Non-split superstrings of dimension $(1|2)$

TL;DR

This paper advances the theory of non-split supermanifolds by applying Palamodov's framework to include odd parameters in the obstruction analysis. It provides a detailed classification of both even (degree-2) and odd (degree-1) obstructions for deformations of the projective superspace $${\mathbb{CP}}^{1|m}$$ with $m=1,2$, and it corrects prior results in the literature, aligning with Manin's conclusions for the $m=2$ case. For $m=1$, odd obstructions arise precisely when the underlying line bundle degree $k\ge 4$, forming a GL$(V)$-orbit structure that yields a projective family $${\mathbb C}{\mathbb P}^{k-4}$$ of obstructions; for $m=2$, a thorough orbit analysis under Aut$(\mathbf{E})$ reveals a rich landscape of both degree-1 and degree-2 obstructions across cases $a=b$ and $a<b$, with explicit descriptions of the orbit spaces. The work highlights the importance of including odd parameters in obstruction theory and shows that the $m>2$ regime presents substantially greater complexity than previously believed.

Abstract

Any supermanifold diffeomorphic to one whose structure sheaf is the sheaf of sections of a~vector bundle over the underlying manifold is called split. Gawȩdzki (1977) and Batchelor (1979) were the first to prove that any smooth supermanifold is split. In 1981, P.~Green, and Palamodov, found examples of non-split analytic supermanifolds and described obstructions to splitness that were further studied by Manin (resp. Onishchik with his students) following Palamodov's (resp. Green's) approach. Following Palamodov, Donagi and Witten demonstrated that some of the moduli supervarieties of superstring theories are non-split. None of the above-mentioned authors considered odd parameters of supervarieties of obstructions to non-splitness. Here, using Palamodov's approach, we classify and describe the even (degree-2) and the odd (degree-1) obstructions to splitness of $(1|2)$-dimensional superstrings. In particular, we correct calculations of degree-2 obstructions due to Bunegina and Onishchik and confirm Manin's answer.

Theorems & Definitions (2)

• proof
• proof : Proof f the lemma. Since $\dim(W\setminus K)=\dim \mathop{\mathrm{{G\space L}}}\nolimits (V)=4$, to describe the stationary subgroup of a point it suffices to prove that the stabilizer of a vector $w\in W\setminus K$ is $\text{St}_{\mathop{\mathrm{{G\space L}}}\nolimits (V)}w=\{\mathop{\mathrm{id}}\nolimits\}~~\text{for any $w\in W\setminus K$}.$ Having selected a basis of $V$ we get an isomorphism $\mathop{\mathrm{{G\space L}}}\nolimits (V)\longrightarrow \mathop{\mathrm{{G\space L}}}\nolimits (2), \ \ g\mapsto A_g.$ Accordingly, fixing the basis $\{e_{ij}:=v_i\otimes u_j\mid 1\leq i,j\leq 2\}$ of $W$ we get a representation $\mathop{\mathrm{{G\space L}}}\nolimits (V)\longrightarrow \mathop{\mathrm{{G\space L}}}\nolimits (4)\simeq \mathop{\mathrm{{G\space L}}}\nolimits (W), \ \ g\mapsto \widetilde{A}_g:=A_g00A_g .$ Let now $w:=(w_{11}, \dots, w_{22})^t$ and $g\in\text{Stab}_{\mathop{\mathrm{{G\space L}}}\nolimits (V)}w$. Then, by \ref{['2']}, we have $\widetilde{A}_g w=w \Longleftrightarrow A_g (w_{11}, w_{12})^t=(w_{11}, w_{12})^t\ \ \text{and}\ \ A_g (w_{21}, w_{22})^t=(w_{21}, w_{22})^t,$ hence $w_1:=(w_{11}, w_{12})^t\ \ \text{and}\ \ w_2:=(w_{21}, w_{22})^t\ \ \text{belong to $\mathop{\mathrm{Ker}}\nolimits (A_g-1_2)$,}$ where $1_2$ is the unit matrix. If $\dim\mathop{\mathrm{Ker}}\nolimits (A_g-1_2)=1$, then the vectors $w_1$ and $w_2$ are collinear, so their coordinates satisfy equation \ref{['eq*']}, i.e., $w\in K$ contrary to our assumption that $w\in W\setminus K$. Hence, $\dim\mathop{\mathrm{Ker}}\nolimits (A_g-1_2)=2$, i.e., $A_g=1_2$, and so equality \ref{['(1)']} is proved. (d) $\dim H_a>2\Longleftrightarrow a>5$. Clearly, the ranks of decomposable tensors $w$ can be equal to either 2 or 1. If the rank is equal to 2, then the non-zero $\mathop{\mathrm{{G\space L}}}\nolimits (V)$-orbits in $V\otimes H_a$ are $\coprod_{V'\in\text{SGr}(2, H_a)} (V\otimes V')^o$, where $(V\otimes V')^o$ is the set of indecomposable tensors in $V\otimes V'$, see case (c). If the rank is equal to 1, then the set of decomposable tensors in $V\otimes H_a$ is the cone over $K^*$, see eq. \ref{['K*']}; the non-zero $\mathop{\mathrm{{G\space L}}}\nolimits (V)$-orbits in this cone form $\{\mathbin{\hbox{\bf.}}\}\times {\mathbb P}(H_a)\simeq {\mathbb P}(H_a)$. (B1) If $d:=b-a>0$, then $\mathop{\mathrm{Hom}}\nolimits({\mathcal{O}}, {\mathcal{O}}(-d))=0$ and $\mathop{\mathrm{End}}\nolimits({\mathcal{O}}(-d)\oplus {\mathcal{O}})\simeq\mathop{\mathrm{End}}\nolimits({\mathcal{O}}(-d))\oplus\mathop{\mathrm{Hom}}\nolimits( {\mathcal{O}}(-d), {\mathcal{O}})\oplus\mathop{\mathrm{End}}\nolimits({\mathcal{O}})\simeq{\mathbb C} \oplus{\mathbb C}^{d+1}\oplus{\mathbb C} .$ The $\text{G}$-action on $\mathcal{E}nd({\mathcal{O}}\oplus {\mathcal{O}}(-d))$ is as follows: $\text{G}\times H_a\oplus H_b \longrightarrow H_a\oplus H_b,\text{G}\times ({\mathbb C}^{a-3}\oplus {\mathbb C}^{b-3})\longrightarrow {\mathbb C}^{a-3}\oplus {\mathbb C}^{b-3}.$ In other words, for $b>a\geq 4$, we have $({\mathbb C}^\times\oplus{\mathbb C}^{d+1}\oplus{\mathbb C}^\times)\times ({\mathbb C}^{a-3}\oplus {\mathbb C}^{b-3})\longrightarrow {\mathbb C}^{a-3}\oplus {\mathbb C}^{b-3},(\lambda, g, \mu)\times (f_{a}, f_{b})\mapsto (\lambda f_{a}, \, gf_{a}+\mu f_{b}).$ Consider the two cases of $\text{G}$-action. (I) $f_{a}=0$. Then, $(0, f_{b})\mapsto (0, \mu f_{b})$ for any $\mu \in{\mathbb C}^\times$, so the $\text{G}$-orbits in $\{0\}\oplus H_b$ form a set isomorphic to ${\mathbb P}(H_b)$. (II) $f_{a}\neq 0$. Then, the stabilizer of the point under the $\text{G}$-action is (clearly, $\lambda=1$) $\mathop{\mathrm{St}}\nolimits_{\text{G}}(f_a, f_{b})= \{(1, g, \mu)\in \text{G}\mid (f_{a}, f_{b})=(f_{a}, gf_{a}+\mu f_{b})\}\simeq \{(g, \mu)\in {\mathbb C}^{d+1}\times {\mathbb C}^\times \mid (1-\mu)f_{b}=gf_{a} \}.$ Let $(f_a, f_{b})\in\Sigma$. Then the relation $(f_{a}, f_{b})=(f_{a}, gf_{a}+\mu f_{b})$ yields $\mathop{\mathrm{St}}\nolimits_{\text{G}}(f_a, f_{b})\simeq {\mathbb C}^\times$. Clearly, $\dim\Sigma=b-2$, see eq. \ref{['Sigma']}, hence (see eq. \ref{['G']}) $\dim (\Sigma/\text{G})= \dim\Sigma-(\dim \text{G} -1)=b-2-(d+2)=a-4.$ In particular, for $a=4$, we have $\Sigma={\mathbb C}^{a-3}\oplus {\mathbb C}^{b-3}$ and $\Sigma/\text{G}=\{\mathbin{\hbox{\bf.}}\}$. For $a>4$ and $(f_a, f_b)\in \overline \Sigma$, see eq. \ref{['Sigma']}, the relation $(1-\mu)f_{b}=gf_{a}$ implies $\mu=1$ and $g=0$, so that $\mathop{\mathrm{St}}\nolimits_{\text{G}}(f_a, f_{b})=\mathop{\mathrm{id}}\nolimits$. Therefore, $\dim (\overline \Sigma/\text{G})=a+b-6-(d+3)=2a-9.$ To describe the set $\overline \Sigma/\text{G}$ explicitly is an interesting open question for experts in algebraic geometry; its study is beyond the scope of this note. No conflict of interest. Data availability: The data used to support the findings of this study are included within the article. Batchelor M., The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979), 329--338Berezin F. A., Introduction to Superanalysis. Edited and with a foreword by A. A. Kirillov. With an appendix by V. I. Ogievetsky. Translated from the Russian by J. Niederle and R. Kotecký. Translation edited by D. Leites. Mathematical Physics and Applied Mathematics, 9. D. Reidel Publishing Co., Dordrecht (1987) xii+424 pp.Berezin F. A., Introduction to Superanalysis. 2nd edition revised and edited by D. Leites and with appendix by D. Leites, V. Shander, I. Shchepochkina "Seminar on Supersymmetries v. 1$\frac{1}{2}$", MCCME, Moscow, (2011) 432 pp. (in Russian) https://staff.math.su.se/mleites/books/berezin-2013-vvedenie-2nd-ed.pdfBunegina V. A., Onishchik A. L., Homogeneous supermanifolds associated with the complex projective line. J. Math. Sci. 82 (1996) 3503--3527Donagi R., Witten E., Super Atiyah classes and obstructions to splitting of supermoduli space. Pure Appl. Math. Q. 9 (2013), 739--788; http://arxiv.org/pdf/1404.6257.Gawȩdzki K., Supersymmetries --- mathematics of supergeometry. Ann. Inst. H. Poincaré, Sect. a (N.S.) 27 (1977), no. 4, 335--366Green P., On holomorphic graded manifolds. Proc. Amer. Math. Soc. 85 (1982), no. 4, 587--590Hartshorn R., Algebraic Geometry, Springer New York, NY, (1977) xvi+496pp.Hazewinkel M., Martin C., A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line. J. Pure and Applied Algebra 25 (1982), 207 -- 211.Leites D., On odd parameters in geometry. J. Lie Theory. 33:4 (2023) 965--1004; http://arxiv.org/pdf/2210.17096Leites D., Bouarroudj S. (eds.) Special issue in memory of Arkady Onishchik. Commun. in Math. 30:3 (2022)Manin Yu. Gauge Field Theory and Complex Geometry. Second edition. Springer-Verlag, Berlin, (1997) xii+346 pp.Okonek Ch., Schneider M., Spindler H., Vector Bundles on Complex Projective Spaces. Corrected reprint of the 1988 edition. With an appendix by S. I. Gelfand. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, (2011) viii+239 pp.Palamodov V.P., Invariants of analytic ${\mathbb Z}_2$-manifolds. Funct. Anal. Appl., 17:1 (1983), 68--69