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Reading the book$^{\dagger}$ Chern-Simons Theory, Matrix Models, and Topological Strings by Marcos Marino, I'm trying to understand the argument in 7.3.2: here are my main questions which can also be seen as some general string theory questions.

1. Why is the string field theory description of the open B-model topological string called a "spacetime description?" By SUSY localization, the action is $$S = \int_X \Omega \wedge \mathrm{Tr} (A \wedge \bar{\partial} A + \frac{2}{3} A \wedge A \wedge A) $$ where the string field $A$ is just a $\mathrm{End} (E)$ valued 1 form (vector bundle connection) on spacetime $X$. So is it a "spacetime description" just because we have recognized the degrees of freedom of the open string theory as a spacetime QFT?

On the other hand, I understand that in string theory, the usual idea is that the massless part of the spectrum of open strings ending on some d-branes is identified as the spectrum of a worldvolume qft which is some Yang-Mills type theory; what is the relation of that YM theory with the string field A?

2. In particular, in a case with some $N$ d-branes, we have "the spacetime description can be obtained by considering the dimensional reduction of the original string field theory (above action). As usual in D-brane physics, the gauge potential splits into a gauge potential on the world volume of the brane and Higgs fields describing the motion along non-compact, transverse directions." Can someone clarify what is going on here? This is reminiscent of the usual SSB argument with D-brane field theory, but having trouble piecing it together.

${}^{\dagger}$ There is an arXiv version: https://arxiv.org/abs/hep-th/0410165. See 3.3 in particular.

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1. Spacetime Description: The action is "spacetime" because the string field $A$ is fundamentally a field on the target Calabi-Yau $X$, not on the string worldsheet. The B-model localization shows that physical open string states correspond exactly to $\bar{\partial}$-closed $(0,1)$-forms valued in $\mathrm{End}(E)$.

2. Dimensional Reduction: When D-branes wrap a submanifold $S \subset X$, we decompose the connection via the splitting of the cotangent bundle: $$T^*X|_S = T^*S \oplus N^*S$$ This gives $A = A^\parallel + \Phi^\perp$, where:

  • $A^\parallel$ is the gauge field on the brane worldvolume
  • $\Phi^\perp$ are Higgs fields describing brane deformations

The Higgs mechanism occurs because a vacuum expectation value $\langle\Phi^\perp\rangle \neq 0$ corresponds to the brane moving away from $S$, breaking the gauge symmetry from the stabilizer of $S$ to that of the deformed brane.

The dimensional reduction of the holomorphic Chern-Simons action to $S$ yields precisely the expected worldvolume theory (e.g., holomorphic Chern-Simons on $S$ plus a potential for $\Phi^\perp$).

Note that, in contemporary terms, this entire setup is recognized as a prime example of formal deformation theory in the sense of Lurie and Pridham, where the D-brane configuration is a formal moduli problem controlled by the differential graded Lie algebra (or $L_\infty$-algebra) of the holomorphic Chern-Simons action.

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  • $\begingroup$ Thanks for this nice answer, this gets to some of the underlying points; is then the definition of branes wrapping a submanifold that the (co)normal sequence splits? And do we mean by dimensional reduction simply restricting the A field to a field on S? What does this mean physically? Finally thanks for pointing out the connection to formal deformation theory - what is the DGA associated to CS action? $\endgroup$ Commented Nov 6 at 16:19
  • $\begingroup$ 1) Yes, the splitting of the conormal exact sequence $0 \to N^*S \to T^*X|_S \to T^*S \to 0$ is essentially the definition of the decomposition $A = A^\parallel + \Phi^\perp$. 2) Dimensional reduction means enforcing the brane's locality by expanding the action around a solution localized on $S$ and integrating out massive normal directions, not merely restricting the field. $\endgroup$ Commented Nov 7 at 10:37
  • $\begingroup$ 3) The DGLA controlling the holomorphic Chern-Simons theory is $(\mathcal{A}^{0,\bullet}(X, \mathrm{End}(E)),\ \bar{\partial},\ [\cdot,\cdot])$ where the Maurer-Cartan equation $\bar{\partial} A + \frac{1}{2}[A, A] = 0$ is the equation of motion. This DGLA encodes the formal deformation theory of holomorphic bundles (the D-branes). $\endgroup$ Commented Nov 7 at 10:43
  • $\begingroup$ For references on this, see Kevin Costello's seminal paper arxiv.org/abs/math/0412149 (specifically in section 2.2), where we have an explicit construction starting with the dg-category of perfect complexes $\mathrm{Perf}(X)$ and produces a deformed $A_{\infty}$-category $\mathcal{D}_{\infty}^{b}(X)$ whose deformation theory, governed by a DGLA, defines the B-model topological string. $\endgroup$ Commented Nov 7 at 10:44

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