ON INTEGRATION AND VOLUMES OF SUPERMANIFOLDS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2021 Thomas M. Honey Department of Mathematics Contents Abstract 6 Declaration 7 Copyright Statement 8 Acknowledgements 9 1 Introduction and Review 10 1.1 The Usual Case . 11 1.1.1 The Grassmannian Manifolds . 11 1.1.2 The Grassmannian as a Homogeneous space . 13 1.1.3 The Unitary Group . 15 1.1.4 The Stiefel Manifolds and the Grassmannian . 16 1.1.5 Other Homogeneous Spaces . 17 1.1.6 Principal Bundle Structures . 18 1.1.7 Volumes of Homogeneous Spaces . 20 1.1.8 Volume of the Unitary Group . 21 1.1.9 Volume of the Flag Manifolds . 23 1.2 The Super Case: A Summary . 23 2 Supermanifolds 27 2.1 Smooth Supermanifolds . 28 2.1.1 Superdomains and Supermanifolds . 30 2.1.2 Integration on Supermanifolds . 36 2.2 The Grassmannian Supermanifold . 39 2.2.1 Complex Supermanifolds . 39 2 2.2.2 The Grassmannian supermanifolds . 41 3 Hermitian Forms and the Kronecker Product 44 3.1 Hermitian Forms . 45 3.1.1 Bilinear forms . 45 3.1.2 Superinvolutions . 48 3.1.3 Sesquilinear forms . 54 3.1.4 Hermitian forms over supercommutative rings . 56 3.1.5 Hermitian Form on hom(U; V )................... 59 3.1.6 Coordinates . 60 3.2 Linear Superalgebra . 69 3.2.1 Vectorisation and the Kronecker Product . 69 3.2.2 Vectorisation . 70 3.2.3 The Kronecker Product . 71 3.3 The Super Case . 74 3.3.1 The Kronecker Product and the Berezinian . 75 3.3.2 Super Vectorisation and the Trace . 82 3.4 The Hermitian Form on hom(U; V ).................... 83 4 The Volume Element 85 4.1 Complex structures and Hermitian manifolds . 86 4.1.1 Cpjq as a Hermitian space . 86 4.1.2 Cpjq the Hermitian supermanifold . 88 4.2 Grassmannian Supermanifolds as Hermitian Manifolds . 90 4.2.1 The Usual Case . 91 4.3 The Super Case and the Volume Element . 95 5 Calculations 99 5.1 The Usual Case . 100 5.2 The Super Case . 104 5.3 1j0 × (p + 1)jq ................................ 105 5.4 0j1 × pj(q +1)................................ 105 pjp 5.5 Grrjs(C ).................................. 105 3 5.6 r; s > 0; q < r ................................ 106 2j1 5.7 Gr1j1(C ).................................. 107 3j1 5.8 Gr1j1(C ).................................. 108 3j1 5.9 Gr2j0(C ).................................. 109 2j1 5.10 Gr2j0(C ).................................. 111 3j1 5.11 Gr2j1(C ).................................. 112 3j2 5.12 Gr2j0(C ).................................. 112 3j2 5.13 Gr1j1(C ).................................. 114 3j2 5.14 Gr2j1(C ).................................. 115 3j2 5.15 Gr2j2(C ).................................. 115 3j2 5.16 Gr3j1(C ).................................. 116 4j2 5.17 Gr2j0(C ).................................. 116 4j2 5.18 Gr1j1(C ).................................. 119 6 Conclusions and Discussion 120 Bibliography 124 A An Introduction to Superalgebra and on Conventions 128 A.1 Superalgebra . 129 A.1.1 Superrings . 129 A.1.2 Super Vector Spaces . 132 A.1.3 Modules over supercommutative rings . 135 A.2 Free Finitely Generated Modules . 139 A.2.1 Duality and the Tensor Product . 141 A.2.2 The Double Dual . 141 A.2.3 Trace . 142 A.2.4 The Berezinian . 143 A.2.5 Canonical Ideal of a Superring . 144 A.2.6 The Berezinian Module . 144 A.3 Coordinates . 145 A.3.1 The Dual Space . 147 A.3.2 Scalar Multiplication and Matrices . 150 4 A.3.3 The Double Dual . 151 A.3.4 The Trace . 151 A.3.5 The Berezinian . 153 Word count 36798 5 The University of Manchester Thomas M. Honey Doctor of Philosophy On Integration and Volumes of Supermanifolds January 11, 2021 In this thesis we investigate the volumes of certain supermanifolds. The volumes of supermanifolds have been studied before in particular in [1]. This thesis builds on that work. We develop the necessary tools to study mainly the volume of the complex Grassmannian supermanifolds. In the first two chapters we review the problem and how it has been solved for ordinary Grassmannian manifolds. We contrast that with the super case and then introduce briefly what a supermanifold is and give an exposition on what integration entails in the super case. In the third chapter we develop the tools we need to calculate the volume of the Grassmannian supermanifolds as Hermitian supermanifolds. We develop Hermitian forms in the super case and we conclude that the natural Hermitian form on the space of matrices isn't positive definite. We then develop the Kronecker product and vectorisation in the super case. With these developed we show the relation between the Berezinian, or superdeterminant, and the Kronecker product. In the fourth chapter we investigate what the volume element of the Grassmannian supermanifolds coming from a natural Hermitian form is and apply the results of the previous chapter so that we can calculate it. In the fifth and last chapter of the main part of the thesis we calculate the volume of the Grassmannian supermanifolds for different values of the relevant parameters. In [1] there is a conjectured formula for the volume of the Grassmannian supermanifolds and we contrast our results with that. We have provided an appendix on superalgebra to provide a guide on the conven- tions and notations used in the main text. 6 Declaration No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 7 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intel- lectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproductions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) and in The University's Policy on Presentation of Theses. 8 Acknowledgements I would first like to thank my supervisors Hovhannes Khudaverdian and Theodore Voronov for their support over the years that it has taken to produce this thesis and for their help to further my mathematics education. I would secondly like to thank all the other mathematics teachers that have con- tributed to my education including others at the University of Manchester, those at the University of Aberdeen where I completed my undergraduate degree in mathematics and finally my teachers at school. I would also like to thank my friends, office mates and the mathematics postgradu- ate students in Manchester in general. They made my time there especially enjoyable. Finally I want to thank my family who have always been there to support me. 9 Chapter 1 Introduction and Review 10 1.1. THE USUAL CASE 11 We are primarily concerned with calculating the volumes of certain Hermitian su- pjq permanifolds. In particular the Grassmannian supermanifolds, Grrjs(C ), the com- plex supermanifolds of rjs planes in Cpjq. The volume of the simplest of these super- manifolds, the complex projective superspaces CPpjq was obtained in [1]. A conjecture pjq on what the volume of Grrjs(C ) was made in that paper and the following is an pjq investigation of whether or when this conjecture holds. Grrjs(C ) are the "super" n version of the complex manifolds Grk(C ), the space of k dimensional subspaces of Cn. In addition to [1], recently the question for the volume of symplectic superman- ifolds was considered in [2]. There a general formula for the volume of symplectic supermanifolds is obtained however we don't use the methods derived from there. In n this chapter we will give the exposition on working with Grk(C ) and how one usually n obtains the volume of Grk(C ) which we'll call the usual case. Then we will contrast pjq that with what happens in the super case when one wants to consider Grrjs(C ). The exposition on the super case will be brief in order to highlight the contrast. Details and precise definitions on supermanifolds will be given in the following chapter. 1.1 The Usual Case 1.1.1 The Grassmannian Manifolds As we stated above, before we embark on laying out the situation for the super case, we will give a detailed summary of what we can say about the volumes of the ordinary Grassmannian manifolds.