Topics on Topology and Superstring Theory Fabio Ferrari Ruffino Ph.D. Thesis International School for Advanced Studies (SISSA/ISAS) Via Beirut 2, I-34151, Trieste, Italy Mathematical Physics sector Supervisors Referee Prof. Loriano Bonora Prof. Daniel S. Freed Prof. Ugo Bruzzo Abstract In this thesis we discuss some topics about topology and superstring backgrounds with D-branes. We start with a mathematical review about generalized homology and cohomology theories and the Atiyah-Hirzebruch spectral sequence, in order to provide an explicit link between such a spectral sequence and the Gysin map. Then we review the basic facts about line bundles and gerbes with connection. In the second part of the thesis we apply the previous material to study the geometry of type II superstring backgrounds. We first present the cohomological discussion about D-brane charges in analogy with classical electromagnetism, then we use the arXiv:0910.4524v2 [math-ph] 19 Mar 2013 geometry of gerbes to discuss the nature of the A-field and the B-field as follows from the Freed-Witten anomaly, finally we discuss the K-theoretical approaches to classify D-brane charges. In the last part we discuss some topics about spinors and pinors with particular attention to non-orientable manifolds. ferrariruffi[email protected] Acknowledgments I would like to thank Loriano Bonora for having introduced me to string theory, and I thank both him and Raffaele Savelli for all the time spent working together. I also thank Ugo Bruzzo for his help with scientific discussions and in organizational matters. I am also really grateful to Jarah Evslin for all the topics he taught me about string theory and for having always replied with patience and precision to all my questions. Moreover, a special thank to all my friends in Trieste for these four years in which I never felt alone even for a moment. Introduction In this thesis we discuss some topics about topology and superstring backgrounds with D-branes. The idea is not to solve one single problem, but to deal with the topo- logical and geometrical nature of some “protagonists” of superstring backgrounds, mainly for type II theories, solving the problems that appear while describing them. Actually, it happens that some topological statements can be generalized in a more abstract context, so that we are led to deal with purely mathematical topics, inde- pendently of their physical meaning. That’s why the first two parts are dedicated to algebraic topology, beyond the physical usefulness of the subject, while in the third part we specialize to the string theory context. The forth part deals with pinors and spinors. Although it may not be evident reading this work, we used Freed-Witten anomaly as a “guide” [31, 20]. In fact, one of the main physical topics is the use of K-theory to study D-brane charges [51, 52, 66], which is strongly motivated by Freed-Witten anomaly [47, 24], since, as we will see, one advantage of K-theory with respect to ordinary cohomology is that we can use the Atiyah-Hirzebruch spectral sequence to cut anomalous world-volumes. There are two main approaches in the literature, i.e. the Gysin map and the Atiyah-Hirzebruch spectral sequence, and we solved the problem of linking these two classifications of the same objects. Since Atiyah- Hirzebruch spectral sequence works for any cohomology theory (not only K-theory), we extended the result to the axiomatic setting. That’s why in the first part we start with a description of homology and cohomology theories in general [23, 65, 35]. Moreover, the Freed-Witten anomaly is a world-sheet anomaly which can be seen from the superstring action, and to state and study it we are forced to consider the geometrical nature of the terms of this action, in particular of the A-field and the B-field. So the Freed-Witten anomaly is at the origin of the other physical topic, the geometrical classification of A-field and B-field configurations which are anomaly- free, using the geometry of gerbes. That’s also the reason why in the second part of the thesis we give a description of line bundles and gerbes with the language of Deligne cohomology [14]. Finally we discuss some topics about pinors and spinors, considering in particular the problem of linking pinors on a non-orientable manifold and spinors on its orientable double-covering. The work is organized as follows: in part I we deal with the foundations of algebraic topology, discussing gen- • eralized homology and cohomology theories and presenting Borel-Moore ho- mology, which we will need to deal with D-brane charges; moreover, we give a 5 6 complete presentation of the topic of spectral sequences, in order to describe the Atiyah-Hirzebruch one, since it will be useful for physical applications; as we already said, the setting of this part is more general than the one needed in string theory; in part II we review the theory of line bundles and gerbes with connection, • using the language of Deligne cohomology; in part III we apply part of the topological preliminaries to study superstring • backgrounds with D-branes; the main topics are the cohomological descrip- tion of D-brane charges in analogy with classical electromagnetism, the K- theoretical classification of D-brane charges, and the description and classifi- cation of A-field and B-field configurations that are anomaly-free; in part IV we discuss pinors and spinors, with particular attention to non- • orientable manifolds. The main results of the thesis are: an explicit link between the Gysin map and the Atiyah-Hirzebruch spectral • sequence for any cohomology theory, which, for the particular case of K- theory, shows how to link the two main K-theoretical classifications of D-brane charges; the use of the relative Deligne cohomology to classify the allowed configurations • of the A-field and the B-field, showing in particular the different nature of the gauge theories on a D-brane; showing the appropriate version of Borel-Moore homology in order to describe • D-brane charges from a cohomological point of view, considering then the analogous discussion within the K-theoretical viewpoint; the explicit link between pinors on a non-orientable manifold and spinors on • its orientable double-cover which are invariant by sheet-exchange. The presentation of all the topics involved is just partial and should be completed in different directions. One relevant example is the fact that, describing D-brane charges via K-theory and cohomology, we always consider the case of vanishing H-flux, both as a form (i.e. as a gerbe curvature) and as an integral cohomology class (i.e. as the first Chern class of a gerbe). We consider the full generality only describing the nature of the A-field and the B-field. When the H-flux is turned on, K-theory and de-Rahm cohomology must be replaced by their twisted version [42, 66]. We will consider in future works these more general situations. P.S. The results of the thesis are contained in the following articles, that I quote from the bibliography: [9] L. Bonora, F. Ferrari Ruffino and R. Savelli, Classifying A-field and B-field • configurations in the presence of D-branes, JHEP 12 (2008) 78, arXiv:0810.4291; 7 [10] L. Bonora, F. Ferrari Ruffino and R. Savelli, Revisiting pinors, spinors and • orientability, Bollettino U.M.I. (9) 5 (2012), expanded version on arXiv:0907.4334; [26] F. Ferrari Ruffino, Gysin map and Atiyah-Hirzebruch spectral sequence, • Bollettino U.M.I. (9) 4 (2011), expanded version on arXiv:0904.4103; [27] F. Ferrari Ruffino, Topics on the geometry of D-brane charges and Ramond- • Ramond fields, JHEP 11 (2009) 012, arXiv:0909.0689; [28] F. Ferrari Ruffino and R. Savelli, Comparing two different K-theoretical • classifications of D-branes, Journal of Geometry and Physics 61 (2011) pp. 191-212, arXiv:hep-th/0805.1009. Such articles are almost entirely reproduced in this thesis. I will explicitly cite them when they appear in the following. 8 Contents I Homology and cohomology theories 15 1 Foundations 17 1.1 Preliminaries . 17 1.1.1 Singular homology and cohomology . 17 1.1.2 Borel-Moore homology and cohomology with compact support 18 1.1.3 CW-complexes .......................... 24 1.1.4 Simplicial complexes . 25 1.1.5 Categoriesoftopologicalspaces . 27 1.1.6 Basic operations on topological spaces . 28 1.2 Eilenberg-Steenrodaxioms . 29 1.2.1 Reducedhomologyandcohomology . 31 1.2.2 Firstproperties .......................... 33 1.2.3 Borel-Moore homology and cohomology with compact support 35 1.2.4 Multiplicative cohomology theories . 35 1.3 ThomisomorphismandGysinmap . 37 1.3.1 Fiberbundlesandmodulestructure. 37 1.3.2 Orientability and Thom isomorphism . 39 1.3.3 Gysinmap............................. 41 1.4 FiniteCW-complexes........................... 42 1.4.1 Whiteheadaxioms ........................ 42 1.4.2 S-Duality ............................. 43 1.4.3 Extension ............................. 44 2 Spectral sequences 47 2.1 Generalsetting .............................. 47 2.2 Finitefiltrations.............................. 49 2.2.1 Preliminaries . 49 2.2.2 First viewpoint . 50 2.2.3 Secondviewpoint ......................... 58 2.3 Gradinganddoublecomplexes. 64 2.3.1 Grading and regular filtrations . 64 2.3.2 Doublecomplexes......................... 69 2.4 Generalization............................... 72 2.4.1 Cohomologyofthequotients. 72 2.4.2 Axiomatization . 77 2.4.3 Genericcohomologytheory . 79 9 10 CONTENTS 3 Atiyah-Hirzebruch spectral sequence 81 3.1 Descriptionofthespectralsequence . 81 3.1.1 Thefirststep ........................... 82 3.1.2 Thesecondstep.......................... 83 3.1.3 Thelaststep ........................... 84 3.1.4 Fromthefirsttothelaststep . 84 3.2 Gysin map and Atiyah-Hirzebruch spectral sequence . .. 85 3.2.1 Unitclass ............................. 87 3.2.2 Generic cohomology class . 91 4 K-theory 93 4.1 BasicnotionsofK-theory .