Dynamics of D-Branes II. the Standard Action---An Analogue of The

February 2017 arXiv:yymm.nnnnn [hep-th] D(13.3): standard

Dynamics of D-branes II. The standard action — an analogue of the Polyakov action for (fundamental, stacked) D-branes

Chien-Hao Liu and Shing-Tung Yau

Abstract

(ρ,h;Φ,g,B,C) We introduce a new action Sstandard for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This ‘standard action’ is abstractly a non-Abelian gauged sigma model — based on maps ϕ :(XAz,E; ∇) → Y from an Azumaya/matrix manifold XAz with a fundamental module E with a connection ∇ to Y — enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples (ϕ, ∇) to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. With the analysis developed in D(13.1) (arXiv:1606.08529 [hep-th]) for such maps and an improved understanding of the hierarchy of various admissible conditions on the pairs (ϕ, ∇) beyond D(13.2.1) (arXiv:1611.09439 [hep-th]) and how they resolve the built-in obstruction to pull- push of covariant tensors under a map from a noncommutative manifold to a commutative manifold, we develop further in this note some covariant differential calculus needed and apply them to work out the first variation — and hence the corresponding equations of motion for D-branes — of the standard action and the second variation of the kinetic term for maps and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along the same line, the current note brings the Nambu-Goto- string-to-Polyakov-string analogue to D-branes. The current bosonic setting is the first step toward the dynamics of fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallel to what happened to the theory of fundamental strings during years 1976–1981. arXiv:1704.03237v1 [hep-th] 11 Apr 2017

Key words: D-brane; admissible condition; standard action, enhanced non-Abelian gauged sigma model; Azumaya manifold, C∞-scheme, harmonic map; ﬁrst and second variation, equations of motion

MSC number 2010: 81T30, 35J20; 16S50, 14A22, 35R01

Acknowledgements. We thank Andrew Strominger, Cumrun Vafa for inﬂuence to our understanding of strings, branes, and gravity. C.-H.L. thanks in addition Pei-Ming Ho for a discussion on Ramond-Ramond ﬁelds and literature guide and Chenglong Yu for a discussion on admissible conditions; Artan Sheshmani, Brooke Ullery, Ashvin Vishwanath for special/topic/basic courses, spring 2017; Ling-Miao Chou for comments that improve the illustrations and moral support. The project is supported by NSF grants DMS-9803347 and DMS-0074329. Chien-Hao Liu dedicates this work to Noel Brady, Hung-Wen Chang, Chongsun Chu, William Grosso, Pei-Ming Ho, Inkang Kim, who enriched his years at U.C. Berkeley tremendously.∗

∗(From C.H.L.) It was an amazing time when I landed at Berkeley in the 1990s, following Thurston’s transition to Mathematical Sciences Research Institute (M.S.R.I.). On the mathematical side, representing figures on geometry and topology — 2- and 3-dimensional geometry and topology and related dynamical system (Andrew Casson, Curtis McMullen, William Thurston), 4-dimensional geometry and topology (Robion Kirby), 5-and-above dimensional topology (Morris Hirsch, Stephen Smale), algebraic geometry (Robin Hartshorne), differential and complex geometry (Wu-Yi Hsiang, Shoshichi Kobayashi, Hong-Shi Wu), symplectic geometry and geometric quantization (Alan Weinstein), combinatorial and geometric group theory (relevant to 3-manifold study via the fundamental groups, John Stallings) — seem to converge at Berkeley. On the physics side, several first-or-second generation string-theorists (Orlando Alvarez, Korkut Bardakci, Martin Halpern) and one of the creators of supersymmetry (Bruno Zumino) were there. It was also a time when enumerative geometry and topology motivated by quantum field and string theory started to emerge and for that there were quantum invariants of 3-manifolds (Nicolai Reshetikhin) and mirror symmetry (Alexandre Givental) at Berkeley. Through the topic courses almost all of them gave during these years on their related subject and the timed homeworks one at least attempted, one may acquire a very broad foundation toward a cross field between mathematics and physics if one is ambitious and diligent enough. However, despite such an amazing time and an intellectually enriching environment, it would be extremely difficult, if not impossible, to learn at least some good part of them without a friend — well, of course, unless one is a genius, which I am painfully not. For that, on the mathematics side, I thank Hung-Wen (Weinstein and Givental’s student) who said hello to me in our first encounter at the elevator at the 9th floor in Evans Hall and influenced my understanding of many topics — particularly symplectic geometry, quantum mechanics and gauge theory — due to his unconventional background from physics, electrical engineering, to mathematics; Bill (Stallings’ student) and Inkang (Casson’s student) for suggesting me a weekly group meeting on Thurston’s lecture notes at Princeton on 3-manifolds

[Th] W.P. Thurston, The geometry and topology of three-manifolds, typed manuscript, Department of Mathematics, Princeton University, 1979. which lasted for one and a half year and tremendously inﬂuenced the depth of my understanding of Thurston’s work; Noel (Stallings’ student) for a reading seminar on the very thought-provoking French book

[Gr] M. Gromov, Structures métriquepour les variétés riemanniennes, rédigépar J. Lafontaine et P. Pansu, Text Math. 1, Cedic/Fernand-Nathan, Paris, 1980. for a summer. The numerous other after-class discussions for classes some of them and I happened to sit in shaped a large part of my knowledge pool, even for today. Incidentally, thanks to Prof. Kirby, who served as the Chair for the Graduate Students in the Department at that time. I remember his remark to his staff when I arrived at Berkeley and reported to him after meeting Prof. Thurston: “We have to treat them [referring to all Thurston’s then students] as nice as our own” On the physics side, thanks to Chongsun and Pei-Ming (both Zumino’s students) for helping me understand the very challenging topic: quantum field theory, first when we all attended Prof. Bardakci’s course Phys 230A, Quantum Field Theory, based closely on the book

[Ry] L.H. Ryder, Quantum field theory, Cambridge Univ. Press, 1985. and a second time a year later when we repeated it through the same course given by Prof. Alvarez, followed by a reading group meeting on Quantum Field Theory guided by Prof. Alvarez — another person who I forever have to thank and another event which changed the course of my life permanently. The former with Prof. Bardakci was a semester I had to spend at least three-to-four days a week just on this course: attending lectures, understanding the notes, reading the corresponding chapters or sections of the book, doing the homeworks, occasionally looking into literatures to figure out some of the homeworks, and correcting the mistakes I had made on the returned homework. I even turned in the take-home final. Amusingly, due to the free style of the Department of Mathematics at Princeton University and the visiting student status at U.C. Berkeley, that is the first of the only three courses and only two semesters throughout my graduate student years for which I ever honestly did like a student: do the homework, turn in to get graded, do the final, and in the end get a semester grade back. Special thanks to Prof. Bardakci and the TA for this course, Bogdan Morariu, for grading whatever I turned in, though I was not an officially registered student in that course. That was a time when I could study something purely for the beauty, mystery, and/or joy of it. That was a time when the future before me seemed unbounded. That was a time when I did not think too much about the less pleasant side of doing research: competitions, publications, credits, ··· . That was a time I was surrounded by friends, though only limitedly many, in all the best senses the word ‘friend’ can carry. Alas, that wonderful time, with such a luxurious leisure, is gone forever! Dynamics of D-branes, II: The Standard Action

0. Introduction and outline

In this sequel to D(11.1) (arXiv:1406.0929 [math.DG]), D(11.3.1) (arXiv:1508.02347 [math.DG]), D(13.1) (arXiv:1606.08529 [hep-th]) and D(13.2.1) (arXiv:1611.09439 [hep-th]) and along the line of our understanding of the basic structures on D-branes in Polchinski’s TASI 1996 Lec- ture Notes from the aspect of Grothendieck’s modern Algebraic Geometry initiated in D(1) (ρ,h;Φ,g,B,C) (arXiv:0709.1515 [math.AG]), we introduce a new action Sstandard for D-branes that is to D-branes as the (Brink-Di Vecchia-Howe/Deser-Zumino/)Polyakov action is to fundamental strings. This action depends both on the (dilaton field ρ, metric h) on the underlying topology X of the D-brane world-volume and on the background (dilaton field Φ, metric g, B-field B, Ramond-Ramond field C) on the target space-time Y ; and is naturally a non-Abelian gauged sigma model — based on maps ϕ :(XAz,E; ∇) → Y from an Azumaya/matrix manifold XAz with a fundamental module E with a connection ∇ to Y — enhanced by the dilaton term that couples (ϕ, ∇) to (ρ, Φ), the B-coupled gauge-theory term that couples ∇ to B, and the Chern- (ρ,h;Φ,g,B,C) Simons/Wess-Zumino term that couples (ϕ, ∇) to (B,C) in our standard action Sstandard . Before one can do so, one needs to resolve the built-in obstruction of pull-push of covariant tensors under a map from a noncommutative manifold to a commutative manifold. Such issue already appeared in the construction of the non-Abelian Dirac-Born-Infeld action (D(13.1) ). In this note, we give a hierarchy of various admissible conditions on the pairs (ϕ, ∇) that are enough to resolve the issue while being open-string compatible (Sec. 2). This improves our understanding of admissible conditions beyond D(13.2.1). With the noncommutative analysis developed in D(13.1), we develop further in this note some covariant differential calculus for such maps (Sec. 3) and use it to define the standard action for D-branes (Sec. 4). After promoting the setting to a family version (Sec. 5), we work out the first variation — and hence the corresponding equations of motion for D-branes — of the standard action (Sec. 6) and the second variation of the kinetic term for maps and the dilaton term in this action (Sec. 7). Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along the same line, the current standard action is clearly much more manageable. Classically and mathematically and in the special case where the background (Φ,B,C) on Y is set to vanish, this new theory is a merging of the theory of harmonic maps and a gauge theory (free to choose either a Yang-Mills theory or other kinds of applicable gauge theory) with a nilpotent type fuzzy extension. The current bosonic setting is the first step toward fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallel to what happened for fundamental superstrings during 1976–1981; (the road-map at the end: ‘Where we are’).

Convention. References for standard notations, terminology, operations and facts are (1) Azumaya/matrix algebra: [Ar], [Az], [A-N-T]; (2) sheaves and bundles: [H-L]; with connection: [Bl], [B-B], [D-K], [Ko]; (3) algebraic geometry: [Ha]; C∞ algebraic geometry: [Jo]; (4) diﬀerential geometry: [Eis], [G-H-L], [Hi], [H-E], [K-N]; (5) noncommutative diﬀerential geometry: [GB-V-F]; (6) string theory and D-branes: [G-S-W], [Po2], [Po3].

1 · For clarity, the real line as a real 1-dimensional manifold is denoted by R , while the field of real numbers is denoted by R. Similarly, the complex line as a complex 1-dimensional 1 manifold is denoted by C , while the field of complex numbers is denoted by C. √ · The inclusion ‘R ⊂ C’ is referred to the field extension of R to C by adding −1, unless otherwise noted.

1 · All manifolds are paracompact, Hausdorff, and admitting a (locally finite) partition of unity. We adopt the index convention for tensors from differential geometry. In particular, the tuple coordinate functions on an n-manifold is denoted by, for example, (y1, ··· yn). However, no up-low index summation convention is used.

· For this note, ‘diﬀerentiable’, ‘smooth’, and C∞ are taken as synonyms.

· matrix m vs. manifold of dimension m

· the Regge slope α0 vs. dummy labelling index α vs. covariant tensor α

· section s of a sheaf or vector bundle vs. dummy labelling index s

· algebra Aϕ vs. connection 1-form Aµ

· ring R vs. k-th remainder R[k] vs. Riemann curvature tensor Rijkl

i · boundary ∂U of an open set U vs. partial diﬀerentiations ∂t, ∂/∂y · Spec R (:= {prime ideals of R}) of a commutative Noetherian ring R in algebraic geometry k k vs. Spec R of a C -ring R (:= Spec RR := {C -ring homomorphisms R → R}) · morphism between schemes in algebraic geometry vs. C∞-map between C∞-manifolds or C∞-schemes in diﬀerential topology and geometry or C∞-algebraic geometry

· group action vs. action functional for D-branes

0 · metric tensor g vs. element g in a group G vs. gauge coupling constant ggauge

· sheaves F, G vs. curvature tensor F∇, gauge-symmetry group Ggauge

· dilaton ﬁeld ρ vs. representation ρgauge of a gauge-symmetry group Ggauge · The ‘support’ Supp (F) of a quasi-coherent sheaf F on a scheme Y in algebraic geometry or on a Ck-scheme in Ck-algebraic geometry means the scheme-theoretical support of F k unless otherwise noted; IZ denotes the ideal sheaf of a (resp. C -)subscheme of Z of a (resp. Ck-)scheme Y ; l(F) denotes the length of a coherent sheaf F of dimension 0.

· For a sheaf F on a topological space X, the notation ‘s ∈ F’ means a local section s ∈ F(U) for some open set U ⊂ X.

· For an OX -module F, the ﬁber of F at x ∈ X is denoted F|x while the stalk of F at x ∈ X is denoted Fx. · coordinate-function index, e.g. (y1, ··· , yn) for a real manifold vs. the exponent of a power, r r−1 e.g. a0y + a1y + ··· + ar−1y + ar ∈ R[y]. · The current Note D(13.3) continues the study in

[L-Y8] Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its ﬁrst variation, and the equations of motion for D-branes — with remarks on the non-Abelian Chern-Simons/Wess-Zumino term, arXiv:1606.08529 [hep-th]. (D(13.1)) Notations and conventions follow ibidem when applicable.

2 Outline

0. Introduction. 1. Azumaya/matrix manifolds with a fundamental module and diﬀerentiable maps therefrom · Azumaya/matrix manifolds with a fundamental module (XAz, E) · When E is equipped with a connection ∇ · Diﬀerentiable maps from (XAz, E) · Compatibility between the map ϕ and the connection ∇ 2. Pull-push of tensors and admissible conditions on (ϕ, ∇) 2.1 Admissible conditions on (ϕ, ∇) and the resolution of the pull-push issue 2.2 Admissible conditions from the aspect of open strings

C 3. The diﬀerential dϕ of ϕ and its decomposition, the three basic OX -modules, induced structures, and some covariant calculus 3.1 The diﬀerential dϕ of ϕ and its decomposition induced by ∇ C 3.2 The three basic OX -modules relevant to Dϕ, with induced structures Az ∗ Az 3.2.1 The OX -valued cotangent sheaf T X ⊗OX OX of X, and beyond ∗ 3.2.2 The pull-back tangent sheaf ϕ T∗Y C ∗ ∗ 3.2.3 The OX -module T X ⊗OX ϕ T∗Y , where Dϕ lives 4. The standard action for D-branes ∞ · The gauge-symmetry group C (Aut C(E)) · The standard action for D-branes · The standard action as an enhanced non-Abelian gauged sigma model

T 5. Admissible family of admissible pairs (ϕT , ∇ ) T · Basic setup and the notion of admissible families of admissible pairs (ϕT , ∇ )

· Three basic OXT -modules with induced structures

· Curvature tensors with ∂t and other order-switching formulae · Two-parameter admissible families of admissible pairs 6. The first variation of the enhanced kinetic term for maps and ...... 6.1 The first variation of the kinetic term for maps 6.2 The first variation of the dilaton term 6.3 The first variation of the gauge/Yang-Mills term and the Chern-Simons/Wess-Zumino term 6.3.1 The first variation of the gauge/Yang-Mills term 6.3.2 The first variation of the Chern-Simons/Wess-Zumino term for lower-dimensional D-branes 6.3.2.1D( −1)-brane world-point (m = 0) 6.3.2.2 D-particle world-line (m = 1) 6.3.2.3 D-string world-sheet (m = 2) 6.3.2.4 D-membrane world-volume (m = 3) 7. The second variation of the enhanced kinetic term for maps 6.1 The second variation of the kinetic term for maps 6.2 The second variation of the dilaton term · Where we are

3 1 Azumaya/matrix manifolds with a fundamental module and diﬀerentiable maps therefrom

Basics of maps from an Azumaya/matrix manifold with a fundamental module needed for the current note are collected in this section to ﬁx terminology, notations, and conventions. Readers are referred to [L-Y1] (D(1)), [L-L-S-Y] (D(2)), [L-Y5] (D(11.1)) and [L-Y7] (D(11.3.1)) for details; in particular, why this is a most natural description of D-branes when Polchinski’s TASI 1996 Lecture Note is read from the aspect of Grothendieck’s modern Algebraic Geometry. See also [H-W] and [Wi2].

Azumaya/matrix manifolds with a fundamental module (XAz, E)

From the viewpoint of Algebraic Geometry, a D-brane world-volume is a manifold equipped with a noncommutative structure sheaf of a special type dictated by (oriented) open strings.

Deﬁnition 1.1. [Azumaya/matrix manifold with fundamental module] Let X be a (real, smooth) manifold and E be a (smooth) complex vector bundle over X. Let

· OX be the structure sheaf of (smooth functions on) X,

C · OX := OX ⊗R C be its complexiﬁcation,

C · E be the sheaf of (smooth) sections of E, (it’s an OX -module), and

C · End C (E) be the endomorphism sheaf of E as an O -module OX X (i.e. the sheaf of sections of the endomorphism bundle End C(E) of E). Then, the (noncommutative-)ringed topological space

Az Az X := (X, O := End C (E)) X OX is called an Azumaya manifold1(or synonymously, a matrix manifold to be more concrete to string-theorists.) It is important to note that non-isomorphic complex vector bundles may give rise to isomorphic endomorphism bundles and from the string-theory origin of the setting, in which E plays the role of a Chan-Paton bundle on a D-brane world-volume, we always want to record E as a part of the data in deﬁning XAz. Thus, we call the pair (XAz, E) (or (XAz,E) in bundle notation) an Azumaya/matrix manifold with a fundamental module.

While it may be hard to visualize XAz geometrically, there in general is an abundant family of commutative OX -subalgebras Az OX ⊂ A ⊂ OX that deﬁne an abundant family of C∞-schemes

R XA := Spec (A) , each ﬁnite and germwise algebraic over X. They may help visualize XAz geometrically.

Az Deﬁnition 1.2. [(commutative) surrogate of X ] Such XA is called a (commutative) surrogate of (the noncommutative manifold) XAz. Cf. Figure 1-1.

4 XAz

(a) (c) (b)

X A (g) (d) ( f )

(e)

Figure 1-1. The noncommutative manifold XAz has an abundant collection of C∞- schemes as its commutative surrogates. See [L-Y8: Figure 2-1-1: caption] (D(13.1)) for more details.

Without loss of generality, one may assume that X is connected. However even so, a surrogate Az XA of X in general is disconnected locally over X (and can be disconnected globally as well; cf. Figure 1-1). To keep track of this algebraically, recall the following deﬁnition:

Deﬁnition 1.3. [complete set of orthogonal idempotents] (Cf. e.g. [Ei].) Let R be an (associative, unital) ring, with the identity element 1. A set of elements {e1, ··· , es} ⊂ R is called a complete set of orthogonal idempotents if the following three conditions are satisﬁed

2 (1) idempotent ei = ei, i = 1, ··· , s.

(2) orthogonal eiej = 0 for i 6= j.

(3) complete e1 + ··· + es = 1.

A complete set orthogonal idempotents {e1, ··· , es} is called maximal if no ei in the set can be 0 00 further decomposed into a summation ei = e + e of two orthogonal idempotents.

Az Az Let OX ⊂ A ⊂ OX be a commutative OX -subalgebra of OX and XA the associate surrogate of XAz. Then, for U ⊂ X an open set, there is a unique maximal complete set of orthogonal ∞ idempotents {e1, ··· , es} of the C -ring A(U) and it corresponds to the set of connected

1Unfamiliar physicists may consult [Ar] for basics of Azumaya algebras; see also [Az] and [A-N-T]. Simply put, an Azumaya manifold is topologically a smooth manifold but with a structure sheaf that has fibers Azumaya algebras over C. These fibers are all isomorphic to a matrix ring Mr×r(C) (and hence the synonym matrix manifold) for some fixed r but the isomorphisms involved are not canonical (and hence why the term ‘Azumaya manifold’ is more appropriate mathematically).

5 components of XA|U = Spec R(A(U)). Up to a relabelling, ei corresponds the function on XA(U) that is constant 1 on the i-th connected component and 0 on all other connected components.

Finally, we recall also the tangent sheaf and the cotangent sheaf of XAz.

Definition 1.4. [tangent sheaf, cotangent sheaf, inner derivations on XAz] The sheaf Az Az Az of (left) derivations on OX is denoted by T∗X and is called the tangent sheaf of X . The Az ∗ Az Az sheaf of Kählerdifferentials of OX is dented by T X and is called the cotangent sheaf of X . Az C ∗ Az Az T∗X is naturally a (left) OX -module while T X is naturally a (left) OX -module. For our C C purpose, we treat both as OX -modules. There is a natural OX -module homomorphism

Az Az OX −→ T∗X m 7−→ [m, · ] ,

Az 0 0 0 0 where [m, · ] acts on OX by m 7→ [m, m ] := mm − m m. The image of this homomorphism C Az Az is called the sheaf/OX -module of inner derivations on OX and is denoted by Inn (OX ) or Az C Inn (X ). The kernel of the above map is exactly the center OX · Id E , canonically identiﬁed C Az Az with OX , of OX . When the choice of a representative of an element of Inn (OX ) by an element Az Az Az in OX is irrelevant to an issue, we’ll represent elements of Inn (OX ) simply by elements in OX .

When E is equipped with a connection ∇

From the stringy origin of the setting with E serving as the Chan-Paton bundle on the D- brane world-volume, E is equipped with a gauge ﬁeld (i.e. a connection) created by massless excitations of open strings. Thus, let ∇ be a connection on E. Then ∇ induces a connection Az D on O := End C (E). With respect to a local trivialization of E, ∇ = d + A, where A X OX Az is an End C (E)-valued 1-form on X. Then D = d + [A, · ] on O under the induced local OX X C Az trivialization. As a consequence, D leaves the center OX of OX invariant and restrict to the C usual diﬀerential d on OX . Az C Once having the induced connection D on OX , one has then OX -module homomorphism

Az T∗XC −→ T∗X ξ 7−→ Dξ .

Az Lemma 1.5.[ D-induced decomposition of T∗X ] ([DV-M].) One has the short exact sequence Az Az C 0 −→ Inn (OX ) −→ T∗X −→ T∗X −→ 0 split by the above map.

Az The following two lemmas address the issue of when an idempotent in OX can be constant Az under a derivation ∈ T∗X .

Lemma 1.6. [(local) idempotent under D ] With the above notations, let U ⊂ X be an open set, ξ a vector ﬁeld on U, and {e1, ··· , es} be a complete set of orthogonal idempotents of Az OX (U). Assume that, say, Dξe1 commutes with all ei, i = 1, ··· , s. Then Dξe1 = 0.

6 2 Proof. Since e1 = e1,(Dξe1)e1 + e1Dξe1 = Dξe1. If in addition Dξe1 and e1 commute, then one has 2(Dξe1)e1 = Dξe1. The multiplication from the left by e1 gives then 2(Dξe1)e1 = (Dξe1)e1; i.e. (Dξe1)e1 = 0. If, furthermore, Dξe1 commutes also with all e2, ··· , es, then 0 = Dξ(eje1) = (Dξej)e1 + ejDξe1 = (Dξej)e1 + (Dξe1)ej, for j = 2, ··· , s. The multiplication from the left by ej gives then (Dξe1)ej = 0, for j = 2, ··· , s. It follows that Dξe1 = (Dξe1)(e1 + ··· + es) = 0.

Lemma 1.7. [(local) idempotent under inner derivation] With the above notations, let Az Az U ⊂ X be an open set, m ∈ OX (U) represent an inner derivation of OX (U), and {e1, ··· , es} Az be a complete set of orthogonal idempotents of OX (U). Assume that, say, [m, e1] commutes with all ei, i = 1, ··· , s. Then [m, e1] = 0.

Az Proof. Note that the proof of Lemma 1.6 uses only the Leibnitz rule property of Dξ on OX (U) and the commutativity property of Dξe1 with e1, ··· , es. Since [m, · ] satisﬁes also the Leibniz Az rule property on OX (U) and by assumption [m, e1] commutes with e1, ··· , es, the same proof goes through.

∨ C The contraction End C (E) = E ⊗ C E → O deﬁnes a trace map OX OX X

Az C Tr : OX −→ OX .

One has dTr = Tr D,

C where d is the ordinary diﬀerential on OX .

Diﬀerentiable maps from (XAz, E)

As a dynamical object in space-time, a D-brane moving in a space-time Y is realized by a map from a D-brane world-volume to Y . Back to our language, we need thus a notion of a ‘map from (XAz, E; ∇) to Y ’ that is compatible with the behavior of D-branes in string theory.

Definition 1.8. [map from Azumaya/matrix manifold] Let X be a (real, smooth) mani- Az ∞ fold, E be a complex vector bundle of rank r over X, and (X ,E) := (X,C (End C(E)),E) be the associated Azumaya/matrix manifold with a fundamental module. A map (synonymously, differentiable map, smooth map) ϕ :(XAz,E) −→ Y from (XAz,E) to a (real, smooth) manifold Y is defined contravariantly by a ring-homomorphism

] ∞ ∞ ϕ : C (Y ) −→ C (End C(E)) .

Equivalently in terms of sheaf language, let OY be the structure sheaf of Y . Regard both Az OY and OX as equivalence classes of gluing system of rings over the topological space Y and X respectively. Then the above ϕ] speciﬁes an equivalence class of gluing systems of ring- homomorphisms over R ⊂ C Az OY −→ OX , which we will still denote by ϕ].

7 Through the Generalized Division Lemma `ala Malgrange, one can show that ϕ] extends to a commutative diagram ] Az ϕ O o OY X h _ O ϕ˜] ] prY ? O O , X ] / X×Y prX of equivalence classes of ring-homomorphisms (over R or R ⊂ C, whichever is applicable) between equivalence classes of gluing systems of rings, with

] fϕ Aϕ o O Y h ] _ O f˜ϕ π] ] ϕ prY ? O O , X ] / X×Y prX a commutative diagram of equivalence classes of ring-homomorphisms between equivalence ∞ classes of gluing systems of C -rings. Here, prX : X × Y → X and prY : X × Y → Y are Az C Az the projection maps, OX ,→ OX follows from the inclusion of the center OX of OX , and ] ] Aϕ := OX hIm ϕ i = Im ϕ˜ . (Cf. [L-Y7: Theorem 3.1.1] (D(11.3.1)).) In terms of spaces, one has the following equivalent diagram of maps

! XAz ϕ σϕ ) Xϕ w / Y fϕ O πϕ prY f˜ ϕ ) XXo o × Y , prX

∞ where Xϕ is the C -scheme R Xϕ := Spec Aϕ associated to Aϕ.

Deﬁnition 1.9. [graph of ϕ] The push-forwardϕ ˜∗E =: E˜ϕ of E underϕ ˜ is called the graph of C ∞ ˜ ϕ. It is an OX×Y -module. Its C -scheme-theoretical support is denoted by Supp (Eϕ).

Az ∞ Definition 1.10. [surrogate of X specified by ϕ] The C -scheme Xϕ is called the surrogate of XAz specified by ϕ.

Xϕ is ﬁnite and germwise algebraic over X and, by construction, it admits a canonical ∞ embedding f˜ϕ : Xϕ → X × Y into X × Y as a C -subscheme. The image is identical to Supp (E˜ϕ). Cf. Figure 1-2 and Figure 1-3.

8 XAz

ϕ Y

Xϕ fϕ

�ϕ Imϕ

Az Az Figure 1-2. A map ϕ :(X ,E) → Y speciﬁes a surrogate Xϕ of X over X. Xϕ is a C∞-scheme that may not be reduced (i.e. it may have some nilpotent fuzzy structure thereon). It on one hand is dominated by XAz and on the other dominates and is ﬁnte and germwise algebraic over X.

Compatibility between the map ϕ and the connection ∇ Up to this point, the map ϕ :(XAz, E) → Y to Y and the connection ∇ on E are quite independent objects. A priori, there doesn’t seem to be any reason why they should constrain or influence each other at the current purely differential-topological level. However, when one moves on to address the issue of constructing an action functional for (ϕ, ∇) as in [L-Y8] (D(13.1)), one immediately realizes that, · Due to a built-in mathematical obstruction in the problem, one needs some compatibility condition between ϕ and ∇ before one can even begin the attempt to construct an action functional for (ϕ, ∇). Furthermore, as a hindsight, that there needs to be a compatibility condition on (ϕ, ∇) is also implied by string theory: · We need a condition on (ϕ, ∇) to encode the stringy fact that the gauge field ∇ on the D-brane world-volume as ‘seen’ by open strings in Y through ϕ should be massless. We address such compatibility condition on (ϕ, ∇) systematically in the next section.

2 Pull-push of tensors and admissible conditions on (ϕ, ∇)

When one attempts to construct an action functional for a theory that involves maps from a world-volume to a target space-time, one unavoidably has to come across the notion of ‘pulling back a (covariant) tensor, for example, the metric tensor or a diﬀerential form on the target space-time to the world-volume’. In the case where only maps from a commutative world- volume to a (commutative) space-time are involved, this is a well-established standard notion from diﬀerential topology. However, in a case, like ours, where maps from a noncommutative world-volume to a (commutative) space-time is involved, ϕ : Space (S) −→ Space(R) ,

9 ( ) nd X Azumaya cloud

ϕ Y

Xϕ fϕ πϕ

map from Azumaya manifold

Fourier-Mukai transform

X Y Y

Supp ( ) = Γϕ pr2 = Xϕ

pr1

Figure 1-3. The equivalence between a map ϕ from an Azumaya manifold with a Az fundamental module (X, O := End C (E), E) to a manifold Y and a special kind of X OX Fourier-Mukai transform E˜ ∈ Mod C(X × Y ) from X to Y . Here, Mod C(X × Y ) is C the category of OX×Y -modules.

10 with the accompanying contravariant ring-homomorphism S ←− R : ϕ] , there is a built-in mathematical obstruction to such a notion. Here, S is an (associative, unital) noncommutative ring, R is a (associative, unital) commutative ring, and Space (S) and Space (R) are the topological spaces whose function rings are S and R respectively. For the noncommutative ring S, its (standard and functorial-in-the-category-rings) bi-S- module of Kählerdifferentials is naturally defined to be

Kähler 0 0 0 0 ΩS := Span (S,S){ds | s ∈ S)}/(d(ss ) − (ds)s − sds | s, s ∈ S) while for the commutative ring R its (standard and functorial-in-the-category-of-commutative- rings) (left) R-module of Kählerdifferentials is naturally defined to be

Kähler 0 0 0 0 ΩR := Span R{dr | r ∈ R}/(d(rr ) − r dr − rdr | r, r ∈ R) , with the convention that rdr0 = (dr0)r to turn it to a bi-R-module as well. Treating R as a ring (that happens to be commutative), it has also the (standard and functorial-in-the-category-rings) bi-R-module of Kählerdifferentials

nc,Kähler 0 0 0 0 ΩR := Span (R,R){dr | r ∈ R)}/(d(rr ) − (dr)r − rdr | r, r ∈ R) Kähler exactly like ΩS for S. There is a built-in tautological quotient homomorphism as bi-R- modules nc,Kähler Kähler ΩR −→−→ ΩR r1(dr)r2 7−→ r1r2 dr , whose kernel is generated by {rdr0 −(dr0)r | r, r0 ∈ R}. Given the map ϕ : Space (S) → Space (R), one has the following built-in diagram

∗ K¨ahler ϕ nc,K¨ahler ΩS o ΩR

K¨ahler ΩR ,

∗ ] ] ] where ϕ (r1(dr)r2) = ϕ (r1)d(ϕ (r))ϕ (r2). The issue is now whether one can extend the above diagram to the following commutative diagram

∗ K¨ahler ϕ nc,K¨ahler ΩS o ΩR h

ϕ∗ ? Kähler ΩR . The answer is No, in general. See, e.g., [L-Y5: Example 4.1.20] (D(11.1)) for an explicit counterexample. When R is a C∞-ring, e.g. the function-ring C∞(Y ) of a smooth manifold Y , then Kähler the R-module ΩR of differentials of R is a further quotient of the above module ΩR of Kähler differentials by additional relations generated by applications of the chain rule on the transcen- dental smooth operations in the C∞-ring structure of R ([Jo]; cf. [L-Y5: Sec. 4.1] (D(11.1))). The issue becomes even more involved. In particular, as the counterexample ibidem shows · [built-in mathematical obstruction of pullback] For a map ϕ :(XAz, E) → Y , there is no way to define functorially a pull-back map T ∗Y → T ∗XAz that takes a (covariant) 1-tensor on Y to a 1-tensor on XAz. As a consequence, there is no functorial way to pull back a (covariant) tensor on Y to a tensor on XAz.

11 Before the attempt to construct an action functional that involves such maps ϕ, one has to resolve the above obstruction ﬁrst. In [L-Y8] (D(13.1)), we learned how to use the connection ∇ to impose a natural admissible condition on ϕ so that the above obstruction is bypassed through Az the surrogate Xϕ of X speciﬁed by ϕ. With the lesson learned therefrom and further thought beyond [L-Y9] (D(13.2.1)), we propose (Sec. 2.1) in this section a still-natural-but-much-weaker admissible condition on (ϕ, ∇) that bypasses even the surrogate Xϕ but is still robust enough to construct naturally a pull-push map we need on tensors. It turns out that this much weaker admissible condition remains to be compatible with open strings (Sec. 2.2).

2.1 Admissible conditions on (ϕ, ∇) and the resolution of the pull-push issue A hierarchy of admissible conditions on (ϕ, ∇) is introduced. A theorem on how even the weakest admissible condition in the hierarchy can resolve the above obstruction in our case is proved.

Three hierarchical admissible conditions

Deﬁnition 2.1.1. [admissible connection on E] Let ϕ :(XAz, E) → Y be a map. For a Az connection ∇ on E, let D be its induced connection on OX . A connection ∇ on E is called

(∗1)-admissible to ϕ if DξAϕ ⊂ Comm (Aϕ); (∗2)-admissible to ϕ if DξComm (Aϕ) ⊂ Comm (Aϕ); (∗3)-admissible to ϕ if DξAϕ ⊂ Aϕ

Az for all ξ ∈ T∗X. Here, Comm (Aϕ) denotes the commutant of Aϕ in OX . When ∇ is (∗1)-admissible to ϕ, we will take the following as synonyms:

· (ϕ, ∇) is an (∗1)-admissible pair, · ϕ is (∗1)-admissible to ∇, Az · ϕ :(X , E; ∇) → Y is (∗1)-admissible.

Similarly, for (∗2)-admissible pair (ϕ, ∇) and (∗3)-admissible pair (ϕ, ∇), ... , etc..

Lemma 2.1.2. [hierarchy of admissible conditions]

Admissible Condition (∗3) =⇒ Admissible Condition (∗2) =⇒ Admissible Condition (∗1) .

Az Proof. Admissible Condition (∗3) says that the OX -subalgebra Aϕ ⊂ OX is invariant under Az D-parallel transports along paths on X. Since D-parallel transports on OX are algebra- Az isomorphisms, if Aϕ is D-invariant, the OX -subalgebra Comm (Aϕ) of OX must also be D- invariant since it is determined by Aϕ ﬁberwise algebraically. In other words, Admissible Con- dition (∗3) =⇒ Admissible Condition (∗2). Since Aϕ is commutative, Aϕ ⊂ Comm (Aϕ.). Thus, the inclusion D·Aϕ ⊂ D·Comm (Aϕ) always holds. This implies that Admissible Condition (∗2) =⇒ Admissible Condition (∗1).

Deﬁnition 2.1.3. [strict admissible connection on E] Continuing Deﬁnition 2.1.1. Let F∇ Az be the curvature tensor of ∇. It is an OX -valued 2-form on X. Then, for ·= 1, 2, 3, ∇ is called strictly (∗·)-admissible to ϕ if

12 Az · ∇ is (∗·)-admissible to ϕ and F∇ takes values in Comm (Aϕ) ⊂ OX . In this case, (ϕ, ∇) is said to be a strictly (∗·)-admissible pair.

Clearly, the same hierarchy holds for strict admissible conditions:

strict (∗3) =⇒ strict (∗2) =⇒ strict (∗1) .

The Strict (∗3)-Admissible Condition on (ϕ, ∇) was introduced in [L-Y8: Deﬁnition 2.2.1] (D(13.1)) to deﬁne the Dirac-Born-Infeld action for (ϕ, ∇).

Lemma 2.1.4. [commutativity under admissible condition] Let ϕ :(XAz, E; ∇) → Y ] ] ∞ be a map. (1) If (ϕ, ∇) is (∗1)-admissible, then [Dξϕ (f1), ϕ (f2)] = 0 for all f1, f2 ∈ C (Y ) ] ] and ξ ∈ T∗X. (2) If (ϕ, ∇) is (∗2)-admissible, then [Dξ1 ϕ (f1),Dξ2 ϕ (f2)] = 0 for all f1, f2 ∈ ∞ C (Y ) and ξ1, ξ2 ∈ T∗X.

Proof. Statement (1) is the (∗1)-Admissible Condition itself. ∞ ] ] For Statement (2), let f1, f2 ∈ C (Y ) and ξ1, ξ2 ∈ T∗X. Then [Dξϕ (f1), ϕ (f2)] = 0 since

Aϕ ⊂ Comm (Aϕ). Thus, applying Dξ2 to both sides,

] ] ] ] [Dξ2 Dξ1 ϕ (f1), ϕ (f2)] + [Dξ1 ϕ (f1),Dξ2 ϕ (f2)] = 0 .

The (∗2)-Admissible Condition implies that

] Dξ2 Dξ1 ϕ (f1) ∈ Comm (Aϕ) .

] ] And, hence, [Dξ2 Dξ1 ϕ (f1), ϕ (f2)] = 0. Statement (2) follows.

Resolution of the pull-push issue under Admissible Condition (∗1)

The current theme is devoted to the proof of the following theorem:

Theorem 2.1.5. [pull-push under (∗1)-admissible (ϕ, ∇)] Let (ϕ, ∇) be (∗1)-admissible. Then the assignment

∞ ϕ :ΩC∞(Y ) −→ ΩC∞(X) ⊗C∞(X) C (End C(E)) ] ] f1df2 7−→ ϕ (f1) Dϕ (f2) is well-deﬁned.

The study in [L-Y8: Sec. 4] (D(13.1)) allows one to express ϕ(df) locally explicit enough so that one can check that ϕ is well-deﬁned when (ϕ, ∇) is (∗1)-admissible. Note that, with Lemma 2.1.2, this implies that if (ϕ, ∇) is either (∗2)- or (∗3)-admissible, then ϕ is also well- deﬁned. We now proceed to prove the theorem.

13 Lemma 2.1.6. [local expression of ϕ (df) for (∗1)-admissible (ϕ, ∇), I] Let (ϕ, ∇) be (∗1)-admissible; i.e. DξAϕ ⊂ Comm (Aϕ) for all ξ ∈ T∗X. Let U ⊂ X be a small enough open set so that ϕ(U Az) is contained in a coordinate chart of Y , with coordinate y = (y1, ··· , yn). For f ∈ C∞(Y ), recall the germwise-over-U polynomial Rf [1] in (y1, ··· , yn) with coeﬃcients Az in OU from [L-Y8: Sec. 4 & Remark/Notation 4.2.3.5] (D.13.1). Then, for ξ a vector ﬁeld on U and f ∈ C∞(Y ), and at the level of germs over U,

f (ϕ (df))(ξ) = R [1] d ] d f . y Dξ(ϕ (y )), for all multi-degree d in R [1]

d 1 d n dn Here, for a multiple degree d = (d1,, ··· , dn), di ∈ Z≥0, y := (y ) 1 ··· (y ) and d ] d d ] d y Dξ(ϕ (y )) means ‘replacing y by Dξ(ϕ (y ))’.

Proof. Denote the coordinate chart of Y in the Statement by V . Let prX : X × Y → X, prY : X × Y → Y be the projection maps. Recall the induced ring-homomorphism ] ∞ ∞ ˜ ˜ ϕ˜ : C (X × Y ) → C (End C(E)) over R ⊂ C and the graph Eϕ of ϕ and its support Supp (Eϕ) ] ∞ on X × Y . Denote prY (f) ∈ C (X × Y ) still by f when there is no confusion. For clarity, we proceed the proof of the Statement in three steps.

Step (1) How Rf [1] is constructed in [L-Y8: Sec. 4] (D(13.1)) For any p ∈ U, let p ∈ U 0 ⊂ U be a neighborhood of p in U over which the Generalized Division Lemma `ala Malgrange is applied 0 0 to f on a neighborhood U × V of ({p} × V ∩ Supp (E˜ϕ))red =: {q1, ··· , qs} in (X × Y )/X with (i) i ] i ∞ 0 1 n respect to the characteristic polynomials χϕ := det(y · Id r×r − ϕ (y )) ∈ C (U )[y , ··· , y ] ∈ C∞(U 0 × V ), i = 1, ··· , n. Passing to a smaller open subset if necessary, one may assume that 0 0 0 0 0 V is a disjoint union V1 ∪ · · · ∪Vs with U ×Vk a neighborhood of qk and the closure V1, ··· , Vs ∞ are all disjoint from each other. Let 1(k) ∈ C (Y ) be a smooth functions on Y that takes the 0 0 0 value 1 on Vk and the value zero on Vk0 , k 6= k, k = 1, ··· , s. (Cf. [L-Y8: Sec. 4.2.3] (D(13.1)).) Then s n X X f;k d X f;k (i) (j) f|U 0×V 0 = 1(k) · cd y + Q(i,j)χϕ χϕ k=1 d i,j=1 for some f;k ∞ 0 P f;k d ∞ 0 1 n · cd ∈ C (U ) for all k and d, and d cd y ∈ C (U )[y , ··· , y ] for all k,

f;k ∞ 0 0 · Q(i,j) ∈ C (U × Vk) for all k and i, j. In terms of this, over U 0, s f X ] X f;k d R [1] = ϕ (1(k)) cd y k=1 d and ] f ϕ (f) 0 = R [1] d ] d f U y ϕ (y ) for all multi-degree d in R [1] ] (i) ] ] ] sinceϕ ˜ (χϕ ) = 0 for i = 1, ··· , n, ϕ (f) =ϕ ˜ (prY (f)).

] s 0 Step (2) {ϕ (1(k))}k=1 as the maximal complete set of orthogonal D-parallel idempotents/U ] Since ϕ (f) depends only on the restriction of f, regarded on X × Y , to Supp (E˜ϕ), one has ϕ](1 + ··· + 1 ) =ϕ ˜](1 ) = Id (1) (s) X×Y E|U0 0 2 0 over U ⊂ X. Since, in addition, 1(k) = 1(k) for all k, 1(k) 1(k0) = 0 for all k 6= k , and 0 s = the number of the connected components of Xϕ|U 0 for U small enough, the collection ] ] {ϕ (1(1)), ··· , ϕ (1(s))} gives the maximal complete set of orthogonal idempotents in Aϕ|U 0 .

14 ] ] Furthermore, since DξAϕ ⊂ Comm (Aϕ) for all ξ ∈ T∗X, Dξϕ (1(k)) and ϕ (1(k0)) commute for all k, k0 = 1, ··· , s. It follows from Lemma 1.6 that

] ] Dξϕ (1(1)) = ··· = Dξϕ (1(s)) = 0

] ] 0 for all ξ. In other words, ϕ (1(1)), ··· , ϕ (1(s)) are D-parallel over U .

0 Az Step (3) The evaluation of (ϕ (df))(ξ) over U We are now ready to evaluate the OX - 0 valued derivation Dξϕ on f, locally and germwise over U . Step (1) and Step (2) together imply 0 that, for ξ ∈ T∗X and over U ,

] (ϕ (df))(ξ) := Dξ(ϕ (f)) s s X ] X f;k ] d X ] X f;k ] d = ϕ (1(k)) (ξcd ) ϕ (y ) + ϕ (1(k)) cd Dξ(ϕ (y )) k=1 d k=1 d = Term (I) + Term (II)

] since Dξϕ (1(k)) = 0 for k = 1, ··· , s. Note that f Term (II) = R [1] d ] d f . y Dξ(ϕ (y )), for all multi-degree d in R [1] It remains to prove that Term (I) vanishes. But this is the situation studied in [L-Y8: Proposition 4.2.3.1:Proof] (D(13.1)). In essence, since

s n X X f;k d X f;k (i) (j) f = 1(k) · cd y + Q(i,j)χϕ χϕ k=1 d i,j=1 on U 0 × V 0 and f on (X × Y )/X is independent of X, one has

] X X f;k d ] Term (I) = ϕ 1(k) ξcd y = ϕ (ξf) = 0 . k d

Here we denote the canonical lifting of ξ ∈ T∗X to T∗(X × Y ), via the product structure of X × Y , by the same notation ξ. This completes the proof.

Lemma 2.1.7. [local expression of ϕ (df) for (∗1)-admissible (ϕ, ∇), II] Let (ϕ, ∇) be (∗1)-admissible. Continuing the setting and notations in Lemma 2.1.6. Then, locally,

n n X X ∂f (ϕ(df))(ξ) = (ϕdyi)(ξ) ⊗ ∂ f = D ϕ](yi) · ϕ] . ∂yi ξ ∂yi i=1 i=1

∞ Here · is the multiplication in the ring C (End C(E)) (and will be omitted later when there is no sacriﬁce to clarity).

15 Proof. This is a consequence of Lemma 2.1.6. Continuing the setup in the Statement and the ] (i) proof thereof. Then, sinceϕ ˜ (χϕ ) = 0 for i = 1, ··· , n, one has

ϕ] ∂ f =ϕ ˜] ∂ Rf [1] = Rf [1] . ∂yi ∂yi yd ϕ] ∂ yd, for all multi-degree d in Rf [1] ∂yi

] i ] j Since (ϕ, ∇) is (∗1)-admissible, Dξϕ (y ) and ϕ (y ) commute for i, j = 1, ··· , n. It follows that

n X ] i ] ∂ f Dξϕ (y ) · ϕ ( i f) = R [1]| d ] d f . ∂y y Dξ(ϕ (y )), for all multi-degree d in R [1] i=1 Which is ϕ(df) by Lemma 2.1.6. This proves the lemma.

Proof of Theorem 2.1.5. We now check in two steps that ϕ is well-deﬁned. Note that we only need to do so locally over X. Thus, let U ⊂ X be the open set in Lemma 2.1.6 such that ϕ(U Az) is contained in a coordinate chart V of Y , with the coordinate (y1, ··· , yn). Lemma 2.1.7 implies then that the following assignment is the restriction of ϕ to over U and hence is independent of the local coordinate (y1, ··· , yn) on V :

∞ ϕ :ΩC∞(V ) −→ ΩC∞(U) ⊗C∞(U) C (End C(E|U )) ] Pn ]∂f2 ] i f1df2 7−→ ϕ (f1) · i=1 ϕ ∂yi · Dϕ (y ) .

] i ] ∂f Here, we use again the fact that (ϕ, ∇) is (∗1)-admissible so that the summand Dϕ (y )·ϕ ∂yi ] ∂f2 ] i in Lemma 2.1.7 is equal to ϕ ∂yi · Dϕ (y ) here, with f replaced by f2. It remains to show that ϕ is compatible with (a) the commutative Leibniz rule and (b) the chain-rule identities from the C∞-ring structure of C∞(V ).

∞ ∞ (a) The commutative Leibniz rule For f1, f2 ∈ C ∈ C (V ), one has

d(f1f2) − f2 df1 − f1 df2 = 0

in ΩC∞(V ). Under ϕ , one has

] ] ] ] ] ϕ d(f1f2) − f2 df1 − f1 df2 = D ϕ (f1f2) − ϕ (f2) Dϕ (f1) − ϕ (f1) Dϕ (f2) = 0 since ] ] ] ] ] ] ] D ϕ (f1f2) = D ϕ (f1)ϕ (f2) = (Dϕ (f1))ϕ (f2) + ϕ (f1)Dϕ (f2) , which is ] ] ] ] ϕ (f2)Dϕ (f1) + ϕ (f1)Dϕ (f2) for (ϕ, ∇)(∗1)-admissible.

∞ ∞ l (b) The chain-rule identities from the C -ring structure Let ζ ∈ C (R ), l ∈ Z≥1 and ∞ f1, ··· , fl ∈ C (V ). Then, one has

l X d ζ(f1, ··· , fl) − (∂kζ)(f1, ··· , fl) dfk = 0 k=1

16 ∞ l in ΩC∞(V ). Here, ∂kζ is the partial derivative of ζ ∈ C (R ) with respect to its k-th argument. Under ϕ, one has

l X ϕ d ζ(f1, ··· , fl) − (∂kζ)(f1, ··· , fl) dfk k=1 l ] X ] ] = Dϕ ζ(f1, ··· , fl) − ϕ (∂kζ)(f1, ··· , fl) Dϕ (fk) = 0 k=1 since, by Lemma 2.1.7 and the (∗1)-admissibility of (ϕ, ∇),

n ] X ] i ∂ Dϕ ζ(f1, ··· , fl) = Dϕ (y ) ⊗ ∂yi ζ(f1, ··· , fl) i=1 n l X ] i X ∂fk = Dϕ (y ) ⊗ (∂kζ)(f1, ··· , fl) ∂yi i=1 k=1 l X ] = Dϕ (fk) ⊗ (∂kζ)(f1, ··· , fl) k=1 l l X ] ] X ] ] = Dϕ (fk)ϕ (∂kζ)(f1, ··· , fl) = ϕ (∂kζ)(f1, ··· , fl) Dϕ (fk) . k=1 k=1 This completes the proof of Theorem 2.1.5

The pull-push ϕ on tensor product ΩC∞(Y ) ⊗C∞(Y ) · · · ⊗C∞(Y ) ΩC∞(Y )

Having the well-deﬁned

∞ ϕ :ΩC∞(Y ) −→ ΩC∞(X) ⊗C∞(X) C (End C(E)) , it is natural to consider the extension of ϕ to a correspondence between tensor products

k k ∞ ϕ : ⊗C∞(Y )ΩC∞(Y ) −→ ⊗C∞(X) ΩC∞(X) ⊗C∞(X) C (End C(E)) ] ] ] f0 df1 ⊗ · · · ⊗ dfk 7−→ ϕ (f0)Dϕ (f1) ⊗ · · · ⊗ Dϕ (fk) .

] ] Here, the tensor Dϕ (f1) ⊗ · · · ⊗ Dϕ (fk) is deﬁned to the tensors of the underlying 1-forms in ∞ ΩC∞(X) and multiplication of the coeﬃcients in C (End C(E)) from each factor. Explicitly, in terms of a local coordinate (x1, ··· , xm) on a chart U ⊂ X,

] ] ] ϕ (f0)Dϕ (f1) ⊗ · · · ⊗ Dϕ (fk) m X ] ] ] µ1 µk = ϕ (f0)D∂/∂xµ1 ϕ (f1) ··· D∂/∂xµk ϕ (fk) dx ⊗ · · · ⊗ dx . µ1, ··· , µk=1

Lemma 2.1.8. [pull-push of (covariant) tensor] For (ϕ, ∇)(∗1)-admissible, the above extension of ϕ to covariant tensors is well-deﬁned.

17 k Proof. In ⊗C∞(Y )ΩC∞(Y ), one has the identities

f0 df1 ⊗ df2 ⊗ · · · ⊗ dfk = df1f0 ⊗ df2 ⊗ · · · ⊗ dfk

= df1 ⊗ f0df2 ⊗ · · · ⊗ dfk = ······

= df1 ⊗ · · · ⊗ dfk−1f0 ⊗ dfk = df1 ⊗ · · · ⊗ f0dfk = df1 ⊗ · · · ⊗ dfkf0 .

] ] Since the (∗1)-Admissible Condition implies that ϕ (f0) commutes with all of Dϕ (f1), ··· , ] Dϕ (fk), the parallel identities

] ] ] ] ] ] ] ] ϕ (f0)Dϕ (f1) ⊗ Dϕ (f2) ⊗ · · · ⊗ Dϕ (fk) = Dϕ (f1)ϕ (f0) ⊗ Dϕ (f2) ⊗ · · · ⊗ Dϕ (fk) ] ] ] ] = Dϕ (f1) ⊗ ϕ (f0)Dϕ (f2) ⊗ · · · ⊗ Dϕ (fk) = ······ ] ] ] ] = Dϕ (f1) ⊗ · · · ⊗ Dϕ (fk−1)ϕ (f0) ⊗ Dϕ (fk) ] ] ] ] ] ] = Dϕ (f1) ⊗ · · · ⊗ ϕ (f0)Dϕ (fk) = Dϕ (f1) ⊗ · · · ⊗ Dϕ (fk)ϕ (f0)

k ∞ hold in ⊗C∞(X) ΩC∞(X) ⊗C∞(X) C (End C(E)). This proves the lemma.

Az Note that for a (∗1)-admissible map ϕ :(X , E; ∇) → Y , since DξAϕ ⊂ Comm(Aϕ) for C Az all ξ ∈ T∗X and Comm (Aϕ) is itself a (possibly noncommutative) OX -subalgebra of OX , the pull-push ϕ α of a (covariant) tensor α on Y to X is indeed Comm (Aϕ)-valued.

Az Example 2.1.9. [pull-push of 2-tensor under (∗1)-admissible (ϕ, ∇)] Let ϕ :(X , E; ∇) → P i j Y be a (∗1)-admissible map and α = i,j αijdy ⊗ y be a 2-tensor on Y . Then, with respect to local coordinates (x1, ··· , xn) on X and (y1, ··· , yn) on Y ,

m n X X ] ] i ] j µ ν ϕ α = ϕ (αij)D ∂ ϕ (y )D ∂ ϕ (y ) dx ⊗ dx . ∂xµ ∂xν µ,ν=1 i,j=1

] i ] j ] j ] i Since in general D∂/∂xµ ϕ (y )D∂/∂xν ϕ (y ) 6= D∂/∂xν ϕ (y )D∂/∂xµ ϕ (y ), ϕ does not take a symmetric 2-tensor on Y to a Comm (Aϕ)-valued symmetric 2-tensor on X, nor an antisymmetric 2-tensor on Y to a Comm (Aϕ)-valued antisymmetric 2-tensor on X. However, after Az C the post-composition with the trace map Tr : OX → OX , Tr ϕ does take a symmetric (resp. C antisymmetric) 2-tensor on X to an OX -valued symmetric (resp. antisymmetric) 2-tensor on X.

Example 2.1.10. [pull-push of higher-rank tensor under (∗1)-admissible (ϕ, ∇)] Con- tinuing Example 2.1.9. For α a (covariant) tensor on Y of rank ≥ 3, the trace map no longer help bring symmetric (resp. antisymmetric) tensors to symmetric (resp. antisymmetric) tensors.

Az The situation gets better for a map ϕ :(X , E; ∇) → Y that satisﬁes the stronger (∗2)- Admissible Condition: DξComm (Aϕ) ⊂ Comm (Aϕ) for ξ ∈ T∗X.

Az Lemma 2.1.11. [pull-push of tensor under (∗2)-admissible (ϕ, ∇)] Let ϕ :(X , E; ∇) → Y be a (∗2)-admissible map. Then ϕ takes a symmetric (resp. antisymmetric) tensor on Y to a Comm (Aϕ)-valued symmetric (resp. antisymmetric) tensor on X.

18 Proof. In terms of local coordinates (x1, ··· , xn) on X and (y1, ··· , yn) on Y , a (covariant) k-tensor X i1 ik α = αi1 ··· ik dy ⊗ · · · ⊗ dy i1, ··· ik on Y is pull-pushed to a Comm (Aϕ)-valued k-tensor

m n X X ] ] i1 ] ik µ1 µk ∂ ∂ ϕ α = ϕ (αi1 ··· ik )D ϕ (y ) ··· D ϕ (y ) dx ⊗ · · · ⊗ dx . ∂xµ1 ∂xµk µ1, ··· , µk=1 i1, ··· , ik=1 on X. It follows from Lemma 2.1.4 that, under the (∗2)-Admissible Condition, all the factors

] ] i1 ] ik ϕ (αi1 ··· ik ) ,D ∂ ϕ (y ) , ··· ,D ∂ ϕ (y ) ∂xµ1 ∂xµk in a summand commute among themselves. This implies, in particular, that ϕ now takes a symmetric (resp. antisymmetric) tensor on Y to a Comm (Aϕ)-valued symmetric (resp. antisymmetric) tensor on X.

Let V• T ∗Y be the sheaf of diﬀerential forms on Y . The same proof of Lemma 2.1.11 gives also

Az Lemma 2.1.12.[ ϕ and ∧] Let ϕ :(X , E; ∇) → Y be a (∗2)-admissible map. For α, β ∈ V• T ∗Y , deﬁne the wedge product ϕα ∧ ϕβ

V• ∗ C Az of ϕ α, ϕ β ∈ ( T X) ⊗ C O by applying the wedge product to the diﬀerential forms on OX X Az X and multiplication to the OX -valued coeﬃcients. Then, ϕ(α ∧ β) = (ϕα) ∧ (ϕβ) .

Remark 2.1.13. [admissible condition and Ramond-Ramond ﬁeld ] While the current note will take (ϕ, ∇) to be (∗1)-admissible most of the time, Example 2.1.9, Example 2.1.10, Lemma 2.1.11 and Lemma 2.1.12 together suggest that when the coupling of D-brane to Ramond-Ramond ﬁelds is taken into account, the more natural admissible condition on (ϕ, ∇) is the stronger (∗2)-Admissible Condition.

2.2 Admissible conditions from the aspect of open strings

We address in this subsection the implication of Admissible Condition (∗1) on (ϕ, ∇) to the mass of the connection ∇ from the aspect of open strings in the target-space Y . Az Let ϕ :(X , E; ∇) → Y be a (∗1)-admissible map. Recall the surrogate Xϕ := Spec RAϕ Az of X speciﬁed by ϕ and the built-in dominant morphism πϕ : Xϕ → X; cf. Figure 1-2. For x ∈ X, let {e1, ··· , es} ⊂ Aϕ, x the maximal complete set of orthogonal idempotents in the stalk of Aϕ at x. Then, by (∗1)-admissibility of (ϕ, ∇) and Lemma 1.6,

Dξe1 = ······ = Dξes = 0 for all ξ ∈ (T∗X)x. It follows that

19 Lemma 2.2.1. [covariantly invariant decomposition of stalks of E] For any x ∈ X, the decomposition Ex = e1Ex + ··· + esEx is invariant under ∇. I.e. ∇ξ(ekEx) ⊂ ekEx, for k = 1, ··· , s and all ξ ∈ (T∗X)x.

(k) As a consequence, the connection ∇ on Ex induces a connection ∇ on each direct summand ekEx of Ex and one has the direct-sum decomposition (1) (s) (Ex, ∇) = (e1Ex, ∇ ) ⊕ · · · ⊕ (esEx, ∇ ) .

On the other hand, the maximal complete set of orthogonal idempotents {e1, ··· , es} ⊂ Aϕ, x corresponds canonically and bijectively to the set of connected components of the germ Xϕ, x of Xϕ over x ∈ X: (1) (s) Xϕ, x = Xϕ, x t · · · t Xϕ, x . Az Az Through the built-in inclusion Aϕ, x ⊂ OX x, Ex as the fundamental OX, x-module is canonically an Aϕ, x-module as well. Since ekel = 0 for k 6= l, as an Aϕ, x-module the direct summand ekEx (k) is supported exactly on Xϕ, x, for k = 1, ··· , s. The above decomposition of (E, ∇) says then geometrically and in terms of physics terminology that the gauge field ∇ on E has no components (k) (l) that mixes ekEx on Xϕ, x and elE on Xϕ, x for some k 6= l; cf. Figure 2-2-1. Recall now the string-theory origin of D-branes: · A D-brane is where the end-points of an open string stick to. · Excitations of open strings create fields on the D-brane. · As the tension of open strings are constant, the mass of an open string — and hene fields it creates on the D-brane— is proportional to its length. Open strings with arbitrarily small length create massless fields on the brane while open strings with length bounded away from zero create massive fields on the brane.

That the germ (Xϕ, x, Ex; ∇) over any x ∈ X is decomposable in accordance with the connected- component decomposition of Xϕ, x says that ∇ must be created by open-strings of arbitrarily small length, rather than by those of length bounded away from zero. In other words, ∇ is massless. In summary:

Corollary 2.2.2.[ (∗1)-Admissible Condition implies massless of ∇] For a (∗1)-admissible map ϕ :(XAz, E; ∇) → Y , the gauge ﬁeld ∇ on the Chan-Paton sheaf E on the D-brane (or D-brane world-volume) XAz is massless from the aspect of open strings in the taget-space (or target-space-time) Y .

By Lemma 2.1.2, the same holds for (∗2)-admissible maps and (∗3)-admissible maps as well.

3 The diﬀerential dϕ of ϕ and its decomposition, the three basic C OX -modules, induced structures, and some covariant calculus At the classical level Polyakov string or its generalization, a sigma model, is a theory of harmonic maps on the mathematical side. In this section we construct all the building blocks to generalize the existing theory of harmonic maps to a theory of maps ϕ :(XAz, E; ∇) → Y , which describe D-branes. It will turn out that both the connection ∇ and the Admissible Condition (∗1) chosen are needed to build up a mathematically sound theory for such maps ϕ.

20 Y

Imφ general (φ , ∇ )

XAz Image brane Im φ in space-time Y φ with the push-forward gauge field via φ

X ( φ , ∇ ) that satisfies Admissible Condition (∗1 ) D-brane world-volume with a gauge field

Y Imφ

Figure 2-2-1. When (ϕ, ∇) is (∗1)-admissible, the gauge field ∇ on the Chan-Paton sheaf E on any small neighborhood U of x ∈ X localizes at each connected branch of ϕ(UAz) from the viewpoint of open strings in Y . In other words, ∇ is massless from the open-string aspect. In the illustration, the noncommutative space XAz is expressed as a noncommutative cloud shadowing over its underlying topology X, the connection ∇ on E over X is indicated by a gauge field on X. Both the gauge field on X and how open strings “see” it in Y are indicated by squiggling arrows . The situation for a general (ϕ, ∇) (cf. top) and a (ϕ, ∇) satisfying Admissible Condition (∗1) (cf. bottom) are compared. From the open-string aspect, in the former situation ∇ can have both massless components (which are local fields from the open-string and target-space viewpoint) and massive components (which become nonlocal fields from the open-string and target-space viewpoint), while in the latter situation ∇ has only massless components.

21 3.1 The differential dϕ of ϕ and its decomposition induced by ∇ Three kinds of differentials, dϕ, Dϕ, and ad ϕ, of a map ϕ that naturally appear in the setting are defined and their local expressions are worked out in this subsection.

The differential, the covariant differential, and the inner differential of a map ϕ

Let Az ϕ :(X, OX , E) −→ Y be a map deﬁned contravariantly by an equivalence class of gluing systems of ring-homomorphisms

] Az ϕ : OY −→ OX Az over R ⊂ C. Then, for any derivation η on OX , the correspondence Az OY −→ OX f 7−→ η(ϕ](f))

Az deﬁnes an OX -valued derivation on OY . It follows that ϕ induces a correspondence Az Az T X −→ O ⊗ ] T Y ∗ X ϕ ,OY ∗ η 7−→ dηϕ

C that is OX -linear.

C Definition 3.1.1. [differential dϕ of ϕ] The above OX -linear correspondence is denoted by dϕ and called the differential of ϕ.

Recall from Sec. 1 that when E is equipped with a connection ∇, ∇ induces a connection D Az on O := End C (E), which in turn induces a splitting X OX Az T∗XC −→ T∗X

ξ 7−→ Dξ of the exact sequence

Az Az C 0 −→ Inn (OX ) −→ T∗X −→ T∗X −→ 0 .

Deﬁnition 3.1.2. [covariant diﬀerential Dϕ of ϕ] Let

∗ Az ϕ T Y := O ⊗ ] T Y, ∗ X ϕ ,OY ∗ Az regarded as a (left) OX -module via the built-in inclusion OX ,→ OX , be the pull-push of the tangent sheaf T∗Y of X to X. The covariant differential ∞ ∗ ∗ Dϕ ∈ C (T X ⊗OX ϕ T∗Y ) Az of ϕ is the (OX -valued-derivation-on-OY )-valued 1-form on X defined by ] ∞ (Dξϕ)f := Dξ(ϕ (f)) ∈ C (End C(E)) ∞ ∞ ∞ for ξ ∈ C (T∗X) = Der (C (X)) and f ∈ C (Y ). In other words, Dϕ takes a tangent vector ∞ ∞ field on X to a C (End C(E))-valued derivation on C (Y ). In the equivalent sheaf format and ∗ notations, Dξϕ ∈ ϕ T∗Y for ξ ∈ T∗X.

22 Definition 3.1.3. [inner differential ad ϕ of ϕ] Continuing Definition 3.1.2. Represent Az Az elements in Inn (OX ) by elements m ∈ OX . The inner differential ad ϕ of ϕ is defined by the C OX -linear correspondence Az ∗ adϕ : Inn (OX ) −→ ϕ T∗Y ] m 7−→ ad mϕ := [m , ϕ (· )] .

C Since [OX , · ] = 0, ad mϕ depends only on the inner derivation m represents.

By construction,

dηϕ = Dξη + admη ϕ , Az C Az for η = ξη + mη ∈ T∗X 'T∗X ⊕ Inn (OX ) induced by D.

Local expressions of the covariant diﬀerential Dϕ of ϕ for (∗1)-admissible (ϕ, ∇)

Az Note that if ϕ :(X , E; ∇) → Y is (∗1)-admissible, then Dξϕ is a Comm (Aϕ)-valued derivation on OY for ξ ∈ T∗X. In this case, one has the following lemma and corollary that are simply re-writings of Lemma 2.1.6 and Lemma 2.1.7 respectively.

Lemma 3.1.4. [local expression of Dϕ for (∗1)-admissible (ϕ, ∇), I] Let (ϕ, ∇) be (∗1)- admissible; i.e. DξAϕ ⊂ Comm (Aϕ) for all ξ ∈ T∗X. Let U ⊂ X be a small enough open set so that ϕ(U Az) is contained in a coordinate chart of Y , with coordinates y = (y1, ··· , yn). For f ∈ C∞(Y ), recall the germwise-over-U polynomial Rf [1] in (y1, ··· , yn) with coeﬃcients in Az OU from [L-Y8: Sec. 4 & Remark/Notation 4.2.3.5] (D.13.1). Then, for ξ a vector ﬁeld on U and f ∈ C∞(Y ), and at the level of germs over U,

f (Dξϕ) f = R [1] d ] d f . y Dξ(ϕ (y )), for all multi-degree d in R [1]

d 1 d n dn Here, for a multiple degree d = (d1,, ··· , dn), di ∈ Z≥0, y := (y ) 1 ··· (y ) and d ] d d ] d y Dξ(ϕ (y )) means ‘replacing y by Dξ(ϕ (y ))’.

Corollary 3.1.5. [local expression of Dϕ for (∗1)-admissible (ϕ, ∇), II] Assume that (ϕ, ∇) is (∗1)-admissible. Let dY be the exterior diﬀerential on Y . Then

Dϕ = ϕ dY .

i i Locally explicitly, let (e )i=1, ··· , n be a local frame on Y and (e )i=1, ··· , n its dual co-frame. In Pn i terms of these dual pair of local frames, dY = i=1 e ⊗ ei under the canonical isomorphism ∗ ∗ T Y ⊗OY OY 'T Y . Then n n X i X i ] Az (Dξϕ) f = (ϕ e )(ξ) ⊗ eif = (ϕ e )(ξ) ϕ (eif) ∈ OX i=1 i=1

Az Az 1 n under the canonical isomorphism O ⊗ ] O 'O . In particular, let (y , ··· , y ) be X ϕ ,OY Y X coordinates of a local chart on Y . Then, locally, n n X i ∂ X ] i ] ∂f (Dξϕ) f = (ϕ dy )(ξ) ⊗ ∂yi f = Dξϕ (y ) · ϕ ( ∂yi ) . i=1 i=1

Az Here · is the multiplication in OX (and will be omitted later when there is no sacriﬁce to clarity).

23 Local expressions of the inner diﬀerential adϕ of ϕ for admissible inner derivations

Though for the purpose of defining an action functional for D-branes, (ϕ, ∇) is the dynamical Az field of the focus and, hence, the induced covariant derivation D on OX may look to play more roles in our discussion, mathematically results related to D like Lemma 1.6, Lemma 3.1.4, and Corollary 3.1.5 involve only the fact that, for ξ ∈ TξX, Dξ satisfies the Leibniz rule on Az OX and some additional commutativity assumption. This suggest that similar statements hold Az if one considers inner derivations on OX that are compatible to ϕ in an appropriate sense. Cf. Lemma 1.6 vs. Lemma 1.7. This motivates the setting of the current theme.

Az Deﬁnition 3.1.6. [admissible inner derivation] (Cf. Deﬁnition 2.1.1.) Let m ∈ Inn (OX ) Az be an inner derivation represented by an element of OX . Then m is called

(∗1)-admissible to ϕ if [m, Aϕ] ⊂ Comm (Aϕ); (∗2)-admissible to ϕ if [m, Comm (Aϕ)] ⊂ Comm (Aϕ); (∗3)-admissible to ϕ if [m, Aϕ] ⊂ Aϕ .

Az Note that these conditions are independent of the representative chosen in OX of the inner Az C derivation. The set of all (∗1)-admissible-to-ϕ inner derivations on OX form an OX -module, which will be denoted by Inn ϕ (OAz). Similarly, for Inn ϕ (OAz) and Inn ϕ (OAz). (∗1) X (∗2) X (∗3) X

Lemma 3.1.7. [hierarchy of admissible conditions on inner derivation] (Cf. Lemma 2.1.2.)

Inn ϕ (OAz) ⊂ Inn ϕ (OAz) ⊂ Inn ϕ (OAz) . (∗3) X (∗2) X (∗1) X

ϕ Az ϕ Az Proof. Since Aϕ ⊂ Comm (Aϕ), it is immediate that Inn (O ) ⊂ Inn (O ). For the (∗2) X (∗1) X inclusion Inn ϕ (OAz) ⊂ Inn ϕ (OAz), let m ∈ Inn ϕ (OAz) represented by an element in OAz, (∗3) X (∗2) X (∗3) X X 0 00 m ∈ Comm (Aϕ), and m ∈ Aϕ. Then,

[[m, m0], m00] = [[m, m00], m0] + [m, [m0, m00]] from either the Jacobi identity of Lie bracket or the Leibniz rule for a derivation. The ﬁrst term 00 0 0 00 vanishes since [m, m ] ∈ Aϕ and [Aϕ, m ] = 0. The second term also vanishes since [m , m ] = 0 00 0. Since m ∈ Comm (Aϕ) and m ∈ Aϕ are arbitrary, this shows that [m, Comm (Aϕ)] ⊂ Comm (Aϕ). This proves the lemma.

Remark 3.1.8. [Lemma 2.1.2 vs. Lemma 3.1.7 ] Note that in Lemma 2.1.2, the implication (∗3) ⇒ (∗2) uses D-parallel transport properties implied by (∗3), which is an analytic technique. Indeed, the proof of Lemma 3.1.7 applies there. Which says that in both situations, the hierarchy is an algebraic consequence.

In terms of this setting and with arguments parallel to the proof of Lemma 3.1.4 and Corol- lary 3.1.5, one has the following lemma and corollary:

24 Lemma 3.1.9. [local expression of ad ϕ for Inn ϕ (OAz), I] (Cf. Lemma 3.1.4.) (∗1) X Let ϕ :(XAz, E) → Y be a map and m ∈ Inn ϕ (OAz) represented by an element of OAz i.e. (∗1) X X Az [m, Aϕ] ⊂ Comm (Aϕ). Let U ⊂ X be a small enough open set so that ϕ(U ) is contained in a coordinate chart of Y , with coordinates y = (y1, ··· , yn). For f ∈ C∞(Y ), recall the f 1 n Az germwise-over-U polynomial R [1] in (y , ··· , y ) with coeﬃcients in OU from [L-Y8: Sec. 4 & Remark/Notation 4.2.3.5] (D.13.1). Then, at the level of germs over U,

f (ad mϕ) f = R [1] d ] d f . y adm(ϕ (y )), for all multi-degree d in R [1]

d 1 d n dn Here, for a multiple degree d = (d1,, ··· , dn), di ∈ Z≥0, y := (y ) 1 ··· (y ) and d ] d d ] d y ad m(ϕ (y )) means ‘replacing y by ad m(ϕ (y ))’.

Proof. Recall the proof of Lemma 3.1.4 through the proof of Lemma 2.1.6. . With the same setup and notations there, note that for m ∈ Inn ϕ (OAz), (∗1) X

] [m, ϕ (1(k))] = 0 , for k = 1, ··· , s by Lemma 1.7. Since [m, OX ] = 0 holds automatically, the lemma follows.

Corollary 3.1.10. [local expression of ad ϕ for Inn ϕ (OAz), II] (Cf. Corollary 3.1.5.) (∗1) X Continuing the setting of Lemma 3.1.9. Recall that m ∈ Inn ϕ (OAz) is represented by an (∗1) X Az 1 n element of OX . Let (y , ··· , y ) be coordinates of a local chart on Y . Then, locally,

n n X ] i ] ∂f X ] i ] ∂f (ad mϕ) f = ad mϕ (y ) · ϕ ( ∂yi ) = [m, ϕ (y )] · ϕ ( ∂yi ) , i=1 i=1

Az where · is the multiplication in OX (and will be omitted later when there is no sacriﬁce to clarity). In other words, n X ] i ∂ ad mϕ = [m, ϕ (y )] ⊗ ∂yi . i=1

Proof. The related last part of the proof of Corollary 3.1.5 through the proof of Lemma 2.1.7 works in verbitum with Dξ replaced by ad m = [m, · ]. This is simply a re-writing of Lemma 3.1.9 above.

Remark 3.1.11. [comparison with differential of ordinary map] As a comparison, let u : X → Y ∗ be a map between manifolds. Then, du defines a bundle map T∗X → u T∗Y that satisfies

] du(ξ) f = u∗(ξ) f = ξ(u ◦ f) = ξ(u (f))

∞ µ for any ξ ∈ T∗X and f ∈ C (Y ). In terms of local coordinates x= (x )µ=1, ··· , m on X and i y= (y )i=1, ··· , n on Y , n ∂ X dyi ∂f u∗ ∂xµ = dxµ ∂yi (u(x)) . i=1

25 The above notion of covariant differential Dϕ of ϕ is exactly the generalization of the ordinary differential du of u, taking into account the fact that ϕ is now only defined contravariantly ] Az Az through ϕ and that the noncommutative structure sheaf OX of X is no longer naturally trivial as an OX -module but, rather, is endowed with a natural induced connection D from ∇. ϕ Az Furthermore, when (ϕ, ∇) is (∗1)-admissible or when considering only Inn (O ), both the (∗1) X covariant differential Dϕ and the inner differential ad ϕ takes the same form as the chain rule in the commutative case.

C 3.2 The three basic OX -modules relevant to Dϕ, with induced structures C Underlying the notion of covariant diﬀerential Dϕ of ϕ are two basic OX -modules ∗ Az Az the pull-back tangent sheaf ϕ T Y := O ⊗ ] T Y , a left O -module but now regarded · ∗ X ϕ ,OY ∗ Y C C Az as a (left) OX -module through the built-in inclusion OX ,→ OX , and

C ∗ ∗ · the OX -module T X ⊗OX ϕ T∗Y , where Dϕ lives. C We study them in this subsection after taking a look at another basic but simpler OX -module ∗ Az T X ⊗OXOX . They play fundamental roles in our variational problem.

C Az Remark 3.2.0.1. [structures on the OX -algebra OX ] Recall the connection D on the noncom- Az Az mutative structure sheaf O := End C (E) of X induced by the connection ∇ on E. The X OX C Az Az multiplication · in the OX -algebra structure of OX deﬁnes a nonsymmetric OX -valued inner Az C product on OX that is OX -bilinear. This inner product is D-invariant in the sense that D(m1 · m2) = (Dm1) · m2 + m1 · Dm2 ,

1 2 Az for m , m ∈ OX . Together with the built-in trace map Az C Tr ;: OX −→ OX ,

C C Az as an OX -module-homomorphism, one has a symmetric OX -valued inner product on OX deﬁned by the assignment (m1, m2) 7−→ Tr (m1 · m2) =: Tr (m1m2) ,

1 2 Az C for m , m ∈ OX . This inner product is OX -bilinear; and is covariantly constant over X in the sense that 1 2 1 2 1 2 dX (Tr (m m )) = Tr ((Dm )m ) + Tr (m Dm ) , where dX is the exterior diﬀerential on X.

Az ∗ Az 3.2.1 The OX -valued cotangent sheaf T X ⊗OXOX of X, and beyond Let X be endowed with a (Riemannian or Lorentzian) metric h and ∇h be the Levi-Civita ∗ connection on T∗X induced by h. The corresponding inner product on T∗X, its dual T X, and ∗ their tensor products will be denoted h · , · ih. The induced connection on the dual T X and on ∗ h the tensor product of copies of T∗X and copies of T X will be denoted also by ∇ . The deﬁning features of ∇h are h h h h ∇ hξ1, ξ2ih = h∇ ξ1, ξ2ih + hξ1, ∇ ξ2ih (h be ∇ -covariantly constant) , h h h Tor h (ξ , ξ ) := ∇ ξ − ∇ ξ − [ξ , ξ ] = 0 (∇ be torsionless) , ∇ 1 2 ξ1 2 ξ2 1 1 2 for all ξ1, ξ2 ∈ T∗X.

26 Az ∗ Az The OX -valued cotangent sheaf T X ⊗OXOX of X h ∗ Az The connection ∇ on T X and the connection D on OX together induce a connection (h,D) h ∇ := ∇ ⊗ Id Az + Id T ∗X ⊗ D OX ∗ Az Az on T X ⊗OX OX . The inner product h · , · ih on T∗X and the inner product · on OX together Az C ∗ Az induce an OX -valued, OX -bilinear (nonsymmetric) inner product on T X ⊗OXOX by extending C OX -bilinearly 1 1 2 2 1 2 1 2 hω ⊗ m , ω ⊗ m ih := hω , ω ih · (mT mT ) , 1 2 ∗ 1 2 Az Az C where ω , ω ∈ T X and m , m ∈ OX . The trace map Tr : OX → OX turns this further to C C ∗ Az an OX -valued, OX -bilinear (symmetric) inner product on T X ⊗OXOX by the post composition with the above inner product 0 0 0 ( · , · ) 7−→ Tr (h · , · ih) =: Tr h · , · ih 0 ∗ Az for ·, · ∈ T X ⊗OXOX . By construction, both inner products are covariantly constant with respect to ∇(h,D) and they satisfy the Leibniz rules 0 (h,D) 0 (h,D) 0 Dh · , · ih = h ∇ · , · ih + h · , ∇ · ig , 0 0 (h,D) 0 (h,D) 0 dTr h · , · ih = Tr (Dh · , · ih) = Tr h ∇ · , · ih + Tr h · , ∇ · ih 0 ∗ Az for ·, · ∈ T X ⊗OXOX .

V• ∗ Az Az The sheaf ( T X) ⊗OXOX of OX -valued differential forms on X V• ∗ Az Az The setting in the previous theme generalizes to the sheaf ( T X) ⊗OX OX of OX -valued differential forms on X, with the 1-forms ω1, ω2 on X there replaced by general differential forms α1, α2 on X. We will use the same notations h (h,D) ∇ , ∇ , h · , · ih , Tr h · , · ih

V• ∗ V• ∗ Az Az C to denote the connection on T X, the connection on ( T X)⊗OXOX , the OX -valued, OX - V• ∗ Az C C bilinear (nonsymmetric) inner product on ( T X) ⊗OXOX , and the OX -valued, OX -bilinear V• ∗ Az (symmetric) inner product on ( T X) ⊗OXOX respectively. They satisfy the same Leibniz ∗ Az rule as in the case of T X ⊗OXOX

∗ 3.2.2 The pull-back tangent sheaf ϕ T∗Y

C ∗ This is the main character among the three basic OX -modules and is slightly subtler than u T∗Y ∗ Az in the commutative case (cf. Remark 3.1.11) or T X ⊗OXOX in Sec. 3.2.1.

The induced connection and the induced partially-deﬁned inner products

Let Y be endowed with a (Riemannian or Lorentzian) metric g and ∇g be the Levi-Civita ∗ connection on T∗Y induced by g. The corresponding inner product on T∗Y or its dual T Y ∗ will be denoted h · , · ig. The induced connection on the dual T Y and on the tensor product of ∗ g g copies of T∗Y and copies of T Y will be denoted also by ∇ . The deﬁning features of ∇ are g g g g ∇ hv1, v2ig = h∇ v1, v2ig + hv1, ∇ v2ig (g be ∇ -covariantly constant) , g g g g Tor ∇ (v1, v2) := ∇v1 v2 − ∇v2 v1 − [v1, v2] = 0 (∇ be torsionless) , for all v1, v2 ∈ T∗Y .

27 g ∗ Lemma 3.2.2.1.[ (D, ∇ )-induced connection on ϕ T∗Y for (∗)1-admissible (ϕ, ∇)] As- Az g sume that (ϕ, ∇) is (∗1)-admissible. Then, the connection D on OX and the connection ∇ on (ϕ,g) ∗ T∗Y together induce a connection ∇ on ϕ T∗Y , locally of the form

n (ϕ,g) X ] i g ∇ = D ⊗ Id T∗Y + Id OAz · Dϕ (y ) ⊗ ∇ ∂ . X i i=1 ∂y

∗ Proof. Our construction of an induced connection on ϕ T∗Y is local in nature. As long as a construction is independent of coordinates chosen, the local construction glues to a global construction. Let U ⊂ X be a small enough open set so that ϕ(U Az) is contained in a coordinate chart of Y , with coordinate y := (y1, ··· , yn). Then the local expression

n (ϕ,g);y X ] i g ∇ := D ⊗ Id T∗Y + Id OAz · Dϕ (y ) ⊗ ∇ ∂ . X i i=1 ∂y

∗ in the Statement deﬁnes a connection on ϕ T∗Y |U . We only need to show that it is independent of the coordinate (y1, ··· , yn) chosen. 1 n Let z := (z , ··· , z ) be another coordinate on the chart. Then, for (ϕ, ∇)(∗1)-admissible, Dϕ](yi) =: (Dϕ)yi has a local expression in terms of z

n i i ] i X ] j ∂y (Dϕ)y := D(ϕ (y )) = D(ϕ (z )) ⊗ ∂zj j=1 by Corollary 3.1.5, for i = 1, ··· , n. It follows that

n (ϕ,g);z X ] j g ∇ := D ⊗ Id T∗Y + Id OAz · Dϕ (z ) ⊗ ∇ ∂ X j j=1 ∂z

n n i X ] j X ∂y g = D ⊗ Id T∗Y + Id OAz · Dϕ (z ) ⊗ j ∇ ∂ X ∂z i j=1 i=1 ∂y n X ] i g = D ⊗ Id T∗Y + Id OAz · Dϕ (y ) ⊗ ∇ ∂ X i i=1 ∂y =: ∇(ϕ,g);y .

This completes the proof.

∗ Consider next the induced inner products on ϕ T∗Y . Completely naturally, one may attemp Az Az to combine the multiplication in OX and the inner product h · , · ig on T∗Y to an OX -valued, C ∗ C OX -bilinear (nonsymmetric) inner product on ϕ T∗Y by extending OX -bilinearly

1 2 1 2 1 2 ] hm ⊗ v1 , m ⊗ v2ig := m m ⊗ hv1 , v2ig = m m ϕ (hv1 , v2ig) ,

1 2 Az where m , m ∈ Comm (Aϕ) ⊂ OX and v1, v2 ∈ T∗Y and the last equality follows from the canonical isomorphism Az Az O ⊗ ] O 'O . X ϕ , OY Y X However, for this to be well-deﬁned, it is required that

1 2 1 ] 2 ] hm ⊗ f1v1 , m ⊗ f2v2ig = hm ϕ (f1) ⊗ v1 , m ϕ (f2) ⊗ v2ig ,

28 i.e. 1 2 ] 1 ] 2 ] ] m m ϕ f1f2hv1, v2ig = m ϕ (f1)m ϕ (f2)ϕ (hv1, v2ig) , 1 2 Az for all m , m ∈ OX , v1, v2 ∈ T∗Y , and f1, f2 ∈ OY . Which holds if and only if

2 m ∈ Comm (Aϕ) .

Az C What happens if one brings in the trace map Tr : OX → OX ? In this case,

1 2 1 2 ] ] 1 2 ] Tr hm ⊗ f1v1 , m ⊗ f2v2ig = Tr m m ϕ (f1f2hv1, v2ig) = Tr ϕ (f1)m m ϕ (f2hv1, v2ig) by the cyclic-invariance property of Tr while

1 ] 2 ] 1 ] 2 ] Tr hm ϕ (f1) ⊗ v1 , m ϕ (f2) ⊗ v2ig = Tr m ϕ (f1)m ϕ (f2hv1, v2ig) .

The two equal if

either m1 ∈ Comm (Aϕ) or m2 ∈ Comm (Aϕ).

∗ Definition 3.2.2.2. [partially-defined inner product h · , · ig on ϕ T∗Y ] The multiplica- Az Az tion in OX and the inner product h · , · ig on T∗Y together induce a partially defined, OX -valued, C ∗ C OX -bilinear (nonsymmetric) inner product on ϕ T∗Y by extending OX -bilinearly

1 2 1 2 1 2 ] hm ⊗ v1 , m ⊗ v2ig := m m ⊗ hv1 , v2ig = m m ϕ (hv1 , v2ig) ,

1 Az 2 Az where m ∈ OX , m ∈ Comm (Aϕ) ⊂ OX and v1, v2 ∈ T∗Y and the last equality follows from Az Az the canonical isomorphism O ⊗ ] O 'O . X ϕ , OY Y X

∗ Definition 3.2.2.3. [partially-defined inner product Tr h · , · ig on ϕ T∗Y ] The multipli- Az Az C cation in OX , the inner product h · , · ig on T∗Y , and the tarce map Tr : OX → OX together C C ∗ induce a partially defined, OX -valued, OX -bilinear (symmetric) inner product on ϕ T∗Y by ex- C tending OX -bilinearly

1 2 1 2 1 2 ] Tr hm ⊗ v1 , m ⊗ v2ig := Tr m m ⊗ hv1 , v2ig = Tr m m ϕ (hv1 , v2ig) ,

1 2 where either m or m is in Comm (Aϕ), v1, v2 ∈ T∗Y , and the last equality follows from the Az Az canonical isomorphism O ⊗ ] O 'O . X ϕ , OY Y X

By construction, both inner products, when deﬁned, are covariantly constant with respect to ∇(ϕ,g) and one has the Leibniz rules

0 (ϕ,g) 0 (ϕ,g) 0 Dh – , – ig = h ∇ – , – ig + h – , ∇ – ig , 0 0 (ϕ,g) 0 (ϕ,g) 0 dX Tr h – , – ig = Tr (Dh – , – ig) = Tr h ∇ – , – ig + Tr h – , ∇ – ig ,

00 000 00 000 whenever the h – , – ig or Tr h – , – ig involved are deﬁned. The followng lemma is an immediate consequence of Corollary 3.1.5:

29 Lemma 3.2.2.4. [sample list of deﬁned inner products for admissible (ϕ, ∇)] (1) For (ϕ, ∇)(∗1)-admissible, the following list of inner products

h – ,Dξϕig , Tr h – ,Dξϕig , Tr hDξϕ , – ig are deﬁned for ξ ∈ T∗X. (2) For (ϕ, ∇)(∗2)-admissible, the following additional list of inner products

h – , ∇(ϕ,g) · · · ∇(ϕ,g)D ϕi , Tr h – , ∇(ϕ,g) · · · ∇(ϕ,g)D ϕi , Tr h∇(ϕ,g) · · · ∇(ϕ,g)D ϕ , , – i ξ1 ξk ξ g ξ1 ξk ξ g ξ1 ξk ξ g are also deﬁned for ξ, ξ1, ··· , ξk ∈ T∗X, k ∈ Z≥0.

(ϕ,g) ∗ The symmetry properties of the curvature tensor of ∇ on ϕ T∗Y

Az Let ϕ :(X , E; ∇) → Y be a (∗2)-admissible map. Let F∇(ϕ,g) be the curvature tensor of the (ϕ,g) ∗ ∗ induced connection ∇ on ϕ T∗Y — the End C (ϕ T∗Y )-valued 2-form on X deﬁned by OX

(ϕ,g) (ϕ,g) (ϕ,g) (ϕ,g) (ϕ,g) ∗ F (ϕ,g) (ξ1, ξ2) s := ∇ ∇ − ∇ ∇ − ∇ s ∈ ϕ T∗Y ∇ ξ1 ξ2 ξ2 ξ1 [ξ1,ξ2]

∗ for ξ1, ξ2 ∈ T∗X and s ∈ ϕ T∗Y . (This is OX -linear in ξ1, ξ2, and s and, hence, a tensor on X). By construction,

F∇(ϕ,g) (ξ1, ξ2) = − F∇(ϕ,g) (ξ2, ξ1) for ξ1, ξ2 ∈ T∗X. From Lemma 3.2.2.4, the inner products

h∇(ϕ,g)D ϕ , D ϕi , h∇(ϕ,g)D , ∇(ϕ,g)D ϕi , ξ1 ξ3 ξ4 g ξ1 ξ3 ξ2 ξ4 g (ϕ,g) (ϕ,g) h∇ ∇ D ϕ , D ϕi , hF (ϕ,g) (ξ , ξ )D ϕ , D ϕi ξ1 ξ2 ξ3 ξ4 g ∇ 1 2 ξ3 ξ4 g are all deﬁned.

(ϕ,g) ∗ Lemma 3.2.2.5. [symmetry property of the curvature tensor of ∇ on ϕ T∗Y ] For a (∗2)-admissible pair (ϕ, ∇),

hF∇(ϕ,g) (ξ1, ξ2)Dξ3 ϕ , Dξ4 ϕig

= − hDξ3 ϕ , F∇(ϕ,g) (ξ1, ξ2)Dξ4 ϕig + [F∇(ξ1, ξ2) , hDξ3 ϕ , Dξ4 ϕig] .

And, hence,

Tr hF∇(ϕ,g) (ξ1, ξ2)Dξ3 ϕ , Dξ4 ϕig = − Tr hDξ3 ϕ , F∇(ϕ,g) (ξ1, ξ2)Dξ4 ϕig

= − Tr hF∇(ϕ,g) (ξ1, ξ2)Dξ4 ϕ , Dξ3 ϕig = Tr hF∇(ϕ,g) (ξ2, ξ1)Dξ4 ϕ , Dξ3 ϕig .

Proof. This is a consequence of the Leibniz rule

0 (ϕ,g) 0 (ϕ,g) 0 Dh – , – ig = h ∇ – , – ig + h – , ∇ – ig ,

30 0 ∗ for –, – ∈ ϕ T∗Y , when all inner products involved are deﬁned. In detail,

hF∇(ϕ,g) (ξ1, ξ2)Dξ3 ϕ , Dξ4 ϕig (ϕ,g) (ϕ,g) (ϕ,g) (ϕ,g) (ϕ,g) = h(∇ ∇ − ∇ ∇ − ∇ )D ϕ , D ϕig ξ1 ξ2 ξ2 ξ1 [ξ1,ξ2] ξ3 ξ4 = D h∇(ϕ,g)D ϕ , D ϕi − h∇(ϕ,g)D ϕ , ∇(ϕ,g)D ϕi ξ1 ξ2 ξ3 ξ4 g ξ2 ξ3 ξ1 ξ4 g − D h∇(ϕ,g)D ϕ , D ϕi + h∇(ϕ,g)D ϕ , ∇(ϕ,g)D ϕi ξ2 ξ1 ξ3 ξ4 g ξ1 ξ3 ξ2 ξ4 g (ϕ,g) − D hD ϕ , D ϕig + hD ϕ , ∇ D ϕig [ξ1,ξ2] ξ3 ξ4 ξ3 [ξ1,ξ2] ξ4 (ϕ,g) (ϕ,g) (ϕ,g) (ϕ,g) (ϕ,g) = hD ϕ , (∇ ∇ − ∇ ∇ + ∇ )D ϕig ξ3 ξ2 ξ1 ξ1 ξ2 [ξ1,ξ2] ξ4

+ (Dξ1 Dξ2 − Dξ2 Dξ1 − D[ξ1,ξ2])hDξ3 ϕ , Dξ4 ϕig after repeatedly applying the Leibniz rule,

= − hDξ3 ϕ , F∇(ϕ,g) (ξ1, ξ2)Dξ4 ϕig + FD(ξ1, ξ2)hDξ3 ϕ , Dξ4 ϕig .

Az Note that FD = [F∇, · ] on OX . The lemma follows.

Covariant diﬀerentiation and evaluation

(ϕ,g) ∗ Lemma 3.2.2.6.[ Dξ(m⊗vf) vs. ∇ξ (m⊗v) f ] Let ξ ∈ T∗X, f ∈ OY , and m⊗v ∈ ϕ T∗Y . Then,

n (ϕ,g) X ] i ∂ g Dξ(m ⊗ vf) = ∇ξ (m ⊗ v) f + m Dξϕ (y ) ⊗ i (vf) − ∇ ∂ v f . ∂y i i=1 ∂y

Proof. ] X ] i ∂ Dξ(m ⊗ vf) = Dξ mϕ (vf) = Dξm ⊗ vf + Dξϕ (y ) ⊗ ∂yi (vf) i while (ϕ,g) X ] i g ∇ξ (m ⊗ v) f = Dξm ⊗ v + Dξϕ (y ) ⊗ ∇ ∂ v f . i i ∂y The lemma follows.

C ∗ ∗ 3.2.3 The OX -module T X ⊗OX ϕ T∗Y , where Dϕ lives

Assume that (ϕ, ∇) is (∗1)-admissible and recall the metric h on X and the metric g on Y . Then the construction in this subsubsection is a combination of the constructions in Sec. 3.2.1 and Sec. 3.2.2. h ∗ Az g The connection ∇ on T X, the connection D on OX , and the connection ∇ on T∗Y (h,ϕ,g) ∗ ∗ together induce a connection ∇ on T X ⊗OX ϕ T∗Y , locally of the form

n (h,ϕ,g) h T X ] i g Az ∗ ∗ Az ∇ = ∇ ⊗ Id O ⊗ Id T∗Y + Id T X ⊗ D ⊗ Id T∗Y + Id T X ⊗ Id O · Dϕ (y ) ⊗ ∇ ∂ . X X i i=1 ∂y

31 Lemma 3.2.2.1 implies that this is independent of the coordinate (y1, ··· , yn) on coordinate charts chosen and hence well-defined ∗ Az The inner product h · , · ih on T X, the multiplication in OX , and the inner product h · , · ig on Az C T∗Y together induce a partially defined, OX -valued, OX -bilinear (nonsymmetric) inner product ∗ ∗ C on T X ⊗OX ϕT T∗Y by extending OX -bilinearly 1 1 2 2 hω ⊗ mT ⊗ v1 , ω m ⊗ v2i(h,g) 1 2 1 2 1 2 1 2 ] := hω , ω ih · mT mT ⊗ hv1 , v2ig = hω , ω ih m m ϕ (hv1 , v2ig) , 1 2 ∗ 1 Az 2 Az where ω , ω ∈ T X, m ∈ OX , m ∈ Comm (Aϕ) ⊂ OX , v1, v2 ∈ T∗Y and the last equality Az Az Az follows from the canonical isomorphism O ⊗ ] O 'O . The trace map Tr : O → O C X ϕ , OY Y X X X C C ∗ gives another partially defined, OX -valued, OX -bilinear (symmetric) inner product on T X ⊗OX ∗ C ϕ T∗Y by extending OX -bilinearly 1 1 2 2 Tr hω ⊗ mT ⊗ v1 , ω m ⊗ v2i(h,g) 1 2 1 2 1 2 1 2 ] := Tr hω , ω ih · mT mT ⊗ hv1 , v2ig = Tr hω , ω ih m m ϕ (hv1 , v2ig) , 1 2 ∗ 1 2 where ω , ω ∈ T X, either m or m is in Comm (Aϕ), v1, v2 ∈ T∗Y . By construction, both inner products, when defined, are covariantly constant with respect to ∇(h,ϕ,g) and one has the Leibniz rules

0 (h,ϕ,g) 0 (h,ϕ,g) 0 Dh ∼ , ∼ i(h,g) = h ∇ ∼ , ∼ i(h,g) + h ∼ , ∇ ∼ i(h,g) , 0 0 dTr h ∼ , ∼ i(h,g) = Tr (Dh ∼ , ∼ i(h,g)) (h,ϕ,g) 0 (h,ϕ,g) 0 = Tr h ∇ ∼ , ∼ i(h,g) + Tr h ∼ , ∇ ∼ i(h,g) , 00 000 00 000 whenever the h ∼ , ∼ i(h,g) or Tr h ∼ , ∼ i(h,g) involved are deﬁned.

With all the preparations in Sec. 1–Sec. 3, we are ﬁnally ready to construct and study the standard action for D-branes along our line of pursuit.

4 The standard action for D-branes

We introduce in this section the standard action, which is to D-branes as the (Brink-Di Vecchia- Howe/Deser-Zumino/)Polyakov action is to fundamental strings. Abstractly, it is an enhanced non-Abelian gauged sigma model based on maps ϕ :(XAz, E; ∇) → Y .

∞ The gauge-symmetry group C (Aut C(E))

Let Aut C(E) be the automorphism bundle of the complex vector bundle E (of rank r) over E. Aut C(E) ⊂ End C(E) canonically as the bundle of invertible endomorphisms; it is a principal GLr(C)-bundle over X. The set ∞ Ggauge := C (Aut C(E)) of smooth sections of Aut C(E) forms an inﬁnite-dimensional Lie group and acts on the space of pairs (ϕ, ∇) as a gauge-symmetry group:

0 g0 g0 g0 g ∈ Ggauge :(ϕ, ∇ = d + A) 7−→ ( ϕ , ∇ = d + A) := g0ϕg0−1 , d − (dg0)g0−1 + g0Ag0−1 .

The induced action of Ggauge on other basic objects are listed in the lemma below:

32 Lemma 4.1. [induced action of Ggauge on other basic objects] (All the Ggauge -actions are denoted by a representation ρgauge of Ggauge , if in need.)

Az 0 0 0−1 Az (01) on OX : ρgauge (g )(m) = g mg for m ∈ OX . g0 g0 (02) on induced connections : D = d + [A, · ] 7−→ D := d + [ A, · ].

∗ C Az 0 0 0−1 0 0−1 (1) on T X ⊗ C O : ρgauge (g )(ω ⊗ m) = ω ⊗ (g mg ) =: g (ω ⊗ m)g . OX X ∗ (2) for ϕ T∗Y :

∗ g0 ∗ ϕ T∗Y −→ ϕ T∗Y m ⊗ v 7−→ (g0mg0−1) ⊗ v =: g0(m ⊗ v)g0−1 . .

∗ ∗ (3) for T X ⊗OXϕ T∗Y :

∗ ∗ ∗ g0 ∗ T X ⊗OXϕ T∗Y −→ T X ⊗OX ϕ T∗Y ω ⊗ m ⊗ v 7−→ ω ⊗ (g0mg0−1) ⊗ v =: g0(ω ⊗ m ⊗ v)g0−1 . .

(4) for covariant diﬀerential : Dϕ 7−→ g0D gϕ0 = g0Dϕ g0−1.

(5) for pull-push :(gϕ0 )α = g0 ϕα g0−1.

Proof. The proof is elementary. Let us demonstrate Item (2) as an example. ∗ Az For m ⊗ v ∈ ϕ T Y := O ⊗ ] T Y , ∗ X ϕ ,OY ∗

0 0 0 0−1 ρgauge (g )(m ⊗ v) = ρgauge (g )(m) ⊗ v = (g mg ) ⊗ v since Ggauge acts on T∗Y trivially (i.e. by by the identity map Id Y ). The only issue is: Where does (g0mg0−1) ⊗ v now live? To answer this, note that, for f ∈ C∞(Y ), on one hand

0 0 ] 0 ] 0−1 ρgauge (g )(m ⊗ fv) = ρgauge (g )(mϕ (f) ⊗ v) = (g mϕ (f)g ) ⊗ v , while on the other hand

0 0 0−1 ρgauge (g )(m ⊗ fv) = (g mg ) ⊗ fv ,

It follows that

(g0mg0−1) ⊗ fv 0 ] = g0mϕ](f)g0−1 ⊗ v = g0mg0−1 · g0ϕ](f)g0−1 ⊗ v = g0mg0−1 · gϕ (f) ⊗ v .

0 0−1 g0 ∗ Which says that our section (g mg ) ⊗ v now lives in ϕ T∗Y .

33 The standard action for D-branes

Fix a (dilaton field ρ, metric h) on the underlying smooth manifold X (of dimension m) of the Azumaya/matrix manifold with a fundamental module (XAz, E). Fix a background (dilaton field Φ, metric g , B-field B, Ramond-Ramond field C) on the target space(-time) Y (of dimension n).2 Here, h and g can be either Riemannian or Lorentzian.

Definition 4.2. [standard action = enhanced non-Abelian gauged sigma model] With the given background fields (ρ, h) on X and (Φ, g, B, C) on Y , the standard action (ρ,h;Φ,g,B,C) Sstandard (ϕ, ∇) for (∗1)-admissible pairs (ϕ, ∇) is defined to be the functional

S(ρ,h;Φ,g,B,C)(ϕ, ∇) := S(ρ,h;Φ,g,B,C)(ϕ, ∇) standard nAGSM+

:= S(ρ,h;Φ,g) (ϕ, ∇) + S(h;B) (ϕ, ∇) + S(C,B) (ϕ, ∇) map:kinetic+ gauge /YM CS/WZ with the enhanced kinetic term for maps Z Z S(ρ,h;Φ,g) (ϕ, ∇) := 1 T Re Tr hDϕ , Dϕi vol + Re Tr hdρ, ϕdΦi vol , map:kinetic+ 2 m−1 (h,g) h h h X X the gauge/Yang-Mills term Z S(h;B) (ϕ, ∇) := − 1 Re Tr k2πα0F + ϕBk2 vol gauge /YM 2 ∇ h h X and the Chern-Simons/Wess-Zumino term (if (ϕ, ∇) is furthermore (∗2)-admissible, cf. Remark 2.1.13)

formally Z 0 q (C,B) 2πα F∇+ϕ B ˆ Az ˆ SCS/WZ (ϕ, ∇) = Tm−1 Re Tr ϕ C ∧ e ∧ A(X )/A(NXAz /Y ) . X (m) Here,

(0) On Re Note that while eigenvalues of ϕ](f) are all real ([L-Y5: Sec. 3.1] (D(11.1))) ] for f ∈ OY , the eigenvalues of Dξϕ (f), ξ ∈ T∗X, may not be so under the (∗1)-Admissible (ρ,h;Φ,g,B,C) Condition. Thus, Tr (······ ) in the integrand of terms in Sstandard (ϕ, ∇) are in general C-valued and we take the real part Re Tr (······ ) of it. (1) The enhanced kinetic term for maps The ﬁrst summand of S(ρ,h;Φ,g) deﬁnes the map:kinetic+ kinetic energy Z ∇ (h;g) 1 E (ϕ) := Smap:kinetic(ϕ, ∇) := 2 Tm−1 Re Tr hDϕ , Dϕi(h,g) vol h X of the map ϕ for a given ∇ and, hence, will be called the kinetic term for maps in the (ρ,h;Φ,g,B,C) standard action Sstandard (ϕ, ∇). When the metric g on Y is Lorentzian, then depending on the convention of its signature (−, +, ··· +) vs. (+, −, · · · −), one needs to add an overall minus − vs. plus + sign. In this note, for simplicity of presentation, we choose h and g to be both Riemannian (i.e. for Euclideanized/Wick-rotated D-branes and space- time).

2For mathematicians, ρ is a smooth function on X, Φ is a smooth function on Y , B is a 2-form and C is a general differential form on Y . Such background fields (Φ, g, B, C) on Y are created by massless excitations of closed superstrings on Y . The notations for these particular fields are almost already carved into stone in string-theory literature. Which we adopt here.

34 Az · The world-volume X of D-brane is m-dimensional; Tm−1 is the tension of (m − 1)- dimensional D-branes. Like the tension of the fundamental string, it is a ﬁxed constant of nature.

The second summand of S(ρ,h;Φ,g) · map:kinetic+ Z (ρ,h;Φ) Sdilaton (ϕ) := Re Tr hdρ, ϕ dΦih vol h , X

(ρ,h;Φ,g,B,C) will be called the dilaton term of the standard action Sstandard (ϕ, ∇).

Note that if let U be small enough and ﬁx a local trivialization of E|U . and assume that ∇ = d + A with respect to this local trivialization. Then D = d + [A, · ] and, over U with an orthonormal frame (eµ)µ,

X ] Tr hdρ, ϕ dΦih = Tr dρ(eµ)Deµ ϕ (Φ) µ X ] ] X ] = Tr dρ(eµ) eµϕ (Φ) + [A(eµ), ϕ (Φ)] = Tr dρ(eµ) eµϕ (Φ) . µ µ

Thus, while ϕ dΦ depends on the connection ∇, the integrand Tr hdρ, ϕ dΦih vol h does not. This justiﬁes the dilaton term as a functional of ϕ alone.

In contrast, over U with the above setting, Tr hDϕ, Dϕi(h,g) contains summand

X X ] i ] j ] Tr [A(eµ), ϕ (y )] [A(eµ), ϕ (y )] ϕ (gij) , µ i,j

which does not vanish in general. Thus, Tr hDϕ, Dϕi(h,g) does depend on the pair (ϕ, ∇).

(h;B) 0 0 (2) The gauge/Yang-Mills term Sgauge /YM(ϕ, ∇) α is the Regge slope; 2πα is the inverse to the tension of a fundamental string.

0 Az · F∇ is the curvature tensor of the connection ∇ on E; 2πα F∇ + ϕ B is an OX -valued 2-tensor on X; and

0 2 0 0 k2πα F∇ + ϕ Bkh := h2πα F∇ + ϕ B, 2πα F∇ + ϕ Bih from Sec. 3.2.1. Up to the shift by ϕB, this is a norm-squared of the field strength of (h;B) the gauge field, and hence the name Yang-Mills term. Note that in Sgauge /YM(ϕ, ∇), ∇ couples with ϕ only through the background B-field B. When B = 0, this is simply a (h) functional Sgauge /YM(∇) of ∇ alone.

(h;B) · In the current bosonic case, the Yang-Mills functional for the gauge term Sgauge /YM(ϕ, ∇) can be replaced any other standard action functional, e.g. Chern-Simons functional, in gauge theories.

(C,B) (3) The Chern-Simons/Wess-Zumino term SCS/WZ (ϕ, ∇) The coupling constant of Ramond-Ramond ﬁelds with D-branes is taken to be equal to the D-brane tension Tm−1. This choice is adopted from the situation of the Dirac-Born-Infeld action. However, in (C,B) the current bosonic case, one may take a diﬀerent constant. As given here, SCS/WZ (ϕ, ∇) q ˆ Az ˆ is only formal; the anomaly factor A(X )/A(NXAz/Y ) in its integrand remains to be understood in the current situation.

35 Az · The wedge product of OX -valued differential forms was discussed in [L-Y8: Sec.6.1] (D(13.1)). An Ansatz was proposed there in accordance with the notion of ‘symmetrized Az determinant’ for an OX -valued 2-tensor on X in the construction of the non-Abelian Dirac-Born-Infeld action ibidem. Here, we no longer have a direct guide from the con- (h;g) struction of the kinetic term Smap:kinetic(ϕ, ∇) for maps as to how to define such wedge products. However, just like Polyakov string should be thought of as being equivalent to Nambu-Goto string (at least at the classical level) but technically more robust, here we would think that ‘standard D-branes’ should be equivalent to ‘Dirac-Born-Infeld D-branes’ (at least classically) and, hence, will take the same Ansatz: Ansatz [wedge product in the Chern-Simons/Wess-Zumino action] We in- terpret the wedge products that appear in the formal expression for the Chern- (C,B) Simons/Wess-Zumino term SCS/WZ (ϕ, ∇) through the symmetrized determinant that applies to the above defining identities for wedge product; namely, we require that

1 s i (ω ∧ · · · ∧ ω )(e1 ∧ · · · ∧ es) = SymDet (ω (ej))

Az 1 s for OX -valued 1-forms ω , ··· , ω on X. Denote this generalized wedge product by ∧. Then, for lower-dimensional D-branes m = 0, 1, 2, 3, it is reasonable to assume that the (C,B) anomaly factor is 1 (i.e. no anomaly) and SCS/WZ (ϕ, ∇) can be written out precisely.

Locally in terms of a local frame (eµ)µ on an open set U ⊂ X and a coordinate 1 n P i j (y , ··· , y ) on a local chart of Y , one has: (Assuming that B = i,j Bijdy ⊗ dy , Bji = −Bij.) · For D(−1)-brane world-point (m = 0) :

(C(0)) ] SCS/WZ (ϕ) = T−1 · Tr (ϕ C(0)) = T−1 · Tr (ϕ (C(0))) .

Pn i · For D-particle world-line (m = 1) : Assume that C(1) = i=1 Ci dy locally; then Z Z n (C(1)) locally X ] ] i 1 SCS/WZ (ϕ) = T0 Re Tr (ϕ C(1)) = T0 Re Tr ϕ (Ci) · De1 ϕ (y ) de . X U i=1 (ρ,h;Φ) Note that as in the case of the dilaton term Sdilaton (ϕ), this is a functional of ϕ alone. Pn i j · For D-string world-sheet (m = 2) : Assume that C(2) = i,j=1 Cij dy ⊗ dy locally, with Cij = −Cji; then Z (C(0),C(2),B) 0 ] SCS/WZ (ϕ, ∇) = T1 Re (Tr (ϕ C(2) + ϕ (C(0)B) + 2πα ϕ (C(0)) F∇)) X Z 0 ] 0 ] = T1 Re (Tr (ϕ (C(2) + C(0)B) + πα ϕ (C(0))F∇ + πα F∇ϕ (C(0)))) X Z n locally X ] ] i ] j = T1 Re Tr ϕ (Cij + C(0)Bij) De1 ϕ (y ) De2 ϕ (y ) U i,j=1 0 ] 0 ] 1 2 + πα ϕ (C(0)) F∇(e1, e2) + πα F∇(e1, e2) ϕ (C(0)) e ∧ e Z n X ] ] i ] j = T1 Re Tr ϕ (Cij + C(0)Bij) De1 ϕ (y ) De2 ϕ (y ) U i,j=1 0 ] 1 2 + 2 πα ϕ (C(0)) F∇(e1, e2) e ∧ e . Here, the last identity comes from the eﬀect of the trace map Tr .

36 Pn i · For D-membrane world-volume (m = 3) : Assume that C(1) = i=1 Ci dy and C(3) = Pn i j k i,j,k=1 Cijk dy ⊗ dy ⊗ dy locally, with Cijk alternating with respect to ijk; then

Z (C(1),C(3),B) 0 SCS/WZ (ϕ, ∇) = T2 Re (Tr (ϕ C(3) + ϕ (C(1) ∧ B) + 2πα ϕ C(1) ∧ F∇ )) X Z n locally X ] = T2 Re Tr ϕ (Cijk + CiBjk + CjBki + CkBij) U i,j,k=1 ] i ] j ] k · De1 ϕ (y ) De2 ϕ (y ) De3 ϕ (y ) n 0 X X (λµν) ] ] i + πα (−1) ϕ (Ci) Deλ (ϕ (y )) F∇(eµ, eν ) (λµν)∈Sym i=1 3 ] ] i 1 2 3 + F∇(eµ, eν ) ϕ (Ci) Deλ ϕ (y ) e ∧ e ∧ e Z n X ] = T2 Re Tr ϕ (Cijk + CiBjk + CjBki + CkBij) U i,j,k=1 ] i ] j ] k · De1 ϕ (y ) De2 ϕ (y ) De3 ϕ (y ) n 0 X X (λµν) ] ] i 1 2 3 + 2πα (−1) ϕ (Ci) Deλ (ϕ (y )) F∇(eµ, eν ) e ∧ e ∧ e . i=1 (λµν)∈Sym3 Here, the last identity comes from the eﬀect of the trace map Tr .

Their partial study was done in [L-Y8 : Sec. 6.2] (D(13.1)).

(4) The background B-ﬁeld The coupling of (ϕ, ∇) with the background B-ﬁeld B on Y in the part (h;B) (C,B) Sgauge /YM(ϕ, ∇) + SCS/WZ (ϕ, ∇) of the standard action means that we have to adjust the fundamental module E on X by a compatible “twisting” governed by ϕ and B. With this “twisting”, E now lives on a gerb over X. See [L-Y2] (D(5)) for details and further references. However, since the study of the variational problems in this note is mainly local and focuses on the enhanced kinetic term for maps S(ρ,h;Φ,g) , we’ll ignore this twisting for the current note to keep map:kinetic+ the language and expressions simple.

Remark 4.3. [other effects from B-field and Ramond-Ramond field ] There are other effects to D-branes beyond just mentioned above from the background B-field and Ramond-Ramond field that have not yet been taken into account in this project so far; e.g. [H-M1], [H-M2], and [H-Y]. They can influence the action for D-branes as well. Such additional effects should be investigated in the future.

Theorem 4.4. [well-defined gauge-symmetry-invariant action] Except the anomaly factor in the Chen-Simons/Wess-Zumino term, which is yet to be understood, the standard action (ρ,h;Φ,g,B,C) (C,B) Sstandard (ϕ, ∇) as given in Definition 4.2 for (∗1)-admissible pairs (ϕ, ∇) (and SCS/WZ (ϕ, ∇) for (∗2)-admissible (ϕ, ∇)) is well-defined. Assume that the anomaly factor in the Chen-Simons/ Az Wess-Zumino term transforms also by conjugation as for OX under a gauge symmetry, then (ρ,h;Φ,g,B,C) Sstandard (ϕ, ∇) is invariant under gauge symmetries:

(ρ,h;Φ,g,B,C) (ρ,h;Φ,g,B,C) g0 g0 Sstandard (ϕ, ∇) = Sstandard ( ϕ, ∇)

0 ∞ for g ∈ Ggauge := C (Aut C(E)).

37 Proof. For the kinetic term for maps Z (h;g) 1 Smap:kinetic(ϕ, ∇) := 2 Tm−1 Re Tr hDϕ , Dϕi(h,g) vol h , X

0 that it is well-deﬁned follows Lemma 3.2.2.4. Under a gauge transformation g ∈ Ggauge := ∞ 1 m 1 n C (Aut C(E)) and in terms of local coordinates (x , ··· , x ) on X and (y , ··· , y ) on Y ,

g0 g0 X µ X g0 g0 ] ∂ ∂ X µ X 0 ] ∂ 0−1 ∂ D ϕ = dx ⊗ D ∂ ϕ i ⊗g0 i = dx ⊗ g D ∂ ϕ i g ⊗g0 i . ∂xµ ∂y ϕ ∂y ∂xµ ∂y ϕ ∂y µ i µ i Thus,

g0 g0 g0 g0 h D ϕ , D ϕi(h,g) X X µν 0 ] ∂ 0−1 0 ] ∂ 0−1 = h ⊗ g D ∂ ϕ i g · g D ∂ ϕ j g ⊗g0 gij ∂xµ ∂y ∂xν ∂y ϕ µ,ν i,j X X µν 0 ] ∂ 0−1 0 ] ∂ 0−1 0 ] 0−1 = h · g D ∂ ϕ i g · g D ∂ ϕ j g · g ϕ (gij)g ∂xµ ∂y ∂xν ∂y µ,ν i,j 0 X X µν ] ∂ ] ∂ ] 0−1 = g h · D ∂ ϕ i · D ∂ ϕ j · ϕ (gij) g ∂xµ ∂y ∂xν ∂y µ,ν i,j 0 0−1 = g hDϕ , Dϕi(h,g) g .

g0 g0 g0 g0 It follows that Tr h D ϕ , D ϕi(h,g) = Tr hDϕ , Dϕi(h,g) and, hence,

(h;g) g0 g0 (h;g) Smap:kinetic( ϕ, ∇) = Smap:kinetic(ϕ, ∇) .

(ρ,h;Φ,g,B,C) The other terms in Sstandard (ϕ, ∇) do not involve a partially-deﬁned inner product and hence are all deﬁned. That the integrand inside Tr all transform by conjugation under a gauge Az symmetry as for OX follows Lemma 4.1. This proves the theorem.

Remark 4.5. [gauge-fixing condition] As in any gauge field theory (e.g. [P-S]), understanding (ρ,h;Φ,g,B,C) how to fix the gauge is an important part of understanding Sstandard (ϕ, ∇).

The standard action as an enhanced non-Abelian gauged sigma model

Recall that, in an updated language and in a form for easy comparison, a sigma model (σ-model, SM) on a (Riemannian or Lorentzian) manifold (Y, g) (of dimension n) is a ﬁeld theory on a (Riemannian or Lorentzian) manifold (X, h) (of some dimension m) with · Field : diﬀerentiable maps f : X → Y , · Action functional : Z Z (h,g) 1 1 ∗ 2 Ssigma model (f) := ± 2 hdf , dfi(g,h) vol h = ±2 kf gkh vol h X X m n Z i j 1 X X µν ∂f ∂f p m := ± 2 h (x)gij(f(x))∂xµ (x) ∂xν (x) |det h(x)| d x , X µ,ν=1 i,j=1

38 in terms of local coordinates x = (x1, ··· , xm) on X and y = (y1, ··· , yn) on Y ; cf. [GM-L] and see e.g. [C-T] for modern update and further references. (The ± sign depends on the signature of the metric.) At the classical level, this is a theory of harmonic maps; cf. [E-L], [E-S], [Ma], [Sm]. Back to our situation. To begin with, the kinetic term Z (h;g) 1 Smap:kinetic(ϕ, ∇) := 2 Tm−1 Re Tr hDϕ , Dϕi(h,g) vol h X

(ρ,h;Φ,g,B,C) qualiﬁes the standard action Sstandard (ϕ, ∇) to be regarded as a sigma model, now based on Az · Field :(∗1)-admissible diﬀerentiable maps ϕ :(X , E; ∇) → Y .

(ρ,h;Φ,g,B,C) The fact that Sstandard (ϕ, ∇) is invariant under the gauge symmetry group ∞ Ggauge := C (Aut C(E)) and that the latter is non-Abelian justify that this sigma model is indeed a non-Abelian gauged sigma model (nAGSM). However, compared with, for example, the well-studied d = 2, N = (2, 2) (Abelian) gauged linear sigma model, e.g. [H-V] and [Wi1], (ρ,h;Φ,g,B,C) the gauge symmetry of Sstandard (ϕ, ∇) does not arise from gauging a global group-action (ρ,h;Φ,g,B,C) on the target space Y . (For this reason, one may call Sstandard (ϕ, ∇) a sigma model with non-Abelian gauge symmetry as well.) For D-branes, its additional coupling to the background Ramond-Ramond field C on Y is essential ([Po1]) and, hence, the Chern-Simons/Wess-Zumino (C,B) term SCS/WZ (ϕ, ∇). Also, we like our dynamical field (ϕ, ∇) coupled to the background dilaton field Φ on Y as well so that the essence of the other important action — the Dirac-Born-Infeld action — for D-branes can be retained as much as we can. This motivates the dilaton term (ρ,h;Φ) Sdilaton (ϕ). In summary,

(ρ,h;Φ,g,B,C) (ρ,h;Φ,g,B) (C,B) (ρ,h;Φ) Sstandard (ϕ, ∇) := SnAGSM (ϕ, ∇) + SCS/WZ (ϕ, ∇) + Sdilaton (ϕ)

=: S(ρ,h;Φ,g,B,C)(ϕ, ∇) , nAGSM+ which explains the name enhanced non-Abelian gauged sigma model (nAGSM+).

T 5 Admissible family of admissible pairs (ϕT , ∇ )

In this section we introduce the notion of one-parameter admissible families of admissible pairs and rephrase the basic settings and results in Sec. 3.2 in a relative format for such a family. Some curvature tensor computations are given for later use. The natural generalization (without work) to two-parameter admissible families of admissible pairs is remarked in the last theme of the section. This prepares us for the study of the variational problem of the enhanced kinetic term for maps S(ρ,h;Φ,g) (ϕ, ∇) in the standard action S(ρ,h;Φ,g,B,C)(ϕ, ∇) for D-branes. map:kinetic+ standard

T Basic setup and the notion of admissible families of admissible pairs (ϕT , ∇ )

1 Let T = (−ε, ε) ⊂ R , with coordinate t and ε > 0 small, be the one-parameter space and ∂t := ∂/∂t and dt be respectively the tangent vector ﬁeld and the 1-form determined by the coordinate t on T . Let (X,E) be a manifold X of dimension m with a complex vector bundle E of rank r. Recall the structure sheaf OX of X and the OX -module E from E. Consider the following families of objects over T :

39 · XT := X × T , with the structure sheaf OXT and regarded as the constant family of manifolds over T determined by X. XT is equipped with the built-in projection maps prX : XT → X and prT : XT × T → T . For U ⊂ X an open set, we will denote by UT the corresponding open set U × T ⊂ X × T over T .

· T∗XT := the tangent bundle of XT and T∗XT := the tangent sheaf of XT ; ∗ ∗ T XT := the cotangent bundle of XT and T XT := the cotangent sheaf of XT ; T∗(XT /T ) := the relative tangent bundle of XT over T and T∗(XT /T ) := the relative tangent sheaf of XT over T ; ∗ T (XT /T ) := the relative cotangent bundle of XT over T and ∗ T (XT /T ) := the relative cotangent sheaf of XT over T . When X is endowed with a (Riemannain or Lorentzian) metric h, h induces canonically ∗ an inner-product structure on ﬁbers of T∗(XT /T ) and its dual, T (XT /T ), over T . These induced inner-product structure will be denoted by h · , ·ih.

∗ · ET := prX E the pull-back vector bundle of E to XT , regarded as the constant T -family of ∗ vector bundles over X determined by E; and ET := prX E the corresponding OXT -module, regarded as the constant T -family of OX -modules determined by E. The projection map prX : XT → X induces a projection map prE : ET → E between the total space of bundles in question. T∗ET (resp. T∗ET ) denotes the tangent space (resp. the tangent sheaf) of the total space of ET . (XAz, E ) := (X , OAz := End (E ), E ), regarded as the constant T -family of Azu- · T T T XT O C T T XT maya/matrix manifolds with a fundamental module determined by (XAz, E). There is a trace map Tr : OAz −→ O C XT XT

as OXT -modules, which takes Id ET to r. and take the following notational conventions:

· Through the product structure XT = X × T , a vector field ξ (resp. 1-form ω) on X and the vector field ∂t on T lift canonically to a vector field (resp. 1-form) on XT , which will still be denoted by ξ (resp. ω) and ∂t respectively.

Az Az · For referral, the restriction of XT ,XT , ET , ···T to over t ∈ T will be denoted Xt, Xt , Et, ···t respectively.

Definition 5.1. [connection/covariant derivation trivially flat over T ] A connection T T ∇ on ET (equivalently, connection/covariant derivative ∇ on ET ) is said to be trivially flat over T if the horizontal lifting of ∂t to T∗ET lies in the kernel of the map prE∗ : T∗ET → T∗E. For such a ∇T , we will denote the covariant derivative ∇T simply by ∂ . The curvature tensor ∂t t T of ∇ will be denoted by F∇T .

Note that any connection on ET is flat over T and hence, due to the topology of T , can be made trivially flat over T after a bundle-isomorphism. Thus the notion of ‘trivially flat’ is only a notational convenience for our variational problem, not a true constraint. However, caution T t that while ∇ is always flat over T , its restriction ∇ to Xt varies as t varies in T . Thus, in general, F∇T (∂t, · ) 6= 0.

40 T Deﬁnition 5.2. [admissible family of admissible pairs (ϕT , ∇ )] A T-family of maps with Az T varying connections from (X , E) to Y is a pair (ϕT , ∇ ), where

Az ϕT :(X , ET ) −→ Y

Az is a map from (XT , ET ) to Y deﬁned contravariantly by a ring-homomorphism

] ∞ ∞ ϕT : C (Y ) −→ C (End C(ET ))

T ] over R ⊂ C and ∇ is a connection on ET that is trivially ﬂat over T . ϕT induces a homomorphism O −→ OAz Y XT ] between equivalence classes of gluing systems of rings, which will still be denoted by ϕT . Let A ⊂ OAz = O hIm ϕ] i. Then (ϕ , ∇T ) is said to be a (∗ )-admissible T -family ϕT XT XT T T i T of (∗j)-admissible pairs if (ϕT , ∇ ) satisﬁes Admissible Condition (∗i) along T and Admissible Condition (∗j) along X, for i, j = 1, 2, 3.

Example 5.3.[ (∗2)-admissible T -family of (∗1)-admissible pairs] A(∗2)-admissible T - T T family of (∗1)-admissible pairs (ϕT , ∇ ) is a T -family of maps ϕT with a varying connection ∇ trivially ﬂat over T such that

T (∗2): ∂tComm (AϕT ) ⊂ Comm (AϕT ) and (∗1): ∇ξ AϕT ⊂ Comm (AϕT ) for all ξ ∈ T (X /T ). Here, Comm (A ) is the commutant of A in OAz . ∗ T ϕT ϕT XT

Three basic OXT -modules with induced structures

Let X be endowed with a (Riemannian or Lorentzian) metric h and Y be endowed with a (Riemannian or Lorentzian) metric g. Denote the canonically induced inner-product structure from h and g on whatever bundle applicable by h · , · ih and h · , · ig respectively. Denote the ∗ h induced connection on T∗(XT /T ) and T (XT /T ) by ∇ and the Levi-Civita connection on T∗Y by ∇g. The associated Riemann curvature tensor is denoted by Rh and Rg respectively. Let (ϕ , ∇T ) be a (∗ )-admissible T -family of (∗ )-admissible pairs. The basic O C -modules T 1 1 XT with induced structures from the setting, as in Sec. 3.2, are listed below to ﬁx notations.

(0) OAz : the noncommutative structure sheaf on X XT T · The induced connection DT from ∇T , which is also trivially ﬂat over T , An OAz -valued, O C-bilinear (nonsymmetric) inner product from the multiplication · XT X in OAz ; XT C C an OX -valued, OX -bilinear (symmetric) inner product after the post-composition with Tr . · Both inner products are covariantly constant with respect to DT and one has the Leizniz rules

T 1 2 T 1 2 1 T 2 D (mT mT ) = (D mT )mT + mT D mT ; 1 2 T 1 2 dTr (mT mT ) = Tr D (mT mT ) T 1 2 1 T 2 = Tr (D mT )mT + Tr mT D mT .

41 (1) T ∗(X /T ) ⊗ OAz : OAz -valued relative 1-forms on X /T T OXT XT XT T · The induced connection ∇T,(h,DT ) := ∇h ⊗ Id + Id ⊗ DT , trivially ﬂat over T . An OAz -valued, O C -bilinear (nonsymmetric) inner product h , i ; · XT XT · · h C C an OX -valued, OX -bilinear (symmetric) inner product Tr h · , · ih. · Both inner products are covariantly constant with respect to ∇T,(h,DT ) and one has the Leibniz rules

T 0 T,(h,DT ) 0 T,(h,DT ) 0 D h · , · ih = h ∇ · , · ih + h · , ∇ · ig , 0 T 0 T,(h,DT ) 0 T,(h,DT ) 0 dTr h · , · ih = Tr (D h · , · ih) = Tr h ∇ · , · ih + Tr h · , ∇ · ih for , 0 ∈ T ∗(X /T ) ⊗ OAz . · · T OXT XT ∗ Az Az (2) ϕT T∗Y := OX ⊗ ] T∗Y : OX -valued derivations on OY T ϕT , OY T

T,(ϕT ,g) T Pn T ] i g · The induced connection ∇ := D ⊗ Id + Id · i=1 D ϕT (y ) ⊗ ∇ ∂ (in local ∂yi expression), trivially flat over T . A partially defined OAz -valued, O C-bilinear (nonsymmetric) inner product h , i ; · XT X · · g C C a partially defined OX -valued, OX -bilinear (symmetric) inner product Tr h · , · ig. · Both inner products, when defined, are covariantly constant with respect to ∇T,(ϕT ,g) and one has the Leibniz rules

T 0 T,(ϕT ,g) 0 T,(ϕT ,g) 0 D h – , – ig = h ∇ – , – ig + h – , ∇ – ig ,

0 T 0 T,(ϕT ,g) 0 T,(ϕT ,g) 0 dTr h – , – ig = Tr (D h – , – ig) = Tr h ∇ – , – ig + Tr h – , ∇ – ig ,

00 000 00 000 whenever all h – , – ig and Tr h – , – ig involved are defined. ∗ ∗ Az (3) T (X /T ) ⊗ ϕ T∗Y : (O -valued relative 1-form)-valued derivations on O T OXT T XT Y This is a combination of the construction in Item (1) and in Item (2). · The induced connection n T,(h,ϕT ,g) h T X T ] i g ∇ = ∇ ⊗ Id ⊗ Id + Id ⊗ D ⊗ Id + Id ⊗ Id · D ϕT (y ) ⊗ ∇ ∂ i i=1 ∂y (in local expression), trivially flat over T . A partially defined OAz -valued, O C-bilinear (nonsymmetric) inner product h , i ; · XT X · · (h,g) C C a partially defined OX -valued, OX -bilinear (symmetric) inner product Tr h · , · i(h,g). · Both inner products, when defined, are covariantly constant with respect to ∇T,(h,ϕT ,g) and one has the Leibniz rules

T 0 T,(h,ϕT ,g) 0 T,(h,ϕT ,g) 0 D h ∼ , ∼ i(h,g) = h ∇ ∼ , ∼ i(h,g) + h ∼ , ∇ ∼ i(h,g) , 0 T 0 dTr h ∼ , ∼ i(h,g) = Tr (D h ∼ , ∼ i(h,g))

T,(h,ϕT ,g) 0 T,(h,ϕT ,g) 0 = Tr h ∇ ∼ , ∼ i(h,g) + Tr h ∼ , ∇ ∼ i(h,g) , 00 000 00 000 whenever the h ∼ , ∼ i(h,g) and Tr h ∼ , ∼ i(h,g) involved are deﬁned.

42 Curvature tensors with ∂t and other order-switching formulae

T Let (ϕT , ∇ ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. A very basic step in (particularly the second) variational problem involves passing ∂t over a diﬀerential operator on Xt’s. In general, a curvature term appears whenever such passing occurs. In this theme, we collect and prove such formulae we need.

First, passing ∂t over a diﬀerential operator usually means the appearance of a curvature term by the very deﬁnition of a curvature tensor:

T Lemma 5.4. [curvature tensor with ∂t] Let (ϕT , ∇ ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. Let ξ be a vector ﬁeld on an open set U ⊂ X small enough so that Az 1 n ϕT (UT ) is contained in a coordinate chart on Y , with coordinates (y , ··· , y ). The standard lifting of ξ to UT is denoted also by ξ. Note that, by construction, [∂t, ξ] = 0 and all our · Az connection ∇ in Theme ‘Three basic OX -modules with induced structures’ are trivially ﬂat; hence, F (∂ , ξ) = ∂ ∇· − ∇·∂ . One has the following curvature expressions with ∂ on the ∇· t t ξ ξ t t basic OXT -modules: (Below we adopt the convention that the Riemann curvature tensor from a metric is denoted by R while the curvature tensor of a connection in all other bundle situations is denoted by F .)

∗ h h (01) For sections ωT of T (XT /T ) : R∇h (∂t, ξ) ωT = ∂t∇ξ ωT − ∇ξ ∂tωT = 0.

Az T T T (0 ) For sections m of O : F T (∂ , ξ) m = ∂ D m − D ∂ m = [(∂ ∇ )(ξ), m ] . 2 T XT D t T t ξ T ξ t T t T T As a consequence of this, if (ϕT , ∇ ) is furthermore a (∗2)-admissible T -family of (∗2)-admissible pairs, then

T ϕ Az T (∂t∇ )(ξ) ∈ Inn (O ) i.e. [(∂t∇ )(ξ), Aϕ ] ⊂ Comm (Aϕ ) . (∗1) X T T

(1) For sections ω ⊗ m of T ∗(X /T ) ⊗ OAz : T T T OXT XT

F∇T,(h,DT ) (∂t, ξ)(ωT ⊗ mT ) T,(h,DT ) T,(h,DT ) T = ∂t∇ξ (ωT ⊗ mT ) − ∇ξ ∂t(ωT ⊗ mT ) = ωT ⊗ [(∂t∇ )(ξ), mT ] .

∗ Az (2) For sections mT ⊗ v of ϕT T∗Y := OX ⊗ ] T∗Y : T ϕT ,OY (v on the coordinate chart of Y above, with coordinates (y1, ··· , yn))

T,(ϕT ,g) T,(ϕT ,g) F∇T,(ϕT ,g) (∂t, ξ)(mT ⊗ v) = ∂t∇ξ (mT ⊗ v) − ∇ξ ∂t(mT ⊗ v) n T X T ] i g = [(∂t∇ )(ξ), mT ] ⊗ v + mT [(∂t∇ )(ξ), ϕT (y )] ⊗ ∇ ∂ v i i=1 ∂y n X T ] i ] j g g ] j T ] i g g + mT Dξ ϕT (y ) ∂tϕT (y ) ⊗ ∇ ∂ ∇ ∂ v − ∂tϕT (y ) Dξ ϕT (y ) ⊗ ∇ ∂ ∇ ∂ v . j i i j i,j=1 ∂y ∂y ∂y ∂y

T If (ϕT , ∇ ) is furthermore a (∗2)-admissible T -family of (∗1)-admissible pairs, then the last term has a Y -coordinate-free form

X ] j ] i g ∂ ∂ g The last term = mT ∂tϕT (y ) DξϕT (y ) ⊗ R ∂yj , ∂yi v = mT (ϕT R )(∂t, ξ) v . i,j

43 ∗ ∗ ∗ Az (3) For sections ωT ⊗ mT ⊗ v of T (XT /T ) ⊗ ϕT T∗Y := T (XT /T ) ⊗OX OX ⊗ ] T∗Y : T T ϕT ,OY (v on the coordinate chart of Y above, with coordinates (y1, ··· , yn) )

F∇T,(h,ϕT ,g) (∂t, ξ)(ωT ⊗ mT ⊗ v)

T,(h,ϕT ,g) T,(h,ϕT ,g) = ∂t∇ξ (ωT ⊗ mT ⊗ v) − ∇ ∂t(ωT ⊗ mT ⊗ v) = ωT ⊗ F∇T,(ϕT ,g) (∂t, ξ)(mT ⊗ v) .

Proof. Statement (01) follows from the fact that XT is a constant family over T . Statement (02), First Part, follows from a computation with respect to an induced local trivialization of ET from a local trivialization of E

T ∂tDξ mT = ∂t ξmT + [A∇T (ξ), mT ] T T = ξ∂tmT + [∂tA∇T (ξ), mT ] + [A∇T (ξ), ∂tmT ] = Dξ ∂tmT + [(∂t∇ )(ξ), mT ] .

T For Second Part, if (ϕT , ∇ ) is furthermore a (∗2)-admissible T-family of (∗2)-admissible pairs, then for f1, f2 ∈ OY , by First Part and the (∗2)-Admissible Condition,

T ] ] T ] ] T ] ] [(∂t∇ )(ξ), ϕT (f1)], ϕT (f2) = [∂tDξ ϕT (f1), ϕT (f2)] − [Dξ ∂tϕT (f1), ϕT (f2)] = 0 .

T ϕT Az Which says that (∂t∇ )(ξ) ∈ Inn (O ). (∗1) XT Statement (1) is a consequence of Statement (01) and Statement (02). Statement (3) is a consequence of Statement (01) and a property of the induced connection on a tensor product of O C -modules with a connection. Let us carry out Statement (2) as a demonstration of the XT covariant diﬀerential calculus involved. ∗ Let mT ⊗ v ∈ ϕT T∗Y . Then, by Statement (02),

T,(ϕT ,g) T X T ] i g ∂t∇ξ (mT ⊗ v) = ∂t Dξ mT ⊗ v + mT Dξ ϕT (y ) ⊗ ∇ ∂ v i i ∂y T T T X ] i g = Dξ ∂tmT + [(∂t∇ )(ξ), mT ] ⊗ v + (Dξ mT ) ∂tϕT (y ) ⊗ ∇ ∂ v i i ∂y X T ] i g X T ] i T ] i g + (∂tmT ) Dξ ϕT (y ) ⊗ ∇ ∂ v + mT Dξ ∂tϕT (y ) + [(∂t∇ )(ξ), ϕT (y )] ⊗ ∇ ∂ v i i i ∂y i ∂y X T ] i ] j g g + mT Dξ ϕT (y )∂tϕT (y ) ⊗ ∇ ∂ ∇ ∂ v j i i,j ∂y ∂y while

T,(ϕT ,g) T,(ϕT ,g) X ] i g ∇ξ ∂t(mT ⊗ v) = ∇ξ ∂tmT ⊗ v + mT ∂tϕT (y ) ⊗ ∇ ∂ v i i ∂y T X T ] i g T X ] i = Dξ ∂tmT ⊗ v + (∂tmT ) Dξ ϕT (y ) ⊗ ∇ ∂ v + (Dξ mT ) ∂tϕT (y ) ⊗ ∇ ∂ v i ∂yi i ∂y i X T ] i g X ] i T ] j g g + mT Dξ ∂tϕT (y ) ⊗ ∇ ∂ v + mT ∂tϕT (y )Dξ ϕT (y ) ⊗ ∇ ∂ ∇ ∂ v . i j i i ∂y i,j ∂y ∂y Thus,

T,(ϕT ,g) T,(ϕT ,g) F∇T,(ϕT ,g) (∂t, ξ)(mT ⊗ v) = (∂t∇ − ∇ ∂t)(mT ⊗ v) T X T ] i g = [(∂t∇ )(ξ), mT ] ⊗ v + mT [(∂t∇ )(ξ), ϕT (y )] ⊗ ∇ ∂ v i i ∂y X T ] i ] j g g ] j T ] i g g + mT Dξ ϕT (y )∂tϕT (y ) ⊗ ∇ ∂ ∇ ∂ v − ∂tϕT (y )Dξ ϕT (y ) ⊗ ∇ ∂ ∇ ∂ v j i i j i,j ∂y ∂y ∂y ∂y

44 as claimed, after a relabeling of i, j. T T ] i If (ϕT , ∇ ) is furthermore a (∗2)-admissible T -family of (∗1)-admissible pairs, then Dξ ϕT (y ) ] j T ] i ] j and ∂tϕT (y ) commute since [Dξ ϕT (y ), ϕT (y )] = 0 by the (∗1)-Admissible Condition along X and, hence,

T ] i ] j 0 = ∂t[Dξ ϕT (y ), ϕT (y )] T ] i ] j T ] i ] j T ] i ] j = [∂tDξ ϕT (y ), ϕT (y )] + [Dξ ϕT (y ), ∂tϕT (y )] = [Dξ ϕT (y ), ∂tϕT (y )] by the (∗1)-Admissible Condition along X and the (∗2)-Admissible Condition along T . The last summand of F∇T,(ϕT ,g) (∂t, ξ)(mT ⊗ v) is then equal to

X ] j T ] g g g g g mT ∂tϕT (y )Dξ ϕT ⊗ ∇ ∂ ∇ ∂ − ∇ ∂ ∇ ∂ v = mT (ϕT R )(∂t, ξ) v . j i i j i,j ∂y ∂y ∂y ∂y This proves the lemma.

The following lemma addresses the issue of passing ∂t over the covariant diﬀerential DϕT of ϕT . Though such passing is not a curvature issue in the conventional sense, it does carry a taste of curvature calculations.

T T,(ϕ , g) T Lemma 5.5.[ ∂tD ϕT versus ∇ T ∂tϕT ] Let (ϕT , ∇ ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. With the above notation and convention, let ξ be a vector ﬁeld on X. Then, for a chart of Y with coordinates (y1, ··· , yn), one has

n T T,(ϕT , g) g T X T ] i ∂ ∂tDξ ϕT = ∇ξ ∂tϕT − (ad ⊗ ∇ )∂tϕT Dξ ϕT + [(∂t∇ )(ξ), ϕT (y )] ⊗ ∂yi . i=1 Here, only as a compact notation, n g T X ] i T ] j g ∂ (ad ⊗ ∇ )∂tϕT Dξ ϕT := [∂tϕ (y ),Dξ ϕ (y )] ⊗ ∇ ∂ j i ∂y i,j=1 ∂y n X T ] j ] i g ∂ g = − [D ϕ (y ), ∂ ϕ (y )] ⊗ ∇ =: − (ad ⊗ ∇ ) T ∂ ϕ . ξ t ∂ i D ϕT t T j ∂y ξ i,j=1 ∂y

T If (ϕT , ∇ ) is furthermore a (∗2)-admissible T -family of (∗2)-admissible pairs, then the last term has a Y -coordinate-free expression

T ad (∂t∇ )(ξ)ϕT .

Proof. Under the given setting and by Lemma 5.4 (02),

T X T ] i ∂ ∂tDξ ϕT = ∂t Dξ ϕT (y ) ⊗ ∂yi i X T ] i T ] i ∂ X T ] i ] j g ∂ = Dξ ∂tϕT (y ) + [(∂t∇ )(ξ), ϕT (y )] ⊗ i + Dξ ϕT (y )∂tϕT (y ) ⊗ ∇ ∂ i ∂y j ∂y i i,j ∂y while

T,(ϕT ,g) T,(ϕT ,g) X ] i ∂ ∇ξ ∂tϕT = ∇ξ ∂tϕT (y ) ⊗ ∂yi i X T ] i ∂ X ] i T ] j g ∂ = Dξ ∂tϕT (y ) ⊗ i + ∂tϕT (y )Dξ ϕT (y ) ⊗ ∇ ∂ i . ∂y j ∂y i i,j ∂y

45 Thus,

T T,(ϕT ,g) ∂tDξ ϕ − ∇ξ ∂tϕT X T ] i ∂ = [(∂t∇ )(ξ), ϕT (y )] ⊗ ∂yi i X T ] i ] j g ∂ X ] i T ] j g ∂ + Dξ ϕT (y )∂tϕT (y ) ⊗ ∇ ∂ i − ∂tϕT (y )Dξ ϕT (y ) ⊗ ∇ ∂ i j ∂y j ∂y i,j ∂y i,j ∂y

g ∂ g ∂ Either apply the identity ∇ ∂ ∂yi = ∇ ∂ ∂yj to the second term and relabeling i, j of the third, ∂yj ∂yi g ∂ g ∂ or apply the identity ∇ ∂ ∂yi = ∇ ∂ ∂yj to the third term and relabeling i, j of the second, ∂yj ∂yi

X T ] i ] j g ∂ X ] i T ] j g ∂ Dξ ϕT (y )∂tϕT (y ) ⊗ ∇ ∂ i − ∂tϕT (y )Dξ ϕT (y ) ⊗ ∇ ∂ i j ∂y j ∂y i,j ∂y i,j ∂y X T ] i ] j g ∂ g = [Dξ ϕT (y ) , ∂tϕT (y )] ⊗ ∇ ∂ j := (ad ⊗ ∇ )DT ϕ∂tϕT i ∂y ξ i,j ∂y X ] i T ] j g ∂ g T = − [∂tϕT (y ) ,Dξ ϕT (y )] ⊗ ∇ ∂ j := − (ad ⊗ ∇ )∂tϕT Dξ ϕT . i ∂y i,j ∂y This proves the First Statement in Lemma. The Second Statement in Lemma is a consequence of Corollary 3.1.10 and Lemma 5.4 (02). This proves the lemma.

Before continuing the discussion, we introduce a notion that is needed in the next lemma.

1 2 , • Deﬁnition 5.6. [half-torsion tensor Tor ∇g ] Recall the torsion tensor Tor ∇0 of a connection ∇0 on Y 0 0 0 Tor ∇ (v1, v2) := ∇v1 v2 − ∇v2 v1 − [v1, v2] g for v1, v2 ∈ T∗Y . For the Levi-Civita connection ∇ associated to a metric g on Y , Tor ∇g ≡ 0 by construction. Thus, in this case, for a Φ ∈ C∞(Y ),

g g (∇v1 v2 − v1v2)Φ = (∇v2 v1 − v2v1)Φ for v1, v2 ∈ T∗Y . This deﬁnes a symmetric 2-tensor on Y

1 , Φ Tor 2 : T Y × T Y −→ O ∇g ∗ Y ∗ Y , g (v1, v2) 7−→ (∇v1 v2 − v1v2)Φ called the half-torsion tensor of (the torsion-free connection) ∇g associated to Φ ∈ C∞(Y ).

The following lemma addresses the issue of passing ∂ over ‘evaluation of an OAz -valued t XT derivation on C∞(Y )’, and another similar situation:

T T T T,(ϕT , g) Lemma 5.7.[ ∂t((Dξ ϕT )Φ) versus (∂tDξ ϕT )Φ ; Dξ ((∂tϕT )Φ) versus (∇ξ ∂tϕT )Φ] T Let (ϕT , ∇ ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. Continue the notation and

46 Az Az convention in Lemma 5.4. Under the canonical isomorphism OX ⊗ ] OY 'OX , T ϕT ,OY T

T ∂t (Dξ ϕT )Φ n T X T ] i ] j ∂ ∂ g ∂ = ∂tDξ ϕT Φ + Dξ ϕT (y ) ∂tϕT (y ) ⊗ ∂yi ∂yj Φ − ∇ ∂ ∂yj Φ i i,j=1 ∂y 1 T 2 ,Φ = ∂tDξ ϕT Φ − ϕT Tor ∇g (ξ, ∂t); and

T Dξ (∂tϕT )Φ n T,(ϕT , g) X ] i T ] j ∂ ∂ g ∂ = ∇ξ ∂tϕT Φ + ∂tϕT (y ) Dξ ϕT (y ) ⊗ ∂yi ∂yj Φ − ∇ ∂ ∂yj Φ i i,j=1 ∂y 1 T,(ϕT , g) 2 ,Φ = ∇ξ ∂tϕT Φ − ϕT Tor ∇g (∂t, ξ) .

Proof. For the ﬁrst identity,

T X T ] i ∂ ∂t (Dξ ϕT )Φ = ∂t Dξ ϕT (y ) ⊗ ∂yi Φ i X T ] ∂ X T ] i ] j ∂ ∂ = ∂tDξ ϕT ⊗ ∂yi Φ + Dξ ϕT (y )∂tϕT (y ) ⊗ ∂yj ∂yi Φ i i,j while

T X T ] i ∂ ∂tDξ ϕT Φ = ∂t Dξ ϕT (y ) ⊗ ∂yi Φ i X T ] i ∂ X T ] i ] j g ∂ = ∂tDξ ϕT (y ) ⊗ i + Dξ ϕT (y )∂tϕT (y ) ⊗ ∇ ∂ i Φ . ∂y j ∂y i i,j ∂y Thus,

T T ∂t (Dξ ϕT )Φ − ∂tDξ ϕT Φ X T ] i ] j ∂ ∂ g ∂ = Dξ ϕT (y )∂tϕT (y ) ⊗ j i − ∇ ∂ i Φ ∂y ∂y j ∂y i,j ∂y 1 X T ] i ] j ∂ ∂ g ∂ 2 ,Φ = Dξ ϕT (y )∂tϕT (y ) ⊗ i j − ∇ ∂ j Φ = − ϕT Tor ∇g (ξ, ∂t) ∂y ∂y i ∂y i,j ∂y and the ﬁrst identity follows. For the second identity,

T T X ] i ∂ Dξ (∂tϕT )Φ = Dξ ∂tϕT (y ) ⊗ ∂yi Φ i X T ] i ∂ X ] i T ] j ∂ ∂ = Dξ ∂tϕT (y ) ⊗ ∂yi Φ + ∂tϕT (y )Dξ ϕT (y ) ⊗ ∂yj ∂yi Φ i i,j while

T,(ϕT , g) T,(ϕT ,g) X ] i ∂ ∇ξ ∂tϕT Φ = ∇ξ ∂tϕT (y ) ⊗ ∂yi Φ i X T ] i ∂ X ] i T ] j g ∂ = Dξ ∂tϕT (y ) ⊗ i + ∂tϕT (y )Dξ ϕT (y ) ⊗ ∇ ∂ i Φ . ∂y j ∂y i i,j ∂y

47 Thus,

T T,(ϕT , g) Dξ (∂tϕT )Φ − ∇ξ ∂tϕT Φ X ] i T ] j ∂ ∂ g ∂ = ∂tϕT (y )Dξ ϕT (y ) ⊗ j i − ∇ ∂ i Φ ∂y ∂y j ∂y i,j ∂y 1 X ] i T ] j ∂ ∂ g ∂ 2 ,Φ = ∂tϕT (y )Dξ ϕT (y ) ⊗ i j − ∇ ∂ j Φ = − ϕT Tor ∇g (∂t, ξ) ∂y ∂y i ∂y i,j ∂y and the second identity follows. This proves the lemma.

T Remark 5.8. [for (∗2)-admissible family of (∗1)-admissible pairs ] If (ϕT , ∇ ) is furthermore a (∗2)-admissible T -family of (∗1)-admissible pairs, then, as in the proof of Lemma 5.4 (2), T ] i ] j Dξ ϕT (y ) and ∂tϕT (y ) commute for all i, j. In this case,

1 1 2 ,Φ 2 ,Φ ϕ Tor ∇g (ξ, ∂t) = ϕ Tor ∇g (∂t, ξ) .

Two-parameter admissible families of admissible pairs

2 2 Let T = (−ε, ε) ⊂ R , ε > 0 small, be a two-parameter space with coordinates (s, t). The setting and results above for one-parameter admissible families of admissible pairs generalizes without work to two-parameter admissible of admissible pairs. In particular,

Deﬁnition 5.9. [two-parameter admissible family of admissible pairs] A(∗2)-admissible Az T T -family of (∗1)-admissible maps is a (∗1)-admissible map ϕT :(XT , ET ; ∇ ) → Y , where ET is trivially ﬂat over T , such that ∂sComm AϕT ⊂ Comm (AϕT ) and ∂tComm AϕT ⊂ Comm (AϕT ).

The following is a consequence of the proof of Lemma 3.2.2.5:

T Lemma 5.10. [symmetry property of Tr hF T,(ϕ ,g) (∂ , ξ )∂ ϕ ,D ϕ i] Let ∇ T s 2 t T ξ4 T Az T ϕT :(XT , ET ; ∇ ) → Y be a (∗2)-admissible T -family of (∗1)-admissible maps. Let ξ2, ξ4 ∈ T∗X and denote the same for their respective lifting to T∗(XT /T ). Then,

T T,(ϕ ,g) T,(ϕ ,g) Tr hF∇ T (∂s, ξ2)∂tϕT ,Dξ4 ϕT ig = − Tr h∂tϕT ,F∇ T (∂s, ξ2)Dξ4 ϕT ig T T T,(ϕ ,g) T,(ϕ ,g) = − Tr hF∇ T (∂s, ξ2)Dξ4 ϕT , ∂tϕT ig = Tr hF∇ T (ξ2, ∂s)Dξ4 ϕT , ∂tϕT ig .

Proof. Let ξ be ξ2 or ξ4. Since ∂sComm (AϕT ) ⊂ Comm (AϕT ) and and ∂ξAϕT ⊂ Comm (AϕT ), T both ∂ D ϕ and ∂ ∂ ϕ lie in Comm (A ) ⊗ ] T Y . Locally explicitly, s ξ T s t T ϕT ϕ ,OY ∗

T X T ] i ∂ X T ] i ] j g ∂ ∂sDξ ϕT = ∂sDξ ϕT (y ) ⊗ i + Dξ ϕT (y ) ∂sϕT (y ) ⊗ ∇ ∂ i ; ∂y j ∂y i i,j ∂y X ] i ∂ X ] i ] j g ∂ ∂s∂tϕT = ∂s∂tϕT (y ) ⊗ i + ∂tϕT (y ) ∂sϕT (y ) ⊗ ∇ ∂ i . ∂y j ∂y i i,j ∂y

48 Now follow the proof of Lemma 3.2.2.5, but under only the (∗1)-Admissible Condition on T T (ϕ , ∇ ), to convert Tr hF T,(ϕ ,g) (∂ , ξ )∂ ϕ ,D ϕ i to Tr h∂ ϕ ,F T,(ϕ ,g) (∂ , ξ )D ϕ i . T ∇ T s 2 t T ξ4 T g t T ∇ T s 2 ξ4 T g 0 0 Since Tr h – , – ig is deﬁned as long as one of –, – is in Comm (AϕT ) ⊗ ] T∗Y , one realizes ϕT ,OY that all the terms that appear in the process via the Leibniz rule are deﬁned except

− Tr h∇T,(ϕT ,g)∂ ϕ , ∂ DT ϕ i + Tr h∂ ∂ ϕ , ∇T,(ϕT ,g)DT ϕ i . ξ2 t T s ξ4 T g s t T ξ2 ξ4 T g

Under the additional (∗ )-Admissible Condition on (ϕ , ∇T ) along T , both ∂ DT ϕ and ∂ ∂ ϕ 2 T s ξ4 T s t T now lie in Comm (A ) ⊗ ] T Y ; and the above two exceptional terms become deﬁned. ϕT ϕ ,OY ∗ The lemma follows.

6 The ﬁrst variation of the enhanced kinetic term for maps and ......

1 Let (ϕ, ∇) be a (∗1)-admissible pair. Recall the setup in Sec. 5. Let T = (−ε, ε) ⊂ R , for some T ε > 0 small, and (ϕT , ∇ ) be a (∗1)-admissible T -family of (∗1)-admissible pairs that deforms T (ϕ, ∇) = (ϕT , ∇ )|t=0. We derive in Sec. 6.1 and Sec. 6.2 the first variation formula of the newly introduced enhanced kinetic term for maps 1 Z Z S(ρ,h;Φ,g) (ϕ, ∇) := T Re Tr hDϕ , Dϕi vol + Re Tr hdρ, ϕdΦi vol map:kinetic+ m−1 (h,g) h h h 2 X X in the standard action for D-branes. As the ‘taking the real part’ operation Re (······ ) is a OX -linear operation and can always be added back in the end, we will consider 1 Z Z S(ρ,h;Φ,g) (ϕ, ∇)C := T Tr hDϕ , Dϕi vol + Tr hdρ, ϕdΦi vol map:kinetic+ m−1 (h,g) h h h 2 X X so that we don’t have to carry Re around. The first variation of the gauge/Yang-Mills term is analogous to that in the ordinary Yang- Mills theory and the first variation of the Chern-Simons/Wess-Zumino term is an update from [L-Y8: Sec. 6] (D(13.1)). Both are given in Sec. 6.3 under the stronger (∗2)-Admissible Condition.

6.1 The ﬁrst variation of the kinetic term for maps ∇t t Recall the (complexiﬁed) kinetic energy E (ϕt)C of ϕt for a given ∇ , t ∈ T := (−ε, ε),

t 1 Z ∇ C (h;g) t C t t t E (ϕt) := Smap:kinetic(ϕt, ∇ ) := Tm−1 Tr hD ϕt ,D ϕ i(h,g) vol h . 2 X As t varies, with a slight abuse of notation, denote the resulting function of t by

T 1 Z ∇ C (h;g) T C T T E (ϕT ) := Smap:kinetic(ϕT , ∇ ) := Tm−1 Tr hD ϕT ,D ϕT i(h,g) vol h , 2 X with the understanding that all expressions are taken on Xt with t varying in T .

49 µ Let U ⊂ X be an open set with an orthonormal frame (eµ)µ=1, ··· , m. Let (e )µ=1, ··· , m be the Az dual co-frame. Assume that U is small enough so that ϕT (UT ) is contained in a coordinate chart of Y , with coordinates (y1, ··· , yn). Then, over U,

d T 1 Z ∇ C T T E (ϕT ) = Tm−1 ∂tTr hD ϕT ,D ϕT i(h,g) vol h dt 2 U Z 1 T T = Tm−1 Tr ∂thD ϕT ,D ϕT i(h,g)vol h 2 U Z m 1 X T T = Tm−1 Tr ∂t hD ϕT ,D ϕT igvol h 2 eµ eµ U µ=1 m Z X = T Tr h∂ DT ϕ ,DT ϕ i vol m−1 t eµ T eµ T g h U µ=1 Z X = T Tr h∇T,(ϕT ,g)∂ ϕ ,DT ϕ i vol m−1 eµ t T eµ T g h U µ Z X g T + Tm−1 Tr h(ad ⊗∇ )DT ϕ ∂tϕT ,D ϕT ig vol h eµ T eµ U µ n Z X X + T h [(∂ ∇T )(e ), ϕ] (yi)] ⊗ ∂ ,DT ϕ i vol m−1 t µ T ∂yi eµ T g h U µ i=1 = (I.1) + (I.2) + (I.3) .

Z X (I.1) = T Tr DT h∂ ϕ ,DT ϕ i − h∂ ϕ , ∇T,(ϕT ,g)DT ϕ i vol m−1 eµ t T eµ T g t T eµ eµ T g h U µ Z X Z = T e Tr h∂ ϕ ,DT ϕ i vol + T Tr h∂ ϕ , − P ∇T,(ϕT ,g)DT ϕ i vol m−1 µ t T eµ T g h m−1 t T µ eµ eµ T g h U µ U = (I.1.1) + (I.1.2) . Summand (I.1.1) suggests a boundary term. To really extract the boundary term from it, consider the T -family of C-valued 1-forms on U αT := Tr h∂ ϕ ,DT ϕ i , (I, ∂tϕT ) t T T g ∞ which depends C (U)C-linearly on ∂tϕT . Let m X ξT := Tr h∂ ϕ ,DT ϕ i e (I, ∂tϕT ) t T eµ T g µ µ=1 be the T -family of dual -valued vector ﬁelds of αT on U with respect to the metric h. C (I, ∂tϕT ) T ∞ Note that ξ depends C (U)C-linearly on ∂tϕT as well. Then (I, ∂tϕT ) Z X (I.1.1) = T e hξT , e i vol m−1 µ (I, ∂tϕT ) µ h h U µ Z X Z = T h∇h ξT , e i vol + T hξT , P ∇h e i vol . m−1 eµ (I, ∂tϕT ) µ h h m−1 (I, ∂tϕT ) µ eµ µ h h U µ U The ﬁrst term is equal to Z Z Z T T (− div ξ ) vol = T d i T vol = T i T vol , m−1 (I, ∂tϕT ) h m−1 ξ h m−1 ξ h U U (I, ∂tϕT ) ∂U (I, ∂tϕT )

50 which is the sought-for boundary term, whose integrand satisﬁes the requirement that it be ∞ C (U)C-linear on ∂tϕT . The second term is equal to Z T T Tr h∂ ϕ ,DP h ϕ i vol m−1 t T ∇ eµ T g h U µ eµ

∞ by construction, which is C (U)C-linear in ∂tϕT and hence in a ﬁnal form. ∞ The integrand of Summand (I.1.2) is already C (U)C-linear in ∂tϕT and hence in a ﬁnal form. Summand (I.2) can be re-written as Z X g T T (I.2) = − Tm−1 Tr h(ad ⊗∇ )∂tϕT Deµ ϕT ,Deµ ϕT ig vol h . U µ

∞ Thus, its integrand is already C (U)C-linear in ∂tϕT and hence in a final form. C Az C Az Finally, since the built-in inclusion OU ⊂ OU identifies OU with the center of OU , Summand (I.3) is C∞(U)C-linear and hence in its final fom.

Altogether, we almost complete the calculation except the issue of whether all the inner products Tr h · , · ig that appear in the procedure are truly deﬁned. For this, one notices that wherever such an inner product appears above, at least one of its arguments is either ∂tϕT or T Deµ ϕT , for some µ. It follows from Lemma 3.2.2.4 that they are indeed deﬁned. In summary,

T Proposition 6.1.1. [ﬁrst variation of kinetic term for maps] Let (ϕT , ∇ ) be a (∗1)- admissible T -family of (∗1)-admissible pairs. Then, d T d 1 Z ∇ C T T E (ϕT ) = Tm−1 Tr hD ϕT ,D ϕT i(h,g) vol h dt dt 2 U Z = Tm−1 iξT vol h ∂U (I, ∂tϕT ) Z T Pm T,(ϕT ,g) T + T Tr ∂ ϕ , DPm h − ∇ D ϕ vol m−1 t T ∇ eµ µ=1 eµ eµ T g h U µ=1 eµ Z m X g T T − Tm−1 Tr h(ad ⊗∇ )∂tϕT Deµ ϕT ,Deµ ϕT ig vol h U µ=1 m n Z X X + T h [(∂ ∇T )(e ), ϕ] (yi)] ⊗ ∂ ,DT ϕ i vol . m−1 t µ T ∂yi eµ T g h U µ=1 i=1

T Pm T Here, the first summand is the boundary term with ξ := (Tr h∂tϕT ,D ϕT ig)eµ (I, ∂tϕT ) µ=1 eµ ∞ ∞ C (U)C-linear in ∂tϕT ; the integrand of the second and the third terms are C (U)C-linear in ∂tϕT and their real part contribute first-order and second-order terms to the equations of motion ∞ T for (ϕ, ∇); the integrand of the last term is C (U)C-linear in ∂t∇ and its real part contributes terms, first order in ϕ but zeroth order in the connection 1-from of ∇, to the equations of motion (ρ,h;Φ,g,B,C) for (ϕ, ∇) in addition to those from the first variation of the rest part of Sstandard (ϕ, ∇). These lower-order terms contribute to the equations of motion for (ϕ, ∇) but do not change the signature of the system.

51 Remark 6.1.2. [for (∗2)-admissible T -family of (∗2)-admissible pairs] If furtheremore (ϕ, ∇) is T (∗2)-admissible and (ϕT , ∇ ) is a (∗2)-admissible T -family of (∗2)-admissible pairs that deforms (ϕ, ∇), then the third summand of the ﬁrst variation formula in Proposition 6.1.1 vanishes and the fourth/last summand has a Y -coordinate-free form Z m X T T Tm−1 had (∂t∇ )(eµ)ϕT ,Deµ ϕT ig vol h . U µ=1

T In this case, the first variation with respect to ϕ alone (i.e. setting ∂t∇ = 0), cf. the first two summands, takes the form of a direct formal generalization of the first variation formula in the study of harmonic maps; e.g. [E-L], [E-S], [Ma], [Sm].

6.2 The ﬁrst variation of the dilaton term

(ρ,h;Φ,g,B,C) C We now turn to the (complexiﬁed) dilaton term in Sstandard (ϕ, ∇) . Az T Let ϕT :(X , ET ; ∇ ) → Y be a (∗1)-admissible T -family of (∗1)-admissible pairs. Then, over an open set U ⊂ X, Z (ρ,h;Φ) C Sdilaton (ϕT ) = Tr hdρ , ϕT dΦih vol h U m n ! Z X X ∂Φ = Tr dρ(e ) DT ϕ] (yi) ϕ] vol µ eµ T T ∂yi h U µ=1 i=1 Z ! X T = Tr dρ(eµ) ((Deµ ϕT )Φ) vol h . U µ

Z m d (ρ,h;Φ) X T S (ϕT )C = Tr dρ(eµ) ∂t (D ϕT )Φ vol h dt dilaton eµ U µ=1 Z X = Tr dρ(e ) (∂ DT ϕ )Φ vol µ t eµ T h U µ n Z X X + Tr dρ(e ) DT ϕ] (yi)∂ ϕ] (yj) ⊗ ∂ ∂ Φ − ∇g ∂ Φ vol µ eµ T t T ∂yj ∂yi ∂ ∂yi h j U µ i,j=1 ∂y = (II.1) + (II.2) .

∞ The integrand of Summand (II.2) is C (U)C-linear in ∂tϕT and hence in a ﬁnal form.

Z X (II.1) = Tr dρ(e ) ∇T,(ϕT , g)∂ ϕ Φ vol µ eµ t T h U µ Z X − Tr dρ(e ) (ad ⊗ ∇g) DT ϕ Φ vol µ ∂tϕT eµ T h U µ Z X X T ] i ∂ + Tr dρ(eµ) (∂t∇ )(eµ) , ϕT (y ) ⊗ ∂yi Φ vol h U µ i = (II.1.1) + (II.1.2) + (II.1.3) .

Both Summand (II.1.2) and Summand (II.1.3) vanish since

Tr ([a, b]c) = 0 if [b, c] = 0

52 for r × r matrices a, b, c.

Z X (II.1.1) = Tr dρ(e ) DT ((∂ ϕ )Φ)) vol µ eµ t T h U µ n Z X X − Tr dρ(e ) ∂ ϕ] (yi) DT ϕ] (yj) ⊗ ∂ ∂ Φ − ∇g ∂ Φ vol µ t T eµ T ∂yi ∂yj ∂ ∂yj h i U µ i,j=1 ∂y = (II.1.1.1) + (II.1.1.2) .

∞ The integrand of Summand (II.1.1.2) is C (U)C-linear in ∂tϕT and hence in a ﬁnal form. It can be combined with Summand (II.2) to give

(II.1.1.2) + (II.2) n Z X X = − Tr dρ(e ) [∂ ϕ] (yi) ,DT ϕ] (yj)] ⊗ ∂ ∂ Φ − ∇g ∂ Φ vol , µ t T eµ T ∂yi ∂yj ∂ ∂yj h i U µ i,j=1 ∂y which again vanishes due to Tr .

Z X (II.1.1.1) = dρ(e )Tr DT ((∂ ϕ )Φ) vol µ eµ t T h U µ Z X = dρ(eµ) eµTr ((∂tϕT )Φ) vol h U µ Z X Z X = eµ dρ(eµ) Tr ((∂tϕT )Φ) vol h − eµdρ(eµ) Tr ((∂tϕT )Φ) vol h U µ U µ = (II.1.1.1.1) + (II.1.1.1.2)

∞ The integrand of Summand (II.1.1.1.2) is C (U)C-linear in ∂tϕT and hence in a ﬁnal form. To extract the boundary term from Summand (II.1.1.1.1), consider the T -family of C-valued 1-forms on U αT := dρ Tr ((∂ ϕ )Φ) , (II,∂tϕT ) t T ∞ which depends C (U)C-linearly on ∂tϕT . Let

m X ξT = dρ(e ) Tr ((∂ ϕ )Φ) e (II,∂tϕT ) µ t T µ µ=1 be the T-family of dual -valued vector ﬁelds of αT on U with respect to the metric h. C (II,∂tϕT ) T ∞ Note that ξ depends C (U)C-linearly on ∂tϕT as well. Then (II,∂tϕT )

Z X (II.1.1.1.1) = e hξT , e i vol µ (II,∂tϕT ) µ h h U µ Z X Z = h∇h ξT , e i vol + hξT , P ∇h e i vol eµ (II,∂tϕT ) µ h h (II,∂tϕT ) µ eµ µ h h U µ U

The ﬁrst term is equal to Z Z Z T (− div ξ ) vol = d i T vol = i T vol , (II, ∂tϕT ) h ξ h ξ h U U (II, ∂tϕT ) ∂U (II, ∂tϕT )

53 which is the sought-for boundary term, whose integrand satisﬁes the requirement that it be ∞ C (U)C-linear in ∂tϕT . The second term is equal to Z P h dρ µ∇eµ eµ Tr ((∂tϕT )Φ) vol h U ∞ by construction, which is C (U)C-linear in ∂tϕT and hence in a ﬁnal form. In summary,

T Proposition 6.2.1. [first variation of dilaton term] Let (ϕT , ∇ ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. Then, d d Z (ρ,h;Φ) C Sdilaton(ϕT ) = Tr hdρ , ϕT dΦih vol h dt dt U Z = iξT vol h ∂U (II, ∂tϕT ) Z + dρPm ∇h e − Pm e dρ(e ) Tr ((∂ ϕ )Φ) vol . µ=1 eµ µ µ=1 µ µ t T h U T Pm Here, the first summand is the boundary term with ξ := (dρ(eµ) Tr ((∂tϕT )Φ))eµ (II,∂tϕT ) µ=1 ∞ ∞ C (U)C-linear in ∂tϕT ; the integrand of the second summand C (U)C-linear in ∂tϕT and they contribute additional zeroth-order terms to the equations of motion for (ϕ, ∇). In particular, while the dilaton term of the standard action modifies the equations of motion for (ϕ, ∇), it does not change the signature of the system.

6.3 The first variation of the gauge/Yang-Mills term and the Chern-Simons/ Wess-Zumino term Az To make sure that differential forms on Y of rank ≥ 2 are pull-pushed to (OX -valued-)differential Az T forms on X (cf. Lemma 2.1.11), we assume in this subsection that ϕT :(XT , ET ; ∇ ) → Y is a (∗2)-family of (∗2)-admissible maps. (Note that as the gauge/Yang-Mills term is defined through T a norm-squared, (∗1)-admissible family of (∗1)-admissible (ϕT , ∇ ) is enough for the derivation of the first variation formula of the gauge/Yang-Mills term but the result will be slightly messier.)

6.3.1 The ﬁrst variation of the gauge/Yang-Mills term

Let (e1, ··· , em) be an orthonormal frame on U. Then, over U, Z (h;B) T 1 0 2 S (ϕ , ∇ )C := − Tr k2πα F T + ϕ Bk vol gauge /YM T 2 ∇ T h h U Z 2 1 X 0 = − 2 Tr (2πα F∇T + ϕT B)(eµ, eν) vol h . U µ,ν Applying the following basic identities: T T T T T ∂ F T (e , e ) = D (∂ ∇ )(e ) − D (∂ ∇ )(e ) − (∂ ∇ )([e , e ]) , t ∇ µ ν eµ t ν eν t µ t µ ν X ∂ (ϕ B)(e , e ) = ∂ ϕ] (B )DT ϕ] (yi)DT ϕ] (yj) t T µ ν t T ij eµ T eν T i,j X + ϕ] (B ) DT ∂ ϕ] (yi) + (∂ ∇T )(e ), ϕ] (yi) DT ϕ] (yj) T ij eµ t T t µ T eν T i,j X + ϕ] (B )DT ϕ] (yi) DT ∂ ϕ] (yj) + (∂ ∇T )(e ), ϕ] (yj) . T ij eµ T eν t T t ν T i,j

54 and proceeding similarly to Sec. 6.1, one has the following results.

Z 2 d (h;B) T 1 X 0 S (ϕ , ∇ )C = − Tr ∂ (2πα F T + ϕ B)(e , e ) vol dt gauge /YM T 2 t ∇ T µ ν h U µ,ν Z X 0 0 = − Tr ∂t (2πα F∇T + ϕT B)(eµ, eν ) · (2πα F∇T + ϕT B)(eµ, eν ) vol h U µ,ν Z X 0 0 = − Tr 2πα ∂t F∇T (eµ, eν ) · (2πα F∇T + ϕT B)(eµ, eν ) vol h U µ,ν Z X 0 − Tr ∂t ϕT B(eµ, eν ) · (2πα F∇T + ϕT B)(eµ, eν ) vol h U µ,ν = (III.1) + (III.2) .

Z X 0 0 (III.1) := − Tr 2πα ∂t F∇T (eµ, eν ) · (2πα F∇T + ϕT B)(eµ, eν ) vol h U µ,ν Z X = − 2πα0 Tr DT (∂ ∇T )(e ) − DT (∂ ∇T )(e ) − (∂ ∇T )([e , e ]) eµ t ν eν t µ t µ ν U µ,ν 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h Z 0 = − 4πα i T vol h ξ T ∂U (III,∂t∇ ) Z 0 X T 0 P h − 4πα Tr (∂ ∇ )(e ) · (2πα F T + ϕ B)( ∇ e , e ) t ν ∇ T µ eµ µ ν U ν X T 0 − D (2πα F T + ϕ B)(e , e ) eµ ∇ T µ ν µ 1 X ν 0 − 2 e ([eµ, eλ])(2πα F∇T + ϕT B)(eµ, eλ) vol h . µ,λ Here, T X T 0 C ξ T := Tr (∂ ∇ )(e ) · (2πα F T + ϕ B)(e , e ) e ∈ T (U /T ) (III,∂t∇ ) t ν ∇ T µ ν µ ∗ T µ,ν

C T is OU -linear in ∂t∇ ; and the second summand contributes to the equations of motion for (ϕ, ∇). The latter are standard terms from non-Abelian Yang-Mills theory with additional terms from ϕB.

Z X 0 (III.2) := − Tr ∂t ϕT B(eµ, eν ) · (2πα F∇T + ϕT B)(eµ, eν ) vol h U µ,ν Z X X = − Tr ∂ ϕ] (B )DT ϕ] (yi)DT ϕ] (yj) t T ij eµ T eν T U µ,ν i,j X + ϕ] (B ) DT ∂ ϕ] (yi) + (∂ ∇T )(e ), ϕ] (yi) DT ϕ] (yj) T ij eµ t T t µ T eν T i,j X + ϕ] (B )DT ϕ] (yi) DT ∂ ϕ] (yj) + (∂ ∇T )(e ), ϕ] (yj) T ij eµ T eν t T t ν T i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h = (III.2.1) + + (III.2.2.1) + (III.2.2.2) + (III.2.3.1) + (III.2.3.2) in the order of the appearance of the ﬁve summands after the expansion.

55 Z X X (III.2.1) := − Tr ∂ ϕ] (B )DT ϕ] (yi)DT ϕ] (yj) t T ij eµ T eν T U µ,ν i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h Z X X := − Tr (∂ ϕ )B DT ϕ] (yi)DT ϕ] (yj) t T ij eµ T eν T U µ,ν i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h

C has an integrand OU -linear in ∂tϕT and hence in a ﬁnal form.

(III.2.2.1) + (III.2.3.1) Z X X := − Tr ϕ] (B )DT ∂ ϕ] (yi)DT ϕ] (yj) T ij eµ t T eν T U µ,ν i,j X + ϕ] (B )DT ϕ] (yi)DT ∂ ϕ] (yj) T ij eµ T eν t T i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h Z X X = − 2 Tr DT ∂ ϕ] (yi) ϕ] (B ) DT ϕ] (yj) eµ t T T ij eν T U µ,ν i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h Z = − 2 iξT vol h ∂U (III,∂tϕT ) Z X X ] i − 2 Tr ∂tϕT (y ) U ν i,j ] T ] j 0 P h ϕ (B ) D ϕ (y ) · (2πα F T + ϕ B)( ∇ e , e ) T ij eν T ∇ T µ eµ µ ν X T ] T ] j 0 − D ϕ (B ) D ϕ (y ) · (2πα F T + ϕ B)(e , e ) vol . eµ T ij eν T ∇ T µ ν h µ Here,

T X X X ] i ] T ] j 0 ξ := ∂ ϕ (y ) ϕ (B ) D ϕ (y ) · (2πα F T + ϕ B)(e , e ) e (III,∂tϕT ) t T T ij eν T ∇ T µ ν µ µ ν i,j

C C in T∗(UT /T ) is OU -linear in ∂tϕT ; and the second summand contributes to (ρ,h;Φ,g,B,C) δSstandard (ϕ, ∇)/δϕ-part of the equations of motion for (ϕ, ∇).

(III.2.2.2) + (III.2.3.2) Z X X := − Tr ϕ] (B )(∂ ∇T )(e ), ϕ] (yi)DT ϕ] (yj) T ij t µ T eν T U µ,ν i,j X + ϕ] (B )DT ϕ] (yi)(∂ ∇T )(e ), ϕ] (yj) T ij eµ T t ν T i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h .

C T has an integrand OU -linear in ∂t∇ and hence in a ﬁnal form. In summary,

56 T Proposition 6.3.1.1. [ﬁrst variation of gauge/Yang-Mills term] Let (ϕT , ∇ ) be a (∗2)- admissible family of (∗2)-admissible pairs. Then Z d (h;B) T 1 d 0 2 S (ϕ , ∇ )C = − Tr k2πα F T + ϕ Bk vol dt gauge /YM T 2 dt ∇ T h h U Z Z 0 = − 4πα i T vol h − 2 i T vol h ξ T ξ ∂U (III,∂t∇ ) ∂U (III,∂tϕT ) Z 0 X T 0 P h − 4πα Tr (∂ ∇ )(e ) · (2πα F T + ϕ B)( ∇ e , e ) t ν ∇ T µ eµ µ ν U ν X T 0 − D (2πα F T + ϕ B)(e , e ) eµ ∇ T µ ν µ 1 X ν 0 − 2 e ([eµ, eλ])(2πα F∇T + ϕT B)(eµ, eλ) vol h . µ,λ Z X X − Tr ϕ] (B )(∂ ∇T )(e ), ϕ] (yi)DT ϕ] (yj) T ij t µ T eν T U µ,ν i,j X + ϕ] (B )DT ϕ] (yi)(∂ ∇T )(e ), ϕ] (yj) T ij eµ T t ν T i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h Z X X − Tr (∂ ϕ )B DT ϕ] (yi)DT ϕ] (yj) t T ij eµ T eν T U µ,ν i,j 0 · (2πα F∇T + ϕT B)(eµ, eν ) vol h Z X X ] i − 2 Tr ∂tϕT (y ) U ν i,j ] T ] j 0 P h ϕ (B ) D ϕ (y ) · (2πα F T + ϕ B)( ∇ e , e ) T ij eν T ∇ T µ eµ µ ν X T ] T ] j 0 − D ϕ (B ) D ϕ (y ) · (2πα F T + ϕ B)(e , e ) vol . eµ T ij eν T ∇ T µ ν h µ Here,

T X T 0 ξ T := Tr (∂ ∇ )(e ) · (2πα F T + ϕ B)(e , e ) e , (III,∂t∇ ) t ν ∇ T µ ν µ µ,ν T X X X ] i ] T ] j 0 ξ := ∂ ϕ (y ) ϕ (B ) D ϕ (y ) · (2πα F T + ϕ B)(e , e ) e (III,∂tϕT ) t T T ij eν T ∇ T µ ν µ µ ν i,j

C C T C in T∗(UT /T ) , with the ﬁrst OU -linear in ∂t∇ and the second OU -linear in ∂tϕT .

6.3.2 The ﬁrst variation of the Chern-Simons/Wess-Zumino term for lower dimensional D-branes

Az T This is an update of [L-Y8: Sec.6.2] (D(13.1)) in the current setting. Let ϕT :(X , ET ; ∇ ) → Y be an (∗2)-family of (∗2)-admissible maps. We work out the ﬁrst variation of the Chern- (C,B) Simons/Wess-Zumino term SCS/WZ (ϕ, ∇) for the cases where m := dim X = 0, 1, 2, 3. As the details involve no identities or techniques that have not yet been used in Sec. 6.1, Sec. 6.2, and/or Sec. 6.3.1, we only summarize the ﬁnal results below.

57 6.3.2.1 D(−1)-brane world-point (m = 0)

(C(0)) ] For a D(−1)-brane world-point, dim X = 0, ∇ = 0, and SCS/WZ (ϕT ) = T−1 · Tr (ϕT (C(0))). It follows that d (C ) S (0) (ϕ ) = T Tr ∂ (ϕ] (C )) = T Tr (∂ ϕ )C . dt CS/WZ T −1 t T (0) −1 t T (0)

6.3.2.2 D-particle world-line (m = 1)

For a D-particle world-line, dim X = 1. Let e1 be the orthonormal frame on an open set U ⊂ X; e1 its dual co-frame. Then, over U, Z Z n (C(1)) X S (ϕ )C = T Tr ϕ C = T Tr ϕ] (C ) · DT ϕ] (yi) e1 . CS/WZ T 0 T (1) 0 T i e1 T U U i=1 It follows that

d (C(1)) X S (ϕ) = T Tr ∂ ϕ] (yi)ϕ] (C ) dt CS/WZ 0 t T T i ∂U i Z X Z X − T Tr ∂ ϕ] (yi)DT ϕ] (C ) e1 + T Tr D ϕ] (yi) · (∂ ϕ )C e1 . 0 t T e1 T i 0 e1 T t T i U i U i

6.3.2.3 D-string world-sheet (m = 2)

Denote

˘ X i j X ˘ i j C(2) := C(2) + C(0)B = (Cij + C(0)Bij) dy ⊗ dy = Cijdy ⊗ dy ij i,j

1 n in a local coordinate (y , ··· , y ) of Y . For a D-string world-sheet, dim X = 2. Let (e1, e2) be an orthonormal frame on an open set U ⊂ X;(e1, e2) its dual co-frame. Then, over U,

(C ,C ,B) (0) (2) T C SCS/WZ (ϕT , ∇ ) n Z X = T Tr ϕ] (C˘ ) DT ϕ] (yi) DT ϕ] (yj) 1 T ij e1 T e2 T U i,j=1 0 ] 0 ] 1 2 + πα ϕT (C(0)) F∇T (e1, e2) + πα F∇T (e1, e2) ϕT (C(0)) e ∧ e Z n X ] T ] i T ] j 0 ] 1 2 = T Tr ϕ (C˘ ) D ϕ (y ) D ϕ (y ) + 2 πα ϕ (C ) F T (e , e ) e ∧ e . 1 T ij e1 T e2 T T (0) ∇ 1 2 U i,j=1

58 It follows that

(C ,C ,B) d S (0) (2) (ϕ , ∇T )C dt CS/WZ T Z n X ] ˘ T ] i T ] j 0 ] 1 2 = T1 Tr ∂t ϕT (Cij ) De1 ϕT (y ) De2 ϕT (y ) + 2 πα ϕT (C(0)) F∇T (e1, e2) e ∧ e U i,j=1 Z Z 1 2 0 1 2 = T1 i T (e ∧ e ) + 2πα T1 i T (e ∧ e ) ξ ξ T ∂U (IV,∂tϕT ) ∂U (IV,∂t∇ ) Z n X ] i T ] j ] ˘ 1 T ] j ] ˘ 2 h h 1 2 + T1 Tr ∂tϕ (y ) De2 ϕT (y )ϕT (Cij )e − De1 ϕT (y )ϕT (Cij )e (∇e1 e1 + ∇e2 e2) e ∧ e U i,j=1 ZZ X ] i T T ] j ] ˘ T T ] j ] ˘ 1 2 − T1 Tr ∂tϕT (y ) De1 De2 ϕT (y ) · ϕT (Cij ) − De2 De1 ϕT (y ) · ϕT (Cij ) e ∧ e U i,j Z X ] ˘ T ] i T ] j 1 2 + T1 Tr ∂tϕT (Cij )De1 ϕT (y )De2 ϕT (y ) e ∧ e U i,j Z 0 ] 1 2 + 2πα T1 Tr ∂tϕT (C(0)) · F∇T (e1, e2) e ∧ e U Z 0 ] T 1 T 2 h h T 1 2 + 2πα T1 Tr ϕT (C(0)) (∂t∇ )(e2) e − (∂t∇ )(e1) e (∇e1 e1 + ∇e2 e2) − (∂t∇ )([e1, e2]) e ∧ e U Z 0 T ] T T ] T 1 2 − 2πα T1 Tr De1 ϕT (C(0)) · (∂t∇ )(e2) − De2 ϕT (C(0)) · (∂t∇ )(e1) e ∧ e . U Here,

ξT := Tr P ∂ ϕ] (yi)DT ϕ] (yj)ϕ] (C˘ ) e − Tr P ∂ ϕ] (yi)DT ϕ] (yj)ϕ] (C˘ ) e , (IV,∂tϕT ) i,j t T e2 T T ij 1 i,j t T e1 T T ij 2 T ] T ] T ξ T := Tr ϕ (C ) · (∂ ∇ )(e ) e − Tr ϕ (C ) · (∂ ∇ )(e ) e (IV,∂t∇ ) T (0) t 2 1 T (0) t 1 2

C C C T in T∗(UT /T ) , with the ﬁrst OU -linear in ∂tϕT and the second OU -linear in ∂t∇ .

6.3.2.4 D-membrane world-volume (m = 3)

Denote ˘ C(3) := C(3) + C(1) ∧ B X i j k X i j k = (Cijk + CiBjk + CjBki + CkBij) dy ⊗ dy ⊗ dy = C˘ijkdy ⊗ dy ⊗ dy i,j,k i,j,k in a local coordinate (y1, ··· , yn) of Y . For D-membrane world-volume, dim X = 3. Let 1 2 3 (e1, e2, e3) be an orthonormal frame on an open set U ⊂ X;(e , e , e ) its dual co-frame. Then, over U,

(C ,C ,B) (1) (3) T C SCS/WZ (ϕT , ∇ ) n Z X = T Tr ϕ](C˘ ) DT ϕ] (yi) DT ϕ] (yj) DT ϕ] (yk) 2 ijk e1 T e2 T e3 T U i,j,k=1 n 0 X X (λµν) ] T ] i 1 2 3 + 2πα (−1) ϕT (Ci) Dλ ϕT (y ) F∇T (eµ, eν ) e ∧ e ∧ e i=1 (λµν)∈Sym3 It follows that

59 (C ,C ,B) d S (1) (3) (ϕ , ∇T )C dt CS/WZ T Z n X ] ˘ T ] i T ] j T ] k = T2 Tr ∂t ϕ (Cijk) De1 ϕT (y ) De2 ϕT (y ) De3 ϕT (y ) U i,j,k=1 n 0 X X (λµν) ] T ] i 1 2 3 T + 2πα (−1) ϕT (Ci) Deλ ϕT (y ) F∇ (eµ, eν ) e ∧ e ∧ e i=1 (λµν)∈Sym3 Z Z 1 2 3 0 1 2 3 = T2 iξT (e ∧ e ∧ e ) + 4πα T2 iξT (e ∧ e ∧ e ) ˘ (IV,∂ ϕ ;C ) ∂U (IV,∂tϕT ;C(3)) ∂U t T (1) Z 0 1 2 3 + 2πα T2 i T (e ∧ e ∧ e ) ξ T ∂U (IV,∂t∇ ) Z X ] ˘ ] i T ] j T ] k 1 + T2 Tr ϕT (Cijk)∂tϕT (y )De2 ϕT (y )De3 ϕT (y ) e U i,j,k X ] ˘ ] i T ] j T ] k 2 − Tr ϕT (Cijk)∂tϕT (y )De1 ϕT (y )De3 ϕT (y ) e i,j,k X ] ˘ ] i T ] j T ] k 3 P3 h 1 2 3 − Tr ϕT (Cijk)∂tϕT (y )De2 ϕT (y )De1 ϕT (y ) e ( µ=1∇eµ eµ) e ∧ e ∧ e i,j,k Z X ] i T ] ˘ T ] j T ] k T ] ˘ T ] j T ] k − T2 Tr ∂tϕT (y ) De1 ϕT (Cijk)De2 ϕT (y )De3 ϕT (y ) − De2 ϕT (Cijk)De1 ϕT (y )De3 ϕT (y ) U i,j,k T ] ˘ T ] j T ] k 1 2 3 − De3 ϕT (Cijk)De2 ϕT (y )De1 ϕT (y ) e ∧ e ∧ e Z X ] ˘ T ] i T ] j T ] k 1 2 3 + T2 Tr ∂tϕT (Cijk)De1 ϕT (y )De2 ϕT (y )De3 ϕT (y ) e ∧ e ∧ e U i,j,k Z 0 X ] ] i 1 + 4πα T2 Tr ϕT (Ci)∂tϕT (y )F∇T (e2, e3) e U i X ] ] i 2 − Tr ϕT (Ci)∂tϕT (y )F∇T (e1, e3) e i X ] ] i 3 P3 h 1 2 3 + Tr ϕT (Ci)∂tϕT (y )F∇T (e1, e2) e ( µ=1∇eµ eµ) e ∧ e ∧ e i Z 0 X ] i T ] T ] − 4πα T2 Tr ∂tϕT (y ) De1 ϕT (Ci)F∇T (e2, e3) − De2 ϕT (Ci)F∇T (e1, e3) U i T ] 1 2 3 + De3 ϕT (Ci)F∇T (e1, e2) e ∧ e ∧ e Z 0 X X (λµν) ] T ] i 1 2 3 T + 2πα T2 Tr (−1) ∂tϕT (Ci) Deλ ϕT (y ) F∇ (eµ, eν ) e ∧ e ∧ e U i (λµν)∈Sym3 Z 0 X ] T ] i T T ] i T 1 + 2πα T2 Tr ϕT (Ci) De3 ϕT (y )(∂t∇ )(e2) − De2 ϕT (y )(∂t∇ )(e3) e U i X ] T ] i T T ] i T 2 + Tr ϕT (Ci) De3 ϕT (y )(∂t∇ )(e1) + De1 ϕT (y )(∂t∇ )(e3) e i X ] T ] i T T ] i T 3 P3 h + Tr ϕT (Ci) De1 ϕT (y )(∂t∇ )(e2) − De2 ϕT (y )(∂t∇ )(e1) e ( µ=1∇eµ eµ) i e1 ∧ e2 ∧ e3 Z 0 X X (λµν) ] T ] i + 2πα T2 Tr (−1) ϕT (Ci) (∂t∇ )(eλ), ϕT (y ) F∇T (eµ, eν ) U i (λµν)∈Sym3 ] T ] i 1 2 3 − ϕT (Ci) Deλ ϕT (y )(∂tϕT )([eµ, eν ]) e ∧ e ∧ e .

60 Here,

T X ] ˘ ] i T ] j T ] k ξ ˘ := Tr ϕ (Cijk)∂tϕ (y )De ϕ (y )De ϕ (y ) e1 (IV,∂tϕT ;C(3)) T T 2 T 3 T i,j,k X ] ˘ ] i T ] j T ] k − Tr ϕT (Cijk)∂tϕT (y )De1 ϕT (y )De3 ϕT (y ) e2 i,j,k X ] ˘ ] i T ] j T ] k − Tr ϕT (Cijk)∂tϕT (y )De2 ϕT (y )De1 ϕT (y ) e3 , i,j,k

T X ] ] i ξ := Tr ϕ (C )∂ ϕ (y )F T (e , e ) e (IV,∂tϕT ;C(1)) T i t T ∇ 2 3 1 i X ] ] i X ] ] i − Tr ϕT (Ci)∂tϕT (y )F∇T (e1, e3) e2 + Tr ϕT (Ci)∂tϕT (y )F∇T (e1, e2) e3 , i i T X ] T ] i T T ] i T ξ T := Tr ϕ (C ) D ϕ (y )(∂ ∇ )(e ) − D ϕ (y )(∂ ∇ )(e ) e (IV,∂t∇ ) T i e3 T t 2 e2 T t 3 1 i X ] T ] i T T ] i T + Tr ϕT (Ci) De3 ϕT (y )(∂t∇ )(e1) + De1 ϕT (y )(∂t∇ )(e3) e2 i X ] T ] i T T ] i T + Tr ϕT (Ci) De1 ϕT (y )(∂t∇ )(e2) − De2 ϕT (y )(∂t∇ )(e1) e3 i

C C C T in T∗(UT /T ) , with the ﬁrst two OU -linear in ∂tϕT and the third OU -linear in ∂t∇ .

7 The second variation of the enhanced kinetic term for maps

2 2 T Let T = (−ε, ε) ⊂ R , with coordinate (s, t), and (ϕT , ∇ ) be an (∗2)-admissible family of (∗1)-admissible pairs, with (ϕ(0,0), ∇(0,0)) = (ϕ, ∇). Assume further that

Dξ∂sAϕT ⊂ Comm (AϕT ) for all ξ ∈ T∗(XT /T ) .

We work out in this section the second variation formula of the enhanced kinetic term S(ρ,h;Φ,g) (ϕ, ∇) in the standard action S(ρ,h;Φ,g,B,C)(ϕ, ∇). map:kinetic+ standard

7.1 The second variation of the kinetic term for maps Recall Z ∇T (h;g) T 1 T T E (ϕT ) := Smap:kinetic(ϕT , ∇ ) := Tm−1 Tr D ϕT ,D ϕT (h,g) vol h , 2 X with the understanding that all expressions are taken on X(s,t) with (s, t) varying in T . µ Let U ⊂ X be an open set with an orthonormal frame (eµ)µ=1, ··· , m. Let (e )µ=1, ··· , m be the Az dual co-frame. Assume that U is small enough so that ϕT (UT ) is contained in a coordinate chart of Y , with coordinates (y1, ··· , yn). Then, as in Sec. 6.1, over U,

m ∂ T Z D E ∇ X T,(ϕT ,g) T E (ϕT ) = Tm−1 Tr ∇e ∂tϕT ,De ϕT vol h ∂t µ µ g U µ=1 Z m X D g T E + Tm−1 Tr (ad ⊗∇ )DT ϕ ∂tϕT ,De ϕT vol h eµ T µ g U µ=1 m n Z X D X E + T (∂ ∇T )(e ), ϕ] (yi) ⊗ ∂ ,DT ϕ vol m−1 t µ T ∂yi eµ T h g U µ=1 i=1 =: (I 2.1) + (I 2.2) + (I 2.3) ;

61 and 2 ∂ T ∂ ∂ ∂ E∇ (ϕ ) = (I 2.1) + (I 2.2) + (I 2.3) . ∂s ∂t T ∂s ∂s ∂s Which we now compute term by term.

∂ 2 The term ∂s (I .1)

∂ ∂ Z m D E 2 X T,(ϕT ,g) T (I .1) = Tm−1 Tr ∇e ∂tϕT ,De ϕT vol h ∂s ∂s µ µ g U µ=1 Z D E X T,(ϕT ,g) T = Tm−1 Tr ∂s ∇e ∂tϕT ,De ϕT vol h µ µ g U µ Z D E X T,(ϕT ,g) T = Tm−1 Tr ∂s∇e ∂tϕT ,De ϕT vol h µ µ g U µ Z D E X T,(ϕT ,g) T + Tm−1 Tr ∇e ∂tϕT , ∂sDe ϕT vol h µ µ g U µ = (I 2.1.1) + (I 2.1.2) .

(a) Term (I 2.1.1) Z D E 2 X T,(ϕT ,g) T (I .1.1) := Tm−1 Tr ∂s∇e ∂tϕT ,De ϕT vol h µ µ g U µ Z D E X T,(ϕT ,g) T = Tm−1 Tr ∇e ∂s∂tϕT ,De ϕT vol h µ µ g U µ Z X D T E + Tm−1 Tr F T,(ϕT ,g) (∂s, eµ) ∂tϕT ,De ϕT vol h ∇ µ g U µ = (I 2.1.1.1) + (I 2.1.1.2) . 2 For Term (I .1.1.1), as in Sec. 6.1 for Summand (I.1.1), consider the 1-form on UT /T T T α 2 := Tr ∂ ∂ ϕ ,D ϕ (I ,∂s∂tϕT ) s t T T g and let n T X T ξ 2 := Tr ∂ ∂ ϕ ,D ϕ e (I ,∂s∂tϕT ) s t T eµ T g µ µ=1 be its dual on UT /T with respect to h. Then, Z 2 (I .1.1.1) = Tm−1 i T vol h ξ 2 ∂U (I ,∂s∂tϕT ) Z D E T P T,(ϕT ,g) T + T Tr ∂ ∂ ϕ , DP h − ∇ D ϕ vol . m−1 s t T ∇ eµ µ eµ eµ T h U µ eµ g For Term (I 2.1.1.2), recall Lemma 3.2.2.5. Then, Z 2 D X T E (I .1.1.2) = − Tm−1 Tr ∂tϕT , F T,(ϕT ,g) (∂s, eµ)De ϕT vol h ∇ µ g U µ Z X T + Tm−1 Tr F∇(∂s, eµ) , h∂tϕT ,Deµ ϕT ig vol h U µ Z D X T E = − Tm−1 Tr ∂tϕT , F T,(ϕT ,g) (∂s, eµ)De ϕT vol h . ∇ µ g U µ

62 Here, n T T,(ϕT ,g) T,(ϕT ,g) P ] i ∂ F T,(ϕ ,g) (∂ , e ) D ϕ = ∂ ∇ − ∇ ∂ D ϕ (y ) ⊗ ∇ T s µ eµ T s eµ eµ s i=1 eµ T ∂yi n n n X T ] i ∂ X ] i X T ] j g ∂ = [(∂s∇ )(eµ),Deµ ϕT (y )] ⊗ i + Deµ ϕT (y ) [(∂s∇ )(eµ), ϕT (y )] ⊗ ∇ ∂ i ∂y j ∂y i=1 i=1 j=1 ∂y n n X X + D ϕ] (yi) DT ϕ] (yj) ∂ ϕ] (yk) ⊗ ∇g ∇g ∂ eµ T eµ T s T ∂ ∂ ∂yi ∂yk ∂yj i=1 j,k=1 − ∂ ϕ] (yk) DT ϕ] (yj) ⊗ ∇g ∇g ∂ s T eµ T ∂ ∂ ∂yi ∂yj ∂yk explicitly.

(b) Term (I 2.1.2) Z D E 2 X T,(ϕT ,g) T (I .1.2) := Tm−1 Tr ∇e ∂tϕT , ∂sDe ϕT vol h µ µ g U µ Z D E X T,(ϕT ,g) T,(ϕT ,g) = Tm−1 Tr ∇e ∂tϕT , ∇e ∂sϕT vol h µ µ g U µ Z D E X T,(ϕT ,g) g + Tm−1 Tr ∇e ∂tϕT , (ad ⊗ ∇ )DT ϕ ∂sϕT vol h µ eµ T g U µ Z D E X T,(ϕT ,g) Pn T ] i ∂ + Tm−1 Tr ∇eµ ∂tϕT , i=1 (∂s∇ )(eµ), ϕT (y ) ⊗ i vol h . ∂y g U µ

As in Sec. 6.1, consider the 1-forms on UT /T ,

T T,(ϕT ,g) α 2 T,(ϕ ,g) = Tr ∂tϕT , ∇ ∂sϕT , (I ,∂tϕT ,∇ T ) g T g α 2 T = Tr ∂ ϕ , (ad ⊗ ∇ ) T ∂ ϕ , (I ,∂tϕT ,D ϕT ) t T D ϕT s T g T Pn T ] i ∂ α 2 T = Tr ∂ ϕ , [∂ ∇ , ϕ (y )] ⊗ i (I ,∂tϕT ,∂s∇ ) t T i=1 s T ∂y g and let

T X T,(ϕT ,g) ξ 2 T,(ϕ ,g) = Tr ∂tϕT , ∇ ∂sϕT eµ , (I ,∂tϕT ,∇ T ) eµ g µ T X g ξ 2 T = Tr ∂tϕT , (ad ⊗ ∇ ) T ∂sϕT eµ , (I ,∂tϕT ,D ϕT ) Deµ ϕT g µ T X Pn T ] i ∂ ξ 2 T = Tr ∂ ϕ , [(∂ ∇ )(e ), ϕ (y )] ⊗ i e (I ,∂tϕT ,∂s∇ ) t T i=1 s µ T ∂y g µ µ be their respective dual on UT /T with respect to h. Then, Z 2 (I .1.2) = Tm−1 i ξT + ξT + ξT vol h 2 T,(ϕ ,g) (I2,∂ ϕ ,DT ϕ ) (I2,∂ ϕ ,∂ ∇T ) ∂U (I ,∂tϕT ,∇ T ) t T T t T s Z D E T,(ϕT ,g) P T,(ϕT ,g) T,(ϕT ,g) + Tm−1 Tr ∂tϕT , ∇P h − ∇ ∇ ∂sϕT vol h ∇ eµ µ eµ eµ U µ eµ g Z D g T + Tm−1 Tr ∂tϕT , (ad ⊗ ∇ )D ϕT P ∇h e U µ eµ µ E P T,(ϕT ,g) g − ∇ (ad ⊗ ∇ ) T ∂ ϕ vol µ eµ D ϕT s T h eµ g Z D n + T Tr ∂ ϕ , P (∂ ∇T )(P ∇h e ), ϕ] (yi) m−1 t T i=1 s µ eµ µ T U ] ∂ E − P ∇T,(ϕT ,g)(∂ ∇T )(e ), ϕ (yi) ⊗ vol . µ eµ s µ T i h ∂y g

63 ∂ 2 The term ∂s (I .2)

Z m ∂ 2 ∂ X D g T E (I .2) = Tm−1 Tr (ad ⊗∇ )DT ϕ ∂tϕT ,De ϕT vol h ∂s ∂s eµ T µ g U µ=1 Z X D g T E = Tm−1 Tr ∂s (ad ⊗ ∇ )De ϕT ∂tϕT ,De ϕT vol h µ µ g U µ Z X D g T E + Tm−1 Tr (ad ⊗ ∇ )De ϕT ∂tϕT , ∂sDe ϕT vol h µ µ g U µ = (I 2.2.1) + (I 2.2.2) .

(a) Term (I 2.2.1) Z 2 X D g T E (I .2.1) = Tm−1 Tr (ad ⊗ ∇ )DT (∂s∂tϕT ) ,De ϕT vol h eµ µ g U µ Z X D X ∂ E + T Tr DT ∂ ϕ] (yj), ∂ ϕ] (yi) ⊗ ∇g ,DT ϕ vol m−1 eµ s T t T ∂ i eµ T h j ∂y g U µ i,j ∂y Z X D X + T Tr DT ϕ] (yj), ∂ ϕ] (yi)∂ ϕ] (yk) ⊗ Rg ∂ , ∂ ∂ m−1 eµ T t T s T ∂yk ∂yj ∂yi U µ i,j,k − ∂ ϕ] (yi)DT ϕ] (yj), ∂ ϕ] (yk) ⊗ ∇g ∇g ∂ , t T eµ T s T ∂ ∂ ∂yi ∂yj ∂yk T E De ϕT vol h µ g Z X D X ] j ] i g ∂ T E + T Tr ad T ϕ (y ), ∂ ϕ (y ) ⊗ ∇ ,D ϕ vol . m−1 (∂s∇ )(eµ) T t T ∂ i eµ T h j ∂y g U µ i,j ∂y The integrand of the first summand captures a related part in the system of equations of motion for (ϕ, ∇). The integrand of the second summand is tensorial in ∂tϕT and first-order differential operatorial in ∂sϕT . The integrand of the third summand is tensorial in both ∂tϕT and ∂sϕT . T The integrand of the fourth summand is tensorial in ∂tϕT and ∂s∇ .

(b) Term (I 2.2.2) Z 2 X D g T E (I .2.2) = Tm−1 Tr (ad ⊗ ∇ )De ϕT ∂tϕT , ∂sDe ϕT vol h µ µ g U µ Z D E X g T,(ϕT ,g) = Tm−1 Tr (ad ⊗ ∇ )De ϕT ∂tϕT , ∇e ∂sϕT vol h µ µ g U µ Z X D g g T E − Tm−1 Tr (ad ⊗ ∇ )De ϕT ∂tϕT , (ad ⊗ ∇ )∂sϕT De ϕT vol h µ µ g U µ Z X D g X T ] i ∂ E + Tm−1 Tr (ad ⊗ ∇ )De ϕT ∂tϕT , (∂s∇ )(eµ), ϕT (y ) ⊗ i vol h . µ ∂y g U µ i

The integrand of the first summand is tensorial in ∂tϕT and first-order differential operatorial in ∂sϕT . The integrand of the second summand is tensorial in both ∂tϕT and ∂sϕT . The integrand T of the third summand is tensorial in ∂tϕT and ∂s∇ .

∂ 2 The term ∂s (I .3)

64 Z m n ∂ 2 ∂ X D X T ] i ∂ T E (I .3) = Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i ,Deµ ϕT vol h ∂s ∂s ∂y g U µ=1 i=1 Z m n X D X T ] i ∂ T E = Tm−1 ∂s (∂t∇ )(eµ), ϕT (y ) ⊗ i ,Deµ ϕT vol h ∂y g U µ=1 i=1 Z m n X D X T ] i ∂ T E + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , ∂sDeµ ϕT vol h ∂y g U µ=1 i=1 = (I 2.3.1) + (I 2.3.2) .

(a) Term (I 2.3.1)

Z m n 2 X D X T ] i ∂ T E (I .3.1) = Tm−1 ∂s (∂t∇ )(eµ), ϕT (y ) ⊗ i ,De ϕT vol h ∂y µ g U µ=1 i=1 Z m n X D X T ] i ∂ T E = Tm−1 (∂s∂t∇ )(eµ), ϕT (y ) ⊗ i ,De ϕT vol h ∂y µ g U µ=1 i=1 Z m n X D X T ] i ∂ + Tm−1 (∂t∇ )(eµ), ∂sϕT (y ) ⊗ ∂yi U µ=1 i=1 E + (∂ ∇T )(e ), ϕ] (yi)P ∂ ϕ] (yj) ⊗ ∇g ∂ ,DT ϕ vol . t µ T j s T ∂ ∂yi eµ T h ∂yj g The integrand of the ﬁrst summand captures a related part in the system of equations of motion T for (ϕ, ∇). The integrand of the second summand is tensorial in ∂sϕT and ∂t∇ .

(b) Term (I 2.3.2)

Z m n 2 X D X T ] i ∂ T E (I .3.2) = Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , ∂sDe ϕT vol h ∂y µ g U µ=1 i=1 Z m D n E X X T ] i ∂ T,(ϕT ,g) = Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , ∇e ∂sϕT vol h ∂y µ g U µ=1 i=1 m Z X D n E − T P (∂ ∇T )(e ), ϕ] (yi) ⊗ ∂ , (ad ⊗ ∇g) DT ϕ vol m−1 i=1 t µ T i ∂sϕT eµ T h ∂y g U µ=1 Z m n X X D T ] i ∂ T ] j ∂ E + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , (∂s∇ )(eµ), ϕT (y ) ⊗ j vol h . ∂y ∂y g U µ=1 i,j=1

T The integrand of the first summand is tensorial in ∂t∇ and first-order differential operatorial T in ∂sϕT . The integrand of the second summand is tensorial in ∂sϕT and ∂t∇ . The integrand T T of the third summand is tensorial in both ∂t∇ and ∂s∇ .

Finally, recall Lemma 3.2.2.4 and note that with the additional assumption at the beginning of this section, all the inner products Tr h · , · ig that appear in the calculation above are deﬁned.

In summary,

T Proposition 7.1.1. [second variation of kinetic term for maps] Let (ϕT , ∇ ) be a (∗2)- admissible T -family of (∗1)-admissible pairs with the additional assumption that

65 Dξ∂sAϕT ⊂ Comm (AϕT ) for all ξ ∈ T∗(XT /T ). Then,

T Z ∂ ∂ ∇ C ∂ ∂ 1 T T ∂s ∂t E (ϕT ) = ∂s ∂t 2 Tm−1 Tr hD ϕT ,D ϕT i(h,g) vol h U Z = Tm−1 i T vol h ξ 2 ∂U (I ,∂s∂tϕT ) Z + Tm−1 i ξT + ξT + ξT vol h 2 T,(ϕ ,g) (I2,∂ ϕ ,DT ϕ ) (I2,∂ ϕ ,∂ ∇T ) ∂U (I ,∂tϕT ,∇ T ) t T T t T s Z D E T P T,(ϕT ,g) T + Tm−1 Tr ∂s∂tϕT , DP h − ∇e De ϕT vol h µ ∇e eµ µ µ µ U µ g Z X D g T E + Tm−1 Tr (ad ⊗ ∇ )DT (∂s∂tϕT ) ,Deµ ϕT vol h eµ g U µ Z m n X D X T ] i ∂ T E + Tm−1 (∂s∂t∇ )(eµ), ϕT (y ) ⊗ i ,Deµ ϕT vol h ∂y g U µ=1 i=1 Z D E T,(ϕT ,g) P T,(ϕT ,g) T,(ϕT ,g) + Tm−1 Tr ∂tϕT , ∇P h − ∇e ∇e ∂sϕT vol h ∇ eµ µ µ µ U µ eµ g Z D g + Tm−1 Tr ∂tϕT , (ad ⊗ ∇ ) T D ϕT P ∇h e U µ eµ µ E P T,(ϕT ,g) g − ∇eµ (ad ⊗ ∇ )DT ϕ ∂sϕT vol h µ eµ T g Z X D X T ] j ] i g ∂ T E + Tm−1 Tr Deµ ∂sϕT (y ), ∂tϕT (y ) ⊗ ∇ ∂ i ,Deµ ϕT vol h j ∂y g U µ i,j ∂y Z X D X T ] j ] i ] k g ∂ ∂ ∂ + T Tr D ϕ (y ), ∂ ϕ (y )∂ ϕ (y ) ⊗ R , m−1 eµ T t T s T ∂yk ∂yj ∂yi U µ i,j,k − ∂ ϕ] (yi)DT ϕ] (yj ), ∂ ϕ] (yk) ⊗ ∇g ∇g ∂ , t T eµ T s T ∂ ∂ ∂yi ∂yj ∂yk T E Deµ ϕT vol h g Z D E X g T,(ϕT ,g) + Tm−1 Tr (ad ⊗ ∇ )Deµ ϕT ∂tϕT , ∇eµ ∂sϕT vol h g U µ Z X D g g T E − Tm−1 Tr (ad ⊗ ∇ )Deµ ϕT ∂tϕT , (ad ⊗ ∇ )∂sϕT Deµ ϕT vol h g U µ Z D X T E − Tm−1 Tr ∂tϕT , F∇T,(ϕT ,g) (∂s, eµ)Deµ ϕT vol h g U µ Z D n + T Tr ∂ ϕ , P (∂ ∇T )(P ∇h e ), ϕ] (yi) m−1 t T i=1 s µ eµ µ T U E P T,(ϕT ,g) T ] i ∂ − ∇eµ (∂s∇ )(eµ), ϕT (y ) ⊗ i vol h . µ ∂y g Z X D X ] j ] i g ∂ T E T + Tm−1 Tr ad (∂s∇ )(eµ)ϕT (y ), ∂tϕT (y ) ⊗ ∇ ∂ i ,Deµ ϕT vol h j ∂y g U µ i,j ∂y Z X D g X T ] i ∂ E + Tm−1 Tr (ad ⊗ ∇ )Deµ ϕT ∂tϕT , (∂s∇ )(eµ), ϕT (y ) ⊗ i vol h ∂y g U µ i Z m n X D X T ] i ∂ + Tm−1 (∂t∇ )(eµ), ∂sϕT (y ) ⊗ ∂yi U µ=1 i=1 E + (∂ ∇T )(e ), ϕ] (yi)P ∂ ϕ] (yj ) ⊗ ∇g ∂ ,DT ϕ vol t µ T j s T ∂ ∂yi eµ T h ∂yj g Z m D n E X X T ] i ∂ T,(ϕT ,g) + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , ∇eµ ∂sϕT vol h ∂y g U µ=1 i=1 Z m X DPn T ] i ∂ g T E − Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , (ad ⊗ ∇ )∂sϕT Deµ ϕT vol h i=1 ∂y g U µ=1 Z m n X X D T ] i ∂ T ] j ∂ E + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , (∂s∇ )(eµ), ϕT (y ) ⊗ j vol h . ∂y ∂y g U µ=1 i,j=1

66 Here,

n T X T ξ 2 := Tr ∂ ∂ ϕ ,D ϕ e , (I ,∂s∂tϕT ) s t T eµ T g µ µ=1

T X T,(ϕT ,g) ξ 2 T,(ϕ ,g) = Tr ∂tϕT , ∇ ∂sϕT eµ , (I ,∂tϕT ,∇ T ) eµ g µ T X g ξ 2 T = Tr ∂tϕT , (ad ⊗ ∇ )DT ϕ ∂sϕT eµ , (I ,∂tϕT ,D ϕT ) eµ T g µ T X Pn T ] i ∂ ξ 2 T = Tr ∂ ϕ , [(∂ ∇ )(e ), ϕ (y )] ⊗ e (I ,∂tϕT ,∂s∇ ) t T i=1 s µ T ∂yi g µ µ and

n T T,(ϕT ,g) T,(ϕT ,g) P ] i ∂ F T,(ϕ ,g) (∂ , e ) D ϕ = ∂ ∇ − ∇ ∂ D ϕ (y ) ⊗ ∇ T s µ eµ T s eµ eµ s i=1 eµ T ∂yi n n n X T ] i ∂ X ] i X T ] j g ∂ = [(∂s∇ )(eµ),Deµ ϕT (y )] ⊗ i + Deµ ϕT (y ) [(∂s∇ )(eµ), ϕT (y )] ⊗ ∇ ∂ i ∂y j ∂y i=1 i=1 j=1 ∂y n n X X + D ϕ] (yi) DT ϕ] (yj) ∂ ϕ] (yk) ⊗ ∇g ∇g ∂ eµ T eµ T s T ∂ ∂ ∂yi ∂yk ∂yj i=1 j,k=1 − ∂ ϕ] (yk) DT ϕ] (yj) ⊗ ∇g ∇g ∂ s T eµ T ∂ ∂ ∂yi ∂yj ∂yk The summands Z D E T P T,(ϕT ,g) T + T Tr ∂ ∂ ϕ , DP h − ∇ D ϕ vol m−1 s t T ∇ eµ µ eµ eµ T h U µ eµ g Z X D g T E + Tm−1 Tr (ad ⊗ ∇ )DT (∂s∂tϕT ) ,De ϕT vol h eµ µ g U µ Z m n X D X T ] i ∂ T E + Tm−1 (∂s∂t∇ )(eµ), ϕT (y ) ⊗ i ,Deµ ϕT vol h ∂y g U µ=1 i=1 will vanish when imposing the equations of motion for (ϕ, ∇).

67 T If (ϕT , ∇ ) is furthermore a (∗2)-admissible T -family of (∗2)-admissible pairs, Then, the above expression reduces to

T Z ∂ ∂ ∇ C ∂ ∂ 1 T T ∂s ∂t E (ϕT ) = ∂s ∂t 2 Tm−1 Tr hD ϕT ,D ϕT i(h,g) vol h U Z = Tm−1 i T vol h ξ 2 ∂U (I ,∂s∂tϕT ) Z + Tm−1 i ξT + ξT + ξT vol h 2 T,(ϕ ,g) (I2,∂ ϕ ,DT ϕ ) (I2,∂ ϕ ,∂ ∇T ) ∂U (I ,∂tϕT ,∇ T ) t T T t T s Z D E T P T,(ϕT ,g) T + Tm−1 Tr ∂s∂tϕT , DP h − ∇ D ϕT vol h ∇e eµ µ eµ eµ U µ µ g Z m n X D X T ] i ∂ T E + Tm−1 (∂s∂t∇ )(eµ), ϕT (y ) ⊗ i ,De ϕT vol h ∂y µ g U µ=1 i=1 Z D E T,(ϕT ,g) P T,(ϕT ,g) T,(ϕT ,g) + Tm−1 Tr ∂tϕT , ∇P h − ∇ ∇ ∂sϕT vol h ∇ eµ µ eµ eµ U µ eµ g Z D g T + Tm−1 Tr ∂tϕT , (ad ⊗ ∇ )D ϕT P ∇h e U µ eµ µ E P T,(ϕT ,g) g − ∇ (ad ⊗ ∇ ) T ∂ ϕ vol µ eµ D ϕT s T h eµ g Z D X T E − Tm−1 Tr ∂tϕT , F∇T,(ϕT ,g) (∂s, eµ)De ϕT vol h µ g U µ Z D n + T Tr ∂ ϕ , P (∂ ∇T )(P ∇h e ), ϕ] (yi) m−1 t T i=1 s µ eµ µ T U ] ∂ E − P ∇T,(ϕT ,g)(∂ ∇T )(e ), ϕ (yi) ⊗ vol . µ eµ s µ T i h ∂y g Z m n X D X T ] i ∂ + Tm−1 (∂t∇ )(eµ), ∂sϕT (y ) ⊗ ∂yi U µ=1 i=1 E + (∂ ∇T )(e ), ϕ] (yi)P ∂ ϕ] (yj) ⊗ ∇g ∂ ,DT ϕ vol t µ T j s T ∂ ∂yi eµ T h ∂yj g Z m D n E X X T ] i ∂ T,(ϕT ,g) + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , ∇e ∂sϕT vol h ∂y µ g U µ=1 i=1 m Z X D n E − T P (∂ ∇T )(e ), ϕ] (yi) ⊗ ∂ , (ad ⊗ ∇g) DT ϕ vol m−1 i=1 t µ T i ∂sϕT eµ T h ∂y g U µ=1 Z m n X X D T ] i ∂ T ] j ∂ E + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , (∂s∇ )(eµ), ϕT (y ) ⊗ j vol h . ∂y ∂y g U µ=1 i,j=1

68 If one further imposes the equations of motion on (ϕ, ∇), then the expression reduces further to

T Z ∂ ∂ ∇ C ∂ ∂ 1 T T ∂s ∂t E (ϕT ) = ∂s ∂t 2 Tm−1 Tr hD ϕT ,D ϕT i(h,g) vol h U Z = Tm−1 i T vol h ξ 2 ∂U (I ,∂s∂tϕT ) Z + Tm−1 i ξT + ξT + ξT vol h 2 T,(ϕ ,g) (I2,∂ ϕ ,DT ϕ ) (I2,∂ ϕ ,∂ ∇T ) ∂U (I ,∂tϕT ,∇ T ) t T T t T s Z D E T,(ϕT ,g) P T,(ϕT ,g) T,(ϕT ,g) + Tm−1 Tr ∂tϕT , ∇P h − ∇ ∇ ∂sϕT vol h ∇ eµ µ eµ eµ U µ eµ g Z D g T + Tm−1 Tr ∂tϕT , (ad ⊗ ∇ )D ϕT P ∇h e U µ eµ µ E P T,(ϕT ,g) g − ∇ (ad ⊗ ∇ ) T ∂ ϕ vol µ eµ D ϕT s T h eµ g Z D X T E − Tm−1 Tr ∂tϕT , F∇T,(ϕT ,g) (∂s, eµ)De ϕT vol h µ g U µ Z D n + T Tr ∂ ϕ , P (∂ ∇T )(P ∇h e ), ϕ] (yi) m−1 t T i=1 s µ eµ µ T U ] ∂ E − P ∇T,(ϕT ,g)(∂ ∇T )(e ), ϕ (yi) ⊗ vol . µ eµ s µ T i h ∂y g Z m n X D X T ] i ∂ + Tm−1 (∂t∇ )(eµ), ∂sϕT (y ) ⊗ ∂yi U µ=1 i=1 E + (∂ ∇T )(e ), ϕ] (yi)P ∂ ϕ] (yj) ⊗ ∇g ∂ ,DT ϕ vol t µ T j s T ∂ ∂yi eµ T h ∂yj g Z m D n E X X T ] i ∂ T,(ϕT ,g) + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , ∇e ∂sϕT vol h ∂y µ g U µ=1 i=1 m Z X D n E − T P (∂ ∇T )(e ), ϕ] (yi) ⊗ ∂ , (ad ⊗ ∇g) DT ϕ vol m−1 i=1 t µ T i ∂sϕT eµ T h ∂y g U µ=1 Z m n X X D T ] i ∂ T ] j ∂ E + Tm−1 (∂t∇ )(eµ), ϕT (y ) ⊗ i , (∂s∇ )(eµ), ϕT (y ) ⊗ j vol h . ∂y ∂y g U µ=1 i,j=1

7.2 The second variation of the dilaton term We now work out the second variation of the (complexiﬁed) dilaton term Z (ρ,h;Φ) C Sdilaton (ϕT ) = Tr hdρ , ϕT dΦih vol h U Z X T = Tr dρ eµ) ((Deµ ϕT )Φ vol h . U µ

T 2 2 for an (∗1)-admissible family of (∗1)-admissible pairs (ϕT , ∇ ), T := (−ε, ε) ⊂ R with coordinate (s, t). It follows from Sec. 6.2 that, due to the eﬀect of the trace map Tr , Z m ∂ (ρ,h;Φ) X S (ϕ )C = Tr dρ(e ) ∂ (DT ϕ )Φvol ∂t dilaton T µ t eµ T h U µ=1 Z X T = Tr dρ(eµ)Deµ (∂tϕT )Φ vol h . U µ

69 Thus, due to the eﬀect of the trace map Tr again, Z ∂ ∂ (ρ,h;Φ) X T S (ϕT )C = Tr dρ(eµ)∂sD (∂tϕT )Φ vol h ∂s ∂t dilaton eµ U µ Z X = Tr dρ(e )DT ∂ (∂ ϕ )Φvol µ eµ s t T h U µ Z X = Tr dρ(e )DT (∂ ∂ ϕ )Φvol µ eµ s t T h U µ Z X X + Tr dρ(e )DT ∂ ϕ] (yi)∂ ϕ] (yj) ⊗ ∂ ∂ − ∇g ∂ Φ vol µ eµ t T s T i j ∂ j h ∂y ∂y i ∂y U µ i,j ∂y = (II 2.1) + (II 2.2) .

For Summand (II 2.1), repeating the same argument in Sec. 6.1 for Summand (I.1.1), one concludes that Z 2 (II .1) = i T vol h ξ 2 ∂U (II ,∂s∂tϕT ) Z X h X + ∇eµ eµ − eµdρ(eµ) Tr (∂s∂tϕT )Φ vol h , U µ µ where T X ξ 2 := dρ(eµ)Tr (∂s∂tϕT )Φ eµ ∈ T∗(UT /T ) . (II ,∂s∂tϕT ) µ

2 (ρ,h;Φ) The second summand of Summand (II .1) above is the term that captures the Sdilaton (ϕ)- contribution to the system of equations of motion for (ϕ, ∇). P ] i ] j ∂ ∂ g ∂ With ∂s∂tϕT replaced by i,j∂tϕT (y )∂sϕT (y ) ⊗ ∂yi ∂yj − ∇ ∂ ∂yj Φ, one has similarly ∂yi Z 2 (II .2) = i T vol h ξ 2 ∂U (II ,∂tϕT ,∂sϕT ) Z X h X + ∇eµ eµ − eµdρ(eµ) · U µ µ X ] i ] j ∂ ∂ g ∂ Tr ∂tϕT (y )∂sϕT (y ) ⊗ ∂yi ∂yj − ∇ ∂ ∂yj Φ vol h , i i,j ∂y where

T X X ] i ] j ∂ ∂ g ∂ ξ(II2,∂ ϕ ,∂ ϕ ) := dρ(eµ)Tr ∂tϕT (y )∂sϕT (y ) ⊗ ∂yi ∂yj − ∇ ∂ ∂yj Φ eµ t T s T i µ i,j ∂y

2 in T∗(UT /T ). The second summand of Summand (II .2) above contributes to the zeroth order (ρ,h;Φ,g,B,C) terms of the diﬀerential operator on (∂sϕT , ∂tϕT ) from the second variation of Sstandard (ϕ, ∇). In summary,

(ρ,h;Φ) C Proposition 7.2.1. [second variation of Sdilaton (ϕ) ] For the (complexiﬁed) dilaton term Z (ρ,h;Φ) C Sdilaton (ϕ) := Tr hdρ , ϕ dΦih vol h , U

70 T 2 its second variation for a (∗1)-admissible family of (∗1)-admissible pairs (ϕT , ∇ ), T := (−ε, ε) ⊂ 2 R with coordinate (s, t), is given by ∂ ∂ S(ρ,h;Φ)(ϕ )C ∂s ∂t dilaton T Z = i T T vol h ξ 2 +ξ 2 ∂U (II ,∂s∂tϕT ) (II ,∂tϕT ∂sϕT ) Z X X + ∇h e − e dρ(e ) Tr (∂ ∂ ϕ )Φvol eµ µ µ µ s t T h U µ µ Z X X + ∇h e − e dρ(e ) · eµ µ µ µ U µ µ X ] i ] j ∂ ∂ g ∂ Tr ∂tϕT (y )∂sϕT (y ) ⊗ i j − ∇ ∂ j Φ vol h , ∂y ∂y i ∂y i,j ∂y where

T X ξ 2 := dρ(e )Tr (∂ ∂ ϕ )Φ e (II ,∂s∂tϕT ) µ s t T µ µ T X X ] i ] j ∂ ∂ g ∂ ξ(II2,∂ ϕ ,∂ ϕ ) := dρ(eµ)Tr ∂tϕT (y )∂sϕT (y ) ⊗ i j − ∇ ∂ j Φ eµ t T s T ∂y ∂y i ∂y µ i,j ∂y in T∗(UT /T ). The integral Z X h X ∇eµ eµ − eµdρ(eµ) Tr (∂s∂tϕT )Φ vol h U µ µ would vanish when imposing the equations of motion of (ϕ, ∇) after the combination with other (ρ,h;Φ,g,B,C) C Equations-of-Motion capturing parts from the second variation of other terms in Sstandard (ϕ, ∇) .

71 Where we are

The following table summarizes where we are, following the similar steps of fundamental strings

string theory D-brane theory

fundamental objects : fundamental objects : Azumaya/matrix open or closed string , (m − 1)-manifold with a fundamental module with a connection D-brane world-volume : string world-sheet : Azumaya/matrix m-manifold 2-manifold Σ with a fundamental module with a connection (XAz, E; ∇) string moving in space-time Y : D-brane moving in space-time Y : diﬀerentiable map f :Σ → Y (admissible) diﬀerentiable map ϕ :(XAz, E; ∇) → Y

Nambu-Goto action SNG for f’s Dirac-Born-Infeld action SDBI for (ϕ, ∇)’s

Polyakov action SPolyakov for f’s standard action Sstandard for (ϕ, ∇)’s action for Ramond-Neveu-Schwarz superstrings ???, cf. [L-Y6: Sec. 5.1] (D(11.2)) action for Green-Schwarz superstrings ???, cf. [L-Y6: Sec. 5.1] (D(11.2)) quantization ???

(Cf. [L-Y8: Remark 3.2.4: second table]) (D(13.1)). It’s by now a history that as the built-in structure of a string is far richer than that for a point, a physical theory that takes strings as fundamental objects has brought us to where a physical theory that takes only point-particles as fundamental objects cannot reach. Now that a D-brane carries even more built-in structures, are these even-richer-than-string structures all just in vain? Or is a physical theory that takes D-branes as fundamental objects going to lead us to somewhere beyond that from string theories? Besides a theory in its own right, a theory that takes D-branes as fundamental objects has deep connection with other themes outside. In particular, at low dimensions, that there should be the following connections are “obvious” (0)( m = 0) =⇒ a new class of matrix models; cf. [L-Y8: Figure 2-1-2] (D(13.1)) (1)( m = 1) =⇒ nature of non-Abelian Ramond-Ramond ﬁelds; cf. e− vs. EM ﬁeld, [Ja] (2)( m = 2) =⇒ a new Gromov-Witten type theory; cf. [L-Y3] (D(10.1)), [L-Y4] (D(10.2)) but most details to realize these connections remain far from reach at the moment.

• A reﬂection at the end of the ﬁrst decade of the D-project since spring 2007:

我到為種植，我行花未開，豈無佳色在，留待後人來。

～～～弘一法師（李叔同,1880-1942）: 『將離淨峰詠菊誌別』

I came to plant seeds When I departed, the flowers hadn’t yet blossomed Not that there is no beautiful scene Only left to new generations

～～～ Master Hong Yi (1880-1942): “A zen poem on chrysanthemum” (English translation by Ling-Miao Chou)

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