By A.N. Parshin (editor), I.R. Shafarevich (editor), I. Rivin, V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov

This two-part EMS quantity presents a succinct precis of complicated algebraic geometry, coupled with a lucid creation to the hot paintings at the interactions among the classical sector of the geometry of advanced algebraic curves and their Jacobian forms. a very good better half to the older classics at the topic.

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**Extra info for Algebraic geometry 03 Complex algebraic varieties, Algebraic curves and their Jacobians**

**Example text**

32) identify the matrices {N j } as matrices of fusion rules. It is an outstanding open problem to classify all possible fusion rules. Standard examples of fusion rules are those derived from the representation theory of a compact group G, (N1ci i = multiplicity of irrep. k in the tensor product, i ® i, of two irreps. i and i of Gj in this connection see also [6]), or from the representation theory of a quantum group, Uq(Q), q = ezp(21ri/m). For example, if 7j~ IINili < 2 then min IINili = 2 cos (_1r_) , where n == I L #1 n+l I, and N1cji = 1, for Ii - i 1< k < min{i + i, 2(n + 1) - i - j} = 0, otherwise.

We believe that a detailed understanding of how anyons emerge in two-dimensional, highly correlated many-body systems is still missing. 3 Three-dimensional gauge theories with braid statistics Next, we give a mini-review of three-dimensional gauge theories with ChernSimons term describing particles with braid statistics. The simplest examples are abelian gauge theories. They have an action given by S[A] = 1 2 f }'vp [e- 2 F2 + + 2(} 11' e''''>' AI' all A>. - 2jl' AI' (pure matter terms)] . 11) The term proportional to () is the Chern-Simons term, breaking parity, jl' is the matter current.

8). ,(rjp,q)i(P" 0 p'l(A») i(r~ Opq,p,(v)) vfTc(pr) V Tc1 (pk). (v») vj1e(pr)k(A). (v») Vj1e(pr)k(A) Vkl(p'0). -lTc. 59) By choosing as(pk) > as(pp), as(p'l), we find that " R- (J,q, . 60) are consistent? 34). ". 17).