Notes on holomorphic string and superstring theory measures of low genus (Q2786830)

The main goal of this paper is to reconsider the genus 2 superstring measure from the point of view of holomorphy. Such a measure was presented in [\textit{E. D'Hoker} and \textit{D. H. Phong}, ``Lectures on two-loop superstrings'', \url{arXiv:hep-th/0211111}]. The answer in this book involves theta function (Appendix B. (B.3)). The corresponding answer of this paper does not contain theta functions ((B.1), (B.2), cf. \S2, (2.17), \S3, (3.28)). But in Appendix B, the coincidence of these expressions are shown.NEWLINENEWLINEThe prototype of the computation of superstring measures at genus 1 and 2 from the point of view of holomorphy is computation of the bosonic string measures at genus 1 and 2. It is based on the fact Riemann surfaces of genus 1 and 2 are (hyper)elliptic. By using this fact, their moduli spaces are explicitly known. They are described in \S2 as follows:NEWLINENEWLINELet \(\Sigma\) be a Riemann surface of genus \(g\), \(\mathcal L \to \Sigma\) a holomorphic line bundle, and let \(\det H^* (l)\) be \(\det H^0 (\Sigma, \mathcal L) \otimes (\det H^1 (\Sigma.\mathcal L)^{-1}\). Let \(\mathcal M_g\) be the moduli space of Riemann surfaces of genus \(g\), then \(\det H^* (\mathcal L)\) is a holomorphic line bundle over \(\mathcal M_g\). Let \(K\) be the canonical bundle of \(\Sigma\). Then by the Mumford isomorphism NEWLINE\[NEWLINE\det H^* (K^2) \cong (\det H^* (K))^{13},NEWLINE\]NEWLINE there is a global and everywhere nonzero holomorphic section \(\Phi_g\); NEWLINE\[NEWLINE\Phi_g \in H^0 (\mathcal M_g , \det H^* (K^2) \otimes \det{}^{-13} H^* (K)).NEWLINE\]NEWLINE It is unique up to a nonzero complex constant which depends on the string constant \(g_{st}\). \(\Phi_g\) is the holomorphic part of the genus \(g\) vacuum amplitude (the measure of \(\mathcal M_g\), that is the measure on \(\mathcal M_g\) determined by the world volume path integral).NEWLINENEWLINESince \(H^1 (\Sigma, K^2) = \{0\}\) if \(g \geq 2\), we have NEWLINE\[NEWLINE\Phi_g \in H^0 (\mathcal M_g , \det T^* \mathcal M_g \otimes \det{}^{-13} H^0 (\Sigma, K)),\;g \geq 2.NEWLINE\]NEWLINE If \(g = 1\), \(H^1 (\Sigma, K^2) \cong H^0 (\Sigma, K)\) by Serre duality, and we have NEWLINE\[NEWLINE\Phi_1 \in H^0 (\mathcal M_1 , \det T^* \mathcal M_1 \otimes \det{}^{-14} H^0 (\Sigma, K)),NEWLINE\]NEWLINE (\S2.1). After sketching how to get the measure from \(\Phi_g\) (\S2.2), detailed computation of \(\Phi_g\), \(g = 1, 2\) is done by using the fact if \(g = 1\) or 2, then \(\Sigma\) is defined by the equation NEWLINE\[NEWLINEy^2=\prod_{i=1}^s(x-e_i),\quad s = 2(g + 1).NEWLINE\]NEWLINE Here two points \(x \to \infty\), \(y \sim \pm x^{s/2}\) are included and each \(e_i\) is regarded to be a point of \(\mathbb C\mathbb P^1\). Then \(\mathcal M_g\) is \((M/\mathrm{SL}(2,\mathbb C))/\Theta\), where \(M\) is the product of \(s\) copies of \(\mathbb C\mathbb P^1\) with diagonal removed and \(\Theta\) is the group of permutations of \(e_i\). Then by using a way that remove the volume form vol of \(\mathrm{SL}(2,\mathbb C)\), explicit forms of \(\Phi_g\), \(g =1, 2\) are obtained ((2.12), (2.17)). Behavior of \(\Phi_g\) at the separating degeneration is also computed based on the formula \(H^0 (\Sigma, K) \cong H^0 (\Sigma_l, K_l) \oplus H^0 (\Sigma_r, K_r)\) in this Section (\S2.3).NEWLINENEWLINESuperstring measures in genus 1 and 2 are calculated in \S3 following the methods in \S2. Let \(\Sigma\) be a super Riemann surface, \(\mathcal L \to \Sigma\) is a holomorphic line bundle. Then the Berezinian \(\mathrm{Ber}H^* (\mathcal L)\) of the cohomology with coefficients in \(\mathcal L\) (super analog of determinant) is defined by NEWLINE\[NEWLINE\mathrm{Ber}H^* (\mathcal L) = \mathrm{Ber}H^0 (\Sigma, \mathcal L) \otimes \mathrm{Ber}^{-1} H^1 (\Sigma, \mathcal L).NEWLINE\]NEWLINE Let \(\mathrm{Ber}^k (\Sigma) = (\mathrm{Ber}(\Sigma))^{\otimes k}\), \(\mathrm{Ber}_k (\Sigma) = \mathrm{Ber}(H^* (\mathrm{Ber}^k (\Sigma))\) and \(\mathrm{Ber}_k \to \mathcal M_g\) is the line bundle whose fiber at the point corresponding to a given super Riemann surface \(\Sigma\) is \(\mathcal B\rceil\nabla_k (\Sigma)\). Then the super Mumford isomorphism is NEWLINE\[NEWLINE\mathrm{Ber}_3 \cong \mathrm{Ber}_1^5,NEWLINE\]NEWLINE (see [\textit{A. A. Roslyj} et al., Commun. Math. Phys. 120, No. 3, 437--450 (1989; Zbl 0667.58008)]). The holomorphic measure of superstring theory in \(\mathbb R^{10}\) is a holomorphic trivialization \(\Psi_g\) of \(\mathrm{Ber}_3 \otimes \mathrm{Ber}_1^{-5}\). It is uniquely defined from the knowledge of how \(\Psi_g\) should behave at infinity. The author says this knowledge will come from our knowledge of superconformal field theory and string theory.NEWLINENEWLINELet \(\mathfrak{M}_g\) be the moduli of super Riemann surface of genus \(g\), and let \(\mathcal M_{g,\mathrm{spin}}\) is the moduli of Riemann surface of genus \(g\) with a choice of spin structure. Global holomorphic projection \(\pi :\mathfrak{M}_g \to \mathcal M_{g,\mathrm{spin}}\) exists if \(g = 1\) or 2 with an even spin structure. Then the line bundle \(\mathrm{Ber}_1 \to \mathfrak{M}_g\) is the pullback of a line bundle \(\mathcal S \to \mathcal M_{g,\mathrm{spin}}\), and \(\Psi\) is a section of \(\mathrm{Ber}(\mathfrak{M}_g) \otimes \pi^*\mathcal S^{-5}\) and there is a map NEWLINE\[NEWLINE\pi_* : H^0 (\mathfrak{M}_g, \mathrm{Ber}(\mathfrak{M}_g) \otimes \pi^*\mathcal S^{-5}) \to H^0 (\mathcal M_{g,\mathrm{spin}} , \det T^* \mathcal M_{g,\mathrm{spin}} \otimes\mathcal S^{-5}),NEWLINE\]NEWLINE given by integration along the fiber of \(\pi\). Consequently. \(\pi_*(\Psi_g)\) is a global holomorphic section of NEWLINE\[NEWLINE\det T^* \mathcal M_{g,\mathrm{spin}} \otimes \det{}^5 H^* (K^{1/2}) \otimes \det{}^{-5} H^* (K).NEWLINE\]NEWLINE Hence superstring measures of even spin structure of genus 1 and 2 are computed by the same way as bosonic string measure ((3.22), (3.27), (3.28)).NEWLINENEWLINEFor the case of an odd spin structure, this method is not available though in that case, the vacuum amplitude vanishes. But it contribute to certain parity-violating scattering amplitude. The super Mumford isomorphism is valid for an odd spin structure. Denoting splitting of \(\mathfrak{M}_2\), etc., by \(\mathfrak{M}_{2,\pm}\), etc., the restriction \(\Psi_{2,-} |_{\mathcal M_{2,\mathrm{spin}-}}\) is computed by using the formula \(\det H^0 (\Sigma_0 , K^{1/2}) = H^0 (\Sigma_0 , K ^{1/2})\) (\S3, (3.35)).NEWLINENEWLINEThe last two sections of this paper treat behavior of the super string amplitude at a separating (\S4) or non separating (\S5) degeneration.NEWLINENEWLINEAs for the higher genus case, the author says if genus 3, then global meromorphic projection \(\pi : \mathfrak{M}_3 \to\mathcal M_{3,\mathrm{spin}}\) exists. It leads a proposal for a genus 3 superstring measure (see [\textit{S. L. Cacciatori} et al., Nucl. Phys., B 800, No. 3, 565--590 (2008; Zbl 1292.81108)]). While global holomorphic projection \(\pi : \mathfrak{M}_g \to \mathcal M_{g,\mathrm{spin}}\) does not exits if \(g \geq 5\) (see [\textit{R. Donagi} and \textit{E. Witten}, ``Superstring space is not projected'', \url{arXiv:1304.7798}]).NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].