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I have a very hard time to understand something physicists call $A$ or $B$ twists in the context of topological string theory. A canonical reference seems to be this Witten's paper.

Let $\Sigma$ be a Riemann surface, $K$ and $\bar K$ the canonical and anti-canonical line bundles of $\Sigma$ and $K^{1/2}$ and $\bar K^{1/2}$ square roots of these. Let $X$ be a Calabi-Yau manifold, and $TX = T^{1,0}X\oplus T^{0,1}X$ its complexified tangent bundle. Let $\Phi:\Sigma\to X$ be an embedding. We consider $\psi_+ \in \Gamma\left(K^{1/2}\otimes \Phi^\ast(TX)\right)$ and $\psi_-\in \Gamma\left(\bar K^{1/2}\otimes \Phi^\ast(TX))\right)$.

We can now use the decomposition of the complexified tangent bundle of $X$ to decompose $\psi_+$ and $\psi_-$ into holomorphic and anti-holomorphic parts. If $\phi^i$ are complex coordinates on $X$, with $\phi^{\bar i} = \overline{\phi^i}$, we denote $\psi_\pm^i$ and $\psi_{\pm}^{\bar{i}}$ the components of the projections of $\psi_+$ and $\psi_-$ into $\Phi^\ast(T^{1,0}X)$ and $\Phi^\ast(T^{0,1}X)$.

Then physicists do something I simply cannot understand. They “take” $\psi_+^i$ and $\psi_+^{\bar i}$ to be sections of $\Phi^\ast(T^{1,0}X)$ and $K\otimes \Phi^{\ast}(T^{0,1}X)$ respectively. Similarly they say they can “take” $\psi_+^i$ and $\psi_+^{\bar i}$ to be sections of $K\otimes \Phi^\ast(T^{1,0}X)$ and $\Phi^\ast(T^{0,1}X)$ respectively. A similar procedure is then performed on the $\psi_-$.

Now what is this all about? What it means to “take” $\psi_+^i$ and $\psi_+^{\bar i}$ to be sections of some other bundle? I mean, they are already sections of a bundle. I don’t understand what it means to take them to be sections of another bundle. This seems like a very ill-defined procedure to me.

So what are these “twists” physicists do from a mathematically rigorous perspective?

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A physical theory is defined by its Lagrangian. The Lagrangian that Witten is considering is written down in equation 2.4. It is defined on some space of "fields" (space of sections of vector bundles). The Lagrangian in equation 2.4 specifically is defined whenever $\psi_+\in\Gamma(K^{1/2}\otimes\Phi^*(TX))$ and $\psi_-\in\Gamma(\bar{K}^{1/2}\otimes\Phi^*(TX))$.

In the paragraph follow equations 2.5, Witten proposes to define a Lagrangian on a different space of sections. That is what it means to take $\psi_\pm$ from a different bundle: we're going to define a new Lagrangian, say $L_+$, on a different space. This is also why he notes that the derivative operators involved must be interpreted differently. Similarly, we are going to define $L_-$ on a different space of sections.

In other words, it's not that $\psi_\pm$ are held fixed and viewed as objects in different spaces. It's that they are dummy variables in the Lagrangian $L$, and the paper defines an $L_+$ and an $L_-$ which are defined on different spaces of sections, and Witten is describing how to interpret $\psi_\pm$ when they appear in $L_+$ and $L_-$.

Note that the notation $L_+$ and $L_-$ are my invention, just to make it conceptually clear that you might regard them as different functions (because they are defined on different spaces). These do not appear anywhere in the paper. Furthermore, a physicist would likely also take issue with my description of what Witten is exactly doing, and would instead talk about altering the "field content" of the theory, or something to that effect. I hope this high level explanation at least answers your question.

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