Academic Transcript: "Generalized Holomorphic Bundles and the B-field Action" Dr. Nigel Hitchin, Savilian Professor of Geometry Emeritus, University of Oxford [00:00] Dr. Hitchin: Good morning, everyone. It's a pleasure to be here today to discuss some fascinating connections between seemingly disparate areas of geometry, namely complex and symplectic geometry, unified under the umbrella of generalized geometry. Our topic today is "Generalized Holomorphic Bundles and the B-field Action," a subject that has found profound resonance in both pure mathematics and theoretical physics, particularly in string theory.

[00:45] Dr. Hitchin: My aim is to demonstrate how certain transformations, known as B-field transformations, play a crucial role in preserving the underlying structures of generalized geometry. Specifically, we'll see how they preserve exact Courant algebroids and generalized complex structures, and how this offers a formally tractable framework for understanding the interplay between complex and symplectic geometry.

[01:30] Dr. Hitchin: Let's begin by recalling the fundamental building block of generalized geometry: the generalized tangent bundle. For a smooth manifold M , we define the bundle E

T M ⊕ T ∗ M . This bundle carries a natural symmetric bilinear form, often called the standard metric, defined for sections ( X , ξ ) , ( Y , η ) ∈ Γ ( E ) as: $$ \langle (X, \xi), (Y, \eta) \rangle = \frac{1}{2} (\xi(Y) + \eta(X)) $$ This pairing is crucial, as it allows us to identify E with its dual.

[02:15] Dr. Hitchin: More importantly for us today, E is endowed with a canonical Lie algebroid structure, or rather, a Courant algebroid structure. The anchor map is the projection ρ : E → T M , given by ρ ( X , ξ )

X . The bracket, known as the Dorfman bracket, is defined for sections s 1

( X 1 , ξ 1 ) and s 2

( X 2 , ξ 2 ) as: $$ [s_1, s_2]D = ([X_1, X_2], \mathcal{L}{X_1} \xi_2 - i_{X_2} d\xi_1) $$ This bracket, while not fully skew-symmetric, satisfies a modified Jacobi identity and is compatible with the pairing. It makes E into an exact Courant algebroid. The non-skew-symmetric part is precisely d ⟨ s 1 , s 2 ⟩ .

[03:40] Dr. Hitchin: Now, let's introduce generalized complex structures. A generalized almost complex structure is an endomorphism J : E → E such that J 2

− I , where I is the identity on E . It's a geometric structure that mixes vector fields and 1-forms. Such a J is called integrable if the subbundle L J ⊂ E C consisting of its i -eigenvectors, i.e., L J

χ ∈ E C ∣ J χ

i χ , is closed under the Dorfman bracket. That is, for any u , v ∈ Γ ( L J ) , we must have [ u , v ] D ∈ Γ ( L J ) . This condition is the direct analogue of the integrability condition for a standard complex structure via the Nijenhuis tensor.

[04:50] Dr. Hitchin: Let's quickly review the two canonical examples of generalized complex structures:

Complex Manifold: If M is a complex manifold with complex structure J C , then J can be defined as: $$ J(X, \xi) = (J_C X, -J_C^* \xi) $$ where $J_C^$ is the dual action on 1-forms. Here, $L_J = T^{0,1}M \oplus T^{1,0}M$. Symplectic Manifold: If M is a symplectic manifold with symplectic form ω , then J can be defined as: $$ J(X, \xi) = (-\omega^{-1}\xi, \omega X) $$ This uses the isomorphism ω : T M → T ∗ M . Here, L J

( X , ω X ) ∈ E C . The beauty of generalized complex geometry is that it smoothly interpolates and unifies these two distinct geometric settings. [06:30] Dr. Hitchin: What about generalized holomorphic bundles? In classical complex geometry, a holomorphic vector bundle V → M is equipped with a ∂ ¯ -operator, ∂ ¯ V : Ω 0 , k ( M , V ) → Ω 0 , k + 1 ( M , V ) , satisfying the Leibniz rule and ∂ ¯ V 2

0 . The sections of V that are annihilated by ∂ ¯ V are the holomorphic sections. In the generalized context, this ∂ ¯ -operator is replaced by a generalized connection. For a vector bundle V → M , we can define a generalized connection on V as a map ∇ : Γ ( E ) × Γ ( V ) → Γ ( V ) satisfying certain axioms related to the Dorfman bracket and the generalized complex structure J . [07:55] Dr. Hitchin: Specifically, a connection ∇ is called generalized holomorphic with respect to J if its ( 0 , 1 ) -part, ∇ 0 , 1 , defines a flat connection on V along L J . This means that for any s ∈ Γ ( V ) and any u ∈ Γ ( L J ) , ∇ u s

0 . This essentially defines the generalized holomorphic sections as those which are covariantly constant along the generalized anti-holomorphic directions. This framework is particularly relevant for understanding D-branes in string theory, where the "holomorphic" condition for open strings has to be generalized.

[09:10] Dr. Hitchin: Now, let's introduce the protagonist of our second half: the B-field transformation. In physics, the B-field is a closed 2-form B ∈ Ω 2 ( M ) on the spacetime manifold M , meaning d B

0 . For our purposes, we will mostly consider the action of exact B-fields locally, i.e., B

d α for some 1-form α , though the concepts extend to d B

0 . The B-field acts on sections of E in a specific way. For any s

( X , ξ ) ∈ Γ ( E ) , the B-field transformation is given by: $$ e^B(X, \xi) = (X, \xi + i_X B) $$ This is an automorphism of E . Its inverse is e − B ( X , ξ )

( X , ξ − i X B ) .

[10:35] Dr. Hitchin: The crucial point is how this transformation affects the Courant algebroid structure and, subsequently, generalized complex structures. First, consider the Courant algebroid on E . Does e B map the Dorfman bracket to itself? Not quite. But it maps the Courant bracket defined with B to the standard one. What e B does is transform the standard Courant algebroid structure on E to an isomorphic Courant algebroid structure. This is often framed as e B being an isomorphism of Courant algebroids ( E , [ ⋅ , ⋅ ] D , ⟨ ⋅ , ⋅ ⟩ ) to ( E , [ ⋅ , ⋅ ] D B , ⟨ ⋅ , ⋅ ⟩ ) , where [ ⋅ , ⋅ ] D B is a twisted Dorfman bracket incorporating B . A more direct and powerful way to see its action on geometric structures is how it transforms generalized complex structures.

[11:50] Dr. Hitchin: If J is an integrable generalized complex structure, we can define a new structure J B by conjugating J with the B-field transformation: $$ J_B = e^B J e^{-B} $$ Let's unpack this. For any s

( X , ξ ) ∈ Γ ( E ) :

Apply e − B : s ′

e − B s

( X , ξ − i X B ) Apply J : J s ′

J ( X , ξ − i X B ) Apply e B : J B s

e B ( J ( X , ξ − i X B ) ) The remarkable result, which can be shown through direct calculation involving the Dorfman bracket, is that if J is an integrable generalized complex structure, then J B

e B J e − B is also an integrable generalized complex structure. This is a profound statement. It means that the B-field acts as a kind of "gauge transformation" on the space of generalized complex structures, moving between equivalent but distinct forms of generalized complex geometry.

[13:45] Dr. Hitchin: Let's sketch why this holds. The integrability of J relies on the closure of L J under the Dorfman bracket. The transformation e B provides an isomorphism from E to itself. It can be shown that the bracket [ e B s 1 , e B s 2 ] D is related to $e^B [s_1, s_2]D$ by lower order terms involving $B$. If $J$ is integrable, meaning $L_J$ is involutive, then the transformed subbundle $L{J_B} = e^B L_J$ will also be involutive under the Dorfman bracket. The key fact here is that the Dorfman bracket transforms essentially as [ u , v ] D ↦ [ u , v ] D + … under e B , and the integrability condition is preserved. This is a robust property, confirming that B-field transformations act as symmetries of generalized complex geometry.

[15:10] Dr. Hitchin: This preservation extends beyond generalized complex structures to other related structures, such as generalized metrics. If g is a generalized metric (a positive-definite metric on E compatible with J ), then e B g e − B also constitutes a generalized metric. This provides the mathematical backbone for phenomena like T-duality in string theory, where B-field transformations are central to relating different string backgrounds.

[16:00] Dr. Hitchin: Let's now consider the implications for generalized holomorphic bundles. If we have a generalized holomorphic bundle ( V , ∇ ) on ( M , J ) , how does it transform under a B-field action? The action of e B on the generalized complex structure J means that we are now looking at generalized holomorphicity with respect to J B . The generalized connection ∇ also needs to be transformed. The proper definition of a B-field transformation on a generalized connection ∇ (which effectively is a section of T ∗ M ⊗ End ( V ) coupled with a standard connection) is quite intricate. However, the overarching principle is that the B-field transformation induces an equivalence between generalized holomorphic bundles on ( M , J ) and those on ( M , J B ) . This ensures that the concept of generalized holomorphicity itself is robust under these transformations.

[17:35] Dr. Hitchin: Let's briefly touch upon the connection between complex and symplectic geometry. Consider starting with a purely complex manifold M with complex structure J C . Its generalized complex structure is J c o m p ( X , ξ )

( J C X , − J C ∗ ξ ) . If we apply a B-field transformation J B

e B J c o m p e − B , we obtain a new generalized complex structure. This new structure is generally not purely complex or purely symplectic. It represents a "mixed" geometry, where the complex and symplectic aspects are intertwined by the B-field. For instance, a Kahler manifold ( M , g , J C , ω ) is a special case where J C and ω are compatible. The generalized complex structure in this case is a very specific type, and the B-field can deform this into other interesting generalized complex structures. This allows us to unify the study of these geometries, seeing them as different "phases" within the broader space of generalized complex structures connected by B-field gauge transformations.

[19:20] Dr. Hitchin: The beauty here is that the B-field transformations are exact and formally tractable. They provide a precise mechanism for understanding how the complex and symplectic aspects of a manifold can be mixed and transformed while preserving the fundamental integrability conditions. This has direct applications to understanding the moduli space of generalized complex structures and, importantly, the moduli space of D-branes in string theory, which are often described by such bundles on generalized complex manifolds.

[20:10] Dr. Hitchin: The preservation of exact Courant algebroids by these transformations is also fundamental. The Dorfman bracket itself forms an exact Courant algebroid structure on E . When we apply e B , we are essentially performing an isomorphism of this Courant algebroid structure to an equivalent one. This means that the entire algebraic framework underpinning generalized geometry remains consistent under B-field shifts, ensuring the geometric structures derived from it are well-behaved.

[20:55] Dr. Hitchin: To summarize, we've seen that generalized complex structures provide a unified framework for complex and symplectic geometry, based on the bundle E

T M ⊕ T ∗ M and the Dorfman bracket. Integrability of these structures is key, and the B-field, a closed 2-form, acts as a profound symmetry. The transformation J B

e B J e − B ensures that if J is integrable, J B also is. This demonstrates that B-field transformations preserve exact Courant algebroids and generalized complex structures, making them central to the study of these geometries and their applications in areas such as T-duality in string theory and D-brane geometry.

[21:50] Dr. Hitchin: This formal tractability allows mathematicians to explore a richer landscape of geometric structures and physicists to understand the equivalences between different string theory compactifications. It opens doors to new classifications and invariants, bridging the gap between classical differential geometry and its more generalized quantum field theory counterparts.

[22:30] Dr. Hitchin: Thank you for your attention. I am happy to take any questions you may have.

bigsnarfdude/transcript.md

Select an option

No results found