Nezhla Aghaei (South Denmark U.): Combinatorial Quantisation of super Chern Simons theory
Chern-Simons theories with gauge supergroups appear naturally in string theory and they possess interesting applications in mathematics, e.g.
for the construction of knot and link invariants. In this talk we explain the combinatorial quantisation of Chern-Simons theories and also the GL(1|1) generalisation of it,
for punctured Riemann surfaces of arbitrary genus. We construct the algebra of observables, and study their representations and applications to the construction of 3-manifold invariants.
This is based on the joint work https://arxiv.org/abs/1811.09123 and part 2 in progress with A. Gainutdinov, M. Pawelkiewicz, V Schomerus.
Chern-Simons theories with gauge supergroups appear naturally in string theory and they possess interesting applications in mathematics, e.g.
for the construction of knot and link invariants. In this talk we explain the combinatorial quantisation of Chern-Simons theories and also the GL(1|1) generalisation of it,
for punctured Riemann surfaces of arbitrary genus. We construct the algebra of observables, and study their representations and applications to the construction of 3-manifold invariants.
This is based on the joint work https://arxiv.org/abs/1811.09123 and part 2 in progress with A. Gainutdinov, M. Pawelkiewicz, V Schomerus.
Guillaume Baverez (Berlin): A probabilistic view on the algebraic structure of Liouville CFT (slides)
In the 1980's, Segal formulated an axiomatic definition of conformal field theory (CFT) based on (unitary, projective) representations of the "semigroup of annuli". Translated in an analytic language, this means that one should construct semigroups of operators (in the Hille-Yosida sense) on a Hilbert space satisfying suitable commutation relations. In particular, the generators of these semigroups should form a representation (in a suitable sense) of the "Lie algebra" of the semigroup of annuli, the Virasoro algebra. For the Liouville CFT; we were able to realise this programme in a joint work with Guillarmou, Kupiainen, Rhodes and Vargas, and we used this construction to show that the generalised eigenstates of the Liouville Hamiltonian admit an analytic continuation to the whole plane. Moreover, they assemble into highest-weight representations for this Virasoro representation. With Baojun Wu, we show that the representations at the degenerate weights (Kac table) are irreducible, giving a systematic treatment of the famous BPZ equations.
In the 1980's, Segal formulated an axiomatic definition of conformal field theory (CFT) based on (unitary, projective) representations of the "semigroup of annuli". Translated in an analytic language, this means that one should construct semigroups of operators (in the Hille-Yosida sense) on a Hilbert space satisfying suitable commutation relations. In particular, the generators of these semigroups should form a representation (in a suitable sense) of the "Lie algebra" of the semigroup of annuli, the Virasoro algebra. For the Liouville CFT; we were able to realise this programme in a joint work with Guillarmou, Kupiainen, Rhodes and Vargas, and we used this construction to show that the generalised eigenstates of the Liouville Hamiltonian admit an analytic continuation to the whole plane. Moreover, they assemble into highest-weight representations for this Virasoro representation. With Baojun Wu, we show that the representations at the degenerate weights (Kac table) are irreducible, giving a systematic treatment of the famous BPZ equations.
Philip Boalch (Sorbonne): Wild character varieties and quantum groups (slides)
After Riemann's work showing that a meromorphic linear differential equation on the Riemann sphere defined a representation of the fundamental group, Birkhoff 1913 showed there was more topological data that one can attach to such an equation if it had poles of order >1 and was sufficiently generic. This story was extended from the generic to the general case in work of Sibuya and Malgrange in the late 1970s and then redescribed in several different ways (Jurkat, Deligne, Ecalle, Martinet-Ramis, Loday-Richaud etc). This data is called "Stokes data" and forms finite dimensional algebraic varieties known as "wild character varieties" or "wild Betti spaces".
I'll describe some of the repercussions of this revolution in 2d gauge theory particularly in terms of nonlinear differential equations (whose spaces of solutions are wild character varieties). Specifically I will discuss the theorems showing: 1) Quantization of meromorphic Higgs bundles (hyperkahler rotation to meromorphic connections), 2) quantum groups as the quantization of very simple wild character varieties, and 3) general construction of the wild character varieties as algebraic Poisson varieties using a new TQFT type operation called "fission" (which is not inverse to fusion).
References: Three main approaches to Stokes data (and a brief review of the history) are carefully discussed in:
P.B. Topology of the Stokes phenomenon, Proc. Symp. Pure Math. 103 (2021) 55–100, arXiv:1903.12612.
1) was proved in:
O. Biquard and P. B. Wild non-abelian Hodge theory on curves, Compositio Math. 140 (2004), no. 1, 179–204.
2) was proved in :
P.B. Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. math. 146 (2001), 479–506.
3) was proved in a sequence of 4 works 2002-2015:
P.B. Quasi-Hamiltonian geometry of meromorphic connections, (2002), arXiv:0203161.
P.B. Through the analytic halo: Fission via irregular singularities, Ann. Inst. Fourier 59 (2009), no. 7, 2669–2684.
P.B. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014), 301–365.
P. B. and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091, 2015.
After Riemann's work showing that a meromorphic linear differential equation on the Riemann sphere defined a representation of the fundamental group, Birkhoff 1913 showed there was more topological data that one can attach to such an equation if it had poles of order >1 and was sufficiently generic. This story was extended from the generic to the general case in work of Sibuya and Malgrange in the late 1970s and then redescribed in several different ways (Jurkat, Deligne, Ecalle, Martinet-Ramis, Loday-Richaud etc). This data is called "Stokes data" and forms finite dimensional algebraic varieties known as "wild character varieties" or "wild Betti spaces".
I'll describe some of the repercussions of this revolution in 2d gauge theory particularly in terms of nonlinear differential equations (whose spaces of solutions are wild character varieties). Specifically I will discuss the theorems showing: 1) Quantization of meromorphic Higgs bundles (hyperkahler rotation to meromorphic connections), 2) quantum groups as the quantization of very simple wild character varieties, and 3) general construction of the wild character varieties as algebraic Poisson varieties using a new TQFT type operation called "fission" (which is not inverse to fusion).
References: Three main approaches to Stokes data (and a brief review of the history) are carefully discussed in:
P.B. Topology of the Stokes phenomenon, Proc. Symp. Pure Math. 103 (2021) 55–100, arXiv:1903.12612.
1) was proved in:
O. Biquard and P. B. Wild non-abelian Hodge theory on curves, Compositio Math. 140 (2004), no. 1, 179–204.
2) was proved in :
P.B. Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. math. 146 (2001), 479–506.
3) was proved in a sequence of 4 works 2002-2015:
P.B. Quasi-Hamiltonian geometry of meromorphic connections, (2002), arXiv:0203161.
P.B. Through the analytic halo: Fission via irregular singularities, Ann. Inst. Fourier 59 (2009), no. 7, 2669–2684.
P.B. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014), 301–365.
P. B. and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091, 2015.
Tudor Dimofte (Edinburgh): Spark algebras and quantum groups
I’ll begin by reviewing the idea of Tannakian duality (using “fiber functions” to represent categories) especially in the context of tensor and braided tensor categories. Then I’ll discuss what's needed to realize Tannakian duality in QFT and TQFT. Following these ideas through will lead to an explicit construction of Hopf algebras and their Drinfeld doubles within 3d TQFT -- representing categories of line operators -- using extended operators (called (by me) "sparks") on boundary conditions. I'll illustrate the construction for Dijkgraaf-Witten theory (a.k.a. gauge theory with finite gauge group), and then sketch an application to the B-type topological twist of 3d N=4 gauge theories, which initially motivated these investigations. (Work in progress with T. Creutzig and W. Niu.)
I’ll begin by reviewing the idea of Tannakian duality (using “fiber functions” to represent categories) especially in the context of tensor and braided tensor categories. Then I’ll discuss what's needed to realize Tannakian duality in QFT and TQFT. Following these ideas through will lead to an explicit construction of Hopf algebras and their Drinfeld doubles within 3d TQFT -- representing categories of line operators -- using extended operators (called (by me) "sparks") on boundary conditions. I'll illustrate the construction for Dijkgraaf-Witten theory (a.k.a. gauge theory with finite gauge group), and then sketch an application to the B-type topological twist of 3d N=4 gauge theories, which initially motivated these investigations. (Work in progress with T. Creutzig and W. Niu.)
Lorenz Eberhardt (Amsterdam): The Virasoro Minimal String
I will introduce a novel critical string theory in two dimensions and explain that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral (or topological recursion), which provides the holographic description. The worldsheet theory consists of Liouville CFT with central charge c\ge 25 coupled to imaginary Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, which gives the theory its name. The talk is based on joint work with Scott Collier, Beatrix Mühlmann and Victor Rodriguez.
I will introduce a novel critical string theory in two dimensions and explain that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral (or topological recursion), which provides the holographic description. The worldsheet theory consists of Liouville CFT with central charge c\ge 25 coupled to imaginary Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, which gives the theory its name. The talk is based on joint work with Scott Collier, Beatrix Mühlmann and Victor Rodriguez.
Marco de Renzi (Montpellier): Algebraic presentation of cobordisms and TQFTs
It has long been known that the category of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra, the circle. This result allows for a complete classification of TQFTs (Topological Quantum Field Theories) in dimension 2. In this talk I will discuss similar algebraic presentations in dimension 3 and 4 due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. I will also present examples of such algebras and the TQFTs they induce. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.
It has long been known that the category of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra, the circle. This result allows for a complete classification of TQFTs (Topological Quantum Field Theories) in dimension 2. In this talk I will discuss similar algebraic presentations in dimension 3 and 4 due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. I will also present examples of such algebras and the TQFTs they induce. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.
Rémi Rhodes (Aix-Marseille): Coulomb gas and compactified imaginary Liouville theory (slides)
Conformal Field Theories (CFT) play a central role in the description of statistical physics models undergoing a second order phase transition at their critical point. The recent development of the Liouville CFT, which is the scaling limit of random planar maps, has shed some light on the mathematical structure of CFT and has had many applications regarding the derivation of exact formulae for various statistical physics models. In this talk I will present the probabilistic construction of another important CFT, called imaginary Liouville CFT, and explain why it satisfies the axioms of CFT, in particular Segal’s gluing axioms. In physics this path integral is conjectured to describe the scaling limit of critical loop models such as Q-Potts or O(n) models. This CFT has several exotic features: most importantly, it is non unitary and has the structure of a logarithmic CFT. Therefore it provides a playground for the mathematical study of these concepts.
Conformal Field Theories (CFT) play a central role in the description of statistical physics models undergoing a second order phase transition at their critical point. The recent development of the Liouville CFT, which is the scaling limit of random planar maps, has shed some light on the mathematical structure of CFT and has had many applications regarding the derivation of exact formulae for various statistical physics models. In this talk I will present the probabilistic construction of another important CFT, called imaginary Liouville CFT, and explain why it satisfies the axioms of CFT, in particular Segal’s gluing axioms. In physics this path integral is conjectured to describe the scaling limit of critical loop models such as Q-Potts or O(n) models. This CFT has several exotic features: most importantly, it is non unitary and has the structure of a logarithmic CFT. Therefore it provides a playground for the mathematical study of these concepts.
Brian Swingle (Brandeis): A Bosonic Model of Quantum Holography
We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model is generic in that it includes all possible q-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. We will discuss implications for Hamiltonian complexity, the factorization problem in quantum gravity, and quantum simulations of holography. Based on 2311.01516 with Mike Winer.
We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model is generic in that it includes all possible q-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. We will discuss implications for Hamiltonian complexity, the factorization problem in quantum gravity, and quantum simulations of holography. Based on 2311.01516 with Mike Winer.
Valerio Toledano Laredo (Northeastern U.): Stokes phenomena, Poisson-Lie groups and quantum groups.
Let G be a complex reductive group, G* its dual Poisson-Lie group, and g the Lie algebra of G. G-valued Stokes phenomena were exploited by P. Boalch to linearise the Poisson structure on G*. I will explain how Ug-valued Stokes phenomena can be used to give a purely transcendental construction of the quantum group Uhg. I will also show that the semiclassical limit of this construction recovers Boalch’s. The latter result is joint work with Xiaomeng Xu.
Let G be a complex reductive group, G* its dual Poisson-Lie group, and g the Lie algebra of G. G-valued Stokes phenomena were exploited by P. Boalch to linearise the Poisson structure on G*. I will explain how Ug-valued Stokes phenomena can be used to give a purely transcendental construction of the quantum group Uhg. I will also show that the semiclassical limit of this construction recovers Boalch’s. The latter result is joint work with Xiaomeng Xu.
Xiaomeng Xu (Peking): The boundary condition and monodromy problem for some isomonodromy equations
This talk explores some basic properties of the isomonodromy equation of a rank n meromorphic linear system of ODEs with second order pole. In particular, it finds the boundary condition and asymptotic expansion for generic solutions of isomonodromy equation, and then solves the monodromy problem for the linear system. In n=3 case, the results recover Jimbo's formula for Painleve VI. In the end, the talk applies the results to the problem of finding algebraic solutions, to the study of WKB approxiamation and so on. Part of the talk is based on the joint works with Z. Wang, with Q. Tang, and with A. Alekseev, A. Neitzke and Y. Zhou.
This talk explores some basic properties of the isomonodromy equation of a rank n meromorphic linear system of ODEs with second order pole. In particular, it finds the boundary condition and asymptotic expansion for generic solutions of isomonodromy equation, and then solves the monodromy problem for the linear system. In n=3 case, the results recover Jimbo's formula for Painleve VI. In the end, the talk applies the results to the problem of finding algebraic solutions, to the study of WKB approxiamation and so on. Part of the talk is based on the joint works with Z. Wang, with Q. Tang, and with A. Alekseev, A. Neitzke and Y. Zhou.