We introduce coordinates for a principal bundle ST (F ) over the super Teichmüller space ST (F ) of a surface F with s ≥ 1 punctures that extend the lambda length coordinates on the decorated bundleT (F ) = T (F ) × R s + over the usual Teichmüller space T (F ). In effect, the action of a Fuchsian subgroup of P SL(2, R) on Minkowski space R 2,1 is replaced by the action of a super Fuchsian subgroup of OSp(1|2) on the super Minkowski space R 2,1|2 , where OSp(1|2) denotes the orthosymplectic Lie supergroup, and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in R 2,1|2 . As in the bosonic case, there is the analogue of the Ptolemy transformation now on both even and odd coordinates as well as an invariant even two-form on ST (F ) generalizing the Weil-Petersson Kähler form. This finally solves a problem posed in Yuri Ivanovitch Manin's Moscow seminar some thirty years ago to find the super analogue of decorated Teichmüller theory and provides a natural geometric interpretation in R 2,1|2 for the super moduli of ST (F ).
We consider the first order formalism in string theory, providing a new off-shell description of the nontrivial backgrounds around an "infinite metric". The OPE of the vertex operators, corresponding to the background fields in some "twistor representation", and conditions of conformal invariance results in the quadratic equation for the background fields, which appears to be equivalent to the Einstein equations with a Kalb-Ramond B-field and a dilaton. Using a new representation for the Einstein equations with B-field and dilaton we find a new class of solutions including the plane waves for metric (graviton) and the B-field. We discuss the properties of these background equations and main features of the BRST operator in this approach.1 One could expect this generally, since cubic terms in these equations should be of higher order in α ′ .
We define and study the quantum equivariant K-theory of cotangent bundles over Grassmannians. For every tautological bundle in the K-theory we define its one-parametric deformation, referred to as quantum tautological bundle. We prove that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous XXZ spin chain. In addition, we prove that each such operator corresponds to the universal elements of quantum group U ( sl 2 ). In particular, we identify the Baxter operator for the XXZ spin chain with the operator of quantum multiplication by the exterior algebra tautological bundle. The explicit universal combinatorial formula for this operator is found. The relation between quantum line bundles and quantum dynamical Weyl group is shown.
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