By Selman Akbulut, Sema Salur (auth.), Özgür Ceyhan, Yu. I. Manin, Matilde Marcolli (eds.)

In contemporary many years, quantization has ended in attention-grabbing purposes in numerous mathematical branches. This quantity, created from study and survey articles, discusses key subject matters, together with symplectic and algebraic geometry, illustration concept, quantum teams, the geometric Langlands application, quantum ergodicity, and non-commutative geometry. a variety of issues on the topic of quantization are lined, giving a glimpse of the huge topic. The articles are written through uncommon mathematicians within the box and replicate next advancements following the mathematics and Geometry round Quantization convention held in Istanbul.

List of Contributors:

S. Akbulut R. Hadani

S. Arkhipov okay. Kremnizer

Ö. Ceyhan S. Mahanta

E. Frenkel S. Salur

K. Fukaya G. Ben Simon

D. Gaitsgory W. van Suijlekom

S. Gurevich

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Simon b. The constant c equals zero. We first prove a. Since the loop ft is contractible, there is a homotopy Ks (t) : [0, 1] × [0, 1] → Ham such that for every s, Ks (t) is a contractible loop, K0 = ft and K1 = 1l. We denote by Fs (t) the normalized Hamiltonian which generates the loop Ks . As we explained, for every s there is a constant c(s) such that p∗ Fs + c(s) generates a loop, ˆ hs (t), in Q which is a lift of Ks . ˆ s (t) is a lift of the homotopy Ks (t) (which is a homotopy Now since h ˆ and the identity element of between ft and 1l) it is a homotopy between h(t) ˆ Q.

For any two lattices L and L there exists a third lattice L ⊂ L ∩ L . Now we can define a gerbal theory. Definition 13 Let V be a 2-Tate space. A gerbal theory D is • • For each lattice L ⊂ V a Gm -gerbe DL If L ⊂ L are two lattices then we have an equivalence DL φLL ✲ DL ⊗ DL/L (20) 2-Gerbes and 2-Tate Spaces • 33 For V1 ⊂ V2 ⊂ V3 we have a natural transformation ✲ DV1 ⊗ DV /V 3 1 ⇐ == == == == == == == DV1 ⊗ DV2 /V1 ⊗ DV3 /V2 ❄ DV2 ⊗ DV3 /V2 (21) ❄ ✲ DV 3 Given V1 ⊂ V2 ⊂ V3 these natural transformations should commute on a cubical diagram.

Drinfeld, Vladimir Infinite-dimensional vector bundles in algebraic geometry: an introduction. The unity of mathematics, 263–304, Progr. , 244, Birkh¨ auser Boston, Boston, MA, 2006. 2-Gerbes and 2-Tate Spaces 35 7. Kato, Kazuya Existence theorem for higher local fields. Invitation to higher local fields (Mnster, 1999), 165–195 (electronic), Geom. Topol. , 3, Geom. Topol. , Coventry, 2000. 8. Lurie, Jacob Higher topos theory. Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009.

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