at the Conference on Topics in Geometry and Physics at the University of Southern California, November 6, 1992. It is a survey talk, aimed at mathematicians AND physicists, which attempts to bring together the topics in the title without assuming much background in any of them. Closed string field theory leads to a (strong homotopy) generalization of Lie algebra, which is strongly related to the way the moduli spaces CalM0,N+1\Cal M_{0,N+1}CalM0,N+1 fit together as an ``operad''. The latter in turn plays an important role in the understanding of vertex operator algebras.

Communications in Mathematical Physics, 1984

Proceedings of a Symposium in Honor of John C. Moore. (AM-113)

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞al... more We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1)→U(H)→PU(H) to higher categorical central extensions, like the String-extension BU(1)→String(G)→G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe t...

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, ... more Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when ‘gauge parameters’ act in a field dependent way. Such symmetries appear in several field theories, most notably in a ‘Poisson induced’ class due to Schaller and Strobl [SS94] and to Ikeda[Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich’s deformation quantization [Kon97]. Consideration of ‘particles of spin > 2 led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to study ‘field dependent parameters’ in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-lie algebra (L∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries.

Abstract. Traditionally symmetries of field theories are encoded via Lie group actions, or more g... more Abstract. Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when ‘gauge parameters ’ act in a field dependent way. Such symmetries appear in several field theories, most notably in a ‘Poisson induced ’ class due to Schaller and Strobl [SS94] and to Ikeda[Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich’s deformation quanti-zation [Kon97]. Consideration of ‘particles of spin> 2 led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to study ‘field dependent parameters ’ in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-lie algebra (L∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more gen-eral class of theories with field dependent symmetries. 1.

Cohomological physics is a phrase I introduced sometime ago in the context of anomalies in gauge ... more Cohomological physics is a phrase I introduced sometime ago in the context of anomalies in gauge theory, but it all began with Gauss in 1833, if not sooner (cf. Kirchoff's laws, as Nash's "Topology AND Physics - an historical survey" [?] reminded me). The cohomology referred to in Gauss was that of differential forms, div, grad, curl and especially Stokes Theorem (the de Rham complex).

For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H... more For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that gαβgβγ is homotopic to gαγ as a map of Uα ∩ Uβ ∩ Uγ into H. Moreover, on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis [17]. Here we present it using language that has been developed in the interim. We also show how this points a direction ‘on beyond gerbes’.