et al. and Christian Sämann
The Lab Notebook Literature references, Definitions, Theorems, Conventions
Version April 22, 2019
These notes are a collection of literature references relevant to my research and useful as a reference for my PhD and summer project students. The level of explanation ranges from very basic to very advanced. Mostly, I’m bad at keeping notes, and this is material that I need repeatedly for references or didn’t fit in any of my papers. The style is certainly more colloquial than in proper papers, these are notes after all. Some paragraphs may be copied literally from some of my papers with other authors. There- fore, these notes are by no means meant to be original. They should not be cited nor should there be any credit attributed to them. “Everything is in flow,” and in particular these notes are. Statements in this document might be incomplete, contain sloppy formulations or even errors. Sometimes they may be very old and then reflect an outdated understanding of mine. Also, conventions and notations may jump between and even within sections. Therefore, use everything contained in here with great care and consult the original literature. Finally, some sections of these notes may require serious tiding.
Contents
1 Introductory remarks7
2 Foundations 9 2.1 Set theory ...... 9 2.2 Homotopy type theory ...... 9 2.3 Category theory and higher category theory ...... 10 2.3.1 Ordinary category theory ...... 10 2.3.2 Higher category theory ...... 10 2.3.2.1 ∞-categories ...... 11 2.3.2.2 Categorification ...... 11 2.3.3 Other ...... 11
3 Algebra 13 3.1 Group like objects ...... 13 3.1.1 Group theory ...... 13 3.1.2 Groupoids ...... 13 3.1.3 Higher groupoids and groups ...... 13 3.1.3.1 2-groups ...... 13 3.1.3.2 Higher than 2 ...... 14 3.1.3.3 String and Fivebrane groups ...... 15
3.2 L∞-algebroids and related ...... 21 3.2.1 Homotopy Algebras ...... 21 3.2.2 Differential graded algebras ...... 21 3.2.2.1 NQ-manifolds ...... 21 3.2.3 Lie 2-algebras ...... 21 3.2.3.1 Strict Lie 2-algebras ...... 22 3.2.3.2 Semistrict Lie 2-algebras ...... 22
3.2.4 L∞-algebras ...... 23 3.2.4.1 Loop L∞-algebras ...... 25 3.2.5 L∞-algebroids ...... 25 3.2.5.1 Lie algebroids ...... 25 3.2.5.2 Courant algebroids ...... 25 3.2.5.3 Higher Lie n-algebroids ...... 26 3.2.6 2-Crossed modules of Lie algebras ...... 26 3.2.7 Other ...... 26 3.3 Representation theory ...... 27 3.4 Algebraic Topology ...... 27 4 Geometry 29 4.1 Generalities ...... 29 4.1.1 Manifolds ...... 29 4.1.1.1 Spheres ...... 29 4.1.1.2 Loop spaces ...... 30 4.1.2 Differential geometry ...... 31 4.2 Symplectic and Poisson geometry ...... 31 4.2.1 Basics ...... 31 4.2.2 Multisymplectic geometry ...... 31 4.2.3 Poisson and Nambu–Poisson geometry ...... 31 4.3 Complex geometry ...... 32 4.3.1 Calabi–Yau spaces ...... 32 4.3.2 Twistor geometry ...... 32 4.4 Graded geometry ...... 33 4.4.1 Graded vectorspaces ...... 33 4.4.2 N-manifolds ...... 33 4.4.3 Supermanifolds ...... 33 4.4.4 Generalized geometry ...... 33 4.5 Fiber bundles ...... 33 4.5.1 Tangent and cotagent bundles ...... 33 4.5.2 Jet bundles ...... 35 4.5.3 Principal bundles ...... 36 4.5.3.1 Ordinary principal bundles ...... 36 4.5.3.2 Higher principal bundles ...... 36 4.5.3.3 Connections and related ...... 37 4.5.3.4 Characateristic classes ...... 38 4.5.4 Other fibrations ...... 38 4.6 Stacks ...... 38 4.6.1 Lie groupoids and Stacks ...... 38 4.6.2 ∞-stacks ...... 39 4.7 Other areas ...... 39
5 Other areas 41 5.1 Arithmetic ...... 41 5.2 Analysis ...... 41 5.3 Supermathematics ...... 41 5.3.1 Supergeometry ...... 41
6 Classical and Quantum Mechanics 43 6.1 Classical Mechanics ...... 43 6.2 Quantum Mechanics ...... 43 6.2.1 Noncommutative geometry ...... 44 6.2.1.1 Fuzzy geometry ...... 44 6.2.1.2 Higher quantization ...... 44
7 Field Theory 47 7.1 Classical Field Theory ...... 47 7.1.1 Spin and all that ...... 47 7.1.2 Examples of field theories ...... 47 7.1.2.1 General gauge theories ...... 47 7.1.2.2 Yang–Mills theory ...... 47 7.1.2.3 Yang–Mills–Higgs theory ...... 47 7.1.2.4 Chern–Simons theory ...... 47 7.1.2.5 BF-type theories ...... 48 7.1.2.6 General higher gauge theories ...... 48 7.1.2.7 (1,0)- and (2,0)-theory ...... 48 7.1.2.8 General Relativity ...... 48 7.1.3 Kaluza–Klein reduction ...... 48 7.1.4 Classical Integrability ...... 49 7.1.4.1 Integrable field equations ...... 49 7.1.4.2 Penrose–Ward transform and related ...... 49 7.1.4.3 ADHM-construction ...... 50 7.1.4.4 ADHMN-construction ...... 50 7.2 Quantum Field Theory ...... 50 7.2.1 Basics ...... 50 7.2.1.1 Scattering amplitude relations ...... 51 7.2.2 BV formalism ...... 51 7.2.3 Anomalies ...... 52 7.3 Supersymmetry ...... 52 7.3.1 Representations ...... 53 7.3.1.1 Explicit representations ...... 53 7.3.1.2 General statements ...... 53 7.3.2 Superfields ...... 53 7.3.2.1 Superfields for 4 supercharges ...... 53 7.3.2.2 Projective superspace ...... 54 7.3.2.3 Harmonic superspace ...... 54 7.3.3 SUSY field theories ...... 54 7.3.3.1 Super Yang–Mills theories ...... 54 7.3.3.2 M2-brane models: Supersymmetric Chern–Simons matter theories . . . 54 7.3.3.3 Supergravities ...... 55 7.3.4 Other topics ...... 55 7.4 Matrix models ...... 55 7.4.1 Single Matrix Model ...... 55 7.4.1.1 The Hermitian matrix model ...... 55 7.4.2 Hermitian matrix models from scalar field theories on fuzzy spaces ...... 55 7.4.3 String theory inspired Matrix Models ...... 55 7.4.3.1 c < 1 string theory ...... 55 7.4.3.2 BFSS model ...... 55 7.4.3.3 IKKT model ...... 55 7.4.3.4 Higher Lie algebra models ...... 55 7.4.3.5 Other generalizations ...... 56
8 String Theory 57 8.1 Basics ...... 57 8.1.1 T-duality ...... 57 8.1.1.1 Double Field Theory ...... 58 8.1.2 Other dualities ...... 59 8.2 The five superstring theories ...... 59 8.2.1 Type I strings ...... 59 8.2.2 Type II strings ...... 59 8.2.3 Heterotic strings ...... 59 8.3 D-branes ...... 59 8.3.1 D-brane configurations ...... 60 8.3.1.1 Chalmers–Hanany–Witten configurations ...... 60 8.4 String Field Theory ...... 60 8.5 M-theory ...... 60 8.5.1 M2-branes ...... 60 8.5.1.1 Worldvolume ...... 60 8.5.1.2 Target space ...... 60 8.5.2 M5-branes ...... 61 8.5.3 Worldvolume ...... 61 8.5.4 Target space ...... 61 8.5.5 U-duality and EFT ...... 61 8.5.6 Exceptional Field Theory ...... 61
9 Tables 63 9.1 Formulas ...... 63 9.1.1 Various symbols, functions, etc ...... 63 9.1.2 Approximations ...... 63 9.1.3 Expressions ...... 63 9.1.4 Formulas for explicit expressions ...... 65 9.1.5 Integrals ...... 66 9.1.5.1 Elliptic integrals ...... 66 9.1.6 Calculational rules ...... 66 9.1.7 Computational tricks ...... 67 9.1.8 Conventions and identities ...... 67 9.2 Other resources ...... 69 9.2.1 Quotes etc...... 69 9.2.2 Webpages ...... 69 9.2.3 Books ...... 69
Bibliography 71 Chapter 1
Introductory remarks
An excellent source for definitions and theorems on topics touched upon in these notes is the nlab. These notes here are not an attempt to reproduce the nlab, but rather an effort to collect defini- tions/understanding/ literature references to various parts of mathematics in a way that is sometimes different, sometimes more detailed, sometimes a little bit more down-to-earth, and sometimes a little more physicsy than the nlab. In particular, I need a resource that my students and I can use as an easily accessible reference for looking up definitions, results, the relevant literature as well as small computations.
Classification of topics The classification of topics is rather arbitrary and tries to make the best out of a one-dimensional system. It is heavily centred on my own research. Hyperlinks between the topics can hopefully remedy this shortcoming.
Acknowledgements I am very grateful to all my past collaborators and students for interesting discussions, sharing their wisdom, joint suffering through disappointments and in general patience with my often overly optimistic ideas.
To-do-list ◦ Turn into literature references ◦ Collect more references from papers 8 Introductory remarks Chapter 2
Foundations
Three main potential foundations for mathematics:
◦ Set theory, the classical approach. Russel paradox etc. in naive formulation, extendable to Zermelo–Fraenkel set theory ◦ Type theory, and in particular Homotopy Type Theory ◦ Category theory and Higher Category Theory
2.1 Set theory
No one will drive us from the paradise which Cantor created for us. David Hilbert (1862–1943)
§1 Motivation. One of the most elementary notions when developing Philosophy and in particular Mathematics is that of a distinction. For example, we can separate objects having property x from those not having property x, thereby defining a class of mathematical objects. On the other hand, we can split all possible statements into true statements and false statement. For a detailed exploration of this concept, see e.g. G. Spencer-Brown’s “Laws of Form”. A nice way of encoding distinctions is via sets.
2.2 Homotopy type theory
§1 Idea. Type theory was initiated by Bertrand Russell to overcome problems with set theory, such as Russell’s paradox. In type theory, every term has a type and certain operations are only allowed form terms of a specified type. §2 Homotopy type theory. Based on the observation that relevant types in type theory come with additional structure reminiscent of discussions in homotopy, homotopy type theory (HoTT) is now re- garded as the future foundation of mathematics. A key figure in the development of HoTT is Vladimir Voevodsky and a good source for background is [The13]. §3 Links. Type theory also allows us to establish a bridge to computer science via the Curry–Howard correspondence: Proofs, mapping a propositions to a propostion are essentially computer programs, mapping a type to another type. Indeed, proof assistants such as Coq can indeed be used to establish correctness of computer programs. 10 Foundations
2.3 Category theory and higher category theory
2.3.1 Ordinary category theory
§1 Literature. Very quick summary: [JSSW19, Section 3.1]. Quite introductory: [Lei14]. Standard reference, but mostly useless for learning is Mac Lane’s book [ML98]. For physicists: [CP]. Very introductory: [LS97]. §2 What is it good for? link §3 Generalities. Category theory is a meta-mathematical framework that bundles objects and maps between these within a single entity, a category. From this point of view it is possible to absorb many different mathematical structures in a common abstract notion. It also stresses the importance of maps between objects over objects themselves. Most importantly for applications in string theory is its vast power to provide immediate well-motivated and mathematically sound generalizations of traditional mathematical objects. For example, string theory requires various extensions of gauge theories describing the parallel transport of point particles in smooth manifolds to a parallel transport of extended objects on noncommutative spaces. Reformulating conventional gauge theories in a category theoretic language then allows us to directly extract appropriate such generalizations. §4 Quotes
◦ “We will need to use some very simple notions of category theory, an esoteric subject noted for its difficulty and irrelevance.” G. Moore and N. Seiberg, [MS89, Appendix C] ◦ “We’ll only use as much category theory as is necessary. Famous last words...” Roman Abramovich ◦ “Category theory is the subject where you can leave the definitions as exercises.” John Baez
§5 Enriched categories. Enriched categories (monoidal etc.) [Kelb].
2.3.2 Higher category theory
§6 Literature. Outline of higher structures: [JSSW19, Section 3]. n-categories: [Bae]. Higher categories and operads: [Lei03]. Bicategories: [Bena]. Categorification: [BD], good review with many details: [Bor]. §7 Ideas. In a set, two objects are identical or different. In a category, they can also be isomorphic. Morphisms, however, form sets. To allow for isomorphic morphisms, one introduces 2-categories, and iteratively n-categories [Bae]. There are various forms:
◦ Strict higher categories: mostly harmless and less useful, see enriched categories. ◦ Weak higher categories: cumbersome to formulate. Bicategories ok, tricategories are barely doable, tetracategories shouldn’t be touched if you want to keep your sanity. ◦ Better: the general approach via Kan simplicial sets, ∞-categories ◦ Operads: still need to learn this stuff... From here: [BD98]?
§8 Motivation 2-categories. Eckmann and Hilton argument: [EH62] §9 Strict 2-category. A category enriched over the cartesian monoidal category. For each pair of objects in the category, there is a category of morphisms. Appendix in [SW13b]. §10 Bicategory. Original definition: [Bénb]. Good introduction: [Lei98] and in particular [Lei03]. Concise but detailed review [JSW15, Section 2.1]. 2.3 Category theory and higher category theory 11
2.3.2.1 ∞-categories §11 Literature. Detailed and elementary review: [JSW16, Section 2], in particular [BD98]. §12 Simplicial set. Literature: [May93, GJ99], also: “A leisurely introduction to simplicial sets” online §13 Duskin nerve. Nerve of a weak 2-category, weak 2-category as ∞-category [Dus02, Section 6]
2.3.2.2 Categorification §14 Internalization, vertical horizontal categorification. §15 Categorification of vector space. Categorifying a vector as a column of numbers, one arrives at 2-vector spaces as defined by Kapranov and Voevodsky [KV94]. More useful seems to be to categorify the notion of a direction in space, leading to the 2-vector spaces of Baez and Crans [BC04]. Detailed discussion in [BSS18].
2.3.3 Other §16 Multicategories. A multicategory is similar to an ordinary category, but the domain of the arrows is not a single object, but a finite sequence of them. [Lei03]. This goes then in the direction of operads etc. 12 Foundations Chapter 3
Algebra
3.1 Group like objects
3.1.1 Group theory
§1 The second order Casimir operator on irreducible representations of su(n). See [Gro93]. §2 SU(N) orthogonality relation For SU(N), we have the relation
Z 1 dµ (Ω) [ρ(Ω)] [ρ†(Ω)] = δ δ , (3.1.1) H ij kl dim(ρ) il jk where Ω ∈ SU(N), ρ is a finite-dimensional, unitary, irreducible representation and ρ† denotes its complex 1 R conjugate. The measure dµH (Ω) is the Haar measure on SU(N) normalized according to dµH (Ω) = 1. Proof in [Mur]. §3 Loop groups [Fre88, PS88]
3.1.2 Groupoids
§4 Literature. Useful book: [Mac05], see also [Mac87].
3.1.3 Higher groupoids and groups
§5 2-groupoids. 2-groupoids and other 2-things: [Noo07]. Lie 2-groupoids see also [JSW15, Section 2.2]. Bigroupoids: [HKK01] §6 Applications. Infinite-dimensional Lie algebras, that cannot be integrated to a Lie group might be integrated to a Lie 2-group [Woc11a].
3.1.3.1 2-groups
§7 Literature. Definition and exploration: [BL04], categorification: [FB]. Crossed modules [Bro, p. 102-104] and [ML]. §8 Idea. Several descriptions:
◦ As crossed modules of Lie groups, is equivalent to strict Lie groups ◦ As coherent Lie 2-groups (not quite the weakest possible definition) [BL04] ◦ As general weak 2-groups [BL04], see also references in [JSW15] (or Lie 2-groupoid with one object). ◦ As smooth 2-groups [Sch11], see also [DS17]. As shown in [Zhu09b], this definition is equivalent to the canonical definition of a Lie 2-group in terms of simplicial manifolds.
1i.e. the measure which is invariant under left or right multiplication by ρ(Υ), Υ ∈ SU(N) 14 Algebra
◦ Kan simplicial manifolds with single 0-simplex, cf. Zhu’s work and [Hen08].
§9 Equivalence strict 2-groups and crossed modules. It has been shown [BL04] that these strict 2-groups are categorically equivalent to crossed modules of groups. The latter consist of a pairs of groups H, G together with homomorphisms ∂ : H → G and actions B: G n H → H satisfying
−1 ∂(g B h) = g B ∂(h) , ∂(h1) B h2 = h1h2h1 (3.1.2) for all g ∈ G and h, h1,2 ∈ H. To reconstruct the corresponding strict 2-group, put
G0 = G , G1 = G n H , s(g, h) = g , t(g, h) = ∂(h)g , id(g) = (g, 1) , g1 ⊗ g2 = g1g2 , (g1, h1) ⊗ (g2, h2) = (g1g2, h1(g1 B h2)) , (3.1.3) (∂(h1)g, h2) ◦ (g, h1) = (g, h2h1) .
Conversely, a crossed module of groups is derived from the strict 2-group G by putting2 H = ker(s) and G = G0. §10 Remarks on crossed modules of groups. Note that if G = 1, then H is necessarily abelian because of the Peiffer identity.
Note also that the kernel of t is contained in the center of H: Consider a h ∈ H with t(h) = 1G. Then ˜ −1 ˜ 1 ˜ ˜ ˜ hhh = t(h) B h = G B h = h for any h ∈ H. §11 Classification of crossed moduls. Crossed modules classified via central extensions of groups. We consider the exact sequence
1 → ker(t),→H −−−→t G → coker(t) → 1 . (3.1.4)
We assume that ker(t), G and H are connected Lie groups. Note that ker(t) is necessarily abelian, see §10. The pair (H, t) describes an extension of ker(t) by G. Two extensions (H1, t1)(H2, t2) are equivalent, if there is an isomorphism π : H1 → H2, which preserves ker(t) and satisfies t1 = t2π. Classification of such group extensions in [Hoc51]. Statements: t is essentially id+discrete+abelian(?)
3.1.3.2 Higher than 2 §12 Semistrict 3-group, Gray group. [MP11] §13 2-crossed modules of Lie groups. [Con84], also given in [Por], [MP11]. Used in [Jur11b, SW14b]. More on Peiffer lifting, additional useful formulas etc. [BG89, MP11]. §14 Crossed squares and semistrict 3-groups. [Con03] and [MP]. §15 Smooth ∞-group. This is a smooth ∞-groupoid with one object? This is a smooth groupal A∞-space. See 1011.4735. They are to L∞-algebras what groups are to Lie algebras. Note also: an L∞-algebra can be integrated to a smooth ∞-groupoid. [Hen08] §16 Lie n-groupoid. Simplicial form as Kan simplicial set with single 0-simplex [Hen08, Get09, Zhu09a] see also [MT11]. §17 Idea. Just as Lie groups and Lie groupoids can be differentiated to Lie algebras and Lie algebroids, one can differentiate a Lie ∞-groupoid to an L∞-algebra via the Lie functor. §18 Via functors of descent data. An integration of L∞-algebras can be performed [Hen08], but the procedure is very cumbersome. As always, differentiation is easier than integration, and we therefore start with a Lie 2-group G . An n-POV on the Lie algebra Lie(G) of a Lie group G was suggested by Ševera [Sevb]. In this picture, we consider the functor that maps supermanifolds X to descent data for principal G-bundles subordinate to the surjective submersions X × R0|1 → X. As a vector space, the Lie algebra is recovered as the moduli space of such functors. Moreover, its Chevalley–Eilenberg
2look up “Moore complex” 3.1 Group like objects 15 differential is obtained as the action of one of the generators of Hom (R0|1, R0|1) on this moduli space. This description readily categorifies and in his paper [Sevb], Ševera discusses the differentiation of what one calls (∞, 1)-groups.
§19 Lie differentiation. An ∞-groupoid can be Lie differentiated to an L∞-algebroid. A way of differentiating a ∞-group to an L∞-algebra was given in [Sevb]; a detailed explanation is given in [JSW16]. Another approach to Lie differentiation is to take the simplicial description of the (∞, 1)- groupoid to differentiate, apply the tangent functor and construct an NQ-manifold from the pullback along the alternating sum of the differentials of the face maps, see [Sch13][v2, p. 30]. §20 Via tangent functor on simplicial set. Another approach to Lie differentiation is to take the simplicial description of the (∞, 1)-groupoid to differentiate, apply the tangent functor and construct an NQ-manifold from the pullback along the alternating sum of the differentials of the face maps, see [Sch13][v2, p. 30]. §21 Better techniques. Instead of working with intricacies of higher groups and higher functors, use simplicial groups as a description of strict higher groups!
3.1.3.3 String and Fivebrane groups
§22 General procedure. String groups are generalization of Spin groups. Recall that the spin group of O(p, q) arises by going to a double cover of the original group. This implies that we switch to a group ∼ Spin(p, q) that is fiberd over O(p, q) and whose first homotopy group vanishes: π1(O(p, q)) = Z2 for p, q ∼ large enough, while π1(Spin(p, q)) = 0. We can apply the same procedure to eliminate higher homotopy ∼ groups. The next one to eliminate is π3(SO(p, q)) = Z, which gives rise to the string group. Finally, it remains to eliminate π7(SO(p, q)), which gies rise to the fivebrane group. Note that there are lost of models for string groups. Moreover, because every finite-dimensional Lie group G has π3(G) = Z, it is clear that String(p, q) is not a finite-dimensional Lie group. The string group String(n) is a 3-connected cover of the spin group Spin(n). It fits within the Whitehead tower of the orthogonal group O(n). Recall that the Whitehead tower over a space X consists of a sequence of spaces
µ µ µ µ µ ∗ → ... −−−−→i+1 X(i) −−→i ... −−−→3 X(2) −−−→2 X(1) −−−→1 X, (3.1.5)
(i) where the maps µi induce isomorphisms on all homotopy groups in degree k ≥ i and πj(X ) = 0 for j < i. In the case of O(n), we have
· · · → String(n) → Spin(n) → Spin(n) → SO(n) → O(n) . (3.1.6)
The string group is only defined up to homotopy, and therefore the group structure can only be determined up to A∞-equivalence. Moreover the smooth structure on the string group is not determined at all. Therefore, there exist various different models and the first geometric model as a topological group was constructed by Stolz [Sto96] and Stolz and Teichner in [ST]. Because π1 and π3 of String(n) vanish, the string group cannot be modeled by a finite-dimensional Lie group. To escape this issue, one is naturally led to Lie 2-group models of the string group [BL04]. These are Lie 2-groups endowed with a Lie 2-group homomorphism to Spin(n), regarded as a Lie 2-group. A first such model was constructed in [BSCS07], which is a strict but infinite-dimensional Lie 2-group and differentiates to the String Lie 2-algebra. Closely related is the construction of [Hen08], which yields a integration of the String Lie 2-algebra as a simplicial manifold. Moreover, there is an infinite-dimensional model as a strict Lie 2- group [NSW13], which was obtained by smoothing the original Stolz-Teichner construction. The model we shall be mostly interested in in this paper is that of Schommer-Pries [Sch11]. Here, the string group 16 Algebra is modeled by a 2-group object in Bibun, which is semistrict but finite dimensional. We feel that this model should allow for the easiest description of physically interesting solutions to higher gauge theory. §23 Models. Models for the string group: P U(H)-based model as a topological group from Stolz [Sto96] and Stolz and Teichner [ST] (sine P U(H) has no smooth structure in the infinite dimensional case). This was improved to a smooth model in [NSW13], which is one of the models we consider below. Finally, there is the loop space based model of [BSCS07]. §24 Model of Nikolaus et al. Let G be any compact, simple and 1-connected Lie group. Take an infinite dimensional Hilbert space H and the canonical generator 1 ∈ H3(G, Z) and construct the corresponding principal P U(H)-bundle P . We can choose the bundle P to be smooth, as for any such continuous principal bundle, there is an equivalent smooth one, see citations in [NSW13]. Denote by Aut(P ) the P U(H)-equivariant diffeomorphisms of P and Q : Aut(P ) → Diff(M) is the projection onto the base. Gauge transformations of P are therefore elements of the kernel of Q. Define
StringG = {f ∈ Aut(P )|Q(f) ∈ G ⊂ Diff(M)} , (3.1.7) i.e. elements of the string group are bundle automorphisms that cover left-multiplications by elements of G on the base.
Extension to 2-group: For a 2-group G, denote by π0G the group of isomorphism classes of objects in G and by π1 the group of automorphisms of 1 ∈ G0. Moreover, |G| is the geometric realization of the nerve of the category G. A 2-group model for the string group is a smooth 2-group (smoothly metrizable ∼ ∼ and smoothly separable) such that π0G = G and π1G = U(1). Moreover, |G| → G is a 3-connected cover. Starting from the central extension U(1) → U(H) → P U(H) and using the adjoint action of P U(H) onto U(H), we get the sequence C∞(G, U(1)) → C∞(P, U(H))P U(H) → C∞(P,P U(H))P U(H), where C∞(P,P U(H))P U(H) =∼ Gau(P ) is the group of gauge transformations on P . Define Gaud(P ) := C∞(P, U(H))P U(H) × U(1)/ , where (φ · µ, λ) ∼ (φ, (R vol(G)µ)λ), where vol(G) is the Haar measure on G. Then one can conclude that there is an exact sequence
1 t 1 → U(1) → Gaud(P ) −→ StringG → G → . (3.1.8)
This defines a smooth crossed module of Frechet Lie groups. µ1 §25 Model of BCSS. A semistrict model for the string Lie 2-algebra is the 2-term L∞-algebra R −−−→ g with µ1 trivial, µ2 the Lie bracket on g and µ3(x1, x2, x3) := hx1, [x2, x3]i, which turns out to be hard to integrate. Baez et al define a categorically equivalent, but infinite dimensional crossed module of Lie algebras Pkg, which they integrate to a crossed module of Lie groups. In the following, ΩG and P0G denote the based loop and path spaces of G, respectively. Given a Lie π group G, we have the short exact sequence 1 → ΩG → P0G −−→ G → 1, where π is the evaluation map at the endpoint of the loop. There is now a nontrivial central extension of ΩG by U(1), and we have the Kac-Moody group ΩˆG as a nontrivial principal U(1)-bundle over ΩG. At the level of Lie algebras, this central extension is well under control and can be lifted to Lie groups using a theorem3 by Pressley and Segal. The explicit construction for inner product scaled by k ∈ Z of [Mic87] yields Ωˆ kG. The corresponding String 2-group model PkG is a crossed module of Lie groups with (PkG)0 = P0G and (PkG)1 = Ωˆ kG. ∼ The crossed module of Lie algebras Pkg is easier to describe. It is given by (Pkg)1 = Ωˆ kg = Ωg ⊕ R and (Pkg)0 = P0g. The map t :(Pkg)1 → (Pkg)0 is the composition Ωˆ kg → Ωg,→P0g and the commutator on (Pkg)0 is the point wise commutator. The action of (Pkg)0 on (Pkg)1 is p B (`, c) := R 2π 0 ([p, `], 2k 0 dθhp(θ, ` (θ)i). §26 Model of Schommer–Pries. In [Sch11], Schommer-Pries constructed a smooth 2-group model
3Here enters the condition that G is simple, to guarantee that all invariant symmetic bilinear forms on its Lie algebra proportional to each other. 3.1 Group like objects 17 of the string group, and we briefly recall this construction in the following. First, we need to generalize the extension of Lie groups by other Lie groups to the categorified setting, as done in [Sch11, Def. 75]: Def. 3.1.1: An extension of a smooth 2-group G by a smooth 2-group A consists of a smooth 2-group E together with homomorphisms f : A → E, g : E → G and a 2-homomorphism α : g ◦ f → 0 such that E is a principal A-bundle over G. We are interested in extensions of a smooth 2-group G = (G ⇒ G) by a smooth abelian 2-group A = A ⇒ ∗, which form the weak 2-category Ext(G, A). The following theorem gives a way of encoding this weak 2-category in Segal-Mitchison cohomology classes: Theorem 3.1.2 ([Sch11, Thm. 1]): Let G be a Lie group and A be an abelian Lie group, viewed as a trivial G-module. Then there is an (unnatural) equivalence of weak symmetric monoidal 2-categories4:
∼ 3 2 1 Ext(G, BA) = HSM(G, A) × HSM(G, A)[1] × HSM(G, A)[2] . (3.1.9)
For the model Sλ of the string group of SO(n), we are interested in the case G = Spin(n) and 2 1 A = U(1). At least for n ≥ 5, the cohomology groups HSM(G, A) and HSM(G, A) are trivial. Thus, 3,0 2,1 1,2 0,3 3 the corresponding extension is parameterized by an element λ = (λ , λ , λ , λ ) of HSM(G, A), cf. equations (??). We now have the following theorem. 3 ∼ Z Theorem 3.1.3 ([Sch11, Thm. 100]): For n ≥ 5, HSM(Spin(n), U(1)) = and the central extension of smooth 2-groups Sλ corresponding to a generator λ gives a smooth 2-group model for String(n).
Let us now work through the details of this string group model Sλ. Given a simplicial cover V• of Spin(n) as constructed in section ??, the 3-cocycle λ contains the non-trivial smooth maps
0,3 [1] 1,2 [2] 2,1 [3] λ : V3 → A , λ : V2 → A and λ : V1 → A . (3.1.10)
As remarked in section ??, the map λ2,1 is in fact a Čech 2-cocycle and defines an A-bundle gerbe over Spin(n). Identifying bundle gerbes with central groupoid extensions, we obtain the groupoid underlying the smooth 2-group corresponding to λ:
[2] Sλ := V1 × A ⇒ V1 . (3.1.11) Here the source, target and identity maps are given by
s(v0, v1, a) = v1 , t(v0, v1, a) = v0 and id(v0) = (v0, v0, 0) , (3.1.12) and the invertible composition is defined as
2,1 (v0, v1, a0) ◦ (v1, v2, a1) := (v0, v2, a0 + a1 + λ (v0, v1, v2)) (3.1.13) for v0,1,2 ∈ V1 and a0, a1 ∈ A. It remains to specify the 2-group structure on the Lie groupoid Sλ. [2] ×2 Note that there is a Lie groupoid functor (f0, f2) from the Lie groupoid C2 := (V2 × A ⇒ V2) to Sλ × Sλ. This functor is a weak equivalence in MfdCat and upon bundlization, we can invert it. The [2] ×3 ×3 same is true for the functor (f0f0, f2f0, f2f2) from the Lie groupoid C3 := (V3 × A ⇒ V3) to Sλ . This yields bibundles
B2 : Sλ × Sλ → C2 and B3 : Sλ × Sλ × Sλ → C3 . (3.1.14) Furthermore, we have the Lie groupoid functors
m p1 p2 C2 −→ Sλ , C3 −→ Sλ and C3 −→ Sλ , (3.1.15)
4Here, the weak 2-categories M, M[1] and M[2] are the obvious trivial weak 2-categories with objects M, ∗, ∗, morphisms M, M, ∗ and 2-morphisms M, M, M, respectively. 18 Algebra where
1,2 m(y0, y1, a0, a1) := (f1(y0), f1(y1), a0 + a1 + λ (y0, y1)) , ∗ 1,2 ∗ 1,2 p1(z0, z1, a0, a1, a2) := (f1f1(z0), f1f1(z1), a0 + a1 + a2 + f1 λ (z0, z1) + f3 λ (z0, z1)) , (3.1.16) ∗ 1,2 ∗ 1,2 p2(z0, z1, a0, a1, a2) := (f1f2(z0), f1f2(z1), a0 + a1 + a2 + f0 λ (z0, z1) + f2 λ (z0, z1))
[2] [2] [2] for y0,1 ∈ V2 , z0,1 ∈ V3 and a0,1,2 ∈ A. There is a natural isomorphism T : V3 −→ V1 × U(1) defined by
0,3 T (z0) = (f1f1(z0), f1f2(z0), λ (z0)) , z0 ∈ V3 , (3.1.17) which satisfies
p2(z0, z1, a0, a1, a2) ◦ T (z1) = T (z0) ◦ p1(z0, z1, a0, a1, a2) (3.1.18)
0,3 1,2 due to δCˇλ = δN λ . After bundlization and composition with the bibundles (3.1.14), we obtain bibundles
Bm : Sλ × Sλ → Sλ ,
Bp1 : Sλ × Sλ × Sλ → Sλ , (3.1.19)
Bp2 : Sλ × Sλ × Sλ → Sλ ,
and the natural isomorphism T yields a bibundle isomorphism a : Bp1 ⇒ Bp2 . Because the bibundles
Bp1 and Bp2 can be identified with Bm ⊗ (Bm × 1) and Bm ⊗ (1 × Bm), respectively, a is indeed the associator. Here, a is completely determined by T since a is the horizontal composition of T with the identity isomorphism on B3. It remains to define the unit e as well as the left- and right-unitors l and r. Both unitors are trivial (i.e. the identity isomorphism) and up to isomorphism, the unit is uniquely defined as the bundlization e of the Lie groupoid functor (∗ ⇒ ∗) −→ Sλ , (3.1.20) [1] 1 which takes ∗ to a v0 ∈ V1,p with π(v0) = G. Let us now briefly verify that we indeed constructed a smooth 2-group. For this, we need to check that the bibundle (pr1,Bm) is an equivalence and that the internal pentagon identity is satisfied. The former is relatively clear, because B2 and thus also (pr1,B2) are bibundle equivalences. One then readily checks that
(id × mˆ ): Sλ × C2 → Sλ × Sλ (3.1.21) is a bibundle equivalence. It is obvious that the associator only affects the A-part of the Lie groupoids
Sλ, C2 and C3, and therefore the internal pentagon identity reduces to the equation
0,3 0,3 0,3 λ (v1, v2, v3) + λ (v0, v1v2, v3) + λ (v0, v1, v2) (3.1.22) 0,3 0,3 = λ (v0v1, v2, v3) + λ (v0, v1, v2v3) ,
[1] 0,3 where v0,1,2,3 ∈ V1 . This is precisely the equation δN λ = 0, which holds since λ is a Segal-Mitchison 3-cocycle. Finally, note that the interchange law, which is the compatibility condition for the vertical 2,1 1,2 and horizontal multiplications, follows from δN λ = δCˇλ . We conclude this section with the following two remarks: hile we are mostly interested in the smooth 2-group model of the string group Sλ given by the central extension of the smooth 2-group Spin(n) ⇒ Spin(n) by A ⇒ ∗, the above construction of this extension as well as most of our following discussion readily generalizes to arbitrary Lie groups G. ultiplicative bundle gerbes as defined in [CJM+05] are 3.1 Group like objects 19 special cases of the above construction of a 2-group object internal to Bibun from a Segal-Mitchison 3-cocycle. In [DS17], this model was differentiated using the method of Severa to obtain the semistrict string s Z R t Lie 2-algebra gk with k ∈ is given by the sequence −→ g, where t is the trivial representation and µ3(x, y, z) := khx, [y, z]i using the commutator and the Killing form on g, cf. [BR10]. §27 Weblinks. Urs1, Urs2, lots of literature §28 String group. The string group String(n) fits into the following sequence, known as the Whitehead tower of O(n): · · · → String(n) → Spin(n) → Spin(n) → SO(n) → O(n) . (3.1.23) The arrows describe isomorphisms at the level of homotopy groups, with the exception of the lowest homotopy group: π0(O(n)) is removed in the step from O(n) to SO(n), π1(O(n)) in the step to Spin(n) and π2(O(n)) is trivial, anyway. The string group String(n) is obtained by removing π3(O(n)). That is, String(n) is a 3-connected cover of Spin(n). This definition determines String(n) only up to homotopy, and the string group structure is therefore only determined up to A∞-equivalence. §29 Models. Note that String(n) cannot be a finite-dimensional Lie group, since π1 and π3 of String(n) are trivial. First models of the string group were presented in [Sto96,ST]. More convenient models of the string group are given by models which are Lie 2-groups endowed with a Lie 2-group homomorphism to Spin(n). We call such a model a string 2-group model. There are various such string 2-group models but for our considerations, those of [Sch11] and [BSCS07] will suffice. We will refer to these as the skeletal model Stringsk(n) and the loop space model StringΩˆ (n), respectively. The two string 2-group models start from two different but equivalent ways of describing the fun- 3 ∼ ∼ 5 damental bundle gerbe GF over S = SU(2) = Spin(3). The skeletal model uses a suitable , ordinary cover Y1 = U = taUa of SU(2), while the loop space model starts from the surjective submersion ∂ Y2 = P0SU(2) SU(2), where
P0SU(2) := {γ : [0, 1] → SU(2) | γ(0) = 1} , ∂γ := γ(1) . (3.1.24)
[2] Note that Y2 is the space of pairs of paths with the same endpoint. Modulo smoothness at the endpoints, which is a technicality we shall suppress, this yields the space of based, parameterized loops,
ΩSU(2) := {` : [0, 1] → SU(2) | `(0) = `(1) = 1} . (3.1.25)
We then have the following two descriptions of GF:
P Ω\1SU(2)
(3.1.26) ta,bUa ∩ Ub // taUa ΩSU(2) / P0SU(2)
∂ SU(2) SU(2)
Here the horizontal double arrows are the obvious maps from Ua ∩Ub to Ua or Ub as well as the projection 1 from a parameterized loop split at τ = 2 ∈ [0, 1] onto the two resulting based paths. Moreover, P and Ω\1SU(2) are principal U(1)-bundles over ta,bUa ∩ Ub and ΩSU(2), respectively. The string 2-group structure on GF in the skeletal model requires additional elements in the Segal– Mitchison group cohomology complementing the 3-cocycle describing GF geometrically. The loop space model, however, is specified canonically by pointwise multiplication and the canonical product on the
5 A cover of Spin(3) that extends to a simplicial cover of the nerve of BSpin(3) = (Spin(3) ⇒ ∗). 20 Algebra
Kac–Moody central extension Ω\1SU(2). Moreover, the loop space model yields a Lie 2-group which is unital and associative, albeit based on infinite-dimensional spaces. We will not need any further details on the Lie 2-group models and therefore we can directly move on to their Lie 2-algebras. §30 String Lie 2-algebra. A method of differentiating Lie 2-groups to Lie 2-algebras was given by Ševera [Sevb]. This procedure yields corresponding Lie 2-algebras in the form of 2-term L∞-algebra, cf. [BC04] and appendices ?? and ??. In [DS17], Ševera’s method was used to differentiate the skeletal model, and the result is the string Lie 2-algebra R 0 stringsk(3) = −−→ su(2) (3.1.27) with non-trivial products
µ2 : su(2) ∧ su(2) → su(2) , µ2(x1, x2) = [x1, x2] , (3.1.28) µ3 : su(2) ∧ su(2) ∧ su(2) → R , µ3(x1, x2, x3) = (x1, [x2, x3]) , where (−, −) is the appropriately normalized Killing form on su(2). This Lie 2-algebra and closely related ones were first considered in [BC04]. It was named the string Lie 2-algebra since it was shown to integrate to a 2-group model of the string group [Hen08, BSCS07]. 6 The fact that µ1 is trivial might seem very restrictive at first glance. Note, however, that any Lie 2-algebra is categorically equivalent to one with µ1 = 0 [BC04]. The loop space model, on the other hand, can also be differentiated in a straightforward fashion, since it corresponds to a strict Lie 2-group. The result is R µ1 stringΩˆ (3) = Ωsu(2) ⊕ −−−→ P0su(2) ) , (3.1.29) ∼ where µ1 is the concatenation of the projection Ω\1su(2) = Ωsu(2) ⊕ R → Ωsu(2) with the embedding Ωsu(2),→P0su(2) as closed based paths. The remaining non-trivial products are
µ2 : P0su(2) ∧ P0su(2) → P0su(2) , µ2(γ1, γ2) = [γ1, γ2] , µ : P su(2) ⊗ (Ωsu(2) ⊕ R) → Ωsu(2) ⊕ R , 2 0 (3.1.30) Z 1 d µ2 γ, (λ, r) = [γ, λ] , −2 dτ γ(τ), λ(τ) . 0 dτ The categorical equivalence between both Lie 2-algebras was shown in [BSCS07]. Explicitly, we have the morphisms Φ = (φ0, φ1) and Ψ = (ψ0, ψ1) with
Φ Ψ stringsk(3) −−→ stringΩˆ (3) −−→ stringsk(3) . (3.1.31)
The chain maps φ0 and ψ0 are given in the diagram
prR R / Ωsu(2) ⊕ R / R (3.1.32) ·f(τ) ∂ su(2) / P0su(2) / su(2) where prR is the obvious projection, ∂ : P0su(2) → su(2) is the endpoint evaluation and ·f(τ): su(2) → P0su(2) is the embedding of x0 ∈ su(2) as the straight line x(τ) = x0f(τ), where f : [0, 1] → R is a smooth function with f(0) = 0 and f(1) = 1. The maps φ1 and ψ1 read as Z 1 2 φ1(x1, x2) = [x1, x2](f(τ) − f (τ)), 0 , ψ1(x1, x2) = dτ (x ˙ 1, x2) − (x1, x˙ 2) . (3.1.33) 0
6 This also informs the name skeletal, as a L∞-algebra with trivial µ1 corresponds to a category where source and target maps agree, cf. [BC04]. 3.2 L∞-algebroids and related 21
Clearly, (Ψ ◦ Φ)0 is the identity map, and using the composition of morphisms of Lie 2-algebras (??), we readily verify that Ψ ◦ Φ = id . On the other hand, there is a 2-morphism χ :Φ ◦ Ψ → id stringsk stringΩˆ encoded in a map
χ : P0su(2) → Ωsu(2) ⊕ R , χ(γ) = (γ − f(τ)∂γ, 0) , (3.1.34) see appendix ?? for the relevant definitions. Further details are found in [BSCS07, Lemma 37]. Thus, stringsk(3) and stringΩˆ (3) are equivalent as Lie 2-algebras. All this readily generalizes from su(2) to 3 arbitrary Lie algebras g with a preferred element µ3 ∈ H (g, R).
3.2 L∞-algebroids and related
3.2.1 Homotopy Algebras Important examples:
◦ A∞-algebras (think generalisation of matrix algebra to non-associative and beyond) ◦ L∞-algebras (think generalisation of Lie algebra to differential graded Lie algebras and beyond)
§1 A∞-algebra. An A∞-algebra or strong homotopy associative algebra [Sta63a, Sta63b], see also [Kela] Many useful results (decomposition theorem, quasi-isos, etc.) here: [Kaj07]
3.2.2 Differential graded algebras
§2 Chevalley-Eilenberg algebra. Dual description of an L∞-algebra, originally, of a Lie algebra [CE1Z]. §3 Chevalley–Eilenberg resolution. Review and interpretation: [HS97], see [JRSW18]. §4 Koszul–Tate resolution. see [JRSW18].
3.2.2.1 NQ-manifolds
§5 Literature. Original literature/nomenclature: [Sch93a, Sch93b, AKSZ97]. Use in BV: [Sch93a] Definition cf. [DS18a]. Relation to Courant algebroids [Roya]. §6 Vinogradov algebroids. See [DS18a] and references therein. §7 Split NQ-manifolds. Just as all real supermanifolds are diffeomorphic to split ones [Bat79], also all real Z-graded manifolds are diffeomorphic to split ones [BP13]. §8 Symplectic NQ-manifolds. The following examples were first given in [Sev05], see also [Roya] and [CS10] for further details. A symplectic NQ-manifold of degree 0 is simply a symplectic manifold. A symplectic NQ-manifold of degree 1 is a Poisson manifold. Such a manifold can be shown to be symplectomorphic to T ∗[1]M with canonical symplectic structure. A compatible homological vector field Q corresponds to a bi-vector field on M and the condition Q2 = 0 amounts to this bivector being a Poisson tensor. A symplectic NQ-manifold of degree 2 is a Courant algebroid.
3.2.3 Lie 2-algebras §9 Literature. Definition of weak Lie 2-algebras [Royb]. Useful Lie 2-algebras (coherent, semistrict, etc.) in [BC04, BHR10]. 7 §10 Lie 2-algebras. (i.e. weak Lie 2-algebra, “2-term EL∞-algebras”) [Royb] A Lie 2-algebra is a linear category L equipped with a bilinear functor [·, ·]: L ⊗ L → L (the bracket), a bilinear natural transformation S :[f, g] ⇒ −[g, f] (the alternator) and a trilinear natural transformation J :[f, [g, h]] ⇒
7Different, unrelated definition in [HW95]. 22 Algebra
[[f, g], h] + [g, [f, h]], (the Jacobiator). These have to fulfill a number of coherence conditions specified in four diagrams, cf. [Royb]. We will always work with special cases in the following.
3.2.3.1 Strict Lie 2-algebras
§11 Strict Lie 2-algebras. Describe as crossed modules of Lie algebras (“differential crossed modules”), corresponding to strict 2-term L∞-algebras. For alternative inner produces, which are not cyclic in the usual L∞-algebra sense, see [BC04, Zuc13]. §12 Crossed modules of Lie algebras. §13 Equivalence of strict Lie 2-algebras and crossed modules of Lie algebras. cf. [BC04], this is a linearization of the construction for Lie 2-groups and crossed modules. That is, starting from a Lie 2-algebra c = (c1, c0), we take g = c0, h = ker(s) ⊂ c1 and t in the DCM is the restriction of t in the Lie 2-algebra to h. The action of an element in g on h as derivation is defined as
1 g B h = [ g, h] . (3.2.1)
3.2.3.2 Semistrict Lie 2-algebras
§14 Hemistrict Lie 2-algebras. A Lie 2-algebra for which the Jacobiator is the identity chain homotopy is called hemistrict. §15 Semistrict Lie 2-algebras. A Lie 2-algebra for which the alternator is the identity chain homotopy is called semistrict. A semistrict Lie 2-algebra is the same as a 2-term L∞-algebra. §16 Mophisms and 2-mophisms. See discussion in [BC04]. This is natural since Lie 2-algebas form 2-categories. Generalisation to n-term L∞-algebras difficult, but dga picture contained in [SSS]. §17 Classification of semistrict Lie 2-algebras . There is a classification of Lie 2-algebras, which is closely related to the Faulkner construction for 3-Lie algebras [dMFOMER09]. One starts from the observation that any Lie 2-algebra is categorically equivalent to a skeletal one [BC04, Prop. 51]. These Lie 2-algebras correspond to 2-term L∞-algebras with µ1 = 0, which in turn are in one-to-one correspondence with quadruples (g, V, ρ, µ3), where g is a Lie algebra, V the carrier space of the representation ρ of g and µ3 a 3-cocycle on g with values in V . The proof of this statement is relatively straightforward [BC04, Thm. 55]. Integration can be done, confusion in [BC04], the authors showed that Lie 2-groups are classified by a pair of groups G, H, with H abelian, an action of G on H by automorphism and an element of H3(G, H). It is thus tempting to assume that the string Lie 2-algebra can be integrated to such classifying data. As shown in [BL04], however, this cannot be done if the underlying topology is to be respected. The reason behind this problem is that ordinary group cohomology is not the right framework for this integration. As done in [Sch11], one should rather switch to Segal-Mitchison group cohomology [Seg], which we briefly review in the following. §18 Lie 2-algebras and 3-Lie algebras A question often arising is if there is a relationship between the 3-Lie algebras of the M2-brane models [BL08b, Gus09] and the categorified Lie n-algebras discussed in these lectures. Let us therefore summarize the relevant statements. Recall that a 3-Lie algebra [Fil85] is a vector space A endowed with a ternary, totally antisymmetric bracket [−, −, −]: A∧3 → A. This bracket satisfies the fundamental identity
[a, b, [c, d, e]] = [[a, b, c], d, e] + [c, [a, b, d], e] + [c, d, [a, b, e]] (3.2.2) for all a, b, c, d, e ∈ A, which implies that the inner derivations D(a, b), which act on c ∈ A according to
D(a, b)c = [a, b, c] (3.2.3) 3.2 L∞-algebroids and related 23
form a Lie algebra gA. We can further equip A with a metric (−, −) satisfying
([a, b, c], d) + (c, [a, b, d]) = 0 . (3.2.4)
In [CS08], a generalization was defined, in which the 3-bracket is only antisymmetric in its first two slots. It was noted in [dMFOMER09] that the resulting generalized metric 3-Lie algebras are in one-to-one correspondence with metric Lie algebras g and faithful orthogonal g-modules. This observation was then extended in [PS12] to the statement that each generalized metric 3-Lie ∂ algebra has an underlying metric strict Lie 2-algebra A −−→ gA with metrics on A and gA and non-trivial higher products
µ2(D(a, b),D(c, d)) = [D(a, b),D(c, d)] and µ2(D(a, b), c) = [a, b, c] . (3.2.5)
Inversely, on each strict Lie 2-algebra h → g with metrics ((−, −)) and (−, −) on g and h, respectively, there is a bilinear map D : h ∧ h → g such that
((g1,D(a, b)))= −(µ2(g1, a), b) . (3.2.6)
A corresponding 3-bracket is then defined as
[a, b, c] := µ2(D(a, b), c) . (3.2.7)
Altogether, metric 3-Lie algebras and their generalizations are strict metric Lie 2-algebras, and the nomenclature is rather unfortunate. Moreover, the 3-bracket on a 3-Lie algebra is not related to the higher product µ3, which vanishes for strict Lie 2-algebras. There is, however, an interesting class of examples of 3-Lie algebras, in which both ternary maps can be made to agree. Consider the 3-Lie algebra defined originally in [ALMY01], where A = gl(N, C) and
[a, b, c] := tr (a)[b, c] + tr (b)[c, a] + tr (c)[a, b] . (3.2.8)
As observed in [RS14], this 3-Lie algebra can actually be extended to a semistrict Lie 2-algebra on the complex gl(N, C) −−→id gl(N, C) with higher products
µ1(v) = v ,
µ2(w1, w2) = tr (w1)w2 − tr (w2)w1 + [w1, w2] , (3.2.9) µ2(v, w) = −( tr (v)w − tr (w)v + [v, w]) ,
µ3(w1, w2, w3) = tr (w1)[w2, w3] + tr (w2)[w3, w1] + tr (w3)[w1, w2] ,
id where we denoted elements from the left and the right vector space in gl(N, C) −−→ gl(N, C) by v1,2,3 and w1,2,3, respectively. It should be stressed, however that this semistrict Lie 2-algebra is quasi-isomorphic as an L∞-algebra to ∗ → ∗ and therefore trivial for most purposes.
3.2.4 L∞-algebras §19 Idea. Generalisation of Lie algebras to differential graded Lie algebras and beyond. Ideal model to work with for Lie ∞-algebras, which correspond to N-graded L∞-algebras. Z-graded L∞-algebras go beyond that and appear, e.g., in the BV formalism. §20 Literature. A first motivation in mathematical context was given in [SS85], the first explict mention is in the physics literature in the context of string field theory[Zwi93]. Let us briefly recall 24 Algebra
the exact definition [LS93, LM95, MSS]. Based on Stasheffs A∞-algebras [Sta63b, Sta63a].Details and preferred conventions in [JRSW18]. §21 Coalgebra picture, etc. [LM95], helpful may also be the detailed discussions in [Sch, FRZ16, Khu15]. Read also [JRSW18].
§22 Strictification theorem. General strictification theorems for homotopy algebras specialise to L∞- algebras: every L∞-algebra is quasi-isomorphic to a strict one [KM95, Corollary 1.6], see also [BM07]. §23 Minimal model theorem. Every L∞-algebra is quasi-isomorphic to a minimal one [Kad82, Kaj07]. (Note: references are actually for A∞-algebras, but this generalises even to OCHAs (open- closed homotopy algebras) [KS06].) Improved proof for minimal model formulas: [MSW19]
§24 Decomposition theorem. Any L∞-algebra L is L∞-isomorphic to the direct sum of a minimal L∞-algebra (that is, an L∞-algebra with µ1 = 0) and a linearly contractible L∞-algebra (that is, an L∞-algebra with µi = 0 for i > 1 and trivial cohomology), see [Kaj07] for the more general case of A∞-algebras. §25 Tensor products with dgcas. The tensor product between an L∞-algebra and a differential graded commutative algebra carries a natural L∞-algebra structure. This is folklore, cf. also [JSW15]. §26 In deformation theory. Strong homotopy Lie algebras appear in modern deformation theory. Here, the definition of the deformation functor involves the so-called homotopy Maurer-Cartan equations [SS85, Mer00, Laz01]. §27 Gerstenhaber algebras etc. Nice summary in q-alg/9602009v2 §28 Relative Sullivan algebra. A relative Sullivan algebra of some dg-algebra (A, d) is a morphism of dg-algebras which is an inclusion (A, d),→(A ⊗ ∧•V, d0) , (3.2.10) where V is a graded vector space such that there is a basis vα ∈ V of V , α ∈ J, J well-ordered set, such that V<β = span{vα|α < β} and 0 • d vβ ∈ A ⊗ ∧ Vβ . (3.2.11)
It is minimal, if α ≤ β implies that |vα| ≤ |vβ|. Since Sullivan algebras are semifree dgas, they are Chevalley-Eilenberg algebras of L∞-algebras (if the degree-wise duals exist). §29 Sullivan model. Commutative differential graded algebra which are equivalent to dg-algebras of • ∗ Sullivan differential forms on topological spaces. Explicitly, we have a Sullivan algebra (∧ V , dV such that there is a quasi-isomorphism • ∗ ' • (∧ V , dV ) −−→ ΩSull(X). (3.2.12)
§30 Super L∞-algebras. First, dually, as free differential algebras [DFR80]. §31 Action of an L∞-algebra on a space. Rarely discussed in the literature is the notions of 8 an action for L∞-algebras. A very detailed account is found in [MZ12], where the rather evident generalizations of the notions action, extension and semidirect product of Lie algebras to L∞-algebras were explored. §32 Representation on dg-vector spaces. See discussion in [JRSW18, Section 2.5].
§33 Semidirect products of L∞-algebras. Semidirect products of L∞-algebras can be similarly defined, using analogies with Lie algebras [MZ12]. These have an underlying action ρ of an L∞-algebra 0 0 ˜ 0 (L, µ) on another L∞-algebra (L , µ ) such that L = L n L forms an L∞-algebra with brackets
0 µ˜2(X1 + w1,X2 + w2) = µ2(X1,X2) + ρ(X1)w2 − ρ(X2)w1 + µ2(w1, w2) , (3.2.13)
8 with certain restrictions to finite dimensional L∞-algebras 3.2 L∞-algebroids and related 25
0 where X1,2 ∈ L and w1,2 ∈ L . §34 Symmetries of Lie n-algebroids. The dual of the natural notion of L∞-algebra of inner derivations of an L∞-algebroid E is the Weil algebra W(E) [SSS]. If E is a Lie n-algebroid then the inner derivations from an Lie n + 1-algebroid. §35 Cyclic inner product. The most reasonable inner products on strong homotopy algebras seems to be the cyclic one of Kontsevich. Here, vector spaces of different grading (but identical parity) can have non-vanishing inner products between them. Definition see [Zwi93] and [Kon, Pen], see also [Igu04] and [MSS]. Note that grading can mix and indeed has to mix.
§36 Universal enveloping algebra. There is a functor from an associated algebra (or A∞-algebra in general) to a Lie algebra (or L∞-algebra in general), obtained by taking the antisymmetrized product. The universal enveloping algebra is the image of the left adjoint to this functor, see [LM95] for details.
§37 Poincaré–Birkhoff–Witt theorem. Let g be a vector space with an basis (e1, . . . , en). Consider the embedding h of a Lie algebra g into its universal enveloping algebra U(g). Extend it to ordered sequences in the ei via h(ei1 , ei2 , . . . , eik ) := h(ei1 ) ⊗ h(ei2 ) ⊗ ... ⊗ h(eik ). Then the image of h is a basis for U(g). This implies in particular that U(g) =∼ Sym(g)
3.2.4.1 Loop L∞-algebras
§38 Loop L∞-algebra. Original definition in [Zwi93]. See also [Mar01, DJP17]
3.2.5 L∞-algebroids
§39 Symplectic L∞-algebroids. An important feature of symplectic Lie n-algebroids (M, {−, −}, Q) is that they come with an associated Lie n-algebra via a derived bracket construction, see [RW98, FM07, Get, RS15b].
§40 Lie integration. Explicit integration of L∞-algebroids (as of Lie algebroids) is notoriously difficult. Early literature comprises [Get09, Hen08]. This was later improved and specialized in [LBS12,SS].
3.2.5.1 Lie algebroids
Classic book: [Mac05], see also [Mac87]. §41 Literature.
3.2.5.2 Courant algebroids
§42 Literature. [LWX97] §43 Courant algebroid. What is generally known as a Courant algebroid is really a symplectic Lie 2-algebroid [Sev05, Roya]. X §44 Twisting as canonical transformation. See [Roy02] for twisting written as e T , where XT = {T, −}. The flow acts by canonical transformation, which implies that
XT XT XT {e Θ0, e Θ0} = e {Θ0, Θ0} . (3.2.14)
Also discussed in [BHIW16, HIW17]. §45 Generalized reduction. Generalized reduction procedure of Courant algebroids: [BCG07]. Construction of transitive Courant algebroids by reduction due to Ševera [Seva]. §46 Weak Courant-Dorfman algebras. [EZ11], applicable in [PS13]. §47 Courant-Dorfman algebra. A Courant-Dorfman algebra [Roy09], see also [KW] 26 Algebra
3.2.5.3 Higher Lie n-algebroids §48 Vinogradov Lie n-algebroids. A very important hierarchy of symplectic NQ-manifolds are the Vinogradov Lie n-algebroids9 ∗ Vn(M) := T [n]T [1]M, (3.2.15) cf. [GS14, RS15b]. See also [DS18a]. §49 Twisted Vinogradov algebroids Vinogradov Lie n-algebroids Vn(M) may be twisted by a closed n + 1-form T [SW01]. §50 Severa class. of the Vinogradov algebroid [Seva], see also [SW01] and [BC05]. Discussion in [DS18a]
3.2.6 2-Crossed modules of Lie algebras §51 Literature. Original literature: [Con84], see also [MP11],. used in [SW14b] §52 Inner derivation 2-crossed module. [RS08] §53 No semistrict Lie 2-algebra in 2-crossed modules of Lie algebras. We start from a D2CM with notation as above. Define the following structures: let L−1 = l and L0 = h. The products are
µ1(`) = t(`) , µ1(h) = 0 , 1 µ2(`1, `2) = 0 , µ2(h, `) = −µ2(`, h) = {h, t(`)} − {t(`), h} , 2 (3.2.16) 1 µ2(h1, h2) = 2 hh1, h2i − hh2, h1i , µ3(`, ·, ·) = 0 , µ3(h1, h2, h3) = {h1, hh2, h3i} + tot. antisym. .
Check higher Jacobi identities: At level 2, we have µ1(µ1(·)) = 0, which is trivially satisfied. At level 3, we have µ1(µ2(x, y)) = µ2(µ1(x), y) − µ2(µ1(y), x) . (3.2.17)
For x = `1, y = `2 this gives a contradiction:
µ1(µ2(`1, `2)) = µ1(0) = 0 (3.2.18) µ2(µ1(`1), `2) − µ2(µ1(`2), `1) = −{t(`2), t(`1)} + {t(`1), t(`2)} = −[`2, `1] + [`1, `2] 6= 0 .
This forces us to put µ2(h, `) = 0. Then the identity
µ1(µ2(`, h)) = µ2(µ1(`), h) − µ2(µ1(h), `) (3.2.19) gives a contradiction in general. Other idea: modify µ2(h, `) := t(h) B `. Then the first identity at level 3 is satisfied. Then the identity µ1(µ2(`, h)) = µ2(µ1(`), h) − µ2(µ1(h), `) seems to be causing problems. Altogether, it seems that there is no non-trivial semistrict Lie 2-algebra to be extracted from the 2-crossed module of Lie algebras, at least not along the lines above.
3.2.7 Other §54 n-Lie algebras. Original paper [Fil85], generalisations [CS08] and [BL09], full picture and analysis [dMFOMER09]. Further discussion (representations) also in [CDS09]. §55 Metric 3-Lie algebras. It has been shown in [Nag] and later in [Pap08, GG08] that essentially 10 the only 3-Lie algebra admitting u(N) as its associated 2-Lie algebra is isomorphic to A4. In this case, the Lie algebra obtained is su(2), as f˜abc = 4εabc. §56 Nambu–Heisenberg algebra and its twist. (cf. discussion in [DSS10])
9 The essential structure underlying these types of L∞-algebroids was first studied in [Vin90]. 10 Evidently, a similar reduction process from a n + 1-algebra to a n-algebra can be defined. For the series An, this reduction always ends up with A4, as the structure constants are the n-dimensional epsilon-tensors. 3.3 Representation theory 27
3.3 Representation theory
§1 Young symmetrizer. Given a Young tableau λ with rows λi describing a partition of d, we define the symmetrizer sλ and the antisymmetrizer aλ as operations symmetrizing over rows and antisymmetrizing over columns: ⊗d λ1 λd ⊗d sλ : V → Sym V ⊗ ... ⊗ Sym V ⊂ V , (3.3.1) ⊗d µ1 µd ⊗d aλ : V → Λ V ⊗ ... ⊗ Λ V ⊂ V , where V is the fundamental representation and µi are the rows of the conjugate partition to λ. The Young symmetrizer cλ is now defined as cλ := sλaλ , (3.3.2) 2 and forms a projectors cλ = αcλ with α being the product of all hook lengths of the Young tableau λ. For more details, see e.g. [FH91]. Note that for complete projection, need to divide by the αs!
3.4 Algebraic Topology
§1 Künneth’s theorem. see e.g. [Wei94] §2 Homological algebra. [Wei94] §3 Hodge–Kodaira decomposition. [Wei94], see also appendix in [JRSW18] 28 Algebra Chapter 4
Geometry
4.1 Generalities
§1 Derived geometry. In ordinary geometry, one identifies spaces via their algebra of functions. This relates geometry to algebra and ultimately underlies algebraic geometry. keywords: commutative Gelfand–Naimark theorem, Isbell duality (in a very general context). In derived geometry, the algebras are replaced by differential graded algebras, or dgas for short.
4.1.1 Manifolds
§2 Ends of a manifold. Consider a topological space M together with an ascending sequence Ki ⊂ Ki+1, i ∈ N, of compact subsets whose interiors cover M. Then M has an end for every sequence Ui ⊃ Ui+1, where Ui is a connected component of M\Ki. For example, the real line R has two ends, 0 which are obtained from the sequence Ki = [−i, i] with Ui = (i, ∞) and Ui = (−∞, −i). More generally, one defines an end of a manifold M as an element of the inverse limit system {K, components of M\K} indexed by compact subsets K of M, cf. [GS79]. If M is orientable and endowed with a volume form, we say that an end has a finite volume, if there is a compact set K such that the volume of the component of M\K containing the end is finite. Otherwise, we say that the volume is infinite. §3 Eilenberg–Mac Lane spaces. A connected topological space is said to be of type K(G, n), where N G is a group and n ∈ , if πn(X) = G. Example: The delooping of G, BG = G ⇒ ∗, is of type K(G, 1).
4.1.1.1 Spheres
§4 Stereographic projection. We consider a unit n-sphere Sn in Rn+1 centered at the origin. Let σ be the following map σ : Sn → Rn with
p1 pn σ(p) = , ..., . (4.1.1) 1 − pn+1 1 − pn+1
σ maps a point on the sphere to a point in the Rn ⊂ Rn +1- plane. This point is given by the intersection point of a line through the north pole and the point on the sphere with the plane xn+1 = 0, what can be easily checked. σ is called the stereographic projection. The inverse map σ−1 takes the form
2x1 2x2 ||x||2 − 1 σ−1(x) = , , ..., (4.1.2) 1 + ||x||2 1 + ||x||2 1 + ||x||2 30 Geometry
Supposing that the expression for pn+1 is correct, the above formula is readily checked. It remains to 2 n+1 n verify that ||p|| = 1 which is also straight forward. The induced metric on R |xn+1=0 = R is
2δ g = ij . (4.1.3) ij ||x||2 + 1
As an example, consider the case S2 and its relation to R2 and C. We use two stereographic projections, one with lines through the north pole covering the southern hemisphere and denoted by S and one with lines through the south pole covering the northern hemisphere and denoted by N:
s1 s2 s1 s2 ω1 = , ω2 = , ω1 = , ω2 = . (4.1.4) N 1 + s3 N 1 + s3 S 1 − s3 S 1 − s3
2 With the standard complex structure j on S , we can combine them to complex coordinates λN = 1 2 1 2 −1 ωN + iωN and λS = ωS − iωS. We have λN = λS . This makes contact with the complex projective plane CP 1 and shows the isomorphism S2 = CP 1. §5 2-sphere. Formulas for unit 2-sphere: Embedding coordinates S2 ⊂ R3: xˆ0, xˆ1, xˆ2. 0 1 2 Angles: xˆ = cos φ1, xˆ = sin φ1 cos φ2, xˆ = sin φ1 sin φ2, φ1 ∈ [0, π), φ2 ∈ [0, 2π). 1 2 xˆ1 xˆ2 Stereographic projection: (x , x ) = ( 1−xˆ0 , 1−xˆ0 ). 0 1 2 r2−1 2x1 2x2 2 1 2 2 2 Inverse: (ˆx , xˆ , xˆ ) = ( r2+1 , r2+1 , r2+1 ), r = (x ) + (x ) 4(dx1⊗dx1+dx2⊗dx2) Line element: g = 1+r2 4dx1∧dx2 Area element: dA = sin φ1dφ1 ∧ dφ2 = (1+(x1)2+(x2)2)2 R Total area: S2 dA = 4π §6 3-sphere. Formulas for unit 3-sphere: Embedding coordinates S4 ⊂ R4: xˆ0, xˆ1, xˆ2, xˆ3. 0 1 2 2 Angles: xˆ = cos φ1, xˆ = sin φ1 cos φ2, xˆ = sin φ1 sin φ2 cos φ3, xˆ = sin φ1 sin φ2 sin φ3, φ1, φ2 ∈ [0, π), φ3 ∈ [0, 2π). 1 2 3 xˆ1 xˆ2 xˆ3 Stereographic projection: (x , x , x ) = ( 1−xˆ0 , 1−xˆ0 , 1−xˆ0 ). 0 1 2 3 r2−1 2x1 2x2 2x3 2 1 2 2 2 3 2 Inverse: (ˆx , xˆ , xˆ , xˆ ) = ( r2+1 , r2+1 , r2+1 , r2+1 ), r = (x ) + (x ) + (x ) 4(dx1⊗dx1+dx2⊗dx2+dx3⊗dx3) Line element: g = 1+r2 2 8dx1∧dx2∧dx3 Area element: dA = sin φ1 sin φ2dφ1 ∧ dφ2 ∧ dφ3 = (1+(x1)2+(x2)2+(x3)2)3 R 2 Total area: S2 dA = 2π §7 General n-sphere. φ1, . . . , φn−1 ∈ [0, π), φn ∈ [0, 2π). 4(dx1⊗dx1+···+dx3⊗dx3) Line element: g = 1+r2 n−1 2 2ndx1∧···∧dxn Area element: dA = sin φ1 ... sin φn−2 sin φn−1dφ1 ∧ · · · ∧ dφn = (1+(x1)2+···+(xn)2)n n+2 n R 2 2 π 2 Total area: Sn dA = (n + 1) (n+1)!!
4.1.1.2 Loop spaces
§8 Frechét spaces, Loop spaces. Foundations: Frechét spaces as discussed in section 2.4 of [BSCS07]. Literature: Review article: J. Milnor, Remarks on infinite dimensional Lie groups, Relativity, Groups and Topology II, (Les Houches, 1983), North-Holland, Amsterdam, 1984. Book: [PS88]. See also A. Sergeev, “Kähler Geometry of Loop Spaces” for many of the basics. §9 Literature. T. Wurzbacher, “Spinors and twistors on loop spaces”, T. Wurzbacher, “Symplectic geometry of the loop space of a Riemannian manifold”, M. Spera and T. Wurzbacher, “The Dirac-Ramond 4.2 Symplectic and Poisson geometry 31 operator on loops in flat space” [Sta05, Sta] §10 Riemannian geometry of loop space. [MRTA] §11 Forms on loop space. A model for forms on loop space is found in [GJP91]. The connection to higher gauge theory is nicely made in [Hof02], where the transgression map is extended in a non-abelian way. He there also derives strict higher gauge theory. Note, however, that connections on loop space are still more general than the transgressed ones given in this paper (although the action on the most general forms on loop space is discussed). §12 Transgression. Definition, Transgression is a chain map. [Wal12, Walb] On connected spaces: The transgression map T : Hq+1(X,R) → Hq(ΩX,R) is an isomorphism if X is k-connected, q ≤ 2k − 1 and the coefficient ring R is a principal ideal domain [Whi78]. Open issues What does it mean to transgress a self-dual 3-form in six dimensions? §13 Symplectic structure on knot space of 3d 2-plectic manifold. Consider a 3-dimensional 2- plectic manifold (M, $). For simplicity, let M = R3. Using a transgression map [Bry07] (see also [SS10]), we reduce the 2-plectic structure on R3 to a symplectic 2-form on the knot space KR3 of R3; here by KR3 we mean the free loop space LR3 of R3 with reparametrizations factored out.
4.1.2 Differential geometry
§14 Useful literature. Standard: [KN63a, KN63b]. Summaries for physicists: [Nak90, GS89]
4.2 Symplectic and Poisson geometry
4.2.1 Basics
§1 Generalised symplectomorphisms. Lagrangian correspondences, see e.g. [Wei77].
4.2.2 Multisymplectic geometry
§2 Definition. Manifold with non-degenerate, closed p form is (p−1)-plectic or simply multisymplectic.
§3 Multisymplectic phase spaces and L∞-algebra structure. [BHR10] Further literature: observ- ables on such 2-plectic manifolds. This has been developed to various degrees [CIdL96, CIdL99, BHR10]; more details are also found in Ritter & Saemann [RS15b, RS16].
4.2.3 Poisson and Nambu–Poisson geometry
§4 Nambu–Poisson structure. A Nambu-Poisson structure [Nam73, Tak94] on a smooth manifold M is an n-ary, totally antisymmetric linear map {−,..., −} : C∞(M)∧n → C∞(M), which satisfies the generalized Leibniz rule
{f1 f2, f3, . . . , fn+1} = f1 {f2, . . . , fn+1} + {f1, . . . , fn+1} f2 (4.2.1) as well as the fundamental identity
{f1, . . . , fn−1, {g1, . . . , gn}} = {{f1, . . . , fn−1, g1}, . . . , gn} + ··· + {g1,..., {f1, . . . , fn−1, gn}} (4.2.2)
∞ for fi, gi ∈ C (M). The map {−,..., −} is called a Nambu n-bracket, the manifold M is called a Nambu-Poisson manifold, and we call the algebra of smooth functions C∞(M) endowed with the Nambu n-bracket a Nambu-Poisson algebra. 32 Geometry
4.3 Complex geometry
§1 Čech–Dolbeault correspondence. [IP00]
4.3.1 Calabi–Yau spaces §2 Calabi–Yau manifold. A local Calabi–Yau manifold is a Kähler manifold (M, J, g) with vanishing first Chern class. A Calabi–Yau manifold is a compact local Calabi–Yau manifold. The notion of a local Calabi–Yau manifold stems from physicists and using it has essentially two advantages: First, one can consider sources of fluxes on these spaces without worrying about the cor- responding “drains”. Second, one can easily write down metrics on many local Calabi–Yau manifolds, as e.g. on the conifold [PZT00]. We will sometimes drop the word “local” if the context determines the situation. §3 Theorem. (Yau) Yau has proven that for every complex Kähler manifold M with vanishing first Chern class c1 = 0 and Kähler form J, there exists a unique Ricci-flat metric on M in the same Kähler class as J. This theorem is particularly useful, as it links the relatively easily accessible first Chern class to the existence of a Ricci-flat metric. The latter property is hard to check explicitly in most cases, in particular, because no Ricci-flat metric is known on any (compact) Calabi-Yau manifold. Contrary to that, the first Chern class is easily calculated, and we will check the Calabi-Yau property of our manifolds in this way. §4 Equivalent definitions of Calabi-Yau manifolds. Let us summarize all equivalent conditions on a compact complex manifold M of dimension n for being a Calabi-Yau manifold: ◦ M is a Kähler manifold with vanishing first Chern class. ◦ M admits a Levi-Civita connection with SU(n) holonomy. ◦ M admits a nowhere vanishing holomorphic (n, 0)-form Ωn,0. ◦ M admits a Ricci-flat Kähler metric. ◦ M has a trivial canonical bundle. §5 K3 manifold. A K3 manifold is a complex Kähler manifold M of complex dimension 2 with SU(2) holonomy and thus it is a Calabi-Yau manifold. §6 Rigid Calabi–Yau manifold. There is a class of so-called rigid Calabi-Yau manifolds, which do not allow for deformations of the complex structure. This fact causes problems for the mirror conjecture, as it follows that the mirrors of these rigid Calabi-Yau manifolds have no Kähler moduli, which is inconsistent with them being Kähler manifolds. §7 Calabi–Yau supermanifolds. In mathematics, it could be that mirror symmetric partners of rigid Calabi–Yau manifolds are Calabi–Yau supermanifolds, and therefore mirror symmetry might require us to introduce a notion of supersymmetry [Set94]. Definition: holomorphic volume form exists. This definition has become common usage, even if not all such spaces admit a Ricci-flat metric. Counterexamples to Yau’s theorem for Calabi-Yau su- permanifolds can be found in [RW05]. (In a following paper [Zho05], it was conjectured that this was an artifact of supermanifolds with one fermionic dimensions, but in the paper [RW04] published only shortly afterwards, counterexamples to the naïve form of Yau’s theorem with two fermionic dimensions were presented.) Examples: in twistor string theory, many supertwistor spaces are Calabi–Yau supermanifolds. Large class: W CP 3|2(1, 1, 1, 1|p, q) with p + q = 4, which were proposed as target spaces for the topological B-model in [Wit04] and studied in detail in [PW04].
4.3.2 Twistor geometry §8 See. Penrose–Ward transform. 4.4 Graded geometry 33
4.4 Graded geometry
§1 Conventions. Useful conventions for grading, grade-shifts, etc.: [JRSW18]
4.4.1 Graded vectorspaces §2 Literature. See e.g. [CS10, Fai17]
4.4.2 N-manifolds
§3 N-manifold. An N-manifold M is an N0-graded manifold, i.e. a non-negatively graded manifold. We usually denote the body, i.e. the degree 0 part, of M by M0. We shall be interested exclusively in N-manifolds arising from N-graded vector bundles over M0. Such N-manifolds were called split in [SZ16], following the nomenclature for supermanifolds. Note that Batchelor’s theorem [Bat79] for supermanifolds can be extended to N-manifolds, and therefore any smooth N-manifold is diffeomorphic to a split N-manifold[BP13], see also [Roya] for special cases.
4.4.3 Supermanifolds §4 Literature. [Man, CDMIS02, DeW92]. As an example of a Kähler supermanifold see the discussion of the space CP 3|4 in section 5.5. For more details on supergeometry, see e.g. [Var99]. §5 Split supermanifolds. Split supermanifolds are supermanifolds that are grade-shifted vector bundle. All real supermanifolds are diffeomorphic to split ones [Bat79].
4.4.4 Generalized geometry §6 Literature. Lecture notes: [Koe11]. §7 Generalized tangent bundle. [Hit05] §8 Basics. Any target space description of classical strings has to include the massless excitations of the closed string. These consist of the spacetime metric g, the Kalb-Ramond B-field and the dilaton φ. The former two can be elegantly described in Hitchin’s generalized geometry [Hit03, Hit05, Gua03].
4.5 Fiber bundles
§1 Good covers. One can always cover a compact space with a finite cover such that all non-empty intersections are contractible [BT82] or [KL88].
4.5.1 Tangent and cotagent bundles §2 Cartan calculus. By Cartan calculus, we mean the graded Lie algebra cart(M) which describes the graded commutation relations of contractions ιX by vector fields X ∈ X(M), Lie derivatives LX as well as the exterior derivative d, all acting on differential forms. Correspondingly, cart(M) has underlying graded vector space X(M)[−1] ⊕ X(M) ⊕ R[1], and elements (X, Y, r) ∈ cart(M) are regarded as formal sums ιX + LY + rd. The degree follows from the degree shift induced by an element when acting on differential forms. The Lie bracket on cart(M) is simply the graded commutator of compositions of these operations, deg(a1) deg(a2) [[a1, a2]] := a1 ◦ a2 − (−1) a2 ◦ a1 . (4.5.1) We have
[[d, ιX ]] = LX , [[LX , LY ]] = L[X,Y ] , [[LX , ιY ]] = ι[X,Y ] (4.5.2) 34 Geometry and all other Lie brackets vanish. §3 Rules for calculations. With the wedge product, the general form of an r-form in explicit coordinates is: 1 ω = ω dxµ1 ∧ dxµ2 ∧ ... ∧ dxµr , (4.5.3) r! µ1µ2...µr where the ωµ1µ2...µr is totally antisymmetric:
ωµ1µ2...µr = sgn(P )ωP (µ1)P (µ2)...P (µr) (4.5.4)
1 3 1 2 so the 2-form ω = 3dx ∧ dx + 6dx ∧ dx has nonvanishing components ω13 = −ω31 = 3 and ω12 = −ω21 = 6. r r+1 The exterior derivative dr is a map Ω (M) → Ω (M) whose action on a form as in (4.5.3) is defined by: 1 ∂ d ω = ω dxν ∧ dxµ1 ∧ ... ∧ dxµr (4.5.5) r r! ∂xν µ1,...µr
(drω is often called the differential of ω.) An element of ker(dr) is called a closed r-form, an element of im(dr−1) is called an exact r-form. The opposite operation to the exterior derivative which lowers the order of a form is the interior product1: Let X,X1, ..., Xr−1 be vector fields over a manifold M (X,Xi ∈ X (M)). The interior product is r r−1 r a map iX :Ω (M) → Ω (M) acting on a form ω ∈ Ω by the following rule:
iX ω(X1, ..., Xr−1) := ω(X,X1, ..., Xr−1) (4.5.6)
0 2 If ω ∈ Ω , we define explicitely: iX ω = 0. If a vector field is given by X = Xµ∂/∂xµ, then the action of the interior product on a form in coordinates as in (4.5.3) is given by:
1 i ω := Xνω dxµ2 ∧ ... ∧ dxµr (4.5.7) X (r − 1)! νµ2...µr
2 Since the exterior derivative, the interior product is nilpotent: (iX ) = 0. Given an arbitrary r-form ω of the form (4.5.3), then3
1 i (i ω) = i Xνω dxµ2 ∧ ... ∧ dxµr X X X (r − 1)! νµ2...µr 1 1 = XσXνω dxµ3 ∧ ... ∧ dxµr . (4.5.8) (r − 2)! (r − 1)! νσµ3...µr
Again, the last expression has to be symmetric under exchange of the dummy variables σ and ν. The exchange of the vector components is symmetric, but the exchange in ωνσµ3...µr is antisymmetric. This causes iX (iX ω) to vanish. §4 Contraction.
|α| ιX (α ∧ β) = (ιX α) ∧ β + (−1) α ∧ (ιX β) . (4.5.9) §5 Hodge star.
1which must not be confused with the inner product 2This definition is analog to dω = 0 for ω ∈ Ωn (top forms) and yields the desired results for the de Rham complexes in chapter ??. 3 2 Strictly speaking, ω has to be of rank ≥ 2, for smaller rank, we have immediately (iX ) ω = 0. 4.5 Fiber bundles 35
∗ ∗α = (−1)|α|(d−|α|)(−1) tr (g)α , (4.5.10) where dim M = d and g is the metric on M. Two k-forms: α ∧ ∗β = (∗α) ∧ β (4.5.11)
§6 Transgression. More details on transgressions can be found e.g. in [Bry07] and [GT01], see also [Noo]. §7 Torsion and G-manifolds Need: G-structure on manifold M of dimension d, G ⊂ GL(d). Let V = Rd Construct exact sequence of vector space
0 → ker(σ) → g ⊗ V ∗ −−→σ V ⊗ ∧2V ∗ → coker(σ) → 0 , (4.5.12) where σ is the inclusion g ⊂ V ⊗ V ∗, postcomposed with antisymmetrization. For a manifold with G-structure, we have the following associated sequence:
∗ σ 0 → ker(σ) → P2 = ad(P ) ⊗ T M −−→ P3 → coker(σ) → 0 , (4.5.13)
The difference of two connections compatible with the G-structure is an element of P2, so that an arbitrary connection reads as ∇ = ∇0 + A, A ∈ Γ(P2). Thus, the difference of torsions of these connections is a section of P3 and the section of P4 is the intrinsic torsion of the G-structure, the failure of finding a torsion free connection. More: mathoverflow or in [Joy00].
4.5.2 Jet bundles
§8 Jet bundle. Consider a fiber bundle π : E → M over some manifold M. Let Γ(p) with p ∈ M denote the set of local sections of E whose domain contains p. For some r ∈ N, we we define the following equivalence relation on Γ(p):
∂|I| ∂|I| σ ∼ σ ⇔ σ = σ for all 0 ≤ |I| ≤ r . (4.5.14) 1 2 ∂xI 1 ∂xI 2 p p
r N Denote by jpσ the equivalence class represented by σ, where we call r ∈ the order, p ∈ M the source and σ ∈ Γ(p) the target. The jet bundle is the
r r J (π) = {jpσ | p ∈ M, σ ∈ Γ(p)} (4.5.15) endowed with the bundle projection
r r πJ : J (π) → M , πJ (jpσ) = p . (4.5.16)
There is another natural map
r r πE : J (π) → M , πE(jpσ) = σ(p) . (4.5.17)
§9 Example: trivial bundle The first jet bundle of the trivial bundle π : M × R → R is canonically diffeomorphic to T ∗M × R. A section of π is simply a function, and two functions are considered 1 equivalent, if they have the same value and the same differential. Mapping [f] ∈ J (π) to (df|p, f(p)) ∈ π−1(p) for each p ∈ M is therefore bijective. 36 Geometry
4.5.3 Principal bundles
4.5.3.1 Ordinary principal bundles §10 Transition functions. Given a principal G-bundle π : X → M, we can use the G-equivariant homeomorphisms φi to define a transition functions. Note that the homeomorphism is of the form −1 −1 −1 φi(p) = (π(p), gi(p)) for p ∈ π (Ui). Then the expression gij(p) := gi (p)gj(p) for p ∈ π (Ui ∩ Uj) −1 −1 −1 −1 depends only on π(p) since gi (hp)gj(hp) = gi (p)h hgj(p) = gi (p)gj(p). We thus obtain a function gij : Ui ∩ Uj → G, which satisfies the condition gijgjk = gik. This defines a G-valued Čech 1-cochain. One can reconstruct a principal G-bundle from its transition functions, cf. [KN63a, Prop. 5.2]. §11 Hopf fibration. The Hopf bundle is a fiber bundle S3 → S2, which can be generalized to S7 → S4. 3 R4 P 2 2 P 2 R3 Given the S as a submanifold of by xi = 1 and the S by yi = 1 in , the projection π is defined by the Hopf map:
y1 = 2(x1x2 + x3x4) (4.5.18)
y2 = 2(x1x4 − x2x3) (4.5.19) 2 2 2 2 y3 = (x1 + x3) − (x2 + x4) (4.5.20)
The Hopf fibration was the first example of a null-homotopic map from a higher-dimensional sphere to a lower-dimensional one. The preimage of a point in S2 is a circle in S3, called the Hopf-circle. An equivalent description is to take the S3 as |z|2 + |w|2 = 1 in C2 and consider the point on the Riemann sphere corresponding to z/w. The fibration is S1 → S3 → S2 and a principal bundle. The associated vector bundle is L = S3 × C/U(1) with the identification ((z, w), v) ∼ ((eitz, eitw), eitv), a complex line bundle over S2. The sphere S3 is the Lie group of unit quaternions and can be identified with SU(2), the double cover of SO(3). The Hopf bundle is the quotient S2 = SU(2)/U(1). The Hopf fibration underlies the Dirac monopole of charge 1. Both were discovered/defined in 1931, coincidence was only noticed in [GP75], see also references from [Sta98], page 2. 4 §12 Results. Spin(4)-bundles on S are classified by π3(Spin(4))
4.5.3.2 Higher principal bundles §13 Literature. The first non-abelian higher gerbes were defined by Breen & Messing [BM05, Bae02] which were then generalized in various papers [ACJ05, AJ04, Bar06, BS04, Jur11a, SW13b, JSW16]. For the general understanding, it is also very helpful to read up on gerbes [Gir71, Bry07], particularly in the form of Murray’s bundle gerbes [Mur96, Mur07]. Higher gauge theory was probably first studied by Baez [Bae02] and Baez & Schreiber [BS04, BS07]. A very general and useful framework for describing higher groupoids are simplicial sets forming Kan complexes, and the corresponding notion of higher gauge theory can be found in our paper [JSW16]. Particularly important examples of Lie 2-groups are the 2-group models of the String group, a higher version of the spin group. Higher gauge theory with these 2-groups has also been developed [DS17] and the underlying description involves the weak 2-category of bibundles which is the 2-category of presentable stacks mentioned above. Picture using Čech groupoid, as done in [NW13]. A very general framework for studying differential cohomology has been developed by Schreiber [Sch13], which subsumes our above constructions. §14 General definition 2-bundles. [Woc11b] or in [Wala] §15 Example: Principal 2-bundle over a 2-space. One can also replace M by a different manifold, say T ∗[1]M. Then W (M) is the NQ-manifold corresponding to the exact Courant algebroid ∗ ∗ ∗ ∗ TM ⊕ T M. That is, W (T [1]M) = T [2]T [1]M = T [1]T [1]M = V2(M) which is the symplectic Lie 4.5 Fiber bundles 37
2-algebroid V2(M). One readily develops the corresponding higher gauge theory [RSS16], which leads to additional potentials matching quite closely the equations of motion of the M5-brane model from [LP10]. §16 Gerbe. In general, a gerbe is some geometric realization of an element in singular cohomology H3(M, Z), the Dixmier–Douady class of the gerbe. Mostly an abelian principal 2-bundle with structure 2-group U(1) ⇒ ∗. Review in [BSS18]. Naive (cocycle) version: Hitchin–Chatterjee gerbes: Chatterjee’s thesis [Cha98] and the correspond- ing section in Hitchin’s lecture notes [Hit99] are a good starting point. The more sophisticated formu- lation in terms of bundle gerbes is given in [Mur96] and in the lecture notes [Mur07]. The high-brow formulation for gerbes with connective structure is then found in Brylinski’s book [Bry07]. Alternative picture: central groupoid extensions. Symmetries described by associated Courant algebroid whose Ševera class is the Dixmier–Douady class of the gerbe.
4.5.3.3 Connections and related
§17 Cartan’s structure equations. Consider a morphisms of dgas ϕ : W (iso(R1,d)) → W (M) = • 1,d a a (Ω (M), d). On W (iso(R )), we have coordinates ξ and ζ b of degree 1 corresponding to the generators a a of translations and rotations as well as their shifts σξ and σζ b of degree 2. The former are mapped to vielbein and spin connection: a a a a ξ 7→ E and ζ b 7→ ω b . (4.5.21) The shifts are mapped to tensors which by compatibility with the differentials are identified with torsion and curvature: a a a a b a a a a c σξ 7→ T = dE + ω bE , σζ b 7→ R b = dω b + ω c ∧ ω b (4.5.22) Finally, we obtain the Bianchi identities from ϕ,
a a b a b dT + ω bT = R bE = 0 , (4.5.23) a a c a c dR b + ω c ∧ R b − R c ∧ ω b = 0 . sign adjust in R? §18 Cartan formalism Literature. arXiv:1412.2393 (Review) Solder form: Ehresmann, C. (1950). "Les connexions infinitésimales dans un espace fibré différentiel". Colloque de Topologie, Bruxelles: 29–55. Kobayashi, Shoshichi (1957). "Theory of Connections". Ann. Mat. Pura Appl. 43 (1): 119–194. doi:10.1007/BF02411907. Kobayashi, Shoshichi & Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 & 2 (New ed.). Wiley Interscience. ISBN 0-471-15733-3. §19 Link holonomy-connection. [Bar91, MP02] §20 General definition of connections. [Ati57][BKS05, KS15, SSS, FRS13, GS14] §21 Poincaré lemma. Higher: [DS15] §22 History of higher parallel transport. See section 1 of [BS07], Higher-dimensional Wilson surfaces: [Che02]. §23 Fake curvature. It has been shown [GP04, BS04, BS07] that for the parallel transport along surfaces to be reparametrisation invariant, the so-called fake curvature has to vanish:
Fa := Fa − t(Ba) = 0 . (4.5.24)
The kinematical data of an ordinary higher principal bundle with connection requires that all curva- ture forms except for the highest (which are collectively known as fake curvatures) have to vanish. Also visible in the commutator of infinitesimal gauge transformations in homotopy Maurer–Cartan theory, cf. [JRSW18]. 38 Geometry
Way out: use string structures. §24 Further reading. The holonomy functor is explained in great detail in Baez & Huerta [BH11] and Baez & Schreiber [BS04]. A detailed discussion of self-dual strings and the duality can be found e.g. in section 3 of my paper [Sae11]. NQ-manifolds are thoroughly introduced in Roytenberg [Roya]. Their relation to L∞-algebras is reviewed in the papers [LS93, LM95], where also the homotopy Maurer–Cartan equations and their infinitesimal gauge symmetries are found. For a discussion of this in the context of string field theory see Zwiebach [Zwi93]. The construction of local higher gauge theory as done in the previous section was first given in [JSW15]. I chose to follow this route, because it is the shortest way of deriving local higher gauge theory that I am aware of. A more geometrical approach involving morphisms of NQ-manifolds arises from a local version [BKS05, SSS, KS10, GS14] of a construction of Cartan [Car50a, Car50b, Car51] and Atiyah [Ati57]; see also Section 2 of the paper [RSS16] for a concise review and a further extension of this description of local higher gauge theory. A particularly impressive demonstration of the usefulness of higher Lie algebras in physics is the reproduction of the complete brane scan in type II superstring theory and M-theory from considering cocycle extensions of super L∞-algebras [FSS15]. §25 String structure. A string structure is a principal spin bundle over a manifold together with a trivialization of the first Pontrjagin class. First definition: [Kil87], also [Red06, Wal13]. A principal 1 Spin(n)-bundle P admits string classes if and only if 2 p1 = (F,F ) = 0 [ST, Section 5]. Equivalently, it is a principal 2-bundle with structure group a string 2-group model. Connection: Coupling of N = 1, d = 10 supergravity to vector supermultiplets requires inclusion of a Chern–Simons term H = dB + cs(A) to achieve supersymmetry. This has been shown for the abelian case in [BdRdWvN82] and for the non-abelian case in [CM83]. §26 Twisted string structure. Detailed explanation: [SSS, SSS12, FSS14]. Used in (1,0)-models: [SS18, SS17]
4.5.3.4 Characateristic classes
§27 Characteristic classes of higher bundles. Either from invariant polynomials, Chern–Weil theory, or something like here: [GS08]. See also discussion in [KS15].
4.5.4 Other fibrations
§28 Lie algebra bundle. See [Mac87]. Vector bundle whose typical fiber is a Lie algebra.
4.6 Stacks
4.6.1 Lie groupoids and Stacks
§1 Initial remarks. Useful literature: [BX11], [Met] and [Blo08]. Category of Lie groupoids can be extended to a weak 2-category of bibundles. This is the right 2-category to identify Lie groupoids with presentations of smooth stacks. In particular, a bibundle equivalence amounts to Morita equivalence. §2 Bibundles. Reviews in [DS17], see also [Mrc96, Ler10]. For a very detailed review on Lie groupoid bibundles, see also [Blo08]. 4.7 Other areas 39
4.6.2 ∞-stacks §3 Definition. Lie (∞, 1)-groupoids defined via simplicial manifolds, then higher form of morphisms: bisimplicial complex extension of bibundles, see Blohmann and Chu.
4.7 Other areas
§1 Supergeometry See under “supermathematics.” §2 Noncommutative Geometry See under “Quantum mechanics” 40 Geometry Chapter 5
Other areas
5.1 Arithmetic
§1 Fundamental theorem of arithmetic. Every natural number greater than 1 has a unique repre- sentation as a product of primes.
5.2 Analysis
§1 Generalized Jacobi elliptic function. [Paw09] is given by Z s −1 dt S (s, k1, k2) = . (5.2.1) p 2 2 2 2 2 0 (1 − t )(1 − k1t )(1 − k2t ) 5.3 Supermathematics
§1 Supernumbers. See [DeW92, CDMIS02] §2 Complex conjugation convention. First, and most commonly, there is
∗ (θ1θ2) = θ¯2θ¯1 = −θ¯1θ¯2 , (5.3.1) which is used e.g. in [Bar83, DeW92]. Second, there is
∗ (θ1θ2) = −θ¯2θ¯1 = +θ¯1θ¯2 , (5.3.2) which is used in [CDMIS02, DEF+]. The latter convention respects the sign rule that interchanging two Graßmann-odd objects in a monomial should always be accompanied by an additional sign. There is a discussion of this issue in [DEF+]. Manin in his book [Man] also discusses all of these conventions. In this paper, we use the second convention. §3 Supergroups. We often follow [Bar83], see also [Kac] and [Bar83].
5.3.1 Supergeometry §4 Supermanifolds See section in graded geometry. 42 Other areas Chapter 6
Classical and Quantum Mechanics
6.1 Classical Mechanics
6.2 Quantum Mechanics
§1 Quantization axioms. One is naturally led to imposing the following axioms, which yield a full quantization (cf. e.g. [AM]):
Q1. The map f 7→ fˆ is linear over C and maps smooth real functions on M to hermitian linear operators on H . Q2. If f is a constant function, then fˆ is scalar multiplication by the corresponding constant. ˆ ˆ Q3. The correspondence principle: If {f1, f2} = g then [f1, f2] = −i ~ gˆ. µ Q4. The operators xˆ and pˆµ act irreducibly on H .
∞ Here f, fi, g ∈ C (M) and {−, −} and [−, −] denote the Poisson bracket on M and the commutator of elements of End (H ), respectively. But the Grönewold-van Howe theorem now states that there is no such quantization, see [AM] or [GGT96] for details. One can prove an analogous theorem for M = S2. A full quantization of the torus M = T 2 does however exist. §2 Loophole to Grönewold-van Howe theorem. The first two approaches lead to prequantiza- tion and further to the formalism of geometric quantization [Woo92], while the third approach leads to approximate operator representations and eventually to the machinery of deformation quantiza- tion [BFF+78, Kon03]. Recall also that the canonical quantization prescription of Weyl, von Neumann and Dirac is not Q3, but just the corresponding condition on the coordinates of phase space, which further supports the third approach. §3 Review on quantization approaches. For further details on the relations between the various approaches, see e.g. [AE05]. §4 Toeplitz quantization. In Toeplitz quantization (see e.g. [BMS94]). The Toeplitz quantization map is the adjoint of the Berezin quantization map with respect to the Hilbert-Schmidt norm and the L2-measure induced by the Liouville volume form [Sch98]. The ordering prescriptions resulting from Berezin and Toeplitz quantizations of M = CP n correspond to Wick and anti-Wick ordering, respectively, cf. [ILMS08]. Toeplitz quantization is of interest for various reasons. First, it converges towards geometric quanti- zation as shown in [Tuy87]. Second, strict convergence theorems can be deduced, and in particular for M = CP n one has [BMS94]
lim i k TO(k)(f),TO(k)(g) − TO(k) {f, g} = 0 . (6.2.1) k→∞ HS 44 Classical and Quantum Mechanics
6.2.1 Noncommutative geometry
§5 Literature. First mention of NC spaces in literature: [Sny47]. Also, from Riemann’s inaugural lecture [Rie]: “Now it seems that the empirical notions on which the metrical determinations of space are founded, the notion of a solid body and of a ray of light, cease to be valid for the infinitely small. We are therefore quite at liberty to suppose that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena.” Connes book: [Con94] Other useful book: [FGBV01] §6 Examples. See the section on Geometric Quantization. List of quantum spheres: math.QA/0212264. §7 Star products. Literature: [BFF+78]. Observations, e.g. [Kon03]: Gauge equivalence classes of star products on C∞(X) up to first order in ~ are classified by Poisson structures on the manifold X. §8 Formality conjecture. (Kontsevich) Every smooth Poisson manifold can be canonically quantized and the equivalence classes of the resulting algebras are canonically defined. §9 Drinfeld twist. Hopf algebra techniques applied to QM to restore symmetries: Very recently it has been reported in [CKNT04] and in [A+05] that these symmetries can be restored by twisting their coproduct. Such a twist in a general context is due to Drinfeld [Dri90], see also the work of Oeckl [Oec00]. A clear way of understanding these developments is as follows [Wes03, BMPV06].
6.2.1.1 Fuzzy geometry
§10 Idea. Compact Kähler manifolds allow for truncations of their algebra of functions to finite- dimensional matrix algebras. These truncations are identified with the duals of fuzzy spaces. §11 Motivation. These fuzzy spaces are interesting essentially for two reasons: The first one is that the fuzzy framework provides a nice way of regularizing quantum field theories on compact Riemannian spaces without breaking spacetime symmetries. It is therefore considered a useful alternative to the lattice approach. Second, fuzzy spaces arise naturally in string theory when one considers D-brane configurations in certain nontrivial background fields, see e.g. [Mye99]. §12 Fuzzy sphere. Berezin-quantization of the 2-sphere [Ber75], see also [Hop] or [BKV05, ILMS08] §13 Myers effect. Fuzzy sphere arising in D-brane configurations [Mye99], description also in many papers, e.g. [SS10]. §14 Fuzzy flag manifolds.. Also good review with general ideas: [MS08]. §15 Other fuzzy spaces. The fuzzy spaces studied in the literature so far are the fuzzy sphere [Hop] and orbifolds thereof [Mar04], the fuzzy disc [LVZ03], the fuzzy complex projective spaces [BDL+02] and deformations thereof [Ram01], fuzzy tori [Hop88], the fuzzy supersphere [GR98], and fuzzy Graßmannians as well as fuzzy flag manifolds together with their superextensions [DJ03, IMT03, MS08], see also [TV00]. This set of spaces is still very limited, and hence it is desirable to find further examples of fuzzy spaces.
6.2.1.2 Higher quantization
§16 Version based on Nambu. See discussion and review in [DSS10]. In particular Zariski quanti- zation in [DFST97]. §17 Naive versions of higher-dimensional fuzzy spheres. First, the idea of embedding spheres into complex projective space has been used previously to construct fuzzy spheres. In particular, the fuzzy 4-sphere has been constructed from the fact that CP 3 is a sphere bundle over S4, S2 ,→ CP 3 → S4, cf. [MO03, DO03, Abe04]. Second, a purely group theoretic approach was pursued in [GR01, Ram01]. §18 Preparations. Multisymplectic phase spaces: [BHR10] Phase space structure on loop space: 6.2 Quantum Mechanics 45
[Bry07], review in [SS13]. §19 Properly with gerbes. See [BSS18] and references therein. Further literature references: The 2-vector spaces formed by sections of prequantum bundle gerbes, which should underlie categorified Hilbert spaces, were first developed in detail by Waldorf [Wal07] and then technically developed further to prequantum 2-Hilbert spaces [BSS18]; see also the papers [FRS16, FRS14] for a detailed account of higher prequantization. §20 Avoiding gerbes by going to loop spacs. “Transgression,” see [SS13]. See also [BBvS00, KS00, MS01] for M-theory motivated work. 46 Classical and Quantum Mechanics Chapter 7
Field Theory
7.1 Classical Field Theory
7.1.1 Spin and all that §1 Irreps of Poincaré group. We are interested in finding all irreps of the Lorentz subgroup – the extension to the full Poincaré group is trivial. Representations should be linear (i.e. given by linear transformations) and QFT requires unitary representations (up to a phase – recall that to obtain physical information from QFT, we calculate mod–squared amplitudes, which remove phases). Wigner showed [Wig39] that one can reduce such representations up to a phase, to representations up to a sign, and Bargmann showed later [Bar54] that studying all unitary irreps of the Poincaré group up to a sign corresponds to studying all irreps of the universal covering group. §2 Universal cover of O(1, 3) See [FH91] §3 Spinor conventions in various dimensions. [Pol98b, Appendix B]
7.1.2 Examples of field theories §4 Supersymmetric field theories. See also SUSY field theories.
7.1.2.1 General gauge theories §5 Gauge algebroid. [Bar10] and in particular [Hen90], see also [JRSW18].
7.1.2.2 Yang–Mills theory §6 First order formulation. Originally: [OT79], see also [Witten]. §7 Self-Dual Yang–Mills theory. §8 Ward’s conjecture. All integrable field theories arise from dimensional reductions of 4d self-dual Yang–Mills theory. See [MW] for details.
7.1.2.3 Yang–Mills–Higgs theory §9 ’t Hooft–Polyakov monopole. [tH74, Pol74, PS75]
7.1.2.4 Chern–Simons theory §10 SUSY Chern–Simons theory. [Iva91] §11 Localization. [KWY10] 48 Field Theory
7.1.2.5 BF-type theories
§12 BV formulation. [CR01] §13 As higher gauge theories. [GP04, GPP08] as well as [SSS].
7.1.2.6 General higher gauge theories
§14 Literature. Review: [BH11]. Possibly first one based on non-abelian gerbes: [Bae02], but issues with gauge invariance. §15 See also. fake curvature
7.1.2.7 (1,0)- and (2,0)-theory
§16 History. Invented: [Wit95][Str96, Wit96b], see also [SW96]
Taking n M5-branes approaching each other with the Planck length `p going to zero, one finds an interacting field theory decoupled from gravity [Str96, Wit95]. This theory has N = (2, 0) supersymmetry 5n and a moduli space of the form R /Sn, which is superconformal at the origin of the moduli space [Sei98]. §17 Properties of the theory. [SS18, Lam19] and references therein. More details: [CDY15] ADE Classification: [BI97] §18 Arguments against existence. [SS18, Lam19] and references therein. Eckmann–Hilton type arguments: The original paper effectively pointing out abelianity of higher dimensional parallel transport if done outside of 2-categories: [EH62]. Rediscovery of this argument in the physics literature in infinitesimal form: [Tei86]. §19 BPS states. Self-dual strings: [HLW98] §20 PST action. A complete, d = 6 covariant and kappa-symmetric, action for an M-theory five- brane propagating in D = 11 supergravity background [PST95b, PST95a] Follow ups: [PST97b, PST97a, BLN+97, APPS97]. PST action and quantization: [BT14, page 107] §21 Limits, etc. From M2-brane models: [HIMS08, HM08, BT08]. Matrix Models: [ABK+98] §22 Breaking to (1,0)-theories. From a string theory perspective, the breaking of N = (2, 0) supersymmetry to N = (1, 0)-supersymmetry is observed, e.g., in the brane configurations of [BK97]. Also, note that a classification of N = (1, 0) superconformal field theories has been done using F- theory [HMV14, DZHTV15, HMRV15, Bha15].
7.1.2.8 General Relativity
§23 Reviews. [Bla] §24 As dga-morphism. (Super)gravity can be described as a dga-morphism from W(g) to W(M) = d CE(T [1]M), where g = so(d) n R . One gets the vielbein and the spin connection as potentials, torsion and Riemann tensor as curvatures and the obvious Bianchi identities. More under https://ncatlab.org/nlab/show/supergravity
7.1.3 Kaluza–Klein reduction
§25 Over a circle, from 5d to 4d. Take 5d metric ! gµν + AµAν Aν gˆmn = (7.1.1) Aµ 1 7.1 Classical Field Theory 49 with inverse µν µν ! mn g −g Aν gˆ = µν µν . (7.1.2) −Aµg 1 + g AµAν
ˆ 1 µν Then R = R − 4 F Fµν is the Kaluza–Klein miracle.
7.1.4 Classical Integrability
7.1.4.1 Integrable field equations
§26 Monopole. §27 Literature. Ansatz by ’t Hooft and Polyakov [tH74, Pol74], exact solution [PS75], differential equation by Bogomolny [Bog76]: F = ∗∇Φ. §28 Platonic solid monopoles. The platonic solid monopoles [HMM95]. There are explicit examples for charge N-monopoles, which exhibit the symmetry groups of the platonic solids. There is a charge-3 tetrahedrally symmetric one, a charge-4 octahedrally one, charge-5 octahedrally one, and a charge-7 dodecahedrally one [HS95]. They could be the first parts of a sequence of su(2)-monopoles converging towards the spherical magnetic bag. The size of the Platonic monopoles has been shown to be in good agreement with predictions of the bag model [Man12]. §29 Magnetic domains. By magnetic domains, we mean monopole configurations that arise in the limit n → ∞. Such configurations of monopoles are known to exist, and in a certain limit n, v → ∞ one obtains a solution of the Bogomolny equation invariant under a translation group Z [DK05, HW09]. This is an example of a monopole chain [CK01, War05]. A detailed discussion of various aspects of magnetic domains is found in [HPS12]. Similarly, one can consider doubly-periodic monopoles invariant under the action of Z2, given by (x1, x2, x3) 7→ (x1 + i, x2 + j, x3) for i, j ∈ Z2. One has the freedom to impose different boundary conditions as z → ±∞, and configurations satisfying kΦk → A as z → −∞ and kΦk ∼ Bz as z → ∞ for constants A, B are know as monopole walls1 [Lee99, War05, War07]. If a monopole wall has non-zero charge per unit period, then the total charge n is again infinite. Our final examples of monopoles with n → ∞ are magnetic bags [Bol06]. Heuristically, a magnetic bag with finite charge n consists of a finite-area segment of a monopole wall, folded around to form a closed surface. A brane interpretation of magnetic bags is found in [HPS12]. §30 Octonionic instanton. [War84], [FN85] paper3 paper4 Interpretation in string theory: look at discussion in [HT06]
7.1.4.2 Penrose–Ward transform and related
§31 Idea. Certain field equations are equivalent to flatness along certain subspaces of spacetime. Twistor spaces are the moduli spaces of these subspaces and holomorphic bundles over these twistor spaces are then in 1:1 correspondence with solutions of field equations via Penrose–Ward correspondence. §32 Literature. Penrose & Rindler: [PR84, PR86], Ward & Wells [WW90], Mason & Wood- house [MW]. Reviews: [PS05, Wol10] §33 Main theorem. [War77]:
1Configurations for which kΦk ∼ B|z| as z → ±∞ are called monopole sheets. 50 Field Theory
Theorem 7.1.1: Topologically trivial principal bundles over T 3 which become holomorphically trivial when restricted to any CP 1,→T 3 are in one-to-one correspondence with instanton solutions on C4, modulo isomorphisms on both sides. §34 Various twistor spaces. Ambitwistor space [Wit78, IYG78] for N ≤ 3 super Yang–Mills theory, reduces to ordinary twistor space for instantons and Hitchin’s minitwistor space P 2 [Hit82] for monopoles. Self-dual strings: hyperplane twistor space [SW13a]. §35 6d super Yang–Mills. Description of classical solutions to the equations of motion of MSYM theory on M 6 by means of holomorphic data on the ambitwistor space L9|8. The key fact here is that these equations are equivalent to certain constraint equations for a connection on the superspace M 6|16 [HS86] and furthermore, that these constraint equations can in turn be interpreted as integrability conditions along certain null-lines [Dev86, SW10]. Full discussion then in [SWW12]. §36 Higher versions. Various higher versions are found in the following papers: Abelian 6d: [SW13a, MRETC12, MRE12] 4d: [SW13a], Non-abelian 6d and 4d:[SW14a, SW14b, JSW15, JSW16] §37 Twistor string theory. [Wit04] §38 Twistor string theory of 4d super Yang–Mills. No holomorphic Chern–Simons theory on ambitwistor space, first attempts out: [MS06a]. More natural: use a Lie 3-algebra as done in [SW17].
7.1.4.3 ADHM-construction
§39 Idea. Original papers: [AHDM78] §40 ADHM and twistors. Connection twistors to ADHM: [Wit], [Osb82]. §41 Supersymmetric extension of ADHM [Sem82, Vol83]. §42 D-brane interpretation of ADHM. [Wit96a, Dou95, Dou98], see also [DHKM02, Ton05]. §43 ADHM and monads. [WW90] and [Fee]. The technique of obtaining vector bundles from monads stems originally from Horrocks [Hor64], see also [Ati79]. §44 ADHM in higher dimensions. paper1 paper2
7.1.4.4 ADHMN-construction
§45 Idea. Same as ADHM, but for monopoles. Original paper: [Nah80], also see [Ton05] §46 Examples. The ADHMN construction for charges 1 and 2 here is standard and found, e.g. in [GN00]. Spherically symmetric ones: [Ros82] §47 D-brane interpretation. [Dia97], boundary conditions: [Tsi98], also [HT05] §48 Supersymmetric extension. [LS06], §49 Relation to twistors [Hit83] §50 Relation to spectral curve. spectral curve [Hit83, AvM80]
7.2 Quantum Field Theory
7.2.1 Basics
§1 Stationary phase formula. cf. [Mne17, Section 1.2.4] §2 Propagator thory. E.g. [Zei11, Chapter 14] 7.2 Quantum Field Theory 51
7.2.1.1 Scattering amplitude relations
§3 Parke–Taylor formula. conjectured in [PT86], proved using recursion relation by Berends and Giele in [BG88] §4 Berends–Giele recursion relation. [BG88] §5 Double copy. Gravity=(Gauge Theory)2, statement about scattering amplitudes, comes from String Theory (KLT relations). Half Riemann sphere: open string with state operator correspondence, glue two half spheres to sphere to get closed string amplitude. Gluing governed by KLT matrix, which can be computed. Slogan literally true for n = 3, since KLT matrix trivial. Color factor stripped off. §6 Amplitudes and BV. First work on YM (?) [NR18], then [MSW19, Arv]. Original idea to get scattering amplitudes from minimal models: [Kaj02, Kaj07]
7.2.2 BV formalism §7 BRST formalism. [BRS76, Tyu75] §8 Literature. Original literature: Anselmi cites original literature as follows: [BV83, BV85], [BV84], [BV81], [BV77]; [Sch93a]: geometry of BV in terms of graded manifolds; [AKSZ97]: more details in context of topological field theories Reviews: Big, own: [JRSW18]. [HT92]: sections 8 and following; [Hen90]: Useful for open-algebra explanation; [GPS95]: Big review, lots of details, explicit index notation, examples; [BBH00]: also seems to be standard reference by now; [Mne17];[Fio]: Brief review; [Sta98, Sta97]: Classical review by Stasheff; [FKS12] Classical structure; [KLS05]: Lagrangian formalism and ∞-algebras mentioned Other interesting literature:
◦ [FH90]: read, incorporate ◦ [CR01]: superformalism (?) ◦ [Sch] L∞-algebra morphisms, also [FRZ16] ◦ [BFLS98] ◦ [BBvD85]: Higher Spin ◦ [FLS02a]: important, based on [BBvD85], exploring the actual structures ◦ Initial consideration of L∞-algebra Yang–Mills [MS04, MS06b] ◦ [Zei09]: L∞-algebra for Yang–Mills ◦ [LMSZ09]: Zeitlin for ABJM, etc. ◦ [Cos07] Renormalisation ◦ [SZ14, Zuc16]: Higher Chern–Simons ◦ [Zuc17]: BV and higher gauge ◦ [Zuc19, Zuc18]: ERG and BV
BV for higher form fields: [Bau96]: BV for higher forms, Chern–Simons and Donaldson–Witten type; [Alf97]: Non-Abelian BV? Strange... Potentially interesting: [GW96] Renormalizability issues; [FOK91] Geometric Quantization interpre- tation of BRST BV and renormalisation group flow, in particular relation to categorical equivalence: [Sha03], see also [Sha19] §9 BV complex. The BV complex is a combination of a Chevalley–Eilenberg resolution and a Koszul– Tate resolution. For ordinary field theories, this complex has been studied under the name of detour complex [GHW07, GSS08].
The L∞-algebra LYM2 was first given in [MS04, MS06b] in its dual formulation as a differential graded algebra. The same L∞-algebra was then rederived from string field theory considerations and further discussed in [Zei07, Zei09]. 52 Field Theory
For first order Yang–Mills: [Cos07]. Quasi-Iso to 2nd order complex: [RZ18] and [JRSW18] For Nahm, Yang–Mills, BLG and Basu–Harvey–Hoppe: also [LMSZ09] Higher Chern–Simons: [Kim18]
§10 Existence of SBV. [FH90] §11 Gauge fixing. BV triangle: [BV83] §12 BV and Noether Identities. [GPS95, Sta98, FLS02b, HT92] §13 AKSZ formalism. [AKSZ97] More details: [CF01, Roy07, KS15, KS10] and in particular [FRS13] for a modern perspective. Non-topological field theories: [BG11, Gri12] §14 Equivalence of field theories. Two classical field theories are classically equivalent if their L∞-algebras are quasi-isomorphic [JRSW18]. We note that a related notion of equivalence was discussed in [BGST05].
7.2.3 Anomalies
§15 Idea. An anomaly is breaking of a classical symmetry after quantization. This is usually a reflection of the Lagrangian not being a global function but a section of some line bundle. §16 Anomaly cancellation. The tensor product of several line bundles underlying anomalies may be trivializable. This implies that the corresponding Lagrangians can be added and trivialized by some higher gauge potential. §17 Green–Schwarz anomaly cancellation. The Green–Schwarz anomaly cancellation [GS84] amounts to the equation
dH = F1 ∧ F1 − F2 ∧ F2 , (7.2.1)
where F1 and F2 are the field strengths of the spin(n) and E8-valued connections. Application: cancels the 1-loop anomaly in heterotic string theory. Original paper: [GS84] Good introduction: e.g. [Avr06]
7.3 Supersymmetry
§1 Literature and conventions. Recall that the two different choices for Minkowski metric, the mostly plus (−, +, +, +) and the mostly minus (+, −, −, −), are often referred to as the East Coast Metric (ECM) and the West Coast Metric (WCM), respectively.
[WB92] J. Wess and J. Bagger, “Supersymmetry and supergravity”, Princeton, USA: Univ. Pr. (1992) 259 p. ECM, the essentials. [Wei] S. Weinberg, “The quantum theory of fields. Vol. 3: Supersymmetry”, Cambridge, UK: Univ. Pr. (2000) 419 p. ECM, uses Dirac spinors, more physics. [BK] I. L. Buchbinder and S. M. Kuzenko, “Ideas and methods of supersymmetry and supergravity: Or a walk through superspace”, Bristol, UK: IOP (1998) 656 p. ECM, supermathematics, NR theorems, super Feynman rules. [GGRS83] S. J. Gates, Marcus T. Grisaru, M. Rocek and W. Siegel, “Superspace, or one thousand and one lessons in supersymmetry,” Front. Phys. 58 (1983) 1 [hep-th/0108200]. ECM, many useful things, in particular super Feynman rules. [VP99] A. Van Proeyen, “Tools for supersymmetry”, arXiv:hep-th/9910030. ECM, more algebraic. 7.3 Supersymmetry 53
[Mar97a] S. P. Martin, “A Supersymmetry Primer”, arXiv:hep-ph/9709356. ECM, particle physics, MSSM. [Bil01] A. Bilal, “Introduction to supersymmetry”, arXiv:hep-th/0101055. WCM, for particle physics conventions. [Lyk96] J. D. Lykken, “Introduction to supersymmetry”, arXiv:hep-th/9612114. WCM, useful as a reference.
§2 History. At the beginning of the 1970’s, people started looking at SUSY toy models. In this section we will discuss a simple model which will illustrate many of the important physical properties of SUSY theories which are commonly discussed in a field theory context. It will also serve to illustrate the fundamentals of many calculations we will later perform in field theory, in a simple and accessible setting. Much of the following discussion can be found in [Wit81, Wit82] in more detail. §3 Coleman–Mandula theorem. [CM67] We may ask the question: can one extend spacetime symmetries non–trivially beyond the Poincaré group? The answer goes as follows. Assume G is the symmetry group of a theory with S–matrix S such that
◦ G contains the Poincaré group, ◦ all particles have positive energy, with finitely many particles of mass m < m0 for all m0, ◦ S–matrix elements h out |S| in i are analytic and non–trivial,
then the Coleman–Mandula theorem tells us that G =∼ Poincaré group × internal symmetries. So the answer to the above question appears to be no. There was a hidden assumption in this theorem, however – that the Lie algebra of G was generated by commutators. As we saw above, SUSY algebras, however, include anticommutators and therefore provide a loophole to the Coleman–Mandula theorem.
7.3.1 Representations
7.3.1.1 Explicit representations
§4 N = 2 tensor multiplet in 4d .[dWvHVP81], in rigid SUSY: [dWS06] §5 N = (2, 0) tensor multiplet in 6d. [HST83]. Useful for conventions: [CKVP98]
7.3.1.2 General statements
§6 Witten index. Introduce the operator (−)F , which gives +1 on bosonic states and −1 on fermionic states [Wit82]. §7 Theorem of Haag, Sohnius, Lopuszanski. Up to introducing ‘central charges’ Z[i,j] such that
i j [i,j] {Qα,Qβ} = αβZ , (7.3.1)
where the Z are just complex numbers, the N –extended SUSY algebra is the only extension of the Poincaré group which is consistent with the axioms of relativistic quantum field theory [HLS75]. §8 Classification of superconformal multiplets. [Nah78] Maximal dimensions is 6, due to various algebra identities.
7.3.2 Superfields
7.3.2.1 Superfields for 4 supercharges
§9 Literature. Without a doubt: [WB92]. §10 de Wit–Freedman transformations. SUSY transformations break WZ gauge. However, we know that the fields which are being re–introduced can just be gauged back to zero, so there is always 54 Field Theory a compensating gauge transformation which brings us back to WZ gauge (called the de Wit–Freedman transformation [dWF75]) §11 Superfield formulation of Chern–Simons theory. [ZP88, Iva91] §12 Super Feynman graphs. Remarks on regularisation: Unless care is taken to choose a regulator which preserves them, symme- tries of the theory may be violated in perturbative calculations. In particular, one often wishes to use Ward identities, which can be violated by bad choices of regularisation. For SUSY theories, it is neces- sary to keep the number of bosons and fermions equal, which is difficult in dimensional regularisation (DREG). This led Siegel to introduce dimensional reduction, or ‘DRED’ [Sie79]. Here, one splits, for example, the gauge boson Aµ into (Aµˇ, φ) into a 4 − dimensional gauge potential, and an dimen- sional scalar. Although it is known to be correct up to two loops, this regularization has unfortunately problems of its own [Sie80]. A detailed summary of the current situation is found in [JJ97].
7.3.2.2 Projective superspace
§13 N = 2 SYM in projective superspace. [DvU17]
7.3.2.3 Harmonic superspace
7.3.3 SUSY field theories
7.3.3.1 Super Yang–Mills theories
§14 10d SYM. Field equations on superspace equivalent to those of 10d SYM: [HS86]. §15 Reductions of 10d SYM. Reductions to all lower-dimensional SYMs: [BSS77] §16 4d, N = 4 SYM. N = 4 SYM theory is perturbatively finite (the β–function vanishes) to all orders, see e.g. [HST84] and references therein. §17 N = 2 superspace. Interestingly, N = 2 SYM theory is one of the few examples of a theory with extended supersymmetry which admits a manifestly supersymmetric formulation in extended su- perspace2. Consider the superspace R1,3|8 with coordinates xµ, θα, ϑα. On this space, we introduce the chiral coordinates §18 Seiberg–Witten Theory. In this section, we present a sketch of the construction of the exact effective action of N = 2 SYM theory as obtained by Seiberg and Witten [SW94]. For simplicity, we restrict ourselves to the minimal theory without additional flavours and gauge group SU(2). For a more detailed presentation of this material, see e.g. [Bil01, Ler97, AGH97, GH95].
7.3.3.2 M2-brane models: Supersymmetric Chern–Simons matter theories
§19 M2-brane models. General review [BLMP13], see also [Lam19] §20 BLG model. Original M2-brane model: [BL08b, Gus09]. Motivated by Basu–Harvey–Hoppe equation: [BH05, Hop97], which is like Nahm equation for M2-branes. Reduction to D2-branes: [MP08] §21 ABJM model. Original: [ABJM08]. 3-Lie algebra reformulation: [BL09]. Variant: The ABJ model [ABJ08] with gauge group U(n) × U(m). Can be formulated as a higher gauge theory [PS14]. §22 Marginal deformations. (of BLG- and ABJM-type-models) Some in [Ime08]. First detailed study in [ASW10]. §23 BPS equations. The BPS equations of (some of) the M2-brane models are known as the
2See, e.g. [Lyk96] for more details on this formulation in N = 2 superspace. 7.4 Matrix models 55
Hoppe–Basu–Harvey equations [BH05], see also [Hop97]. The analogue of the Basu-Harvey equation in the ABJM model was previously derived in [GRGVRV08, Ter08, HL08] Interpretation of Basu–Harvey and fuzzy S3:[NPR09]
7.3.3.3 Supergravities
§24 N = 8 supergravity. N = 8 supergravity might actually be finite, similarly to N = 4 super Yang–Mills theory [BDR07].
7.3.4 Other topics
§25 Division algebras and supersymmetry. [KT83, BH10a, BH10b, Hue11, Hue14]
7.4 Matrix models
7.4.1 Single Matrix Model
7.4.1.1 The Hermitian matrix model
§1 Literature. Original: [BIPZ78]. Reviews: [DFGZJ95, EKR15]
7.4.2 Hermitian matrix models from scalar field theories on fuzzy spaces
§2 Literature. For detailed expositions of scalar quantum field theory on the fuzzy sphere, the fuzzy disc and fuzzy quantum mechanics, see [OS07a, OS07b, Sae10, ISS11, Sae15, RS15a]. Reviews: [Tek15]
7.4.3 String theory inspired Matrix Models
7.4.3.1 c < 1 string theory
§3 Literature. c < 1 string theory [DFGZJ95]
7.4.3.2 BFSS model
§4 Literature. Original paper: [BFSS97] §5 Idea. Reduce 10d SYM to 1d, M-theory in DLCQ.
7.4.3.3 IKKT model
§6 Literature. Original paper: [IKKT97], see also [AIK+99]. R2d + Yang-Mills theory on noncommutative θ [AII 00] §7 Idea. Reduce 10d SYM to 0d. §8 Deformation. Mass term + cubical interactions: [BMN02]. §9 Proposal as origin for gravity. [Ste10]. Issue: Can only capture Kähler manifolds.
7.4.3.4 Higher Lie algebra models
§10 Literature. [RS14] for Lie 2-algebra models, [RS16] for higher models 56 Field Theory
7.4.3.5 Other generalizations
§11 Nambu–Poisson. [Yon16], reformulating with Lie 2-algebra corresponding to A4 doesn’t work since Lie 2-algebra trivial. §12 3-Lie algebra models. [Sat09a, LP09, Sat09b, Sat10], [FT09] and [DSS11]. Makes sense as regularisation of Schild-type action of the M2-brane [PS09] Chapter 8
String Theory
8.1 Basics
§1 Books. First: [GSW87b, GSW87a] After D-branes: [Pol98a, Pol98b], Mid-2000s: [Zwi04, BBS06], Mid-2010s: [BLT13] §2 Useful lecture notes. Polchinski: [Pol94, Pol96] Various: [LT89, Joh00, Sza02] §3 Kalb–Ramond field. It is well known that the B-field of string theory should really be regarded as part of a Deligne 3-cocycle from a global perspective. That is, it is a part of the connective structure of an abelian gerbe [Gaw87, FW99]. §4 N = 2 string. Besides the bosonic string theory having a 26-dimensional target space (and some consistency problems due to a tachyon in the spectrum) and the super string theory with N = 1 worldsheet supersymmetry having a 10-dimensional target space, the N = 2 string living naturally in 4 dimensions received much attention as a toy model. In our consideration, this string will essentially serve as a model for some D-brane configurations arising in the context of twistor geometry. First papers: [OV90, OV91] For more details see [Mar92a, Mar92b, Lec99, GOS03] and references therein. The underlying worldsheet theory [BS77] is N = 2 supergravity coupled to two N = 2 massless chiral multiplets, the latter forming the ordinary sigma model describing a string. It has been shown in [OV90] that the N = 2 open string is equivalent to self-dual Yang–Mills theory in 2+2 dimensions. It was also proven there that the N = 2 closed string is equivalent to self-dual supergravity. In [Sie92], it was argued that the appropriate field theory is rather a fully supersymmetrized version, and thus the N = 2 critical string should correspond to N = 4 supersymmetric self-dual Yang– Mills theory. D-branes is four-dimensional (supersymmetric) SDYM theory reduced to the appropriate worldvol- ume [Mar97b, GOS03]. As is familiar from the topological models yielding hCS theory, we can introduce A- and B-type boundary conditions for the D-branes in N = 2 critical string theory. For the target space R2,2, the A-type boundary conditions are compatible with D-branes of worldvolume dimension (0,0), (0,2), (2,0) and (2,2) only [JS01, GOS03]. §5 Link to noncommutative geometry. See discussion in [Lüs10]. Details in [SW99].
8.1.1 T-duality
§6 Literature. Complete reviews on T-duality: [GPR94] and [AAGL95] 0 First results: Büscher papers [Bus87]. Continuation to all orders in gs and α :[RV92] §7 As canonical transformation on string phase space. [AAGL94, KS95], should be related to 58 String Theory twists of Courant algebroids obtained by canonical transformations [BHIW16, HIW17]. §8 Torus T 3 example. §9 Courant algebroid approach to T-duality. [CG] see also [DS18b]
8.1.1.1 Double Field Theory
§10 Literature The idea of DFT goes back to the early 90ies [Duf90, Tse91, Sie93a, Sie93b] and the development of double geometry [Hul05, HR09] led to the paper [HZ09], which gave DFT its name and seems to have triggered most of the current interest in this area. A detailed review of DFT is found in the overview papers [Zwi12, BT14, AMN13, HLZ13]. Also very good perspective: [Ber19] Details: Generalized diffeomorphisms in [Sie93a, Sie93b], section condition in [HR09], see section 1.1 in [DLP+15]. §11 Section condition. Berman: Distinguish strong constraint and section condition: strong con- straint without isometries yields unique section condition, otherwise, there is a choice of section condition which amounts to T-duality. §12 Generalized metric. The transformation rule for the metric and the B-field can also be written in a homogeneous way, by introducing the generalized metric [SW89, GRV89, MS93, Gua03]
κλ κν ! gµν − Bµκg Bλν Bµκg HMN = µκ µν . (8.1.1) −g Bκν g
Note that this metric is obtained from the ordinary metric g via a finite B-field transformation with adjoint action ! ! !T 1 B g 0 1 B H = . (8.1.2) 0 1 0 g−1 0 1
This explains the terminology, a B-field transformation modifies linearly the value of the B-field. More- over, we have the relation H−1 = η−1Hη−1. §13 DFT action. The biggest success of double field theory is probably the provision of an action1 [HHZ10] Z 2D −2d SDFT = d x e R (8.1.3) based on the Ricci scalar
1 M N KL 1 M L KN R = HMN ∂ HKL∂ H − HMN ∂ HKL∂ H 8 2 (8.1.4) M N M N − 2∂ d∂ HMN + 4HMN ∂ d∂ d .
This action is invariant under the generalized diffeomorphisms (??). Moreover, upon imposing the strong section condition ∂ = 0 and integrating by parts, it reduces to the usual action for the NS sector of ∂xµ supergravity: Z D √ −2φ 2 1 2 SNS = d x g e R + 4(∂φ) − 12 H . (8.1.5)
A more detailed analysis shows that constructing consistent doubled versions of the torsion, Riemann and Ricci tensors is much more involved. §14 Covariant fluxes. Equation (2.37) in [HL14] §15 Narain moduli space. [Nar86] §16 Heterotic DFT. See, e.g., [HK11] and discussion of [DHS18].
1A first action for double field theory was already suggested in [Sie93a]. 8.2 The five superstring theories 59
8.1.2 Other dualities
§17 Conifold transition. The transition from a deformed conifold through a singular conifold to a resolved conifold is an allowed process in string theory which amounts to a topology change. An application of this transition is found in the famous large N duality in [Vaf01]: In type IIA string theory compactified on the deformed conifold, i.e. on T ∗S3, wrapping ND6-branes around the S3 produces U(N) Yang–Mills theory in the remaining four noncompact directions filling spacetime. In the large N limit, this is equivalent to type IIA string theory on the small resolution, i.e. on O(−1)⊕O(−1) → CP 1. The inverse process is found in the mirror picture of this situation: ND5-branes wrapped around the sphere of the small resolution give rise to U(N) Yang–Mills theory, the large N limit corresponds to type IIB string theory compactified on the deformed conifold T ∗S3.
8.2 The five superstring theories
8.2.1 Type I strings
8.2.2 Type II strings
8.2.3 Heterotic strings
§1 Summary. Heterotic string theory is a theory of closed strings which is a hybrid of a bosonic string theory (the left-moving sector) and a superstring theory (the right-moving sector). The 16 dimensions differing between left- and right-moving sectors are compactified on an even, self-dual lattice and there are two such lattices, leading to different gauge groups in the remaining 10 dimensions. One is the HO string with gauge group SO(32), the other is the HE string with gauge group E8 × E8. There are no open strings in heterotic string theory, since it is not possible to define boundary conditions relate left- and right-moving sectors ?. Original literature: [GHMR85, GHMR86]. §2 Links to other theories. The HO and HE theories are linked by T-duality. S-duality relates the HO string to the type I string. HE theory on T 3 is supposed to be dual to M-theory on K3, see [LPS99] and references therein. §3 Anomaly. Discussion of Anomaly: [HW85, Sen86], resultion via Green–Schwarz anomaly cancella- tion amounts to the equation
dH = F1 ∧ F1 − F2 ∧ F2 , (8.2.1) where F1 and F2 are the field strengths of the spin(n) and E8-valued connections. §4 Compactification. Strominger [Str86] derived the conditions for a compactification of heterotic string theory to four dimensions with a maximally N = 1 supersymmetric vacuum. These equations are called the Strominger system.
8.3 D-branes
§1 Literature. Books: [Joh] (extended version of “D-brane primer”). Reviews: [Joh00] Branes and gauge theory (very helpful!) [GK99]. §2 Brane scan. See Hull hep-th/9705162 (p,q)-branes: 1,5. 3 singlet 7 singlet, but couplings change 9 doublet, but unknown?
Best derivation: Via super L∞-cocycles in “brane bouquet” picture [FSS15]. 60 String Theory
8.3.1 D-brane configurations
8.3.1.1 Chalmers–Hanany–Witten configurations
§3 Basics. The Chalmers–Hanany–Witten D-brane configuration [CH97, HW97], See also [Che11]. Such configurations were thoroughly analyzed using their effective description in terms of three-dimensional gauge theories and mirror symmetry [IS96, AHI+97]. They also proved to be very useful in the explo- ration of singular monopoles [CK98, CK99]. §4 Chern–Simons boundary terms and (p, q)-branes. Chern–Simons boundary terms arise from another type of defect in the CHW configuration: the (p, q)-branes. Their contribution to the field theory p on the D3-branes is a Chern-Simons term with Chern-Simons level k = q , cf. [KOO99] and [BHKK99]. §5 Hanany–Witten configurations and orbifolds. Hanany–Witten setups are equivalent to branes at orbifolds [BK98, HZ98].
8.4 String Field Theory
§1 String field theory. Open bosonic SFT: Witten [Wit86]. Open susy SFT requires introduction of A∞-algebra. Closed SFT: L∞-algebra [Zwi93]. Bla string field theory
8.5 M-theory
§1 Literature for M-branes. [Ber08], also [GK99]
8.5.1 M2-branes
8.5.1.1 Worldvolume
§2 Action. Howe–Tucker form of the action of the supermembrane [BST87, BST88], action (4.1) in [Aha96].
8.5.1.2 Target space
§3 Algebraic outline. The effective description of stacks of M2-branes which has been proposed by Bagger, Lambert, and Gustavsson in [BL07, BL08b, BL08a, Gus09] is based on so-called 3-Lie algebras. These algebras had been introduced in [Fil85] as a generalization of the notion of a Lie algebra: a 3-Lie algebra is characterized by a totally antisymmetric trilinear product satisfying a Jacobi-type identity. It was soon realized, however, that if one demands the positivity of the kinetic terms in the action there remains an essentially unique such 3-algebra. Other notions of 3-algebras were explored in [BL09] and [CS08] and, respectively, N = 6 and N = 2 actions based on them were constructed. These 3-algebras were put into a Lie-theoretic framework in [dMFOMER09], where it was established that they are in one- to-one correspondence with pairs (g,V ) of a metric Lie algebra g and its faithful unitary representation V . This analysis also showed that there are two kinds of relevant 3-algebras: real [CS08] and Hermitian [BL09], depending on whether the representation V is real or complex. §4 See also. BLG model and ABJM model in field theory. 8.5 M-theory 61
8.5.2 M5-branes
8.5.3 Worldvolume §5 Action. Townsend [Tow96], action (4.2) in [Aha96]
8.5.4 Target space §6 (2,0)-theory See: field theory §7 Available models Tensor hierarchy and related: [SSW11] with followup papers [SSW13, SSWW11] as well as [Chu13], which is a special case of previous proposa. 3-Lie algebra based things: [LP10] with various followups. Reinterpretations of latter model via loop spaces [PS11] and higher gauge theory [PS12]. §8 M5 from M2. abelian M5 from BLG-type models: [HM08, HIMS08]: 3-Lie algebra identified with Nambu-bracket in 3 directions yields the desired result. §9 M2 from M5. In 3-Lie algebra model: [LS16]
8.5.5 U-duality and EFT §10 M-theory T-duality. Type IIA superstring theory compactified on T 2 is invariant under T- duality, which suggests that M-theory on T 3 should be invariant under a generalization. See [Sen96] and in particular the introduction in [Aha96]. T-duality in M-theory is scalar-vector duality in the worldvolume of the M2-brane. Büscher Rules: (3.5) in [Aha96]. More details, more recent: [DLP+15, HL16]
8.5.6 Exceptional Field Theory 62 String Theory Chapter 9
Tables
9.1 Formulas
9.1.1 Various symbols, functions, etc §1 ’t Hooft tensors. The ’t Hooft tensors (or eta-symbols) are given by
i(±) ηµν := εiµν4 ± δiµδν4 ∓ δiνδµ4 (9.1.1)
i(±) i(±) and satisfy the relation ηµν = ± ∗ ηµν , where ∗ is the Hodge star operator. They form three Kähler structures, which give rise to a hyper-Kähler structure on the Euclidean spacetime R4. Note furthermore that any space of the form R4m with m ∈ N is evidently a hyper-Kähler manifold.
9.1.2 Approximations §2 Stirling-Formula. x! ≈ exp(x ln x − x).
9.1.3 Expressions §3 Pauli matrices etc. The Pauli matrices are given by: ! ! ! ! 1 0 0 1 0 −i 1 0 σ := σ := σ := σ := 0 0 1 1 1 0 2 i 0 3 0 −1
σµ = (1, σi), σ¯µ = (1, −σi).
The gamma-matrices (Dirac, standard representation) are given by:
1 ! ! 1 ! 0 0 i 0 σi 0 0 1 2 3 γ := γ := γ5 := = iγ γ γ γ 0 −1 −σi 0 1 0
The gamma-matrices (Weyl, chiral representation) are given by:
1 ! ! 1 ! 0 0 i 0 σi 0 0 1 2 3 γ := γ := γ5 := = iγ γ γ γ 1 0 −σi 0 0 −1
In Euclidean space, the Pauli matrices are given by: ! ! ! ! 0 1 0 −i 1 0 i 0 σ := σ := σ := σ := 1 1 0 2 i 0 3 0 −1 4 0 i 64 Tables
σµ = (σi, i), σ¯µ = (−σi, −i).
§4 Euclidean 4d gamma-matrices. We use the following conventions for the gamma matrices generating the Clifford algebras in various dimensions with Euclidean signature: {γµ, γν} = 2δµν , (γµ)† = γµ . (9.1.2) In four dimensions, we work with the following explicit set: ! ! 0 σµ 1 0 γµ := , γ := γ1γ2γ3γ4 = , (9.1.3) σ¯µ 0 5 0 −1 where σµ := (−i~σ, 1), σ¯µ := (i~σ, 1). Note that we use these conventions only in the definition of the 3|8 generalized 3-Lie algebras C2d, while on three-dimensional superspace R , we followed the convention of Wess and Bagger [WB92]. With our conventions in four dimensions, we have (γµ)† = γµ and the following useful formulas: µ µν µν 1 µνρσ {γ5, γ } = 0 , [γ5, γ ] = 0 , γ = − 2 ε γ5γρσ , µ 1 µνκλ [γ5, γ ] = − ε γνγκγλ , 3 (9.1.4) ν ρσ νρσκ µν ρ νρ µ µρ ν {γ , γ } = 2ε γκγ5 , [γ , γ ] = 2(δ γ − δ γ ) , µν σρ µνσρ µρ νσ µσ νρ {γ , γ } = 2ε γ5 − 2(δ δ − δ δ )1 . An explicit embedding of SU(2) is given by
i j ijk 4 [γ , γ ] = 2ε γ5γ γk , i, j, k = 1, 2, 3 . (9.1.5) The full Lorentz algebra reads as usual: [γµν, γσρ] = 2(δµσγνρ + δνργµσ − δµργνσ − δνσγµρ) . (9.1.6) In arbitrary even dimensions, note that we have for multi-indices A, B: [γA, γB] = 0 , ({γA, γB} = 0 , ) (9.1.7) iff A and B have an odd (even) number of common indices. §5 Generators of Clifford algebras If γi, i = 1,..., 2d − 1 generate the Clifford algebra Cl(R2d−1), then the 2d-tuple µ i 2 1 d−1 (γ ) = (γ ⊗ σ , 1s ⊗ σ ) , s = 2 , µ = 1,..., 2d (9.1.8) 2d d 1 2d 2d generates Cl(R ). On the other hand, we just add γch := i γ ··· γ to the generators of Cl(R ) to obtain a set of generators of Cl(R2d+1). We can start the induction from the usual Pauli matrices σi, which generate Cl(R3) and satisfy [σi, σj] = −2 i εijk σk. In this case, all the generators are hermitian and we have γch = diag(1s, −1s). In the main text, we use the basis of Pauli matrices given by ! ! ! 0 1 0 i 1 0 σ1 = , σ2 = , σ3 = . (9.1.9) 1 0 −i 0 0 −1
a 2 d−1 Recall that for even d + 1, there is a set of generators λ , a = 1, . . . , r of u(r), r = 2 2 given by 1 2 2 i 2 i 2 √ 1 , γµ , γµν , γµνρ , γµνρσ ,..., (9.1.10) r r r r r r where γµ1...µk is the normalized antisymmetric product of gamma-matrices γµ1 , . . . , γµk . With this normalization, they satisfy the Fierz identity
a a λαβ λγδ = δαδ δβγ . (9.1.11) As these generators of u(r) form an orthogonal set with respect to the Hilbert-Schmidt norm, we conclude that all of them are traceless except for the identity matrix. 9.1 Formulas 65
9.1.4 Formulas for explicit expressions §6 Representations of su(n). Note that the quantization of n-dimensional complex projective space CP n has algebra of isometries su(n+1), which has (n+1)2 −1 generators. The dimension of the Hilbert space underlying the quantization is denoted by N. The relation between N, the level of quantization ` ∈ N0 and the dimension n of the complex projective space is (n + `)! N = . (9.1.12) n!`!
We have an N-dimensional representation of the generators of su(n + 1), denoted by Li, i = 1,..., (n + 1)2 − 1. These satisfy the algebra
[Li,Lj] =: ifijkLk , (9.1.13) where the fijk are the structure constants of su(n + 1). The normalization of the matrices Li is fixed by demanding that the eigenvalues of the second Casimir C2,
C2 B Φ := [Li, [Li, Φ]] , (9.1.14) are 2k(k + n) for k = 0, . . . , ` with multiplicities n(2k + n)((k + n − 1)!)2 . (9.1.15) (k!)2(n!)2
That is, the sum over the eigenvalues of C2 is ` X n(2k + n)((k + n − 1)!)2 Σ = 2k(k + n) , (9.1.16) 1 (k!)2(n!)2 k=0 and inductively, one can show that 2`(1 + `)2(2 + `)2 ··· (n + `)2((n + 1) + `) Σ = . (9.1.17) 1 (n + 1)!(n − 1)!
The matrices Li now satisfy the equations Σ Σ 1 tr (L ) = 0 ,L2 = 1 1 and tr (L L ) = 1 δ . (9.1.18) i i 2N 2 i j 2N (n + 1)2 − 1 ij Finally, we have the following identity for the structure constants:
fijkfij` = 2(n + 1)δk` . (9.1.19) §7 Lie algebra conventions. Note that our conventions differ slightly from those of [OS07a]. Ev- † 2 erywhere in our discussion, we use orthonormal hermitian generators τµ = τµ, µ = 1,...,N of u(N), which satisfy1 αβ γδ αδ βγ tr (τµτν) = δµν and τµ τµ = δ δ . (9.1.20) 2 1 The generators of u(N) split into the generators τ , m = 1,...,N − 1 of su(N) and τ 2 = √ 1 . For m N N N the former, the Fierz identity reads as 1 τ αβτ γδ = δαδδβγ − δαβδγδ . (9.1.21) m m N The Haar measure of SU(N) satisfies the following orthogonality relation: Z 1 dµ (Ω) [ρ(Ω)] [ρ†(Ω)] = δ δ , (9.1.22) H ij kl dim(ρ) il jk where Ω ∈ SU(N), ρ is a finite-dimensional, unitary, irreducible representation and ρ† denotes its complex conjugate, see e.g. [OS07a] for the proof.
1We always sum over indices which appear twice in a product, irrespective of their positions. 66 Tables
9.1.5 Integrals
9.1.5.1 Elliptic integrals §8 List of examples encountered. The following integrals are relevant for 1d matrix models (Matrix Quantum Mechanics).
Z b Z b p 2 2 2 2 p 2 2 2 2 I1 = dλ (b − λ )(λ − a ) , J1 = dλ (b − λ )(λ + a ) , a 0 Z b Z b 2p 2 2 2 2 2p 2 2 2 2 I2 = dλ λ (b − λ )(λ − a ) , J2 = dλ λ (b − λ )(λ + a ) , (9.1.23) a 0 Z b 3 Z b 3 p 2 2 2 2 p 2 2 2 2 I3 = dλ (b − λ )(λ − a ) , J3 = dλ (b − λ )(λ + a ) , a 0 with b > a ≥ 0. These integrals can be computed explicitly in terms of elliptic functions. We have2 i I = b a2 + b2 E − a2 + b2 E − a2 − b2 (F − K ) , (9.1.24a) 1 3 0 2 2 0 1 J = a −a2 + b2 E + a a2 + b2 K , (9.1.24b) 1 3 1 1
i 4 2 2 4 4 2 2 4 I2 = b 2 a − a b + b E0 − 2 a − a b + b E2 15 (9.1.24c) 4 2 2 4 + a − 3a b + 2b (F2 − K0) , 1 J = a 2 a4 + a2b2 + b4 E − 2a4 + 3a2b2 + b4 K , (9.1.24d) 2 15 1 1
i 6 4 2 2 4 6 6 4 2 2 4 6 I3 = b 2 a − 5a b − 5a b + b E0 − 2 a − 5a b − 5a b + b E2 35 (9.1.24e) 6 4 2 2 4 6 + a + 8a b − 11a b + 2b (F2 − K0) , 1 J = 2a −a6 − 5a4b2 + 5a2b4 + b6 E + a a2 + b2 2a4 + 9a2b2 − b4 K , (9.1.24f) 3 35 1 1 where we abbreviated complete and incomplete elliptical functions as follows:
a2 b2 b a2 E := E , E := E − , E := E arcsin , 0 b2 1 a2 2 a b2 b a2 F := F arcsin , (9.1.25) 2 a b2 a2 b2 K := K , K := K − . 0 b2 1 a2
9.1.6 Calculational rules §9 Antiholomorphic derivative. Note that 1 z¯ z¯ ∂z¯ = ∂z¯ = lim ∂z¯ = lim f(ε, z, z¯) (9.1.26) z zz¯ ε→0 zz¯ + ε ε→0 and Z lim dzd¯z f(ε, z, z¯) = π (9.1.27) ε→0 so 1 1 ∂ = δ(z, z¯) . (9.1.28) z¯ z π 2Our conventions for elliptic functions agree with those of Mathematica. 9.1 Formulas 67
9.1.7 Computational tricks §10 Schwinger parameters Z ∞ 1 −α(k2+m2) 2 2 = dα e (9.1.29) k + m 0 §11 Poisson resummation Z ∞ X X fˆ(y) = dx e2πixyf(x) ⇒ f(n) = fˆ(m) (9.1.30) −∞ n∈Z m∈Z §12 Sum to contour integral +∞ Z X T β βk0 T f(k = iω ) = dk f(k ) coth (9.1.31) 0 n 2πi 0 0 2 2 n=−∞ C f(k0) without poles on the imaginary axis.
9.1.8 Conventions and identities The following details our various conventions and useful relations. All of the relations given here are introduced as and when they are needed in the text, and exercises are provided against which the reader may check his/her understanding of the material. Further useful identities are found in [WB92], appendices A and B.
Metric
µν 2 2 Our metric tensor is ηµν = η = diag(−1, +1, +1, +1). As a result, the mass–shell condition is p +m = µ 0. We write ≡ ∂ ∂µ.
Epsilon tensors The epsilon tensors, which raise and lower spinor indices, are ! 0 −1 = = , (9.1.32) αβ α˙ β˙ 1 0 ! ˙ 0 1 αβ = α˙ β = . (9.1.33) −1 0
Spinor contractions Spinors anticommute. Spinors in the Weyl representation are naturally contracted ‘in the & direction’, α αβ βα βα β ψχ ≡ ψ χα = ψβχα = − ψβχα = + χαψβ = χ ψβ = χψ . (9.1.34) Spinors in the conjugate Weyl representation are naturally contracted in the % direction, ¯ ¯ α˙ ¯β˙ α˙ ¯β˙ α˙ α˙ ¯β˙ ¯β˙ ¯ ψχ¯ ≡ ψα˙ χ¯ = α˙ β˙ ψ χ¯ = −β˙α˙ ψ χ¯ = +β˙α˙ χ¯ ψ =χ ¯β˙ ψ =χ ¯ψ . (9.1.35) Fixing the directions of these contractions means that the inner products are symmetric. The products of spinor components are proportional to the tensors, 1 ˙ θαθβ = − 1 θ2αβ, θ¯α˙ θ¯β˙ = + θ¯2α˙ β , (9.1.36) 2 2 1 θ θ = 1 θ2 , θ¯ θ¯ = − θ¯2 . (9.1.37) α β 2 αβ α˙ β˙ 2 α˙ β˙ We sometimes abbreviate θθ ≡ θ2, θ¯θ¯ = θ2 and θθ θ¯θ¯ = θ4. A useful result is δ4(θ) ≡ θ2θ¯2 . (9.1.38) 68 Tables
Pauli matrices The Pauli sigma matrices are defined with lower Lorentz index, ! ! ! ! 1 0 0 1 0 −i 1 0 σ = , σ = σ = σ = . (9.1.39) 0 0 1 1 1 0 2 i 0 3 0 −1
Their natural spinor indices are σαα˙ . The barred sigma matrices, with their natural spinor indices are
˙ σ¯µ αα˙ := α˙ βαβσµ . (9.1.40) ββ˙ Explicitly σ¯µ = (σ0, −σ1, −σ2, −σ3) . (9.1.41)
Dirac matrices We use the Weyl representation for Dirac matrices, ! 0 σµ γµ = , (9.1.42) σ¯µ 0 which yields the following explicit form:
( −1 0 ! 0 1! 0 −i! 1 0 !) µ 0 0 −1 0 1 0 0 i 0 0 0 −1 γ = −1 0 , 0 −1 , 0 i , −1 0 . (9.1.43) 0 −1 0 −1 0 0 −i 0 0 0 1 0
The Dirac matrices obey the anticommutation relations (Clifford algebra)
{γµ, γν} = −2ηµν . (9.1.44)
Relations between Pauli matrices 1 θσµθ¯ θσνθ¯ = − θ2θ¯2ηµν , (9.1.45) 2 1 (σµν) β ≡ σµ σ¯ν αβ˙ − σν σ¯µ αβ˙ , α 4 αα˙ αα˙ (9.1.46) 1 (¯σµν)α˙ ≡ σ¯µ αα˙ σν − σ¯ν αα˙ σµ , β˙ 4 αβ˙ αβ˙
µ ν ν µ β µν β σ σ¯ + σ σ¯ = −2η δα , α (9.1.47) σ¯µσν +σ ¯νσµα˙ = −2ηµνδα˙ . β˙ β˙ From the last relations, we easily read off that
µ ν µ ναα˙ µν tr (σ σ¯ ) = σαα˙ σ¯ = −2η (9.1.48)
D–algebra In quantum computations using supergraphs, one has to reshuffle the supersymmetric covariant deriva- tives. In the text we needed the following formulæ
4 4 Dα(p, θ1)δ (θ1 − θ2) = −Dα(−p, θ2)δ (θ1 − θ2) , (9.1.49)
2 2 4 D¯ D δ (θ − ϑ) = 16 . (9.1.50) θ=ϑ 9.2 Other resources 69
¯ 2 2 ¯ 2 ¯ 2 2 ¯ 2 2 2 D D D = 16D ,D D D = 16D . (9.1.51) ¯ 2 2 2 ¯ 2 α ¯ 2 D D + D D − 2D D Dα = 16 . (9.1.52) For a chiral superfield Φ: ¯ ¯ α˙ α Dα˙ D DαD Φ = 16Φ . (9.1.53) More helpful identities are summarized in [BK], section 2.5.6 and [GGRS83], eqs. (6.3.28).
9.2 Other resources
9.2.1 Quotes etc. §1 First fundamental confusion of calculus.. [Woo09] Confusing coordinates with coordinate func- tions.
9.2.2 Webpages §2 NLab. An excellent reference for everything concerning modern mathematics and string theory is the nLab. In this wiki, most of the entries are presented from an ∞-categorical point of view. This takes some time to understand, but once this language is mastered, it is hard to go back. §3 Conferences. Stringwiki list
9.2.3 Books §4 Homotopy Type Theory. A useful resource on homotopy type theory, also from a computer science perspective is [The13]. §5 Differential cohomology in a cohesive infinity-topos. This [Sch13] is Urs Schreiber’s Habili- tation thesis, which keeps growing. It is a very useful reference with often more detailed and/or polished explanations than on the nLab. Always read the updated version from the nLab, the arXiv version is older. 70 Tables Bibliography
[A+05] P. Aschieri and others, A gravity theory on noncommutative spaces, Class. Quant. Grav. 22 (2005) 3511[ hep-th/0504183].
[AAGL94] E. Alvarez, L. Alvarez-Gaume, and Y. Lozano, A Canonical Approach to Duality Trans- formations, Phys. Lett. B 336 (1994) 183[ hep-th/9406206].
[AAGL95] E. Alvarez, L. Alvarez-Gaume, and Y. Lozano, An Introduction to T-Duality in String Theory, Nucl. Phys. Proc. Suppl. 41 (1995) 1[ hep-th/9410237].
[Abe04] Y. Abe, A construction of fuzzy S4, Phys. Rev. D 70 (2004) 126004[ hep-th/0406135].
[ABJ08] O. Aharony, O. Bergman, and D. L. Jafferis, Fractional M2-branes, JHEP 0811 (2008) 043[ 0807.4924 [hep-th]].
[ABJM08] O. Aharony, O. Bergman, D. L. Jafferis, and J. M. Maldacena, N = 6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals, JHEP 0810 (2008) 091[ 0806.1218 [hep-th]].
[ABK+98] O. Aharony, M. Berkooz, S. Kachru, N. Seiberg, and E. Silverstein, Matrix descrip- tion of interacting theories in six dimensions, Adv. Theor. Math. Phys. 1 (1998) 148 [hep-th/9707079].
[ACJ05] P. Aschieri, L. Cantini, and B. Jurčo, Nonabelian bundle gerbes, their differential geom- etry and gauge theory, Commun. Math. Phys. 254 (2005) 367[ hep-th/0312154].
[AE05] S. T. Ali and M. Englis, Quantization methods: A guide for physicists and analysts, Rev. Math. Phys. 17 (2005) 391[ math-ph/0405065].
[AGH97] L. Alvarez-Gaume and S. F. Hassan, Introduction to S-duality in N = 2 supersymmetric gauge theories: A pedagogical review of the work of Seiberg and Witten, Fortsch. Phys. 45 (1997) 159 [hep-th/9701069].
[Aha96] O. Aharony, String theory dualities from M theory, Nucl. Phys. B 476 (1996) 470 [hep-th/9604103].
[AHDM78] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185.
[AHI+97] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110]. 72 BIBLIOGRAPHY
[AII+00] H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, and others, Noncommutative Yang-Mills in IIB matrix model, Nucl. Phys. B 565 (2000) 176[ hep-th/9908141].
[AIK+99] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, and A. Tsuchiya, IIB matrix model, Prog. Theor. Phys. Suppl. 134 (1999) 47[ hep-th/9908038].
[AJ04] P. Aschieri and B. Jurco, Gerbes, M5-brane anomalies and E8 gauge theory, JHEP 0410 (2004) 068[ hep-th/0409200].
[AKSZ97] M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A 12 (1997) 1405[ hep-th/9502010].
[Alf97] J. Alfaro, BV gauge theories, hep-th/9702060.
[ALMY01] H. Awata, M. Li, D. Minic, and T. Yoneya, On the quantization of Nambu brackets, JHEP 0102 (2001) 013[ hep-th/9906248].
[AM] R. Abraham and J. Marsden, Foundations of mechanics, American Mathematical Soci- ety, second edition (2008)..
[AMN13] G. Aldazabal, D. Marques, and C. Nunez, Double field theory: A pedagogical review, Class. Quant. Grav. 30 (2013) 163001[ 1305.1907 [hep-th]].
[APPS97] M. Aganagic, J. Park, C. Popescu, and J. H. Schwarz, World-volume action of the M theory five-brane, Nucl. Phys. B 496 (1997) 191[ hep-th/9701166].
[Arv] A. Arvanitakis, TBA, to appear..
[ASW10] N. Akerblom, C. Saemann, and M. Wolf, Marginal deformations and 3-algebra struc- tures, Nucl. Phys. B 826 (2010) 456[ 0906.1705 [hep-th]].
[Ati57] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181.
[Ati79] M. F. Atiyah, Geometry of Yang–Mills fields, Lezioni Fermiane, Pisa, 1979.
[AvM80] M. Adler and P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. Math. 38 (1980) 318.
[Avr06] S. D. Avramis, Anomaly-free supergravities in six dimensions, hep-th/0611133.
[Bae] J. C. Baez, An introduction to n-categories, 7th Conference on Category Theory and Computer Science, pp. 1-33, eds. E. Moggi and G. Rosolini, Lecture Notes in Computer Science vol. 1290, Springer, Berlin, 1997 [q-alg/9705009].
[Bae02] J. C. Baez, Higher Yang–Mills theory, hep-th/0206130.
[Bar54] V. Bargmann, On unitary ray representations of continuous groups, Ann. of Math. 59 (1954) 1.
[Bar83] I. Bars, Supergroups and their representations, Lectures Appl. Math. 21 (1983) 17. BIBLIOGRAPHY 73
[Bar91] J. W. Barrett, Holonomy and path structures in general relativity and Yang-Mills theory, Int. J. Theor. Phys. 30 (1991) 1171.
[Bar06] T. Bartels, Higher gauge theory I: 2-Bundles, PhD thesis, University of California- Riverside (2006) [math.CT/0410328].
[Bar10] G. Barnich, A note on gauge systems from the point of view of Lie algebroids, AIP Conf. Proc. 1307 (2010) 7[ 1010.0899 [math-ph]].
[Bat79] M. Batchelor, The structure of supermanifolds, Trans. Am. Math. Soc. 253 (1979) 329.
[Bau96] L. Baulieu, B-V quantization and field-anti-field duality for p form gauge fields, topolog- ical field theories and 2-D gravity, Nucl. Phys. B 478 (1996) 431.
[BBH00] G. Barnich, F. Brandt, and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439[ hep-th/0002245].
[BBS06] K. Becker, M. Becker, and J. H. Schwarz, String theory and M-theory: A modern intro- duction, Cambridge University Press, 2006.
[BBvD85] F. A. Berends, G. J. H. Burgers, and H. van Dam, On the theoretical problems in constructing interactions involving higher spin massless particles, Nucl. Phys. B 260 (1985) 295.
[BBvS00] E. Bergshoeff, D. S. Berman, J. P. van der Schaar, and P. Sundell, A noncommutative M-theory five-brane, Nucl. Phys. B 590 (2000) 173[ hep-th/0005026].
[BC04] J. Baez and A. S. Crans, Higher-dimensional algebra VI: Lie 2-algebras, Th. App. Cat. 12 (2004) 492[ math.QA/0307263].
[BC05] P. Bressler and A. Chervov, Courant algebroids, J. Math. Sci. 128 (2005) 3030 [hep-th/0212195].
[BCG07] H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007) 726[ math.DG/0509640].
[BD] J. C. Baez and J. Dolan, Categorification, in: ’Higher Category Theory,’ p. 1-36, eds. Ezra Getzler and Mikhail Kapranov, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998 [math.QA/9802029].
[BD98] J. C. Baez and J. Dolan, Higher-dimensional algebra III: n-categories and the algebra of opetopes, Adv. Math. 135 (1998) 145[ q-alg/9702014].
[BDL+02] A. P. Balachandran, B. P. Dolan, J.-H. Lee, X. Martin, and D. O’Connor, Fuzzy complex projective spaces and their star-products, J. Geom. Phys. 43 (2002) 184 [hep-th/0107099].
[BDR07] Z. Bern, L. J. Dixon, and R. Roiban, Is N = 8 supergravity ultraviolet finite?, Phys. Lett. B 644 (2007) 265[ hep-th/0611086].
[BdRdWvN82] E. Bergshoeff, M. de Roo, B. de Wit, and P. van Nieuwenhuizen, Ten-dimensional Maxwell–Einstein supergravity, its currents, and the issue of its auxiliary fields, Nucl. Phys. B 195 (1982) 97. 74 BIBLIOGRAPHY
[Bena] J. Benabou, Introduction to bicategories, Lecture Notes in Math. 47 (Springer-Verlag, 1967)..
[Bénb] J. Bénabou, Introduction to bicategories, in: “Reports of the Midwest Category Semi- nar,” Lecture Notes in Mathematics Volume 47, 1967, pp. 1-77..
[Ber75] F. A. Berezin, General concept of quantization, Commun. Math. Phys. 40 (1975) 153.
[Ber08] D. S. Berman, M-theory branes and their interactions, Phys. Rept. 456 (2008) 89 [0710.1707 [hep-th]].
[Ber19] D. S. Berman, A Kaluza-Klein Approach to Double and Exceptional Field Theory, in: “Higher Structures in M-Theory,” proceedings of the LMS/EPSRC Durham Symposium, 12-18 August 2018 [1903.02860 [hep-th]].
[BFF+78] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation Theory and Quantization. 1. Deformations of Symplectic Structures, Ann. Phys. 111 (1978) 61.
[BFLS98] G. Barnich, R. Fulp, T. Lada, and J. Stasheff, The sh Lie structure of Poisson brackets in field theory, Commun. Math. Phys. 191 (1998) 585[ hep-th/9702176].
[BFSS97] T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55 (1997) 5112[ hep-th/9610043].
[BG88] F. A. Berends and W. T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759.
[BG89] R. Brown and N. D. Gilbert, Algebraic models of 3-types and automorphism structures for crossed modules, Proc. LMS 3 (1989) 51.
[BG11] G. Barnich and M. Grigoriev, First order parent formulation for generic gauge field theories, JHEP 1101 (2011) 122[ 1009.0190 [hep-th]].
[BGST05] G. Barnich, M. Grigoriev, A. Semikhatov, and I. Tipunin, Parent field theory and unfolding in BRST first-quantized terms, Commun. Math. Phys. 260 (2005) 147 [hep-th/0406192].
[BH05] A. Basu and J. A. Harvey, The M2-M5 brane system and a generalized Nahm’s equation, Nucl. Phys. B 713 (2005) 136[ hep-th/0412310].
[BH10a] J. C. Baez and J. Huerta, Division algebras and supersymmetry, Proc. Symp. Pure Maths. 81 (2010) 65[ 0909.0551 [hep-th]].
[BH10b] J. C. Baez and J. Huerta, Division Algebras and Supersymmetry II, 1003.3436 [Unknown].
[BH11] J. C. Baez and J. Huerta, An invitation to higher gauge theory, Gen. Relativ. Gravit. 43 (2011) 2335[ 1003.4485 [hep-th]].
[Bha15] L. Bhardwaj, Classification of 6d N = (1, 0) gauge theories, JHEP 1511 (2015) 002 [1502.06594 [hep-th]]. BIBLIOGRAPHY 75
[BHIW16] T. Bessho, M. A. Heller, N. Ikeda, and S. Watamura, Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids, JHEP 1604 (2016) 170[ 1511.03425 [hep-th]].
[BHKK99] O. Bergman, A. Hanany, A. Karch, and B. Kol, Branes and supersymmetry breaking in 3D gauge theories, JHEP 9910 (1999) 036[ hep-th/9908075].
[BHR10] J. C. Baez, A. E. Hoffnung, and C. L. Rogers, Categorified symplectic geometry and the classical string, Commun. Math. Phys. 293 (2010) 701[ 0808.0246 [math-ph]].
[BI97] J. D. Blum and K. Intriligator, New Phases of String Theory and 6d RG Fixed Points via Branes at Orbifold Singularities, Nucl. Phys. B 506 (1997) 199[ hep-th/9705044].
[Bil01] A. Bilal, Introduction to supersymmetry, hep-th/0101055.
[BIPZ78] E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35.
[BK] I. L. Buchbinder and S. M. Kuzenko, Ideas and methods of supersymmetry and super- gravity: Or a walk through superspace, Bristol, UK: IOP (1998) 656 p..
[BK97] I. Brunner and A. Karch, Branes and six dimensional fixed points, Phys. Lett. B 409 (1997) 109[ hep-th/9705022].
[BK98] I. Brunner and A. Karch, Branes at Orbifolds versus Hanany Witten in Six Dimensions, JHEP 9803:003 (1998)[ hep-th/9712143].
[BKS05] M. Bojowald, A. Kotov, and T. Strobl, Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries, J. Geom. Phys. 54 (2005) 400[ math.DG/0406445].
[BKV05] A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya, Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114.
[BL04] J. C. Baez and A. D. Lauda, Higher-dimensional algebra V: 2-groups, Th. App. Cat. 12 (2004) 423[ math.QA/0307200].
[BL07] J. Bagger and N. Lambert, Modeling multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108].
[BL08a] J. Bagger and N. Lambert, Comments on multiple M2-branes, JHEP 0802 (2008) 105 [0712.3738 [hep-th]].
[BL08b] J. Bagger and N. D. Lambert, Gauge symmetry and supersymmetry of multiple M2- branes, Phys. Rev. D 77 (2008) 065008[ 0711.0955 [hep-th]].
[BL09] J. Bagger and N. Lambert, Three-algebras and N = 6 Chern–Simons gauge theories, Phys. Rev. D 79 (2009) 025002[ 0807.0163 [hep-th]].
[Bla] M. Blau, Lecture notes on General Relativity, online lecture notes..
[BLMP13] J. Bagger, N. Lambert, S. Mukhi, and C. Papageorgakis, Multiple membranes in M- theory, Phys. Rept. 527 (2013) 1[ 1203.3546 [hep-th]]. 76 BIBLIOGRAPHY
[BLN+97] I. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. Sorokin, and M. Tonin, Co- variant action for the super-five-brane of M-theory, Phys. Rev. Lett. 78 (1997) 4332 [hep-th/9701149].
[Blo08] C. Blohmann, Stacky Lie groups, Int. Math. Res. Notices. 2008 (2008) rnn082 [math.DG/0702399].
[BLT13] R. Blumenhagen, D. Lüst, and S. Theisen, Basic concepts of string theory, Springer, 2013, available online[doi].
[BM05] L. Breen and W. Messing, Differential geometry of gerbes, Adv. Math. 198 (2005) 732 [math.AG/0106083].
[BM07] C. Berger and I. Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, Contemp. Math. 431 (2007) 31[ math.AT/0512576].
[BMN02] D. E. Berenstein, J. M. Maldacena, and H. S. Nastase, Strings in flat space and pp waves from N = 4 super Yang Mills, JHEP 0204 (2002) 013 [hep-th/0202021].
[BMPV06] A. P. Balachandran, G. Mangano, A. Pinzul, and S. Vaidya, Spin and statistics on the Groenwald-Moyal plane: Pauli-forbidden levels and transitions, Int. J. Mod. Phys. A 21 (2006) 3111[ hep-th/0508002].
[BMS94] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of Käh- ler manifolds and gl(N), N → ∞ limits, Commun. Math. Phys. 165 (1994) 281 [hep-th/9309134].
[Bog76] E. B. Bogomolny, Stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449.
[Bol06] S. Bolognesi, Multi-monopoles, magnetic bags, bions and the monopole cosmological problem, Nucl. Phys. B 752 (2006) 93[ hep-th/0512133].
[Bor] F. Borceux, Handbook of Categorical Algebra, ..
[BP13] G. Bonavolonta and N. Poncin, On the category of Lie n-algebroids, J Geom. Phys. 73 (2013) 70–90[ 1207.3590 [math.DG]].
[BR10] J. C. Baez and C. L. Rogers, Categorified symplectic geometry and the string Lie 2- algebra, Homol. Homot. Appl. 12 (2010) 221[ 0901.4721 [math-ph]].
[Bro] K. S. Brown, Cohomology of Groups (Graduate Texts in Mathematics, No. 87), Springer (November 18, 1982)..
[BRS76] C. Becchi, A. Rouet, and R. Stora, Renormalization of gauge theories, Annals Phys. 98 (1976) 287.
[Bry07] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser, Boston, 2007.
[BS77] L. Brink and J. H. Schwarz, Local complex supersymmetry in two-dimensions, Nucl. Phys. B 121 (1977) 285. BIBLIOGRAPHY 77
[BS04] J. C. Baez and U. Schreiber, Higher gauge theory: 2-connections on 2-bundles, hep-th/0412325.
[BS07] J. C. Baez and U. Schreiber, Higher gauge theory, Contemp. Math. 431 (2007) 7 [math.DG/0511710].
[BSCS07] J. C. Baez, D. Stevenson, A. S. Crans, and U. Schreiber, From loop groups to 2-groups, Homol. Homot. Appl. 9 (2007) 101[ math.QA/0504123].
[BSS77] L. Brink, J. H. Schwarz, and J. Scherk, Supersymmetric Yang–Mills theories, Nucl. Phys. B 121 (1977) 77.
[BSS18] S. Bunk, C. Saemann, and R. J. Szabo, The 2-Hilbert space of a prequantum bundle gerbe, Rev. Math. Phys. 30 (2018) 1850001[ 1608.08455 [math-ph]].
[BST87] E. Bergshoeff, E. Sezgin, and P. K. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B 189 (1987) 75.
[BST88] E. Bergshoeff, E. Sezgin, and P. K. Townsend, Properties of the Eleven-Dimensional Super Membrane Theory, Annals Phys. 185 (1988) 330.
[BT82] R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer, New York, 1982.
[BT08] I. A. Bandos and P. K. Townsend, Light-cone M5 and multiple M2-branes, Class. Quant. Grav. 25 (2008) 245003[ 0806.4777 [hep-th]].
[BT14] D. S. Berman and D. C. Thompson, Duality symmetric string and M-theory, Phys. Rept. 566 (2014) 1[ 1306.2643 [hep-th]].
[Bus87] T. H. Buscher, A symmetry of the string background field equations, Phys. Lett. B 194 (1987) 59.
[BV77] I. A. Batalin and G. A. Vilkovisky, Relativistic S matrix of dynamical systems with boson and fermion constraints, Phys. Lett. 69B (1977) 309.
[BV81] I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981) 27.
[BV83] I. A. Batalin and G. A. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D 28 (1983) 2567.
[BV84] I. A. Batalin and G. A. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nucl. Phys. B 234 (1984) 106.
[BV85] I. A. Batalin and G. A. Vilkovisky, Existence theorem for gauge algebra, J. Math. Phys. 26 (1985) 172.
[BX11] K. Behrend and P. Xu, Differentiable stacks and gerbes, J. Symp. Geom. 9 (2011) 285 [math.DG/0605694].
[Car50a] H. Cartan, Cohomologie réelle d’un espace fibré principal différentielle I, II,, Séminaire Henri Cartan 2 (1950). 78 BIBLIOGRAPHY
[Car50b] H. Cartan, Cohomologie réelle d’un espace fibré principal différentielle II,, Séminaire Henri Cartan 2 (1950).
[Car51] H. Cartan, Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, in: Colloque de topologie (espaces fibrés), p.15–27, Bruxelles, 1950...
[CDMIS02] P. Cartier, C. DeWitt-Morette, M. Ihl, and C. Saemann, Supermanifolds - Application to supersymmetry, in: “Multiple facets of quantization and supersymmetry: Michael Marinov memorial volume”, Eds. M. Olshanetsky and A. Vainshtein, World Scientific (2002) [math-ph/0202026].
[CDS09] S. Cherkis, V. Dotsenko, and C. Saemann, On superspace actions for multiple M2- branes, metric 3-algebras and their classification, Phys. Rev. D 79 (2009) 086002 [0812.3127 [hep-th]].
[CDY15] C. Cordova, T. T. Dumitrescu, and X. Yin, Higher Derivative Terms, Toroidal Com- pactification, and Weyl Anomalies in Six-Dimensional (2,0) Theories, 1505.03850 [hep-th].
[CE1Z] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Am. Math. Soc. 63 (1948-01-01Z) 85.
[CF01] A. S. Cattaneo and G. Felder, On the AKSZ formulation of the Poisson sigma model, Lett. Math. Phys. 56 (2001) 163[ math.QA/0102108].
[CG] G. R. Cavalcanti and M. Gualtieri, Generalized complex geometry and T-duality, in: “A Celebration of the Mathematical Legacy of Raoul Bott,” CRM Proceedings & Lecture Notes, American Mathematical Society, 2010 [1106.1747 [math.DG]].
[CH97] G. Chalmers and A. Hanany, Three dimensional gauge theories and monopoles, Nucl. Phys. B 489 (1997) 223[ hep-th/9608105].
[Cha98] D. S. Chatterjee, On gerbs, PhD thesis, Trinity College Cambridge (1998).
[Che02] I. Chepelev, Non-abelian Wilson surfaces, JHEP 0202 (2002) 013[ hep-th/0111018].
[Che11] S. A. Cherkis, Instantons on gravitons, Commun. Math. Phys. 306 (2011) 449 [1007.0044 [hep-th]].
[Chu13] C.-S. Chu, A theory of non-abelian tensor gauge field with non-abelian gauge symmetry G × G, Nucl. Phys. B 866 (2013) 43[ 1108.5131 [hep-th]].
[CIdL96] F. Cantrijn, L. A. Ibort, and M. de Leon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino. 54 (1996) 225.
[CIdL99] F. Cantrijn, A. Ibort, and M. de Leon, On the geometry of multisymplectic manifolds, J. Austr. Math. Soc. 66 (1999) 303.
[CJM+05] A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang, Bundle gerbes for Chern–Simons and Wess–Zumino–Witten theories, Commun. Math. Phys. 259 (2005) 577[ math.DG/0410013]. BIBLIOGRAPHY 79
[CK98] S. A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three dimensions, Nucl. Phys. B 525 (1998) 215[ hep-th/9711145].
[CK99] S. A. Cherkis and A. Kapustin, Singular monopoles and gravitational instantons, Com- mun. Math. Phys. 203 (1999) 713[ hep-th/9803160].
[CK01] S. A. Cherkis and A. Kapustin, Nahm transform for periodic monopoles and N = 2 super Yang- Mills theory, Commun. Math. Phys. 218 (2001) 333[ hep-th/0006050].
[CKNT04] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu, On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT, Phys. Lett. B 604 (2004) 98[ hep-th/0408069].
[CKVP98] P. Claus, R. Kallosh, and A. Van Proeyen, M five-brane and superconformal (0,2) tensor multiplet in six-dimensions, Nucl. Phys. B 518 (1998) 117[ hep-th/9711161].
[CM67] S. R. Coleman and J. Mandula, All possible symmetries of the S matrix, Phys. Rev. 159 (1967) 1251.
[CM83] G. F. Chapline and N. S. Manton, Unification of Yang–Mills theory and supergravity in ten dimensions, Phys. Lett. B 120 (1983) 105.
[Con84] D. Conduché, Modules croisés généralisés de longueur 2, J. Pure Appl. Algebra. 34 (1984) 155.
[Con94] A. Connes, Noncommutative geometry, Academic Press, 1994.
[Con03] D. Conduché, Simplicial crossed modules and mapping cones, Georgian Math. J. 10 (2003) 623.
[Cos07] K. J. Costello, Renormalisation and the Batalin–Vilkovisky formalism, 0706.1533 [math.QA].
[CP] B. Coecke and E. O. Paquette, Categories for the practising physicist, 0905.3010 [quant-ph].
[CR01] A. S. Cattaneo and C. A. Rossi, Higher-dimensional BF theories in the Batalin– Vilkovisky formalism: The BV action and generalized Wilson loops, Commun. Math. Phys. 221 (2001) 591[ math.QA/0010172].
[CS08] S. Cherkis and C. Saemann, Multiple M2-branes and generalized 3-Lie algebras, Phys. Rev. D 78 (2008) 066019[ 0807.0808 [hep-th]].
[CS10] A. S. Cattaneo and F. Schaetz, Introduction to supergeometry, 1011.3401 [math-ph].
[DEF+] P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, and E. Witten, Quantum fields and strings: A course for mathematicians. Vol. 1, AMS, Providence (1999)..
[Dev86] C. Devchand, Integrability on lightlike lines in six-dimensional superspace, Z. Phys. C 32 (1986) 233.
[DeW92] B. S. DeWitt, Supermanifolds, Cambridge University Press, 1992. 80 BIBLIOGRAPHY
[DFGZJ95] P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rept. 254 (1995) 1[ hep-th/9306153].
[DFR80] R. D’Auria, P. Fre, and T. Regge, Graded Lie Algebra Cohomology and Supergravity, Riv. Nuovo Cim. 3N12 (1980) 1.
[DFST97] G. Dito, M. Flato, D. Sternheimer, and L. Takhtajan, Deformation quantization and Nambu mechanics, Commun. Math. Phys. 183 (1997) 1[ hep-th/9602016].
[DHKM02] N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, The calculus of many instantons, Phys. Rept. 371 (2002) 231[ hep-th/0206063].
[DHS18] A. Deser, M. A. Heller, and C. Saemann, Extended Riemannian geometry II: Local heterotic double field theory, JHEP 1804 (2018) 106[ 1711.03308 [hep-th]].
[Dia97] D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220[ hep-th/9608163].
[DJ03] B. P. Dolan and O. Jahn, Fuzzy complex Grassmannian spaces and their star products, Int. J. Mod. Phys. A 18 (2003) 1935[ hep-th/0111020].
[DJP17] M. Doubek, B. Jurčo, and J. Pulmann, Quantum L∞ algebras and the homological perturbation lemma, 1712.02696 [math-ph].
[DK05] G. V. Dunne and V. Khemani, Numerical investigation of monopole chains, J. Phys. A 38 (2005) 9359[ hep-th/0506209].
[DLP+15] M. J. Duff, J. X. Lu, R. Percacci, C. N. Pope, H. Samtleben, and E. Sezgin, Membrane Duality Revisited, Nucl. Phys. B 901 (2015) 1[ 1509.02915 [hep-th]].
[dMFOMER09] P. de Medeiros, J. M. Figueroa-O’Farrill, E. Mendez-Escobar, and P. Ritter, On the Lie- algebraic origin of metric 3-algebras, Commun. Math. Phys. 290 (2009) 871[ 0809.1086 [hep-th]].
[DO03] B. P. Dolan and D. O’Connor, A fuzzy three sphere and fuzzy tori, JHEP 0310 (2003) 060[ hep-th/0306231].
[Dou95] M. R. Douglas, Branes within branes, hep-th/9512077.
[Dou98] M. R. Douglas, Gauge Fields and D-branes, J. Geom. Phys. 28 (1998) 255 [hep-th/9604198].
[Dri90] V. G. Drinfeld, Quasi-Hopf algebras, Lengingrad Math. J. 1 (1990) 1419.
[DS15] G. A. Demessie and C. Saemann, Higher Poincaré lemma and integrability, J. Math. Phys. 56 (2015) 082902[ 1406.5342 [hep-th]].
[DS17] G. A. Demessie and C. Saemann, Higher gauge theory with string 2-groups, Adv. Theor. Math. Phys. 21 (2017) 1895[ 1602.03441 [math-ph]].
[DS18a] A. Deser and C. Saemann, Extended Riemannian geometry I: Local double field theory, Ann. H. Poincare. 19 (2018) 2297[ 1611.02772 [hep-th]]. BIBLIOGRAPHY 81
[DS18b] A. Deser and C. Saemann, Extended Riemannian geometry III: Global double field theory with nilmanifolds, 1812.00026 [hep-th].
[DSS10] J. DeBellis, C. Saemann, and R. J. Szabo, Quantized Nambu-Poisson manifolds and n-Lie algebras, J. Math. Phys. 51 (2010) 122303[ 1001.3275 [hep-th]].
[DSS11] J. DeBellis, C. Saemann, and R. J. Szabo, Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model, JHEP 1104 (2011) 075[ 1012.2236 [hep-th]].
[Duf90] M. J. Duff, Duality Rotations in String Theory, Nucl. Phys. B 335 (1990) 610.
[Dus02] J. W. Duskin, Simplicial matrices and the nerves of weak n-categories i: Nerves of bicategories, Th. Appl. Cat. 9 (2002) 198.
[DvU17] A. Davgadorj and R. von Unge, N = 2 super Yang-Mills theory in Projective Superspace, 1706.07000 [hep-th].
[dWF75] B. de Wit and D. Z. Freedman, On combined supersymmetric and gauge invariant field theories, Phys. Rev. D 12 (1975) 2286.
[dWS06] B. de Wit and F. Saueressig, Off-shell N = 2 tensor supermultiplets, JHEP 0609 (2006) 062[ hep-th/0606148].
[dWvHVP81] B. de Wit, J. W. van Holten, and A. Van Proeyen, Structure of N = 2 Supergravity, Nucl. Phys. B 184 (1981) 77.
[DZHTV15] M. Del Zotto, J. J. Heckman, A. Tomasiello, and C. Vafa, 6d conformal matter, JHEP 1502 (2015) 054[ 1407.6359 [hep-th]].
[EH62] B. Eckmann and P. J. Hilton, Group-like structures in general categories I: Multiplica- tions and comultiplications, Math. Ann. 145 (1962) 227.
[EKR15] B. Eynard, T. Kimura, and S. Ribault, Random matrices, 1510.04430 [math-ph].
[EZ11] J. Ekstrand and M. Zabzine, Courant-like brackets and loop spaces, JHEP 1103 (2011) 074[ 0903.3215 [math-ph]].
[Fai17] M. Fairon, Introduction to graded geometry, Eur. J. Math. 3 (2017) 208[ 1512.02810 [math.DG]].
[FB] M. Forrester-Barker, Group objects and internal categories, math.CT/0212065.
[Fee] P. M. N. Feehan, Geometry of the moduli space of self-dual connections over the four- sphere, PhD Thesis, Columbia University (1992)..
[FGBV01] H. Figueroa, J. M. Gracia-Bondía, and J. C. Várilly, Elements of Noncommutative Geometry, Birkäuser, New York, 2001 [doi].
[FH90] J. M. L. Fisch and M. Henneaux, Homological perturbation theory and the algebraic structure of the antifield-antibracket formalism for gauge theories, Commun. Math. Phys. 128 (1990) 627.
[FH91] W. Fulton and J. Harris, Representation theory: A first course (Graduate texts), Springer, 1991. 82 BIBLIOGRAPHY
[Fil85] V. T. Filippov, n-Lie algebras, Sib. Mat. Zh. 26 (1985) 126.
[Fio] D. Fiorenza, An introduction to the Batalin–Vilkovisky formalism, Comptes Rendus des Rencontres Mathematiques de Glanon, 2003. [math.QA/0402057].
[FKS12] G. Felder, D. Kazhdan, and T. M. Schlank, The classical master equation, in: “Perspec- tives in representation theory : a conference in honor of Igor Frenkel’s 60th birthday,” May 12-17, 2012, Yale University, New Haven, CT [1212.1631 [math.AG]].
[FLS02a] R. Fulp, T. Lada, and J. Stasheff, Sh-Lie algebras induced by gauge transformations, Commun. Math. Phys. 231 (2002) 25[ math.QA/0012106].
[FLS02b] R. Fulp, T. Lada, and J. Stasheff, Noether’s variational theorem II and the BV formal- ism, Proceedings, 22nd Winter School on Geometry and Physics: Srní, Czech Republic, Jan 12–19, 2002 [math.QA/0204079].
[FM07] D. Fiorenza and M. Manetti, L∞ structures on mapping cones, Alg. Numb. Th. 1 (2007) 301[ math.QA/0601312].
[FN85] S. Fubini and H. Nicolai, The octonionic instanton, Phys. Lett. B 155 (1985) 369.
[FOK91] J. M. Figueroa-O’Farrill and T. Kimura, Geometric BRST quantization, Commun. Math. Phys. 136 (1991) 209.
[Fre88] D. Freed, The geometry of loop groups, J. Diff. Geom. 28 (1988) 223.
[FRS13] D. Fiorenza, C. L. Rogers, and U. Schreiber, A higher Chern–Weil derivation of AKSZ σ-models, Int. J. Geom. Meth. Mod. Phys. 10 (2013) 1250078[ 1108.4378 [math-ph]].
[FRS14] D. Fiorenza, C. L. Rogers, and U. Schreiber, L∞-algebras of local observables from higher prequantum bundles, Homol. Homot. App. 16 (2014) 107[ 1304.6292 [math-ph]].
[FRS16] D. Fiorenza, C. L. Rogers, and U. Schreiber, Higher U(1)-gerbe connections in geometric prequantization, Rev. Math. Phys. 28 (2016) 1650012[ 1304.0236 [math-ph]].
[FRZ16] Y. Fregier, C. L. Rogers, and M. Zambon, Homotopy moment maps, Adv. Math. 303 (2016) 954[ 1304.2051 [math.DG]].
[FSS14] D. Fiorenza, H. Sati, and U. Schreiber, Multiple M5-branes, string 2-connections, and 7d nonabelian Chern–Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229[ 1201.5277 [hep-th]].
[FSS15] D. Fiorenza, H. Sati, and U. Schreiber, Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields, Int. J. Geom. Meth. Mod. Phys. 12 (2015) 1550018[ 1308.5264 [hep-th]].
[FT09] K. Furuuchi and D. Tomino, A supersymmetric reduced model with a symmetry based on Filippov algebra, JHEP 0905 (2009) 070[ 0902.2041 [hep-th]].
[FW99] D. S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math. 3 (1999) 819[ hep-th/9907189]. BIBLIOGRAPHY 83
[Gaw87] K. Gawedzki, Topological actions in two-dimensional quantum field theories, Nonper- turbative quantum field theory (Cargèse, 1987), 101–141, NATO Adv. Sci. Inst. Ser. B Phys., 185, Plenum, New York, 1988..
[Get] E. Getzler, Higher derived brackets, 1010.5859 [math-ph].
[Get09] E. Getzler, Lie theory for nilpotent L∞-algebras, Ann. Math. 170 (2009) 271 [math.AT/0404003].
[GG08] J. P. Gauntlett and J. B. Gutowski, Constraining maximally supersymmetric membrane actions, JHEP 0806 (2008) 053[ 0804.3078 [hep-th]].
[GGRS83] S. J. Gates, M. T. Grisaru, M. Rocek, and W. Siegel, Superspace, or one thousand and one lessons in supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200].
[GGT96] M. J. Gotay, H. B. Grundling, and G. M. Tuynman, Obstruction results in quantization theory, J. Nonlinear Sci. 6 (1996) 469[ dg-ga/9605001].
[GH95] C. Gomez and R. Hernandez, Electric-magnetic duality and effective field theories, hep-th/9510023.
[GHMR85] D. J. Gross, J. A. Harvey, E. J. Martinec, and R. Rohm, Heterotic string theory. 1. The free heterotic string, Nucl. Phys. B 256 (1985) 253.
[GHMR86] D. J. Gross, J. A. Harvey, E. J. Martinec, and R. Rohm, Heterotic string theory. 2. The interacting heterotic string, Nucl. Phys. B 267 (1986) 75.
[GHW07] A. R. Gover, K. Hallowell, and A. Waldron, Higher spin gravitational couplings and the Yang–Mills detour complex, Phys. Rev. D 75 (2007) 024032[ hep-th/0606160].
[Gir71] J. Giraud, Cohomologie non abélienne, Springer, Berlin, 1971.
[GJ99] P. Goerss and J. Jardine, Simplicial homotopy theory, Birkhäuser, Boston, 1999.
[GJP91] E. Getzler, J. D. Jones, and S. Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology. 30 (1991) 339.
[GK99] A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71 (1999) 983[ hep-th/9802067].
[GN00] D. J. Gross and N. A. Nekrasov, Monopoles and strings in noncommutative gauge theory, JHEP 0007 (2000) 034[ hep-th/0005204].
[GOS03] D. Glück, Y. Oz, and T. Sakai, D-branes in N = 2 strings, JHEP 0308 (2003) 055 [hep-th/0306112].
[GP75] W. Greub and H. Petry, Minimal coupling and complex line bundles, J. Math. Phys. 16 (1975) 1347.
[GP04] F. Girelli and H. Pfeiffer, Higher gauge theory – differential versus integral formulation, J. Math. Phys. 45 (2004) 3949[ hep-th/0309173].
[GPP08] F. Girelli, H. Pfeiffer, and E. M. Popescu, Topological Higher Gauge Theory - from BF to BFCG theory, J. Math. Phys. 49 (2008) 032503[ 0708.3051 [hep-th]]. 84 BIBLIOGRAPHY
[GPR94] A. Giveon, M. Porrati, and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77[ hep-th/9401139].
[GPS95] J. Gomis, J. Paris, and S. Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept. 259 (1995) 1[ hep-th/9412228].
[GR98] H. Grosse and G. Reiter, The fuzzy supersphere, J. Geom. Phys. 28 (1998) 349 [math-ph/9804013].
[GR01] Z. Guralnik and S. Ramgoolam, On the polarization of unstable D0-branes into non- commutative odd spheres, JHEP 0102 (2001) 032[ hep-th/0101001].
[GRGVRV08] J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk, and H. Verlinde, A massive study of M2-brane proposals, JHEP 0809 (2008) 113[ 0807.1074 [hep-th]].
[Gri12] M. Grigoriev, Parent formulations, frame-like Lagrangians, and generalized auxiliary fields, JHEP 1212 (2012) 048[ 1204.1793 [hep-th]].
[Gro93] D. J. Gross, Two-dimensional QCD as a string theory, Nucl. Phys. B 400 (1993) 161 [hep-th/9212149].
[GRV89] A. Giveon, E. Rabinovici, and G. Veneziano, Duality in string background space, Nucl. Phys. B 322 (1989) 167.
[GS79] R. E. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 225 (1979) 403.
[GS84] M. B. Green and J. H. Schwarz, Anomaly cancellation in supersymmetric d=10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117.
[GS89] M. Göckeler and T. Schücker, Differential Geometry, Gauge Theories, And Gravity, Cambridge University Press, 1989.
[GS08] G. Ginot and M. Stienon, G-gerbes, principal 2-group bundles and characteristic classes, 0801.1238 [math.AT].
[GS14] M. Gruetzmann and T. Strobl, General Yang–Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework OR From Bianchi identi- ties to twisted Courant algebroids, Int. J. Geom. Meth. Mod. Phys. 12 (2014) 1550009 [1407.6759 [hep-th]].
[GSS08] A. R. Gover, P. Somberg, and V. Soucek, Yang–Mills detour complexes and conformal geometry, Commun. Math. Phys. 278 (2008) 307[ math.DG/0606401].
[GSW87a] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory. Vol. 1: Introduction, Cambridge University Press, 1987.
[GSW87b] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory. Vol. 2: Loop Ampli- tudes, Anomalies And phenomenology, Cambridge University Press, 1987.
[GT01] K. Gomi and Y. Terashima, Higher-dimensional parallel transports, Math. Res. Lett. 8 (2001) 25. BIBLIOGRAPHY 85
[Gua03] M. Gualtieri, Generalized complex geometry, PhD thesis, Oxford (2003) [math.DG/0401221].
[Gus09] A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [0709.1260 [hep-th]].
[GW96] J. Gomis and S. Weinberg, Are nonrenormalizable gauge theories renormalizable?, Nucl. Phys. B 469 (1996) 473[ hep-th/9510087].
[Hen90] M. Henneaux, Lectures on the antifield-BRST formalism for gauge theories, Nucl. Phys. B, Proc. Supp. 18 (1990) 47.
[Hen08] A. Henriques, Integrating L∞-algebras, Comp. Math. 144 (2008) 1017 [math.CT/0603563].
[HHZ10] O. Hohm, C. Hull, and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 1008 (2010) 8[ 1006.4823 [hep-th]].
[HIMS08] P.-M. Ho, Y. Imamura, Y. Matsuo, and S. Shiba, M5-brane in three-form flux and multiple M2-branes, JHEP 0808 (2008) 014[ 0805.2898 [hep-th]].
[Hit82] N. J. Hitchin, Monopoles and geodesics, Commun. Math. Phys. 83 (1982) 579.
[Hit83] N. J. Hitchin, On the construction of monopoles, Commun. Math. Phys. 89 (1983) 145.
[Hit99] N. J. Hitchin, Lectures on special Lagrangian submanifolds, math.DG/9907034.
[Hit03] N. Hitchin, Generalized Calabi–Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281[ math.DG/0209099].
[Hit05] N. Hitchin, Brackets, forms and invariant functionals, math.DG/0508618.
[HIW17] M. A. Heller, N. Ikeda, and S. Watamura, Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds, JHEP 1702 (2017) 078 [1611.08346 [hep-th]].
[HK11] O. Hohm and S. K. Kwak, Double field theory formulation of heterotic strings, JHEP 1106 (2011) 096[ 1103.2136 [hep-th]].
[HKK01] K. Hardie, K. Kamps, and R. Kieboom, A homotopy bigroupoid of a topological space, App. Cat. Struct. 9 (2001) 311.
[HL08] K. Hanaki and H. Lin, M2-M5 systems in N = 6 Chern-Simons theory, JHEP 0809 (2008) 067[ 0807.2074 [hep-th]].
[HL14] F. Hassler and D. Lust, Consistent Compactification of Double Field Theory on Non- geometric Flux Backgrounds, JHEP 1405 (2014) 085[ 1401.5068 [hep-th]].
[HL16] S. Hu and T. Li, U-duality transformation of membrane on T n revisited, JHEP 1608 (2016) 138[ 1606.00688 [hep-th]].
[HLS75] R. Haag, J. T. Lopuszanski, and M. Sohnius, All possible generators of supersymmetries of the S-matrix, Nucl. Phys. B 88 (1975) 257. 86 BIBLIOGRAPHY
[HLW98] P. S. Howe, N. D. Lambert, and P. C. West, The self-dual string soliton, Nucl. Phys. B 515 (1998) 203[ hep-th/9709014].
[HLZ13] O. Hohm, D. Lüst, and B. Zwiebach, The spacetime of double field theory: Review, remarks, and outlook, Fortsch. Phys. 61 (2013) 926[ 1309.2977 [hep-th]].
[HM08] P.-M. Ho and Y. Matsuo, M5 from M2, JHEP 0806 (2008) 105[ 0804.3629 [hep-th]].
[HMM95] N. J. Hitchin, N. Manton, and M. Murray, Symmetric monopoles, Nonlinearity. 8 (1995) 661[ dg-ga/9503016].
[HMRV15] J. J. Heckman, D. R. Morrison, T. Rudelius, and C. Vafa, Atomic classification of 6d SCFTs, Fortsch. Phys. 63 (2015) 468[ 1502.05405 [hep-th]].
[HMV14] J. J. Heckman, D. R. Morrison, and C. Vafa, On the classification of 6d SCFTs and generalized ADE orbifolds, JHEP 1405 (2014) 028[ 1312.5746 [hep-th]].
[Hoc51] G. Hochschild, Group extensions of Lie groups, Ann. Math. 54 (1951) 96.
[Hof02] C. Hofman, Nonabelian 2-forms, hep-th/0207017.
[Hop] J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, PhD Thesis, MIT, 1982, http://dspace.mit.edu/handle/1721.1/15717..
2 [Hop88] J. Hoppe, DiffAT and the curvature of some infinite dimensional manifolds, Phys. Lett. B 215 (1988) 706.
[Hop97] J. Hoppe, On M-algebras, the quantisation of Nambu-mechanics, and volume preserving diffeomorphisms, Helv. Phys. Acta. 70 (1997) 302 [hep-th/9602020].
[Hor64] G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. LMS 3 (1964) 689.
[HPS12] D. Harland, S. Palmer, and C. Saemann, Magnetic domains, JHEP 1210 (2012) 167 [1204.6685 [hep-th]].
[HR09] C. M. Hull and R. A. Reid–Edwards, Non-geometric backgrounds, doubled geometry and generalised T-duality, JHEP 0909 (2009) 014[ 0902.4032 [hep-th]].
[HS86] J. P. Harnad and S. Shnider, Constraints and field equations for ten-dimensional super Yang-Mills theory, Commun. Math. Phys. 106 (1986) 183.
[HS95] C. J. Houghton and P. M. Sutcliffe, Octahedral and dodecahedral monopoles, Nonlinear- ity.(1995) [hep-th/9601147].
[HS97] P. J. Hilton and U. Stammbach, A course in homological algebra, Springer, 1997.
[HST83] P. S. Howe, G. Sierra, and P. Townsend, Supersymmetry in six dimensions, Nucl. Phys. B 221 (1983) 331.
[HST84] P. S. Howe, K. S. Stelle, and P. K. Townsend, Miraculous ultraviolet cancellations in supersymmetry made manifest, Nucl. Phys. B 236 (1984) 125. BIBLIOGRAPHY 87
[HT92] M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, 1992.
[HT05] K. Hashimoto and S. Terashima, Stringy derivation of Nahm construction of monopoles, JHEP 0509 (2005) 055[ hep-th/0507078].
[HT06] K. Hashimoto and S. Terashima, ADHM is Tachyon Condensation, JHEP 0602 (2006) 018[ hep-th/0511297].
[Hue11] J. Huerta, Division Algebras and Supersymmetry III, 1109.3574 [hep-th].
[Hue14] J. Huerta, Division Algebras and Supersymmetry IV, 1409.4361 [hep-th].
[Hul05] C. M. Hull, A geometry for non-geometric string backgrounds, JHEP 0510 (2005) 065 [hep-th/0406102].
[HW85] C. M. Hull and E. Witten, Supersymmetric Sigma Models and the Heterotic String, Phys. Lett. 160B (1985) 398.
[HW95] P. Hanlon and M. Wachs, On Lie k-algebras, Adv. Math. 113 (1995) 206.
[HW97] A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152[ hep-th/9611230].
[HW09] D. Harland and R. Ward, Dynamics of periodic monopoles, Phys. Lett. B 675 (2009) 262[ 0901.4428 [hep-th]].
[HZ98] A. Hanany and A. Zaffaroni, Branes and six dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180[ hep-th/9712145].
[HZ09] C. Hull and B. Zwiebach, Double field theory, JHEP 0909 (2009) 99[ 0904.4664 [hep-th]].
[Igu04] K. Igusa, Graph cohomology and Kontsevich cycles, Topology. 43 (2004) 1469 [math.AT/0303157].
[IKKT97] N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467[ hep-th/9612115].
[ILMS08] C. Iuliu-Lazaroiu, D. McNamee, and C. Saemann, Generalized Berezin quantization, Bergman metrics and fuzzy Laplacians, JHEP 0809 (2008) 059[ 0804.4555 [hep-th]].
[Ime08] E. Imeroni, On deformed gauge theories and their string/M-theory duals, JHEP 0810 (2008) 026[ 0808.1271 [hep-th]].
[IMT03] E. Ivanov, L. Mezincescu, and P. K. Townsend, Fuzzy CP (n|m) as a quantum superspace, hep-th/0311159.
[IP00] T. A. Ivanova and A. D. Popov, Cech, Dolbeault and de Rham cohomologies in Chern– Simons and BF theories, hep-th/0101150.
[IS96] K. A. Intriligator and N. Seiberg, Mirror symmetry in three dimensional gauge theories, Phys. Lett. B 387 (1996) 513[ hep-th/9607207]. 88 BIBLIOGRAPHY
[ISS11] M. Ihl, C. Sachse, and C. Saemann, Fuzzy scalar field theory as matrix quantum me- chanics, JHEP 1103 (2011) 091[ 1012.3568 [hep-th]].
[Iva91] E. A. Ivanov, Chern–Simons matter systems with manifest N = 2 supersymmetry, Phys. Lett. B 268 (1991) 203.
[IYG78] J. Isenberg, P. B. Yasskin, and P. S. Green, Non-self-dual gauge fields, Phys. Lett. B 78 (1978) 462.
[JJ97] I. Jack and D. R. T. Jones, Regularisation of supersymmetric theories, hep-ph/9707278.
[Joh] C. V. Johnson, D-branes, Cambridge, USA: Univ. Pr. (2003) 548 p..
[Joh00] C. V. Johnson, D-brane primer, hep-th/0007170.
[Joy00] D. D. Joyce, Compact manifolds with special holonomy, Oxford University Press, 2000.
[JRSW18] B. Jurco, L. Raspollini, C. Saemann, and M. Wolf, L∞-algebras of classical field theories and the Batalin–Vilkovisky formalism, 1809.09899 [hep-th].
[JS01] K. Jünemann and B. Spendig, D-brane scattering of N = 2 strings, Phys. Lett. B 520 (2001) 163[ hep-th/0108069].
[JSSW19] B. Jurco, U. Schreiber, C. Saemann, and M. Wolf, Higher Structures in M-Theory, in: “Higher Structures in M-Theory,” proceedings of the LMS/EPSRC Durham Symposium, 12-18 August 2018 [1903.02807 [hep-th]].
[JSW15] B. Jurco, C. Saemann, and M. Wolf, Semistrict higher gauge theory, JHEP 1504 (2015) 087[ 1403.7185 [hep-th]].
[JSW16] B. Jurco, C. Saemann, and M. Wolf, Higher groupoid bundles, higher spaces, and self- dual tensor field equations, Fortschr. Phys. 64 (2016) 674[ 1604.01639 [hep-th]].
[Jur11a] B. Jurčo, Crossed module bundle gerbes; Classification, string group and differential geometry, Int. J. Geom. Meth. Mod. Phys. 08 (2011) 1079[ math.DG/0510078].
[Jur11b] B. Jurčo, Nonabelian bundle 2-gerbes, Int. J. Geom. Meth. Mod. Phys. 08 (2011) 49 [0911.1552 [math.DG]].
[Kac] V. G. Kac, Representations of classical Lie superalgebras, in ”Bonn 1977, Proceedings, Differential Geometrical Methods in Mathematical Physics.II.”, Berlin 1977, 597-626..
[Kad82] T. Kadeishvili, Algebraic structure in the homology of an A∞-algebra, Soobshch. Akad. Nauk. Gruz. SSR 108 (1982) 249.
[Kaj02] H. Kajiura, Homotopy algebra morphism and geometry of classical string field theory, Nucl. Phys. B 630 (2002) 361[ hep-th/0112228].
[Kaj07] H. Kajiura, Noncommutative homotopy algebras associated with open strings, Rev. Math. Phys. 19 (2007) 1[ math.QA/0306332].
[Kela] B. Keller, Introduction to A-infinity algebras and modules, math/9910179. BIBLIOGRAPHY 89
[Kelb] G. Kelly, Basic concepts of enriched category theory, Reprints in Theory and Applica- tions of Categories, No. 10, 2005, available online..
[Khu15] D. Khudaverdyan, Higher Lie and Leibniz algebras, PhD thesis, University of Luxem- bourg (2015) available online[ 1501.01925 [math.CT]].
[Kil87] T. P. Killingback, World sheet anomalies and loop geometry, Nucl. Phys. B 288 (1987) 578.
[Kim18] H. Kim, AKSZ-type topological quantum field theories and rational homotopy theory, 1809.09583 [hep-th].
[KL88] M. Karoubi and C. Leruste, Algebraic Topology via Differential Geometry, Cambridge Univ. Press, Cambridge, 1988.
[KLS05] P. O. Kazinski, S. L. Lyakhovich, and A. A. Sharapov, Lagrange structure and quanti- zation, JHEP 0507 (2005) 076[ hep-th/0506093].
[KM95] I. Kriz and P. May, Operads, algebras, modules and motives, SMF, Paris, 1995, available online.
[KN63a] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Volume I, Inter- science, New York, 1963.
[KN63b] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Volume II, Inter- science, New York, 1963.
[Koe11] P. Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59 (2011) 169 [1006.1536 [hep-th]].
[Kon] M. Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics Paris, 1992, Progr. Math., Vol. 120, p. 97-121, Birkhäuser, Basel, 1994..
[Kon03] M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003) 157[ q-alg/9709040].
[KOO99] T. Kitao, K. Ohta, and N. Ohta, Three-dimensional gauge dynamics from brane config- urations with (p,q)-fivebrane, Nucl. Phys. B 539 (1999) 79[ hep-th/9808111].
[KS95] C. Klimcik and P. Severa, Strings in Spacetime Cotangent Bundle and T-duality, Mod. Phys. Lett. A 10 (1995) 323[ hep-th/9411003].
[KS00] S. Kawamoto and N. Sasakura, Open membranes in a constant C-field background and noncommutative boundary strings, JHEP 0007 (2000) 014[ hep-th/0005123].
[KS06] H. Kajiura and J. Stasheff, Homotopy algebras inspired by classical open-closed string field theory, Commun. Math. Phys. 263 (2006) 553[ math.QA/0410291].
[KS10] A. Kotov and T. Strobl, Generalizing geometry - Algebroids and sigma models, in “Hand- book on Pseudo-Riemannian Geometry and Supersymmetry,” ed. V. Cortes [1004.0632 [hep-th]]. 90 BIBLIOGRAPHY
[KS15] A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles, Int. J. Geom. Meth. Mod. Phys. 12 (2015) 1550006[ 0711.4106 [math.DG]].
[KT83] T. Kugo and P. K. Townsend, Supersymmetry and the Division Algebras, Nucl. Phys. B 221 (1983) 357.
[KV94] M. M. Kapranov and V. A. Voevodsky, 2-categories and Zamolodchikov tetrahedra equa- tions, Proc. Symp. Pure Math. 56 (1994) 177.
[KW] F. Keller and S. Waldmann, Deformation theory of courant algebroids via the Rothstein algebra, 0807.0584 [math.QA].
[KWY10] A. Kapustin, B. Willett, and I. Yaakov, Exact results for Wilson loops in superconformal Chern–Simons theories with matter, JHEP 1003 (2010) 089[ 0909.4559 [hep-th]].
[Lam19] N. Lambert, Lessons from M2’s and hopes for M5’s, in: “Higher Structures in M- Theory,” proceedings of the LMS/EPSRC Durham Symposium, 12-18 August 2018 [1903.02825 [hep-th]].
[Laz01] C. I. Lazaroiu, String field theory and brane superpotentials, JHEP 0110 (2001) 018 [hep-th/0107162].
[LBS12] D. Li-Bland and P. Severa, Integration of exact Courant algebroids, ERAMS 19 (2012) 58 [1101.3996 [math.DG]].
[Lec99] O. Lechtenfeld, Mathematics and physics of N = 2 strings, hep-th/9912281.
[Lee99] K.-M. Lee, Sheets of BPS monopoles and instantons with arbitrary simple gauge group, Phys. Lett. B 445 (1999) 387[ hep-th/9810110].
[Lei98] T. Leinster, Basic bicategories, math.CT/9810017.
[Lei03] T. Leinster, Higher operads, higher categories, Cambridge University Press, Cambridge, 2003 [math.CT/0305049].
[Lei14] T. Leinster, Basic category theory, Cambridge University Press, 2014 [1612.09375 [math.CT]].
[Ler97] W. Lerche, Introduction to Seiberg-Witten theory and its stringy origin, Nucl. Phys. Proc. Suppl. 55B (1997) 83 [hep-th/9611190].
[Ler10] E. Lerman, Orbifolds as stacks?, L’Enseign. Math. 56 (2010) 315 [0806.4160 [math.DG]].
[LM95] T. Lada and M. Markl, Strongly homotopy Lie algebras, Commun. Alg. 23 (1995) 2147 [hep-th/9406095].
[LMSZ09] C. I. Lazaroiu, D. McNamee, C. Saemann, and A. Zejak, Strong homotopy Lie algebras, generalized Nahm equations and multiple M2-branes, 0901.3905 [hep-th].
[LP09] K. Lee and J.-H. Park, Three-algebra for supermembrane and two-algebra for superstring, JHEP 0904 (2009) 012[ 0902.2417 [hep-th]]. BIBLIOGRAPHY 91
[LP10] N. Lambert and C. Papageorgakis, Nonabelian (2,0) tensor multiplets and 3-algebras, JHEP 1008 (2010) 083[ 1007.2982 [hep-th]].
[LPS99] H. Lu, C. N. Pope, and K. S. Stelle, M-theory/heterotic Duality: a Kaluza-Klein Per- spective, Nucl. Phys. B 548 (1999) 87[ hep-th/9810159].
[LS93] T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087[ hep-th/9209099].
[LS97] F. W. Lawvere and S. H. Schanuel, Conceptual mathematics: A first introduction to categories, Cambridge University Press, 1997.
[LS06] O. Lechtenfeld and C. Saemann, Matrix models and D-branes in twistor string theory, JHEP 0603 (2006) 002[ hep-th/0511130].
[LS16] N. Lambert and D. Sacco, M2-branes and the (2,0) superalgebra, JHEP 1609 (2016) 107[ 1608.04748 [hep-th]].
[LT89] D. Lust and S. Theisen, Lectures on string theory, Lect. Notes Phys. 346 (1989) 1.
[Lüs10] D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 1012 (2010) 084[ 1010.1361 [hep-th]].
[LVZ03] F. Lizzi, P. Vitale, and A. Zampini, The fuzzy disc, JHEP 0308 (2003) 057 [hep-th/0306247].
[LWX97] Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom. 45 (1997) 547[ dg-ga/9508013].
[Lyk96] J. D. Lykken, Introduction to supersymmetry, in Fields, strings and duality. Proceedings, Summer School, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI’96, Boulder, USA, June 2-28, 1996 [[hep-th/9612114]].
[Mac87] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, 1987.
[Mac05] K. C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, Cambridge University Press, Cambridge, 2005.
[Man] Y. I. Manin, Gauge field theory and complex geometry, Grundlehren der mathematischen Wissenschaften, 289, Springer, Berlin, 1988..
[Man12] N. Manton, Monopole planets and galaxies, Phys. Rev. D 85 (2012) 045022[ 1111.2934 [hep-th]].
[Mar92a] N. Marcus, A tour through N=2 strings, hep-th/9211059.
[Mar92b] N. Marcus, The N = 2 open string, Nucl. Phys. B 387 (1992) 263[ hep-th/9207024].
[Mar97a] S. P. Martin, A Supersymmetry primer, hep-ph/9709356.
[Mar97b] E. J. Martinec, M-theory and N = 2 strings, hep-th/9710122. 92 BIBLIOGRAPHY
[Mar01] M. Markl, Loop Homotopy Algebras in Closed String Field Theory, Commun.Math.Phys. 221 (2001) 367[ hep-th/9711045].
[Mar04] X. Martin, Fuzzy orbifolds, hep-th/0405060.
[May93] J. P. May, Simplicial objects in algebraic topology, University of Chicago Press, 1993.
[Mer00] S. Merkulov, L∞-algebra of an unobstructed deformation functor, Int. Math. Res. Not. 3 (2000) 147[ math.AG/9907031].
[Met] D. Metzler, Topological and smooth stacks, math.DG/0306176.
[Mic87] J. Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization, Commun. Math. Phys. 110 (1987) 173.
[ML] S. Mac Lane, Homology (Classics in Mathematics), Springer (February 24, 1995)..
[ML98] S. Mac Lane, Categories for the working mathematician, Springer, New York, 1998.
[Mne17] P. Mnev, Lectures on Batalin–Vilkovisky formalism and its applications in topological quantum field theory, 1707.08096 [math-ph].
[MO03] J. Medina and D. O’Connor, Scalar field theory on fuzzy S4, JHEP 0311 (2003) 051 [hep-th/0212170].
[MP] A. Mutlu and T. Porter, Crossed squares and 2-crossed modules, math.KT/0210462.
[MP02] M. Mackaay and R. Picken, The holonomy of gerbes with connections, Adv. Math. 170 (2002) 287[ math.DG/0007053].
[MP08] S. Mukhi and C. Papageorgakis, M2 to D2, JHEP 0805 (2008) 085[ 0803.3218 [hep-th]].
[MP11] J. F. Martins and R. Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, Diff. Geom. App. 29 (2011) 179[ 0907.2566 [math.CT]].
[Mrc96] J. Mrcun, Stability and invariants of Hilsum–Skandalis maps, PhD thesis, Universiteit Utrecht (1996) [math.DG/0506484].
[MRE12] L. J. Mason and R. A. Reid-Edwards, The supersymmetric Penrose transform in six dimensions, 1212.6173 [hep-th].
[MRETC12] L. Mason, R. Reid-Edwards, and A. Taghavi-Chabert, Conformal field theories in six- dimensional twistor space, J. Geom. Phys. 62 (2012) 2353[ 1111.2585 [hep-th]].
[MRTA] Y. Maeda, S. Rosenberg, and F. Torres-Ardila, Riemannian geometry of loop spaces, 0705.1008 [math.DG].
[MS89] G. W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177.
[MS93] J. Maharana and J. H. Schwarz, Noncompact symmetries in string theory, Nucl. Phys. B 390 (1993) 3[ hep-th/9207016]. BIBLIOGRAPHY 93
[MS01] Y. Matsuo and Y. Shibusa, Volume preserving diffeomorphism and noncommutative branes, JHEP 2001 (2001) 006[ hep-th/0010040].
[MS04] M. Movshev and A. Schwarz, On maximally supersymmetric Yang–Mills theories, Nucl. Phys. B 681 (2004) 324[ hep-th/0311132].
[MS06a] L. J. Mason and D. Skinner, An ambitwistor Yang–Mills Lagrangian, Phys. Lett. B 636 (2006) 60[ hep-th/0510262].
[MS06b] M. Movshev and A. Schwarz, Algebraic structure of Yang–Mills theory, Prog. Math. 244 (2006) 473[ hep-th/0404183].
[MS08] S. Murray and C. Saemann, Quantization of flag manifolds and their supersymmetric extensions, Adv. Theor. Math. Phys. 12 (2008) 641[ hep-th/0611328].
[MSS] M. Markl, S. Shnider, and J. Stasheff, Operads in algebra, topology and physics, Math- ematical Surveys and Monographs, American Mathematical Society, 2002..
[MSW19] T. Macrelli, C. Saemann, and M. Wolf, Scattering amplitude recursion relations in BV quantisable theories, 1903.05713 [hep-th].
[MT11] R. A. Mehta and X. Tang, From double Lie groupoids to local Lie 2-groupoids, Bull. Braz. Math. Soc. 42 (2011) 651[ 1012.4103 [math.DG]].
[Mur] F. D. Murnaghan, The theory of group representations, The Johns Hopkins Press, Bal- timore (1938)..
[Mur96] M. K. Murray, Bundle gerbes, J. Lond. Math. Soc. 54 (1996) 403[ dg-ga/9407015].
[Mur07] M. K. Murray, An introduction to bundle gerbes, in: The many facets of geometry: A tribute to Nigel Hitchin, eds. O. Garcia–Prada, J.-P. Bourguignon and S. Salamon, Oxford University Press, Oxford, 2010 [0712.1651 [math.DG]].
[MW] L. J. Mason and N. M. J. Woodhouse, Integrability, self-duality, and twistor theory, Clarendon, Oxford (1996)..
[Mye99] R. C. Myers, Dielectric-branes, JHEP 9912 (1999) 022[ hep-th/9910053].
[MZ12] R. Mehta and M. Zambon, L∞-algebra actions, Diff. Geo. App. 30 (2012) 576 [1202.2607 [math.DG]].
[Nag] P.-A. Nagy, Prolongations of Lie algebras and applications, 0712.1398 [math.DG].
[Nah78] W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149.
[Nah80] W. Nahm, A simple formalism for the BPS monopole, Phys. Lett. B 90 (1980) 413.
[Nak90] M. Nakahara, Geometry, Topology and Physics, Hilger, Bristol, 1990.
[Nam73] Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973) 2405.
[Nar86] K. S. Narain, New heterotic string theories in uncompactified dimensions < 10, Phys. Lett. B 169 (1986) 41. 94 BIBLIOGRAPHY
[Noo] M. Noonan, Calculus on categories, preprint..
[Noo07] B. Noohi, Notes on 2-groupoids, 2-groups and crossed-modules, Homotopy, Homology, and Applications,. 9, (2007) , no. 1, 75 [math.CT/0512106].
[NPR09] H. Nastase, C. Papageorgakis, and S. Ramgoolam, The fuzzy S2 structure of M2-M5 systems in ABJM membrane theories, JHEP 0905 (2009) 123[ 0903.3966 [hep-th]].
[NR18] A. Nuetzi and M. Reiterer, Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization, 1812.06454 [math-ph].
[NSW13] T. Nikolaus, C. Sachse, and C. Wockel, A smooth model for the string group, Int. Math. Res. Notices. 2013 (2013) 3678[ 1104.4288 [math.AT]].
[NW13] T. Nikolaus and K. Waldorf, Four equivalent versions of non-abelian gerbes, Pacific J. Math. 246 (2013) 355 [1103.4815 [math.DG]].
[Oec00] R. Oeckl, Untwisting noncommutative Rd and the equivalence of quantum field theories, Nucl. Phys. B 581 (2000) 559[ hep-th/0003018].
[OS07a] D. O’Connor and C. Saemann, Fuzzy scalar field theory as a multitrace matrix model, JHEP 0708 (2007) 066[ 0706.2493 [hep-th]].
[OS07b] D. O’Connor and C. Saemann, A multitrace matrix model from fuzzy scalar field theory, Proc. of the Int. Workshop on Supersymmetries and Quantum Symmetries, Dubna, 30.7.-4.8.2007 [0709.0387 [hep-th]].
[Osb82] H. Osborn, On the Atiyah-Drinfeld-Hitchin-Manin construction for selfdual gauge fields, Commun. Math. Phys. 86 (1982) 195.
[OT79] S. Okubo and Y. Tosa, Duffin–Kemmer formulation of gauge theories, Phys. Rev. D 20 (1979) 462.
[OV90] H. Ooguri and C. Vafa, Selfduality and N = 2 String MAGIC, Mod. Phys. Lett. A 5 (1990) 1389.
[OV91] H. Ooguri and C. Vafa, Geometry of N=2 strings, Nucl. Phys. B 361 (1991) 469.
[Pap08] G. Papadopoulos, M2-branes, 3-Lie algebras and Plücker relations, JHEP 0805 (2008) 054[ 0804.2662 [hep-th]].
[Paw09] M. Pawellek, On a generalization of Jacobi’s elliptic functions and the double Sine- Gordon kink chain, 0909.2026 [math-ph].
[Pen] M. Penkava, L∞ algebras and their cohomology, q-alg/9512014.
[Pol74] A. M. Polyakov, Particle spectrum in the quantum field theory, JETP Lett. 20 (1974) 194.
[Pol94] J. Polchinski, What is string theory?, hep-th/9411028.
[Pol96] J. Polchinski, Lectures on D-branes, hep-th/9611050. BIBLIOGRAPHY 95
[Pol98a] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press, 1998.
[Pol98b] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond, Cambridge Univer- sity Press, 1998.
[Por] T. Porter, The Crossed Menagerie: an introduction to crossed gadgetry and cohomol- ogy in algebra and topology, online: http://ncatlab.org/timporter/files/menagerie9.pdf, available online..
[PR84] R. Penrose and W. Rindler, Spinors and space-time. Vol. 1: Two spinor calculus and relativistic fields, Cambridge University Press, Cambridge, 1984.
[PR86] R. Penrose and W. Rindler, Spinors and space-time. Vol. 2: Spinor and twistor methods in space-time geometry, Cambridge University Press, Cambridge, 1986.
[PS75] M. Prasad and C. M. Sommerfield, Exact classical solution for the ’t Hooft monopole and the Julia–Zee dyon, Phys. Rev. Lett. 35 (1975) 760.
[PS88] A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, Oxford, 1988.
[PS05] A. D. Popov and C. Saemann, On supertwistors, the Penrose-Ward transform and N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 9 (2005) 931[ hep-th/0405123].
[PS09] J.-H. Park and C. Sochichiu, Taking off the square root of Nambu-Goto action and obtaining Filippov-Lie algebra gauge theory action, Eur. Phys. J. C 64 (2009) 161 [0806.0335 [hep-th]].
[PS11] C. Papageorgakis and C. Saemann, The 3-Lie algebra (2,0) tensor multiplet and equa- tions of motion on loop space, JHEP 1105 (2011) 099[ 1103.6192 [hep-th]].
[PS12] S. Palmer and C. Saemann, M-brane models from non-abelian gerbes, JHEP 1207 (2012) 010[ 1203.5757 [hep-th]].
[PS13] S. Palmer and C. Saemann, Six-dimensional (1,0) superconformal models and higher gauge theory, J. Math. Phys. 54 (2013) 113509[ 1308.2622 [hep-th]].
[PS14] S. Palmer and C. Saemann, The ABJM model is a higher gauge theory, Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1450075[ 1311.1997 [hep-th]].
[PST95a] P. Pasti, D. P. Sorokin, and M. Tonin, Duality symmetric actions with manifest space- time symmetries, Phys. Rev. D 52 (1995) 4277[ hep-th/9506109].
[PST95b] P. Pasti, D. P. Sorokin, and M. Tonin, Note on manifest Lorentz and general coordinate invariance in duality symmetric models, Phys. Lett. B 352 (1995) 59[ hep-th/9503182].
[PST97a] P. Pasti, D. P. Sorokin, and M. Tonin, Covariant action for a D = 11 five-brane with the chiral field, Phys. Lett. B 398 (1997) 41[ hep-th/9701037].
[PST97b] P. Pasti, D. P. Sorokin, and M. Tonin, On Lorentz invariant actions for chiral p-forms, Phys. Rev. D 55 (1997) 6292[ hep-th/9611100]. 96 BIBLIOGRAPHY
[PT86] S. J. Parke and T. R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459.
[PW04] A. D. Popov and M. Wolf, Topological B-model on weighted projective spaces and self- dual models in four dimensions, JHEP 0409 (2004) 007[ hep-th/0406224].
[PZT00] L. A. Pando Zayas and A. A. Tseytlin, 3-branes on resolved conifold, JHEP 0011 (2000) 028 [hep-th/0010088].
[Ram01] S. Ramgoolam, On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B 610 (2001) 461[ hep-th/0105006].
[Red06] D. C. Redden, Canonical metric connections associated to string structures, PhD thesis, Notre Dame, Indiana (2006).
[Rie] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Inaugu- ralvorlesung, english: “On the Hypotheses which lie at the Bases of Geometry,” available online...
[Ros82] P. Rossi, Exact results in the theory of nonabelian magnetic monopoles, Phys. Rept. 86 (1982) 317.
[Roya] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant alge- broids, in: “Quantization, Poisson Brackets and Beyond,” ed. Theodore Voronov, Con- temp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 [math.SG/0203110].
[Royb] D. Roytenberg, On weak Lie 2-algebras, in: “XXVI Workshop on Geometrical Methods in Physics 2007,” ed. Piotr Kielanowski et al., AIP Conference Proceedings volume 956, American Institute of Physics, Melville, NY [0712.3461 [math.QA]].
[Roy02] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002) 123[ math.QA/0112152].
[Roy07] D. Roytenberg, AKSZ-BV formalism and Courant algebroid-induced topological field theories, Lett. Math. Phys. 79 (2007) 143[ hep-th/0608150].
[Roy09] D. Roytenberg, Courant-Dorfman algebras and their cohomology, Lett. Math. Phys. 90 (2009) 311[ 0902.4862 [math.QA]].
[RS08] D. M. Roberts and U. Schreiber, The inner automorphism 3-group of a strict 2-group, J. Homot. Rel. Struct. 3 (2008) 193 [0708.1741 [math.CT]].
[RS14] P. Ritter and C. Saemann, Lie 2-algebra models, JHEP 1404 (2014) 066[ 1308.4892 [hep-th]].
[RS15a] S. Rea and C. Saemann, The Phase Diagram of Scalar Field Theory on the Fuzzy Disc, JHEP 1511 (2015) 115[ 1507.05978 [hep-th]].
[RS15b] P. Ritter and C. Saemann, Automorphisms of strong homotopy Lie algebras of local observables, 1507.00972 [hep-th].
[RS16] P. Ritter and C. Saemann, L∞-algebra models and higher Chern–Simons theories, Rev. Math. Phys. 28 (2016) 1650021[ 1511.08201 [hep-th]]. BIBLIOGRAPHY 97
[RSS16] P. Ritter, C. Saemann, and L. Schmidt, Generalized higher gauge theory, JHEP 1604 (2016) 032[ 1512.07554 [hep-th]].
[RV92] M. Rocek and E. P. Verlinde, Duality, quotients, and currents, Nucl. Phys. B 373 (1992) 630[ hep-th/9110053].
[RW98] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys. 46 (1998) 81[ math.QA/9802118].
[RW04] M. Rocek and N. Wadhwa, On Calabi-Yau supermanifolds. II, hep-th/0410081.
[RW05] M. Rocek and N. Wadhwa, On Calabi-Yau supermanifolds, Adv. Theor. Math. Phys. 9 (2005) 315[ hep-th/0408188].
[RZ18] M. Rocek and A. M. Zeitlin, Homotopy algebras of differential (super)forms in three and four dimensions, Lett. Math. Phys. 108 (2018) 2669[ 1702.03565 [math-ph]].
[Sae10] C. Saemann, The multitrace matrix model of scalar field theory on fuzzy CP n, SIGMA 6 (2010) 50[ 1003.4683 [hep-th]].
[Sae11] C. Saemann, Constructing self-dual strings, Commun. Math. Phys. 305 (2011) 513 [1007.3301 [hep-th]].
[Sae15] C. Saemann, Bootstrapping fuzzy scalar field theory, JHEP 1504 (2015) 044[ 1412.6255 [hep-th]].
[Sat09a] M. Sato, Covariant formulation of M-theory I, Int. J. Mod. Phys. A 24 (2009) 5019 [0902.1333 [hep-th]].
[Sat09b] M. Sato, Covariant formulation of M-theory II, 0902.4102 [hep-th].
[Sat10] M. Sato, Model of M-theory with eleven matrices, JHEP 1007 (2010) 026[ 1003.4694 [hep-th]].
[Sch] F. Schuhmacher, Deformation of L∞-algebras, math.QA/0405485.
[Sch93a] A. Schwarz, Geometry of Batalin–Vilkovisky quantization, Commun. Math. Phys. 155 (1993) 249[ hep-th/9205088].
[Sch93b] A. Schwarz, Semiclassical approximation in Batalin–Vilkovisky formalism, Commun. Math. Phys. 158 (1993) 373[ hep-th/9210115].
[Sch98] M. Schlichenmaier, Berezin-Toeplitz quantization and Berezin symbols for arbitrary com- pact Kähler manifolds, Talk presented at the XVII workshop on geometric methods in physics, Bialowieza, Poland, July 3-July 9, 1998 [math.QA/9902066].
[Sch11] C. Schommer–Pries, Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geom. Top. 15 (2011) 609[ 0911.2483 [math.AT]].
[Sch13] U. Schreiber, Differential cohomology in a cohesive infinity-topos, Habilitation Thesis, 2011 [1310.7930 [math-ph]].
[Seg] G. Segal, Cohomology of topological groups, in “Symposia Mathematica, Vol. IV (IN- DAM, Rome, 1968/69),” p. 377-387, Academic Press, London, 1970.. 98 BIBLIOGRAPHY
[Sei98] N. Seiberg, Notes on theories with 16 supercharges, Nucl. Phys. Proc. Suppl. 67 (1998) 158[ hep-th/9705117].
[Sem82] A. M. Semikhatov, Supersymmetric instanton, JETP Lett. 35 (1982) 560.
[Sen86] A. Sen, Local Gauge and Lorentz Invariance of the Heterotic String Theory, Phys. Lett. 166B (1986) 300.
[Sen96] A. Sen, T-Duality of p-Branes, Mod. Phys. Lett. A 11 (1996) 827[ hep-th/9512203].
[Set94] S. Sethi, Supermanifolds, rigid manifolds and mirror symmetry, Nucl. Phys. B 430 (1994) 31[ hep-th/9404186].
[Seva] P. Severa, Letter No. 5 to Alan Weinstein (1998), available at http://sophia.dtp.fmph.uniba.sk/∼severa/letters/..
[Sevb] P. Severa, L∞-algebras as 1-jets of simplicial manifolds (and a bit beyond), math.DG/0612349.
[Sev05] P. Severa, Some title containing the words ‘homotopy’ and ‘symplectic’, e.g. this one, Trav. math. 16 (2005) 121 [math.SG/0105080].
[Sha03] E. Sharpe, Lectures on D-branes and sheaves, hep-th/0307245.
[Sha19] E. Sharpe, Categorical Equivalence and the Renormalization Group, in: “Higher Struc- tures in M-Theory,” proceedings of the LMS/EPSRC Durham Symposium, 12-18 August 2018 [1903.02880 [hep-th]].
[Sie79] W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193.
[Sie80] W. Siegel, Inconsistency of supersymmetric dimensional regularization, Phys. Lett. B 94 (1980) 37.
[Sie92] W. Siegel, The N = 2 (4) string is selfdual N = 4 Yang-Mills, Phys. Rev. D 46 (1992) 3235 [hep-th/9205075].
[Sie93a] W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073].
[Sie93b] W. Siegel, Two-vierbein formalism for string-inspired axionic gravity, Phys. Rev. D 47 (1993) 5453[ hep-th/9302036].
[Sny47] H. S. Snyder, Quantized space-time, Phys. Rev. 71 (1947) 38.
[SS] P. Severa and M. Siran, Integration of differential graded manifolds, 1506.04898.
[SS85] M. Schlessinger and J. Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra. 38 (1985) 313.
[SS10] C. Saemann and R. J. Szabo, Branes, quantization and fuzzy spheres, PoS CNCFG2010 (2010) 005 [1101.5987 [hep-th]]. BIBLIOGRAPHY 99
[SS13] C. Saemann and R. J. Szabo, Groupoids, loop spaces and quantization of 2-plectic mani- folds, Rev. Math. Phys. 25 (2013) 1330005[ 1211.0395 [hep-th]].
[SS17] C. Saemann and L. Schmidt, The non-abelian self-dual string and the (2,0)-theory, 1705.02353 [hep-th].
[SS18] C. Saemann and L. Schmidt, Towards an M5-brane model I: A 6d superconformal field theory, J. Math. Phys. 59 (2018) 043502[ 1712.06623 [hep-th]].
[SSS] H. Sati, U. Schreiber, and J. Stasheff, L∞-algebra connections and applications to String- and Chern–Simons n-transport, in: “Quantum Field Theory,” eds. B. Fauser, J. Tolks- dorf and E. Zeidler, p. 303, Birkhäuser 2009 [0801.3480 [math.DG]].
[SSS12] H. Sati, U. Schreiber, and J. Stasheff, Differential twisted String and Fivebrane struc- tures, Commun. Math. Phys. 315 (2012) 169[ 0910.4001 [math.AT]].
[SSW11] H. Samtleben, E. Sezgin, and R. Wimmer, (1,0) superconformal models in six dimen- sions, JHEP 1112 (2011) 062[ 1108.4060 [hep-th]].
[SSW13] H. Samtleben, E. Sezgin, and R. Wimmer, Six-dimensional superconformal couplings of non-abelian tensor and hypermultiplets, JHEP 1303 (2013) 068[ 1212.5199 [hep-th]].
[SSWW11] H. Samtleben, E. Sezgin, R. Wimmer, and L. Wulff, New superconformal models in six dimensions: Gauge group and representation structure, PoS CORFU 2011 (2011) 71 [1204.0542 [hep-th]].
[ST] S. Stolz and P. Teichner, What is an elliptic object?, in “Topology, geometry and quan- tum field theory,” volume 308 of London Math. Soc. Lecture Note Ser., pages 247-343. Cambridge Univ. Press, Cambridge, 2004..
[Sta] A. Stacey, The differential topology of loop spaces, math.DG/0510097.
[Sta63a] J. D. Stasheff, On the homotopy associativity of H-spaces, I., Trans. Amer. Math. Soc. 108 (1963) 275.
[Sta63b] J. D. Stasheff, On the homotopy associativity of H-spaces, II., Trans. Amer. Math. Soc. 108 (1963) 293.
[Sta97] J. Stasheff, Deformation theory and the Batalin–Vilkovisky master equation, “Deforma- tion theory and symplectic geometry.” Proceedings, Meeting, Ascona, Switzerland, June 16–22, 1996 [q-alg/9702012].
[Sta98] J. Stasheff, The (secret?) homological algebra of the Batalin–Vilkovisky approach, Con- temp. Math. 219 (1998) 195[ hep-th/9712157].
[Sta05] A. Stacey, The Geometry of the loop space and a construction of a Dirac operator, math.DG/0505077.
[Ste10] H. Steinacker, Emergent geometry and gravity from matrix models: An introduction, Class. Quant. Grav. 27 (2010) 133001[ 1003.4134 [hep-th]].
[Sto96] S. Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304 (1996) 785. 100 BIBLIOGRAPHY
[Str86] A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253.
[Str96] A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44[ hep-th/9512059].
[SW89] A. D. Shapere and F. Wilczek, Selfdual models with theta terms, Nucl. Phys. B 320 (1989) 669.
[SW94] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087].
[SW96] N. Seiberg and E. Witten, Comments on string dynamics in six dimensions, Nucl. Phys. B 471 (1996) 121[ hep-th/9603003].
[SW99] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 9909 (1999) 032[ hep-th/9908142].
[SW01] P. Severa and A. Weinstein, Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001) 145[ math.SG/0107133].
[SW10] H. Samtleben and R. Wimmer, N = 8 superspace constraints for three-dimensional gauge theories, JHEP 1002 (2010) 070[ 0912.1358 [hep-th]].
[SW13a] C. Saemann and M. Wolf, On twistors and conformal field theories from six dimensions, J. Math. Phys. 54 (2013) 013507[ 1111.2539 [hep-th]].
[SW13b] U. Schreiber and K. Waldorf, Connections on non-abelian gerbes and their holonomy, Th. Appl. Cat. 28 (2013) 476[ 0808.1923 [math.DG]].
[SW14a] C. Saemann and M. Wolf, Non-abelian tensor multiplet equations from twistor space, Commun. Math. Phys. 328 (2014) 527[ 1205.3108 [hep-th]].
[SW14b] C. Saemann and M. Wolf, Six-dimensional superconformal field theories from principal 3-bundles over twistor space, Lett. Math. Phys. 104 (2014) 1147[ 1305.4870 [hep-th]].
[SW17] C. Saemann and M. Wolf, Supersymmetric Yang–Mills theory as higher Chern–Simons theory, JHEP 1707 (2017) 111[ 1702.04160 [hep-th]].
[SWW12] C. Saemann, R. Wimmer, and M. Wolf, A twistor description of six-dimensional N = (1, 1) super Yang-Mills theory, JHEP 1205 (2012) 20[ 1201.6285 [hep-th]].
[SZ14] E. Soncini and R. Zucchini, 4-d semistrict higher Chern–Simons theory I, JHEP 1410 (2014) 79[ 1406.2197 [hep-th]].
[SZ16] Y. Sheng and C. Zhu, Higher extensions of Lie algebroids, Comm. Cont. Math. (2016) 1650034[ 1103.5920 [math-ph]].
[Sza02] R. J. Szabo, BUSSTEPP lectures on string theory: An introduction to string theory and D-brane dynamics, hep-th/0207142.
[Tak94] L. Takhtajan, On foundation of the generalized Nambu mechanics (second version), Commun. Math. Phys. 160 (1994) 295[ hep-th/9301111]. BIBLIOGRAPHY 101
[Tei86] C. Teitelboim, Gauge invariance for extended objects, Phys. Lett. B 167 (1986) 63.
[Tek15] J. Tekel, Phase strucutre of fuzzy field theories and multitrace matrix models, Acta Physica Slovaca. 65 (2015) 369 [1512.00689 [hep-th]].
[Ter08] S. Terashima, On M5-branes in N = 6 membrane action, JHEP 0808 (2008) 080 [0807.0197 [hep-th]].
[tH74] G. ’t Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys. B 79 (1974) 276.
[The13] The Univalent Foundations Program, Homotopy type theory - Univalent foundations of mathematics, Institute for Advanced Study, Princeton, 2013, available online.
[Ton05] D. Tong, TASI lectures on solitons, hep-th/0509216.
[Tow96] P. K. Townsend, D-branes from M-branes, Phys. Lett. B 373 (1996) 68 [hep-th/9512062].
[Tse91] A. A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395.
[Tsi98] D. Tsimpis, Nahm equations and boundary conditions, Phys. Lett. B 433 (1998) 287 [hep-th/9804081].
[Tuy87] G. M. Tuynman, Quantization: Towards a comparison between methods, J. Math. Phys. 28 (1987) 2829.
[TV00] S. P. Trivedi and S. Vaidya, Fuzzy cosets and their gravity duals, JHEP 0009 (2000) 041 [hep-th/0007011].
[Tyu75] I. V. Tyutin, Gauge invariance in field theory and statistical physics in operator formal- ism, Lebedev Physics Institute preprint 39 (1975) [0812.0580 [hep-th]].
[Vaf01] C. Vafa, Superstrings and topological strings at large N, J. Math. Phys. 42 (2001) 2798 [hep-th/0008142].
[Var99] S. Varsaie, Kählerian supermanifolds, J. Math. Phys. 40 (1999) 6701.
[Vin90] A. M. Vinogradov, The union of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators, Mat. Zametki. 47 (1990) 138.
[Vol83] I. V. Volovich, Supersymmetric Yang-Mills theories and twistors, Phys. Lett. B 129 (1983) 429.
[VP99] A. Van Proeyen, Tools for supersymmetry, hep-th/9910030.
[Wala] K. Waldorf, A global perspective to connections on principal 2-bundles, 1608.00401 [math.DG].
[Walb] K. Waldorf, Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection, 1004.0031 [math-ph].
[Wal07] K. Waldorf, Algebraic structures for bundle gerbes and the Wess-Zumino term in con- formal field theory, PhD thesis, Universität Hamburg (2007) available online. 102 BIBLIOGRAPHY
[Wal12] K. Waldorf, Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps, Cah. Topol. Géom. Différ. Catég. 53 (2012) 162 [0911.3212 [math.DG]].
[Wal13] K. Waldorf, String connections and Chern–Simons theory, Trans. Amer. Math. Soc. 365 (2013) 4393[ 0906.0117 [math.DG]].
[War77] R. S. Ward, On selfdual gauge fields, Phys. Lett. A 61 (1977) 81.
[War84] R. S. Ward, Completely solvable gauge field equations in dimension greater than four, Nucl. Phys. B 236 (1984) 381.
[War05] R. S. Ward, Periodic monopoles, Phys. Lett. B 619 (2005) 177[ hep-th/0505254].
[War07] R. Ward, A monopole wall, Phys. Rev. D 75 (2007) 021701[ hep-th/0612047].
[WB92] J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, 1992.
[Wei] S. Weinberg, The quantum theory of fields. Vol. 3: Supersymmetry, Cambridge, UK: Univ. Pr. (2000) 419 p..
[Wei77] A. Weinstein, Lectures on symplectic manifolds, American Mathematical Society, 1977.
[Wei94] C. A. Weibel, An introduction to homological algebra, Cambridge University Press, 1994.
[Wes03] J. Wess, Deformed coordinate spaces: Derivatives, hep-th/0408080.
[Whi78] G. W. Whitehead, Elements of homotopy theory, Springer, New York, 1978.
[Wig39] E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. 40 (1939) 149.
[Wit] E. Witten, Some Comments On The Recent Twistor Space Constructions. (TALK), In *Lawrence 1978, Proceedings, Complex Manifold Techniques In Theoretical Physics*, 207-218..
[Wit78] E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. B 77 (1978) 394.
[Wit81] E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513.
[Wit82] E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253.
[Wit86] E. Witten, Noncommutative geometry and string field theory, Nucl. Phys. B 268 (1986) 253.
[Wit95] E. Witten, Some comments on string dynamics, proceedings of “Strings ‘95”, USC, 1995 [hep-th/9507121].
[Wit96a] E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135].
[Wit96b] E. Witten, Five-branes and M-theory on an orbifold, Nucl. Phys. B 463 (1996) 383 [hep-th/9512219]. BIBLIOGRAPHY 103
[Wit04] E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189[ hep-th/0312171].
[Woc11a] C. Wockel, Categorified central extensions, étale Lie 2-groups and Lie’s Third The- orem for locally exponential Lie algebras, Adv. Math. 228 (2011) 2218[ 0812.1673 [math.GR]].
[Woc11b] C. Wockel, Principal 2-bundles and their gauge 2-groups, Forum Math. 23 (2011) 566 [0803.3692 [math.DG]].
[Wol10] M. Wolf, A first course on twistors, integrability and gluon scattering amplitudes, J. Phys. A 43 (2010) 393001[ 1001.3871 [hep-th]].
[Woo92] N. M. J. Woodhouse, Geometric quantization, Clarendon, New York, 1992.
[Woo09] N. Woodhouse, Introduction to analytical dynamics, Springer, London, 2009.
[WW90] R. S. Ward and R. O. Wells, Twistor geometry and field theory, Cambridge University Press, 1990.
[Yon16] T. Yoneya, Covariantized matrix theory for D-particles, JHEP 1606 (2016) 058 [1603.06402 [hep-th]].
[Zei07] A. M. Zeitlin, Homotopy Lie superalgebra in Yang–Mills theory, JHEP 0709 (2007) 068 [0708.1773 [hep-th]].
[Zei09] A. M. Zeitlin, Batalin–Vilkovisky Yang–Mills theory as a homotopy Chern–Simons the- ory via string field theory, Int. J. Mod. Phys. A 24 (2009) 1309[ 0709.1411 [hep-th]].
[Zei11] E. Zeidler, Quantum field theory I: Basics in mathematics and physics: A bridge between mathematicians and physicists, Springer, 2011.
[Zho05] C.-G. Zhou, On Ricci flat supermanifolds, JHEP 0502 (2005) 004[ hep-th/0410047].
[Zhu09a] C. Zhu, Kan replacement of simplicial manifolds, Lett. Math. Phys. 90 (2009) 383 [0812.4150 [math.DG]].
[Zhu09b] C. Zhu, Lie n-groupoids and stacky Lie groupoids, Int. Math. Res. Not. 21 (2009) 4087–4141[ math.DG/0609420].
[ZP88] B. M. Zupnik and D. G. Pak, Superfield formulation of the simplest three-dimensional gauge theories and conformal supergravities, Theor. Math. Phys. 77 (1988) 1070.
[Zuc13] R. Zucchini, AKSZ models of semistrict higher gauge theory, JHEP 1303 (2013) 014 [1112.2819 [hep-th]].
[Zuc16] R. Zucchini, A Lie based 4-dimensional higher Chern–Simons theory, J. Math. Phys. 57 (2016) 052301[ 1512.05977 [hep-th]].
[Zuc17] R. Zucchini, Algebraic formulation of higher gauge theory, J. Math. Phys. 58 (2017) 062301[ 1702.01545 [hep-th]].
[Zuc18] R. Zucchini, Exact renormalization group in Batalin–Vilkovisky theory, JHEP 1803 (2018) 132[ 1711.01213 [hep-th]]. 104 BIBLIOGRAPHY
[Zuc19] R. Zucchini, Exact renormalization group and effective action: a Batalin–Vilkovisky algebraic formulation, J. Math. Phys. 60 (2019) 012303[ 1711.07795 [math-ph]].
[Zwi93] B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33[ hep-th/9206084].
[Zwi04] B. Zwiebach, A first course in string theory, Cambridge University Press, 2004.
[Zwi12] B. Zwiebach, Double field theory, T-duality, and Courant brackets, Lect. Notes Phys. 851 (2012) 265[ 1109.1782 [hep-th]]. Index
Symbols cover A∞-algebra ...... 21 good...... 33 G-structure ...... 35 crossed modules of Lie algebras...... 22 ’t Hooft–Polyakov monopole ...... 47 cyclic inner product ...... 25 Čech–Dolbeault correspondence ...... 32 D 2-sphere ...... 30 decomposition theorem ...... 24 3-sphere ...... 30 degree ...... 15, 33, 37 A derived geometry ...... 29 ABJM model...... 54 DFT action ...... 58 AKSZ formalism ...... 52 Dirac monopole ...... 36 double copy ...... 51 B Drinfeld twist ...... 44 Berends–Giele recursion relation ...... 51 Duskin nerve ...... 11 bibundles...... 38 bicategory...... 10 E BLG model ...... 54 Eilenberg–Mac Lane spaces...... 29 brane scan ...... 59 ends of a manifold...... 29 BV complex ...... 51 Enriched categories...... 10
C F Calabi–Yau ...... 32 fake curvature ...... 37, 48 Calabi–Yau manifold ...... 32 first Chern class ...... 32 Calabi-Yau ...... 32 first fundamental confusion of calculus...... 69 Calabi-Yau manifold ...... 32 Frechét spaces, Loop spaces ...... 30 canonical bundle ...... 32 fundamental theorem of arithmetic ...... 41 Cartan calculus ...... 33 Fuzzy flag manifolds...... 44 Chalmers–Hanany–Witten D-brane fuzzy sphere ...... 44 configuration...... 60 G Chern class...... 32 general n-sphere...... 30 Chevalley–Eilenberg resolution ...... 21, 51 generalized Jacobi elliptic function...... 41 complex generalized metric ...... 58 manifold ...... 32 generalized tangent bundle ...... 33 structure ...... 32 Geometric Quantization ...... 44 conifold ...... 32 gerbe...... 37 conifold transition ...... 59 Green–Schwarz anomaly cancellation. . . . .52, 59 connection ...... 32 Courant algebroid ...... 25, 36 H Courant-Dorfman algebra ...... 25 Heterotic DFT ...... 58 106 INDEX
Hodge–Kodaira decomposition...... 27 Narain moduli space...... 58 homological algebra ...... 27 nLab...... 69 Homotopy type theory...... 9 homotopy type theory ...... 69 O Hopf bundle ...... 36 octonionic instanton ...... 49 Hopf fibration ...... 36 P J Parke–Taylor formula ...... 51 jet bundle ...... 35 Penrose–Ward transform ...... 32 platonic solid monopoles...... 49 K Poincaré lemma ...... 37 K3 manifold...... 32 Poincaré–Birkhoff–Witt theorem...... 25 Kähler propagator thory ...... 50 form ...... 32 manifold ...... 32 R metric ...... 32 Relative Sullivan algebra ...... 24 Kalb–Ramond field ...... 57 Ricci-flat ...... 32 Kaluza–Klein reduction...... 48 rigid ...... 32 KLT matrix ...... 51 rigid Calabi–Yau manifold ...... 32 KLT relations ...... 51 rigid Calabi-Yau manifold ...... 32 Koszul–Tate resolution...... 21, 51 Russell’s paradox ...... 9 Künneth’s theorem ...... 27 S L See also ...... 48 Levi-Civita connection ...... 32 Seiberg–Witten Theory...... 54 Lie n-groupoid ...... 14 Severa class ...... 26 Lie algebra bundle...... 38 simplicial set ...... 11 Lie differentiation ...... 15 smooth ∞-group ...... 14 Lie functor ...... 14 stationary phase formula ...... 50 Lie integration ...... 25 stereographic projection ...... 29 local Calabi–Yau manifold...... 32 Stirling-Formula ...... 63 strict 2-category ...... 10 Loop L∞-algebra ...... 25 strict Lie 2-algebras ...... 22 M strictification theorem ...... 24 M2-brane models ...... 54 string 2-group model...... 19, 38 magnetic bags ...... 49 string field theory...... 23, 60 magnetic domains ...... 49 string group ...... 19 manifold ...... 32 string Lie 2-algebra...... 20 metric...... 32 string structure...... 38 minimal model theorem ...... 24 Strominger system ...... 59 monad ...... 50 Sullivan model ...... 24 monopole ...... 49 super monopole sheets ...... 49 manifold ...... 32 monopole walls ...... 49 supergroups ...... 41 Multicategories ...... 11 surjective...... 14 Myers effect ...... 44 SUSY Chern–Simons theory ...... 47 SUSY field theories...... 47 N N-manifold ...... 33 T Nambu–Poisson structure ...... 31 ’t Hooft tensors ...... 63 INDEX 107
Theorem V Yau...... 32 Vinogradov Lie n-algebroids ...... 26 Toeplitz quantization ...... 43 transgression ...... 31, 35 W twisted string structure...... 38 Witten index ...... 53 twistor string theory...... 50
U Y universal enveloping algebra ...... 25 Young symmetrizer...... 27