PDF superior Modular Forms and Vertex Operator Algebras

Modular Forms and Vertex Operator Algebras

Modular Forms and Vertex Operator Algebras

r such factors, allowing us to switch the order of a(z), b(w) by their locality. The terms with 0 ≤ s ≤ r have (z − u) appearing to a power of at least r, which allows us to move c(u) through a(z) while also still having (w − u) to the rth power, so that we can move c(u) through b(w). Similarly, on the right hand side, the terms with r < s ≤ 2r will vanish, and the other terms give us the same expression as on the left hand side. This establishes (3.4) and the lemma.

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2D sigma models and differential Poisson algebras

2D sigma models and differential Poisson algebras

bracket (2.22) and the quadratic part of the action (3.1) expanded around a constant back- ground suggests that two-point correlation functions of suitable boundary vertex operators yield the extension of Kontsevich’s star product formula to higher forms on a manifestly covariant as well as explicit form, at least in the case of R n target space topology. Moreover, assuming the supercurrent to be anomaly free we expect its charge to be a deformation of the de Rham differential into a nilpotent operator that is compatible with the star product. Clearly, the first steps in this direction are to reproduce the higher-form generalized Poisson bracket (2.22) at order ~ and then verify the bi-differential operator found in [11, 12] at order ~ 2 , which we leave for separate considerations. 5 More precisely, we propose that the BRST cohomology of the model contains a ring generated by the constant modes of (φ α , θ α ) that realizes the star product deformation of the space Ω(N ) of differential forms
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Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

of the space of conformal blocks with chiral insertion of type V i at the ith marked point of Σ g , n . In the particular case that the category C is finitely semisimple, the structure of a modular functor is reasonably well understood. Specifically, precise conditions are known under which the representa- tion category of a vertex algebra V is a modular tensor category. In this case the Reshetikhin–Turaev construction allows one to obtain a modular functor just on the basis of C as an abstract category. In a remarkable development, Lyubashenko and others (see [KL] and references cited there) have ex- tended many aspects of this story to a larger class of monoidal categories that are not necessarily semisimple any longer. In particular, given an abstract monoidal category with adequate additional properties, one can still construct representations of mapping class groups.
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Group actions on algebras and module categories

Group actions on algebras and module categories

In section 4, following [2], we introduce functors relating the module categories of A and A/G when G acts freely on A. The main results in these notes (section 5) relate the representation types of A and A/G (which was the original purpose of the introduction of Galois coverings). Indeed, given a Galois covering F : A → A/G = B with B a finite dimensional k-algebra, then B is representation-finite if and only if A is locally representation-finite (that is, for each i ∈ A 0 , there are only finitely

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Modules and comodules over nonarchimedean Hopf algebras

Modules and comodules over nonarchimedean Hopf algebras

The breakthrough in the p-adic representation theory of p-adic groups was made in papers by P. Schneider and J. Teitelbaum. Motivated by examples of p-adic representations, which they found in their study of cohomologies of Drinfeld upper-half planes, they studied locally analytic p-adic groups and their p-adic representations in general. They studied categories of continuous representations of p-adic groups in Banach spaces over K (the orbit maps are continuous), locally-analytic representations (orbit maps are locally-analytic) of p-adic groups in topological Frechet K-vector spaces of compact type (i.e. inductive limits of p-adic Banach spaces, endowed with inductive limit topology) and relations between these two types of representations. They singled out “good” categories of representations of both types, which they called admissible representations.
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10. OX OPERATOR BASKETS AND AERIAL PLATFORMS

10. OX OPERATOR BASKETS AND AERIAL PLATFORMS

We comply with the provisions in the directives of the European Parlament and from the Council 2006/42/CE, on 17th May 2006, regarding the approach of several legislations of the member states on machinery and 2004/108/CE, on 15th December 2004, regarding the approach of several legislations of the member states about electromagnetic compatibility, and it has been manufactured in accordance with the following norms: - UNE-EN ISO 12100 - 1:2004. Security on machines. Basic concepts, main principles for its design. Part 1: Basic terminology, methodology.

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Combinatorial Hopf algebras and dual graded graphs

Combinatorial Hopf algebras and dual graded graphs

1 . So far we constructed a bialgebra structure on the group algebra kG. Truth is, the construction applies to weaker structures such as monoids. Consider A = k N as a bialgebra and suppose that there exists an antipode over A. We have, by definition of the antipode, that S ( n ) + N n = 0 for each n, this means that S ( n ) = − n which would lead to a contradiction since − n is not in A (We emphasize the fact that “ + N ” is the addition operation of the natural numbers, different from “ + ” the sum opperation in A). Lemma 1.1.16. Let A be a graded connected k-bialgebra with decomposition A = M
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Electoral systems and forms of abstention

Electoral systems and forms of abstention

Second, the introduction of a quorum might either increase or decrease the number of attendees in equilibrium, depending on whether the second meeting is ever reached. More speci cally, if the quorum requirement is higher than the number of members who would attend if there was no quorum, the decision is taken in the rst meeting when individuals are harmed \enough" by delaying the decision. In that case, the quorum is binding, and its introduction yields to an increase in the equilibrium number of attendees with respect to the no-quorum case. Conversely, if the second meeting is ever reached, and provided that individ- uals value strictly less a decision that has been postponed, the introduction of a quorum has the opposite e ect of lowering the number of attendees in equilibrium. Third, given that we focus on symmetric individuals' favorite policies, intro- ducing a quorum has in most cases no e ect on the chosen policy, no matter whether the second meeting is reached or not. In terms of welfare, it turns out that introducing a quorum, while having no direct policy e ects, always yields to a loss of aggregate welfare, even in the cases for which less individuals attend the meeting as compared to the no-quorum equilibrium.
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106 Lee mas

Hall algebras and Donaldson-Thomas invariants

Hall algebras and Donaldson-Thomas invariants

(5) Behrend’s interpretation of Donaldson-Thomas invariants as weighted Euler characteristics [2]. The credit for the development of motivic Hall algebras as a tool for studying moduli spaces of sheaves on Calabi-Yau threefolds is due jointly to Joyce and to Kontsevich and Soibelman. Joyce introduced motivic Hall algebras in a long series of papers [17, 18, 19, 20, 21, 22]. He used this framework to define generalizations of the naive Donaldson-Thomas invariants considered above, which apply to moduli stacks containing strictly semistable sheaves. He also worked out the wall-crossing formula for these invariants and proved a very deep no-poles theorem. Kontsevich and Soibelman [27] constructed an alternative theory which incorporates motivic vanishing cycles, and therefore applies to genuine DT invaraints and motivic versions thereof. They also produced a more conceptual statement of the wall-crossing formula. Some of their work was conjectural and is still being developed today. Joyce and Song [24] later showed how to directly incorporate the Behrend function into Joyce’s framework, and so obtain rigorous results on DT invariants.
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34 Lee mas

Modes and forms of contemporary slavery

Modes and forms of contemporary slavery

23 Formalmente, la abolición del instituto de la esclavitud se produce por parte de las grandes potencias coloniales a lo largo del siglo XIX: Inglaterra con una ley de 1833 que entró en vigor en 1838, Francia en 1848 con un Decreto del gobierno revolucionario, los Estados Unidos en 1865, mientras España y Portugal (países donde non se había desarrollado un movimiento abolicionista consistente) fueron los últimos Estados europeos en declarar ile- gal la esclavitud, en 1876 y en 1878, respectivamente. Cfr., por último, J. ALLAIN, “125 años de abolicíon: el derecho de la esclavitud y la explotación humana”, en e. PÉREZ ALONSO (dir.), El derecho ante las formas contemporáneas de esclavitud, cit., pp. 147-182. Véase también S. DRESCHER, Abolition: A History of Slavery and Antislavery, Cambridge University Press, Cambridge, 2009.
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27 Lee mas

Hopf-Frobenius algebras and a simpler Drinfeld double

Hopf-Frobenius algebras and a simpler Drinfeld double

We have generalised the notions of interacting Frobenius algebras [12, 16] and interacting Hopf algebras [7] to the non-commutative case, and in the process shown that they are rather common structures. This work could be viewed as an extension of classical results showing that concrete Hopf algebras over finite dimensional vector spaces are also Frobenius algebras [26]. Another perspective is that we make precise how much ambient symmetry is required to obtain a Hopf- Frobenius algebra. The original setting of interacting Frobenius algebras [12] was a †-compact category, which provides a lot of duality on top of the commutative algebras themselves. We show that none of this structure is necessary: all that is required is one-sided half-dual for the carrier object. The major question that remains is to pin down exactly when the Frobenius condition holds; as Lemma 3.12 shows, this is tightly related to the existence of integrals. Compact closure
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26 Lee mas

SUSY $N$ supergroups and their real forms

SUSY $N$ supergroups and their real forms

Proof. Suppose f : X → X 0 is an isomorphism. Then X e and X e 0 are isomorphic. Conversely, suppose |f| : X e → X e 0 is an isomorphism. |f | : X e → X e 0 lifts to the universal covers, giving an isomorphism f e : C → C which fixes the origin. By a standard result from one-variable complex analysis, f e is multiplication by a nonzero scalar λ. If N = 1, define the super Lie algebra morphism ϕ : g → g by C 7→ λC 0 , Z 7→ √ λZ 0 . Clearly ϕ| g 0 = d|f |, so F := (|f |, ϕ) is an isomorphism of

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Discourse forms and social categorization in Cha’palaa

Discourse forms and social categorization in Cha’palaa

same referent that has been tracked and maintained throughout all of Manuel’s turns since its initial introduction through the ethnonym peechulla. In this way Humberto confirms his uptake of the pronominal frame and by extension the social validity of offering an ethnonym to stand for a reference to a single individual by way of his social category. At a very high level of granularity, it confirms the status of the category as a social phenomenon. The conversation between Manuel and Humberto continued for several more minutes, all concerning the same referent but never using any other recognitional terms for him, only third person pronouns and, more often, predicates unmarked for person marking of any kind. In Cha’palaa discourse more generally reference is highly underspecified at the clause or sentence level and relies heavily on discourse structure for disambiguation, notably more so than languages with obligatory person marking. Once a referential framework has been established between speakers, that framework is present to be exploited by speakers for disambiguating their underspecified utterances (until they are modified by the introduction of new referents or by the switch reference system of the grammar). Understanding Manuel and Humberto’s conversation in those terms helps to show how Manuel successfully drew on the intersubjective social reality of race to acheive person reference in interaction. Picking up at a later part of the conversation we can obvserve that the same framework has been maintained throughout the duration of the conversation, in which all clauses reference the same Black man under discussion. Throughout much of the conversation not even third person pronouns (ya) are required for tracking co-reference, and person is grammatically unmarked in all but one clause below:
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Hypertrichosis: Etiology, clinical forms and managementCuestionario de recertificacin

Hypertrichosis: Etiology, clinical forms and managementCuestionario de recertificacin

c) Lipoatrofia. La diabetes lipoatrófica frecuentemente se asocia a hipertricosis. La forma congénita autosó- mica recesiva se denomina síndrome de Berardine- lli-Seip. Las manifestaci[r]

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Bettelheim – Economic Calculation and Forms of Property

Bettelheim – Economic Calculation and Forms of Property

For countries that have accomplished a socialist revolution, a positive response to this question is clearly of decisive importance. In particular, it implies that the importing or "reproducing" (or even "perfecting") of the proven techniques of the most industrialized capitalist countries can create an obstacle to the development of the productive forces, since these forces have characteristics that call for new relations of production and enable these relations to really open up a new stage of history by revolutionizing the productive forces. It does seem that, at the level of political technique, what distinguishes Soviet from Chinese industrialization (which is achieved under the slogan: autonomy, independence, and development through one's own resources) is a recognition of the necessity for not taking capitalist technique as a model, while at the same time deriving the maximum of what is utilizable from it in the construction of socialism. This is expressed in an immense economy in the requirements for accumulation, thanks to which the industrialization of China can be achieved without pressure being exerted on the standard of living of the peasant masses; on the contrary, this standard has been greatly improved, and is progressing steadily. In contrast, in the Soviet Union the technical line followed (which was determined by a set of ideological and political conditions) has led to the realization of an onerous "primitive socialist accumulation," whose economic and political consequences have been so considerable that, in the final
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110 Lee mas

Immobilization of solids and mondriga quadratic forms

Immobilization of solids and mondriga quadratic forms

The purpose ofthis section is to prove that ifthe lines €0 , ..• , f 3 are quadratically dependent, then: a the quadratic form Q uhao; zero asan cigenvalue if and only ifthe four points [r]

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Infotechnology:new social forms, digital noometamorphosis
and noomorphosis

Infotechnology:new social forms, digital noometamorphosis and noomorphosis

From the perspective of economic sociology, look- ing back to 40,000 years B.C., Wood (2000) has shown that the developed nations are in the sixth wave (from 1975 to 2010), correspond[r]

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Leavitt path algebras and the IBN property

Leavitt path algebras and the IBN property

A ring has IBN if any two bases of a finitely generated free module have the same number of elements. Fields, division rings, commutative rings, Noetherian rings all have IBN. A classical example of a ring without IBN is the algebra of endomorphisms of a countably infinite dimensional vector space. The free module of rank 1 over this ring has bases of n elements for any positive integer n. In the early 60s William Leavitt asked and then answered this question: Given any positive integers m < n is there a ring R having a free module with a basis of m elements and another basis with n elements but no bases with k elements if k < n and not equal to m? Such a ring is said to be non-IBN of type (m, n).
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Interval-valued algebras and fuzzy logics

Interval-valued algebras and fuzzy logics

mined by just two values: its lower and upper bound. The aim of this work is to develop a logic that has intervals as truth values. The intended semantics are residuated lattices on the set of closed subintervals of the unit interval. We call this set the triangularization of the unit interval. A particular subset of this triangularization is its so-called diagonal, consisting of those intervals for which the lower and upper bound coincide. These intervals are called exact intervals and rep- resent truth values of propositions about which the knowledge is complete. Intuitively, the truth values of formulas constructed with these propositions should be exact intervals as well (because in these cases, the situation is similar to working with formulas in fuzzy logics). The semantics of so-called interval-valued fuzzy logics have already been examined by different authors. Especially interval-valued triangular norms, triangular conorms and implicators have received ample atten- tion. Most of these authors [1, 7, 26, 37, 44] only consider interval-valued operations that map the diagonal on the diagonal, although the most general definitions of triangular norms, triangu- lar conorms and implicators allow other operations as well [15, 21, 19, 48]. Generally speaking, interval-valued operations satisfy not as many properties as operations on the unit interval. For example, standard interval-valued residuated lattices can never satisfy prelinearity [16]. A lot of other properties can hold though. There are even interval-valued implicators that satisfy all the Smets-Magrez axioms [17, 70].
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