nLab Alexander Varchenko

Alexander Varchenko is a mathematician at the University of North Carolina at Chapel Hill; he was a student of Vladimir Arnold. His work is at the boundary of geometry, special functions, combinatorics, representation theory and mathematical physics.

• list
• webpage

Selected writings

On the hypergeometric integral construction of solutions to the Knizhnik-Zamolodchikov equation:

• Vadim Schechtman, Alexander Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$

• Vadim Schechtman, Alexander Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$

• Vadim Schechtman, Alexander Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$

On the hypergeometric integral construction of conformal blocks for the $\mathfrak{sl}(n)$ WZW model:

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$

  (for $\mathfrak{sl}(2, \mathbb{C})$)

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 $[$euclid:cmp/1104272957$]$

On hypergeometric functions, the Knizhnik-Zamolodchikov equation and quantum groups:

• Alexander Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 $[$arXiv:hep-th/9403102, doi:10.1007/BF02103772$]$

• Alexander Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv. Ser. in Math. Phys. 21, World Sci. Publ. 1995. x+371 pp. (doi:10.1142/2467)

• V. Tarasov, Alexander Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pp. (arXiv:q-alg/9703044, numdam:AST_1997__246__R1_0)

On Lie algebra weight systems for the special linear Lie algebra $\mathfrak{sl}(2)$:

• S. Tyurina, Alexander Varchenko, A remark on $\mathfrak{sl}_2$ approximation of the Kontsevich integral of the unknot, Journal of Mathematical Sciences 131 (2005) 5270–5274 (arXiv:math/0111201, doi:10.1007/s10958-005-0399-1)

• Sergei Chmutov, Alexander Varchenko, Remarks on the Vassiliev knot invariants coming from $\mathfrak{sl}(2)$, Topology 36 (1), 153-178, 1997 (doi:10.1016/0040-9383(95)00071-2)

Further on 1-twisted de Rham cohomology related to $\mathfrak{sl}(2)$-Verma modules:

• Alexey Slinkin, Alexander Varchenko, Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules, SIGMA 15 (2019) 075 [arXiv:1812.09791, doi:10.3842/SIGMA.2019.075]

Related $n$Lab entries

• hypergeometric function

• quantum group,

• conformal field theory

• representation theory

• Kac-Moody Lie algebra

• singularity theory

category: people