Papers by Karl-Georg Schlesinger
We review the approach to the geometric Langlands program for algebraic curves via S-duality of a... more We review the approach to the geometric Langlands program for algebraic curves via S-duality of an N=4 supersymmetric four dimensional gauge theory, initiated by Kapustin and Witten in 2006. We sketch some of the central further developments. Placing this four dimensional gauge theory into a six dimensional framework, as advocated by Witten, holds the promise to lead to a formulation which makes geometric Langlands duality a manifest symmetry (like coavariance in differential geometry). Furthermore, it leads to an approach toward geometric Langlands duality for algebraic surfaces, reproducing and extending the recent results of Braverman and Finkelberg.
arXiv (Cornell University), Mar 18, 2014
Irreversible phenomena-such as the production of entropy and heat-arise from fundamental reversib... more Irreversible phenomena-such as the production of entropy and heat-arise from fundamental reversible dynamics because the forward dynamics is too complex, in the sense that it becomes impossible to provide the necessary information to keep track of the dynamics. On a heuristic level, this is well captured by coarse graining. We suggest that on a fundamental level the impossibility to provide the necessary information might be related to the incompleteness results of Gödel. This would hold interesting implications for both, mathematics and physics.
arXiv (Cornell University), Jun 26, 2007
arXiv (Cornell University), Dec 8, 2014
We investigate what it means to apply the solution, proposed to the firewall paradox of [AMPS] by... more We investigate what it means to apply the solution, proposed to the firewall paradox of [AMPS] by [HH], to the famous quantum paradoxes of Schrödinger's Cat and Wigner's Friend if ones views these as posing a thermodynamic decoding problem (as does Hawking radiation in the firewall paradox). The implications might point to a relevance of the firewall paradox for the axiomatic and set theoretic foundations underlying mathematics. We reconsider in this context the results of [Ben1976a] and [Ben1976b] on the foundational challenges posed by the randomness postulate of quantum theory. A central point in our discussion is that one can mathematically not naturally distinguish between computational complexity (as central in [HH] and [Sus]) and proof theoretic complexity (since they represent the same concept on a Turing machine), with the latter being related to a finite bound on Kolmogorov entropy (due to Chaitin incompleteness).
arXiv (Cornell University), Nov 22, 2011
We argue that a special step in the chain of dualities used in [Tan 2008] implicitly suggests to ... more We argue that a special step in the chain of dualities used in [Tan 2008] implicitly suggests to view Langlands duality as being fundamentally rooted in an eight-dimensional theory on the F-theory 7brane. We give further arguments why such an eight-dimensional perspective might be of interest.
arXiv (Cornell University), May 4, 2006
Present day physics rests on two main pillars: General relativity and quantum field theory. We di... more Present day physics rests on two main pillars: General relativity and quantum field theory. We discuss the deep and at the same time problematic interplay between these two theories. Based on an argument by Doplicher, Fredenhagen and Roberts, we propose a possible universality property for noncommutative quantum field theory in the sense that any theory of quantum gravity should involve quantum field theories on noncommutative space-times as a special limit. We propose a mathematical framework to investigate such a universality property and start the discussion of its mathematical properties. The question of its connection to string theory could be a starting point for a new perspective on string theory.
arXiv (Cornell University), May 4, 2006
The fivebrane in M-theory comes equipped with a higher order gauge field which should have a form... more The fivebrane in M-theory comes equipped with a higher order gauge field which should have a formulation in terms of a 2-gerbe on the fivebrane. One can pose the question if the BV-quantization scheme for such a higher order gauge theory should differ from the usual BV-algebra structure. We give an algebraic argument that this should, indeed, be the case and a fourth order equation should appear as Master equation, in this case. We also discover a second order term in this equation which seems to indicate that deformation theory (i.e. solving the Master equation) in this case involves a nonlinear algebraic theory which goes beyond complexes and cohomology.
arXiv (Cornell University), Aug 29, 2000
We start from a quantum computational principle suggested for string theory in a previous paper a... more We start from a quantum computational principle suggested for string theory in a previous paper and discuss how it might lead to a dynamical principle implying the correct classical gravitational limit. Besides this, we briefly look at some structural properties of moduli space.
arXiv (Cornell University), Feb 4, 2004
We find a self-dual noncommutative and noncocommutative Hopf algebra H F acting as a universal sy... more We find a self-dual noncommutative and noncocommutative Hopf algebra H F acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to the Grothendieck-Teichmüller group GT as introduced by Drinfeld as a universal symmetry of quasitriangular quasi-Hopf algebras. We discuss the relationship to a similar self-dual noncommutative and noncocommutative Hopf algebra H GT , previously found as the universal symmetry of trialgebras and three dimensional extended topological quantum field theories. As an application of our result, we get a transitive action of a sub-Hopf algebra H D of H GT on the relative period matrices of holomorphic N = 1 special geometries, i.e. H D appears as a kind of duality Hopf algebra for holomorphic N = 1 special geometries.
arXiv (Cornell University), Mar 20, 2002
In this paper, we arrive from different starting points at the conclusion that the symmetry given... more In this paper, we arrive from different starting points at the conclusion that the symmetry given by an action of the Grothendieck-Teichmüller group GT on the so called extended moduli space of string theory can not be physical-in the sense that it does not survive the inclusion of general nonperturbative vacua given by boundary condition on the level of two dimensional conformal field theory-but has to be extended to a quantum symmetry given by a self-dual, noncommutative, and noncocommutative Hopf algebra H GT. First, we show that a class of two dimensional boundary conformal field theories always uniquely defines a trialgebra and find H GT as the universal symmetry of such trialgebras (in analogy to the definition of GT as the universal symmetry of quasi-triangular quasi-Hopf algebras). Second, we argue in a more heuristic approach that the H GT symmetry can also be found in a more geometric picture using the language of gerbes.
arXiv (Cornell University), Aug 29, 2000
String theory is accused by some of its critics to be a purely abstract mathematical discipline, ... more String theory is accused by some of its critics to be a purely abstract mathematical discipline, having lost the contact to the simple yet deeply rooted questions which physics provided until the beginning of this century. We argue that, in contrary, there are indications that string theory might be linked to a fundamental principle of a quantum computational character. In addition, the nature of this principle can possibly provide some new insight into the question of universality of string theory (string theory as the "theory of everything").
arXiv (Cornell University), Nov 2, 2001
We introduce general q-deformed multiple polylogarithms which even in the dilogarithm case differ... more We introduce general q-deformed multiple polylogarithms which even in the dilogarithm case differ slightly from the deformation usually discussed in the literature. The merit of the deformation as suggested, here, is that q-deformed multiple polylogarithms define an algebra, then (as in the undeformed case). For the special case of qdeformed multiple ζ-values, we show that there exists even a noncommutative and noncocommutative Hopf algebra structure which is a deformation of the commutative Hopf algebra structure which one has in the classical case. Finally, we discuss the possible correspondence between q-deformed multiple polylogarithms and a noncommutative and noncocommutative self-dual Hopf algebra recently introduced by the author as a quantum analog of the Grothendieck-Teichmüller group.
Journal of Mathematical Physics, Oct 1, 2000
We construct a deformation of the function algebra on the quantum group SL q (2) into a trialgebr... more We construct a deformation of the function algebra on the quantum group SL q (2) into a trialgebra in the sense of Crane and Frenkel. We show that this naturally acts on the trialgebraic deformation of the Manin plane, previously introduced by the authors. Alternatively, one can view it as acting on the trialgebraic deformation of the fermionic Manin plane. We prove that the trialgebraic deformation of SL q (2) defines a 2 − C *-category, a structure as needed for the superselection structure of massive two dimensional quantum field theories. Besides this, we investigate another approach to trialgebra deformations of a bialgebra as a deformation of a Fock space construction over the bialgebra.
arXiv (Cornell University), Mar 21, 2001
We suggest that trialgebraic symmetries might be a sensible starting point for a notion of integr... more We suggest that trialgebraic symmetries might be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory
European Physical Journal C, Mar 13, 2006
It was suggested in [Kon 1999] that the Grothendieck-Teichmüller group GT should act on the Duflo... more It was suggested in [Kon 1999] that the Grothendieck-Teichmüller group GT should act on the Duflo isomorphism of su(2) but the corresponding realization of GT turned out to be trivial. We show that a solvable quotient of the motivic Galois group-which is supposed to agree with GT-is closely related to the quantum coadjoint action on U q (sl 2) for q a root of unity, i.e. in the quantum group case one has a nontrivial realization of a quotient of the motivic Galois group. From a discussion of the algebraic properties of this realization we conclude that in more general cases than U q (sl 2) it should be related to a quantum version of the motivic Galois group. Finally, we discuss the relation of our construction to quantum field and string theory and explain what we believe to be the "physical reason" behind this relation between the motiviv Galois group and the quantum coadjoint action. This might be a starting point for the generalization of our construction to more invloved examples.
Physical Review C, Sep 27, 2022
Duals for Nonabelian Lattice Gauge Theories
Springer eBooks, 2000
The usual duality between an abelian group and the group of its characters gives rise to the well... more The usual duality between an abelian group and the group of its characters gives rise to the well known notion of the dual of an abelian lattice gauge theory which is of considerable importance. The development of a concept of a dual for nonabelian lattice gauge theories has long been prevented by the fact that the Doplicher-Roberts theorem shows that the dual of a nonabelian group is no longer a group but a certain monoidal category. After very briefly reviewing the needed concepts from lattice gauge theory (in order to make the talk selfcontained), we present a categorical construction of such duals, making use of a functorial reformulation of the notion of a lattice gauge theory (due to J. Baez). We show that the commutative tetrahedron of 2-category theory immediately leads to a gauge invariant action for the dual theories. As an example, we discuss the case of the gauge group SU(2) where we find that classical connections are labeled by spin networks, i.e. the theory as an “already quantized” form.
Physical Review C, May 10, 2022
arXiv (Cornell University), Dec 15, 2004
We construct a universal envelope for any Poisson-and Gerstenhaber algebra. While the deformation... more We construct a universal envelope for any Poisson-and Gerstenhaber algebra. While the deformation theory of Poisson algebras seems to be partially trivial, results from string and M-theory suggest a rich deformation theory of Gerstenhaber algebras. We apply our construction in this case to well known questions on the topological closed string BRST-complex. Finally, we find a simliar algebraic structure, as for the universal envelope, in the SU (2)-WZW model.
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Papers by Karl-Georg Schlesinger