Download Communications In Mathematical Physics - Volume 283 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Q n ). ,σn (i n )) We denote such a dioperad as (P, γP , IP ) or simply P. 2. 1) (P; IP , . . , IP , Q, IP , . . , IP ). ,in ) can be reobtained from i ◦ j . ,in ) . 32 L. 3. 2) exist. A (partial) nonassociative dioperad consists of the same data as those of a (partial) dioperad satisfying all the axioms of a (partial) dioperad except the associativity. The notion of homomorphism and isomorphism of a (partial pseudo-) dioperad are naturally defined. 4. ,in ) or i ◦ j is defined (see Appendix C in [H2] for more details).

The 2-colored partial dioperad can be naturally defined. If the associativities of γQ , γ B , γ I do not hold, then it is called a 2-colored (partial) nonassociative dioperad. 4. If we restrict to {P(1, m|0, n)}m,n∈N and {Q(1, n)}n∈N , they simply gives a structure of 2-colored (partial) operad [V,Kont,Ko2]. Now we discuss an important example of the 2-colored partial nonassociative dioperad. Let J1 , J2 be two index sets, Ui = ⊕n∈Ji (Ui )(n) , i = 1, 2 be two graded vector spaces. Consider two families of sets, EU2 (m, n) = HomC (U2⊗m , U2⊗n ), ⊗m − EU1 |U2 (m − , m + |n − , n + ) = HomC (U1⊗m + ⊗ U2⊗n + , U1 ⊗n − ⊗ U2 ), 40 L.

4. In particular, the surface (A) in Fig. 4 shows how an annulus is obtained by sewing two oppositely oriented boundary punctures on the same disk with strips and tubes in S(1, 3|0, 0), and surface (C) in Fig. 4, viewed as a propagator of the close string, can be obtained by sewing an element in S(0, 1|1, 0) with an element in S(0, 1|0, 1) along the interior punctures. We only show in Figure 4 a simple case in which there are only two boundary punctures and no interior puncture. In general, the number of boundary punctures and interior punctures can be arbitrary.

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