# Question on Hori, Iqbal and Vafa's 'D-branes and Mirror Symmetry'

+ 1 like - 0 dislike
82 views

In the paper mentioned above, on page 19, the physics of A-type supersymmetry is related to a Lagrangian submanifold $\gamma$ of a Kaehler manifold $X$. In particular, the phrase "...holomorphic components of normal and tangent of $\gamma$..." is used.

What does one mean by the holomorphic component of a tangent vector of a Lagrangian submanifold? A Lagrangian submanifold does not necessarily have complex structure, and does not even need to be even-dimensional, so how can a tangent vector of the Lagrangian submanifold have a holomorphic component?

This post imported from StackExchange MathOverflow at 2017-04-08 22:34 (UTC), posted by SE-user Mtheorist
Because it says "normal and tangent", does it mean the fiber $T_p X \otimes_\mathbb{R} \mathbb{C}$ for $p \in L \subset X$? Just from the ambient complexified tangent bundle.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.