Some mathematical aspects and scattering amplitudes in the pure spinor formalism
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2011-11-10
Autores
Zuñiga, Humberto Gomez [UNESP]
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Universidade Estadual Paulista (Unesp)
Resumo
Primeiro vamos dar uma breve revisão sobre o artigo de Nekrasov “ Lectures on curved beta-gamma systems, pure spinors, and anomalies”, a fim de mostrar que o formalismo de espinor puro é livre de anomalia quando a origem é removido do espaço espinor puro. Desta forma, damos uma nova proposta para os operadores de imagem no formalismo de espinor puro mínimo. Nós calculamos amplitudes de espalhamento a nível de árvore, realizando a integração no espaço espinor puro como uma integral de Cauchy tipo multidimensional. A amplitude é escrita em termos de variáveis do espaço de espinor puro projetivo, o que é muito útil na hora de relacionar rigorosamente as versões mínima e não mínima do formalismo de espinor puro. A linguagem natural para relacionar esses formalismos é o isomorfismo de Cech-Dolbeault. Além disso, o cociclo de Dolbeault correspondente à amplitude de espalhamento a nível de árvore deve ser avaliada no espaço compacto SO(10)/SU(5) em vez de tudo o espaço de espinor puro, o que significa que a origem é removido neste espaço. Nós também obtimos uma relação entre a função de Green para um campo escalar sem massa em dez dimensões e as amplitude de espalhamento a nível de árvore. Os fatores globais constantes nas amplitudes de espalhamento são muito importante, porque eles precisam satisfazer as condições de unitariedade e S-dualidade [66]. Estes coeficientes não tinham sido computados no formalismo espinor puro, devido à dificuldade para resolver as integrais no espaço de espinores puro. Nós calculamos estas integrais usando o formalismo de espinor puro não mínimo. Assim, encontramos os coeficientes das amplitudes de um e dois-“loop” para quatro pontos sem massa. Contrastando com as dificuldades matemáticas no formalismo RNS, em que o desconhecimento das normalizações...
First, we give a brief review about the Nekrasov’s paper “Lectures on curved betagamma systems, pure spinors, and anomalies” in order to show the pure spinor formalism is anomaly free when the origin is removed from the pure spinor space. In this way we give a new proposal for the “picture lowering” operators in the minimal pure spinor formalism. We compute the tree level scattering amplitude by performing the integration over the pure spinor space as a multidimensional Cauchy-type integral. The amplitude is written in terms of the projective pure spinor variables, which turns out to be useful to relate rigorously the minimal and non-minimal versions of the pure spinor formalism. The natural language for relating these formalisms is the Cech-Dolbeault isomorphism. Moreover, the Dolbeault cocycle corresponding to the three-level scattering amplitude must be evaluated in SO(10)/SU(5) instead of the whole pure spinor space, which means that the origin is removed from this space. We also relate the Green’s function for the massless scalar field in ten dimensions to the tree-level scattering amplitude and comment about the scattering amplitude at higher orders. The overall constan factors in the scattering amplitudes are very important because they need to satisfy the unitarity and S-duality conditions [66]. These coefficients have not been computed in the pure spinor formalism due to the difficulty to solve the integrals on the pure spinors space. We compute these integrals by using the non-minimal pure spinor formalism. So, we find the coefficients of the massless one and two-loop four-point amplitude from first principles. Contrasting with the mathematical difficulties in the RNS formalism where unknown normalizations of chiral determinant formulæ force the two-loop coefficient to be... (Complete abstract click electronic access below)
First, we give a brief review about the Nekrasov’s paper “Lectures on curved betagamma systems, pure spinors, and anomalies” in order to show the pure spinor formalism is anomaly free when the origin is removed from the pure spinor space. In this way we give a new proposal for the “picture lowering” operators in the minimal pure spinor formalism. We compute the tree level scattering amplitude by performing the integration over the pure spinor space as a multidimensional Cauchy-type integral. The amplitude is written in terms of the projective pure spinor variables, which turns out to be useful to relate rigorously the minimal and non-minimal versions of the pure spinor formalism. The natural language for relating these formalisms is the Cech-Dolbeault isomorphism. Moreover, the Dolbeault cocycle corresponding to the three-level scattering amplitude must be evaluated in SO(10)/SU(5) instead of the whole pure spinor space, which means that the origin is removed from this space. We also relate the Green’s function for the massless scalar field in ten dimensions to the tree-level scattering amplitude and comment about the scattering amplitude at higher orders. The overall constan factors in the scattering amplitudes are very important because they need to satisfy the unitarity and S-duality conditions [66]. These coefficients have not been computed in the pure spinor formalism due to the difficulty to solve the integrals on the pure spinors space. We compute these integrals by using the non-minimal pure spinor formalism. So, we find the coefficients of the massless one and two-loop four-point amplitude from first principles. Contrasting with the mathematical difficulties in the RNS formalism where unknown normalizations of chiral determinant formulæ force the two-loop coefficient to be... (Complete abstract click electronic access below)
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Supersimetria, Teoria das supercordas, Supersymmetry, Theories, Superstring
Como citar
ZUÑIGA, Humberto Gomez. Some mathematical aspects and scattering amplitudes in the pure spinor formalism. 2011. 165 f. Tese (doutorado) - Universidade Estadual Paulista, Instituto de Física Teórica, 2011.