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Register a new userFermionic superstring loop amplitudes in the pure spinor formalism
أمبليدات حلقة السوبرسترين الفيرميوني في الصيغة الحمضية النقية
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Authors
Christian Stahn
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صيغة السبينور النقية للسوبرسترين العشرية تؤدي إلى أمبيرات حلقة مبينة السوبرسيميترية، والتي تعبر عنها بشكل إحداثيات سوبرسيميترية في السبينور النقي. يستكشف هذا البحث طرقاً مختلفة لتقييم هذه الإحداثيات ويستخدمها ثم لحساب العوامل الكينماتية لأمبيرات الأربعة نقاط الطاقة الصفرية في الحلقة الواحدة والحلقتين تتضمن راموندين وأربعة حالات.
The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.
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We compute the massless five-point amplitude of open superstrings using the non-minimal pure spinor formalism and obtain a simple kinematic factor in pure spinor superspace, which can be viewed as the natural extension of the kinematic factor of the massless four-point amplitude. It encodes bosonic and fermionic external states in supersymmetric form and reduces to existing bosonic amplitudes when expanded in components, therefore proving their equivalence. We also show how to compute the kinematic structures involving fermionic states.
The pure spinor formulation of superstring theory includes an interacting sector of central charge $c_{lambda}=22$, which can be realized as a curved $betagamma$ system on the cone over the orthogonal Grassmannian $text{OG}^{+}(5,10)$. We find that the spectrum of the $betagamma$ system organizes into representations of the $mathfrak{g}=mathfrak{e}_6$ affine algebra at level $-3$, whose $mathfrak{so}(10)_{-3}oplus {mathfrak u}(1)_{-4}$ subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine $mathfrak{e}_6$ characters. We interpret this as an instance of a more general pattern of enhancements in curved $betagamma$ systems, which also includes the cases $mathfrak{g}=mathfrak{so}(8)$ and $mathfrak{e}_7$, corresponding to target spaces that are cones over the complex Grassmannian $text{Gr}(2,4)$ and the complex Cayley plane $mathbb{OP}^2$. We identify these curved $betagamma$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $mathcal{N}=2$ SCFTs on $S^2$.
Mason and Skinner recently constructed a chiral infinite tension limit of the Ramond-Neveu-Schwarz superstring which was shown to compute the Cachazo-He-Yuan formulae for tree-level d=10 Yang-Mills amplitudes and the NS-NS sector of tree-level d=10 supergravity amplitudes. In this letter, their chiral infinite tension limit is generalized to the pure spinor superstring which computes a d=10 superspace version of the Cachazo-He-Yuan formulae for tree-level d=10 super-Yang-Mills and supergravity amplitudes.
The full two-loop amplitudes for five massless states in Type~II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The $alpha to 0$ limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type~II supergravity. Investigations of the $alpha$ expansion of the Type~II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.
In this paper we describe how representation theory of groups can be used to shorten the derivation of two loop partition functions in string theory, giving an intrinsic description of modular forms appearing in the results of DHoker and Phong [1]. Our method has the advantage of using only algebraic properties of modular functions and it can be extended to any genus g.
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