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Special Holonomy and Two-Dimensional Supersymmetric

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Special Holonomy and Two-Dimensional Supersymmetric Special Holonomy and Two-Dimensional Supersymmetric σ-Models Vid Stojevic arXiv:hep-th/0611255v1 23 Nov 2006 Submitted for the Degree of Doctor of Philosophy Department of Mathematics King’s College University of London 2006 Abstract d = 2 σ-models describing superstrings propagating on manifolds of special holonomy are characterized by symmetries related to covariantly constant forms that these manifolds hold, which are generally non-linear and close in a field dependent sense. The thesis explores various aspects of the special holonomy symmetries (SHS). Cohomological equations are set up that enable a calculation of potential quantum anomalies in the SHS. It turns out that it’s necessary to linearize the algebras by treating composite currents as generators of additional symmetries. Surprisingly we find that for most cases the linearization involves a finite number of composite currents, with the exception of SU(3) and G2 which seem to allow no finite linearization. We extend the analysis to cases with torsion, and work out the implications of generalized Nijenhuis forms. SHS are analyzed on boundaries of open strings. Branes wrapping calibrated cycles of special holonomy manifolds are related, from the σ-model point of view, to the preser- vation of special holonomy symmetries on string boundaries. Some technical results are obtained, and specific cases with torsion and non-vanishing gauge fields are worked out. Classical W -string actions obtained by gauging the SHS are analyzed. These are of interest both as a gauge theories and for calculating operator product expansions. For many cases there is an obstruction to obtaining the BRST operator due to relations between SHS that are implied by Jacobi identities. We relate the problem to extra gauge symmetries that exist when only a subalgebra of the SHS on the same special holonomy background is gauged. Such gauge symmetries are infinitely reducible and don’t imply conserved currents. We propose a solution which avoids the Jacobi problem by gauging a subalgebra of SHS with complex conserved currents that involves spectral flow generators in one direction only, unlike the reducible algebra which involves both. Statement of Original Work Chapters 1 to 4 explain the background to the original work, which is contained in Chapters 5 to 8. The only exception to this is the unconventional gauge fixing procedure for the bosonic string in 4.2 and the related observations in 4.6, which to my knowledge are original contributions. Chapters 5, 6, and 8 contain my personal work, except 6.8 which was done in collaboration with my supervisor Paul Howe. Chapter 7 contains work done in collaboration with Paul Howe and Ulf Lindstr¨om, with each of us contributing roughly an equal amount. The above doesn’t account for the important help I received through personal com- munication. This is hopefully rectified in the Acknowledgements. Contents Chapter 1. Introduction 9 Chapter 2. Geometric Background 15 2.1 Differentialmanifolds ........................... 15 2.1.1 Fiberbundles ......................... 16 2.1.2 n-forms............................. 16 2.1.3 Almost complex and complex structures . 19 2.1.4 Symplecticmanifolds . .. .. .. .. .. .. 21 2.1.5 Connections and holonomy groups . 21 2.2 Riemannian manifolds leading to the Calabi-Yau . ..... 24 2.2.1 (Almost) Hermitian manifolds . 26 2.2.2 K¨ahler and Calabi-Yau manifolds . 27 2.3 Manifolds of special holonomy . 29 2.4 Calibrations................................. 32 Chapter 3. Antifield formalism 35 3.1 Basic objects of interest in the path integral approach toQFT .... 36 3.2 Local, global, and conformal-type symmetries . ...... 37 3.3 Faddeev-Popov gauge fixing of Yang-Mills and BRST symmetry ... 40 3.4 BRSTquantization ............................ 43 3.5 BVquantization .............................. 49 3.5.1 Some technicalities about the properness condition . 54 3.6 Global and conformal-type symmetries . 55 3.7 QFTswithbackgroundfields ....................... 59 3.8 Anomalies.................................. 61 Chapter 4. Two-Dimensional σ-models 65 4.1 Lorentz group representations and spinors in d=2 . ..... 66 4.2 Bosonicstringtheory ........................... 68 4.3 d =2supersymmetryandsuperfields. 74 4.4 The (1, 1) σ-model............................. 76 4.5 Superconformalsymmetries . 78 4.6 An aside on the bi-Hamiltonian formulation of string theory...... 80 Chapter 5. The Algebra of Symmetry Transformations Related to Covari- antly Constant Forms 85 5.1 Geometryandsymmetries......................... 85 5.2 Commutation relations for general L-typesymmetries . 86 5.3 The algebra of L-type transformation via antibrackets . 90 7 Chapter 6. Special holonomy algebras 95 6.1 K¨ahlermanifolds.............................. 96 6.2 Calabi-Yau W -algebras .......................... 97 6.3 The linearized Calabi-Yau algebras . 99 6.3.1 D=2............................... 99 6.3.2 D=5............................... 100 6.3.3 D=4............................... 101 6.3.4 D=3............................... 102 6.4 Hyper-K¨ahler and quaternionic K¨ahler manifolds . ........ 103 6.5 G2 manifolds ................................ 104 6.6 Spin(7)manifolds ............................. 106 6.7 GeneralizedNijenhuisforms. 106 6.7.1 U(m); D = 2m ........................ 107 6.7.2 SU(m); D = 2m ........................ 107 6.7.3 Sp(k); D = 4k ......................... 108 6.7.4 Sp(k) Sp(1); D = 4k ..................... 108 · 6.7.5 Spin(7) and G2 ........................ 108 6.8 Discussion.................................. 108 Chapter 7. Special holonomy σ-models with boundaries 111 7.1 Reviewofbasics .............................. 112 7.2 L-typesymmetriesontheboundary . 115 7.3 Consistency................................. 116 7.4 Targetspacegeometry........................... 117 7.4.1 G2 ................................ 117 7.4.2 SU(3).............................. 119 7.4.3 Spin(7)............................. 120 7.5 Examplesofsolutions ........................... 121 7.5.1 U( n ) U(m) ......................... 121 2 ≡ 7.5.2 SU( n ) SU(m)........................ 122 2 ≡ 7.5.3 G2 ................................ 122 7.5.4 Spin(7)............................. 124 Chapter 8. W -superstrings on manifolds of special holonomy 125 8.1 W -algebras and W -strings......................... 125 8.2 Thegeneralsolution ............................ 126 8.3 W3-string .................................. 129 8.4 W 5 -string.................................. 129 2 8.5 W -superstrings on Calabi-Yau 3-folds . 134 8.6 Discussion.................................. 136 Appendices 139 A Conventions................................. 139 B Calculating the commutation relations in the antifield formalism . 139 C Properties of the Poisson bracket and the antibracket . 140 Bibliography 141 8 Acknowledegments 149 Chapter 1 Introduction The biggest challenge that physics has faced in the recent decades is to make sense of gravity at the quantum level. It is difficult to assess how much progress has been made so far by the two main attempts, string theory and loop quantum gravity. Each certainly has its problems. The problem they both share is that at present they are unable to make any testable physical predictions. Loop quantum gravity has an infinite number of undetermined coefficients in the Hamiltonian constraints [141], and is generally unable to reproduce even the most basic qualities of the continuity of space on the large scale. String theory is formulated in a background dependent way, and at present provides no mechanism for singling out the background that describes our universe. Some estimates [50] of the number of vacua in which the standard model can be realized are of the stupendously high order of 10500, which has recently led many string theorists to resort to the anthropic principle. However, arguments have also been put forward that this number is really vastly lower [163], and that most of the effective field theories we see through our perturbative understanding of string theory are in fact inconsistent. When compared to string theory, loop quantum gravity has the disadvantage that it’s derived in a rather contrived way, and while there certainly are some correct ideas within the formalism, progress is slow. On the other hand, progress in string theory is being made at a fast pace, but admittedly not always in a direction directly relevant for physics. On can argue that any attempt to get a handle on quantum gravity using perturba- tive techniques seems to lead to string theory. General relativity is not perturbatively renormalizable for D > 2, and since the uncontrollable ultraviolet divergences can be related to the point particle description of gravitons it is natural to try and work with relativistic extended objects instead. These can be described by the σ-model action ∂φi ∂φj S = dDσG (φ)γµν , (1.1) ij ∂σµ ∂σν Z where σν are the coordinates and γµν a metric on the D-dimensional worldvolume tra- versed by a (D 1)-dimensional object moving through a target space manifold with − coordinates φi. The target space metric can be expanded in Riemann normal coordinates as i k l Gij(φ)= δij + Rijkl(φ )φ φ + (1.2) 0 · · · i D−2 around some point φ . Since [φ] = , we have that [Rijkl] = D 2. So unless the 0 2 − target space is flat, (1.1) is renormalizable only for D = 2, when it describes a string. Of course, this is only the beginning of the story. The ingredients that fix string theory as a virtually unique theory are supersymmetry and conformal invariance. Supersymme- try is related to
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