what is the stringy Kähler moduli space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:17:13Z http://mathoverflow.net/feeds/question/97091 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97091/what-is-the-stringy-kahler-moduli-space what is the stringy Kähler moduli space? Yosemite Sam 2012-05-16T07:39:11Z 2012-05-16T23:06:37Z <p>I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold.</p> <p>Could you suggest a nice paper/blog post where to read some heuristics about it? Some examples? Connections to stability conditions/curve-counting?</p> http://mathoverflow.net/questions/97091/what-is-the-stringy-kahler-moduli-space/97164#97164 Answer by Satoshi Nawata for what is the stringy Kähler moduli space? Satoshi Nawata 2012-05-16T23:06:37Z 2012-05-16T23:06:37Z <p>I have written an very brief introduction to <a href="http://physics.stackexchange.com/questions/28197/cy-moduli-fields/28272#28272" rel="nofollow">Calabi-Yau moduli</a>. This is covered by standard textbooks of string theory. (See the chapter 9 of <a href="http://www.amazon.com/String-Theory-M-Theory-Modern-Introduction/dp/0521860695" rel="nofollow">BBS</a>.) In addition, there are many good reviews such as <a href="http://arxiv.org/abs/hep-th/9702155v1" rel="nofollow">this</a> on this topic.</p> <p>The metric deformations $\delta g$ of a Calabi-Yau 3-fold $M$ are classified by the one of type $(1,1)$ and type $(2,1)$. The Ricci flat condition $$R_{mn}(g+\delta g)=0 \ ,$$ requires the infinitesimal $(1,1)$-form $\delta g$ to be harmonic. Therefore the K\"ahler moduli space is locally described by the vector space of $H_{1,1}$. The K\"ahler metric defines the K\"ahler form $J = ig_{ij}dz^i\wedge dz^j$, and positivity of the metric is equivalent to $$\int_M J\wedge J\wedge J >0$$ The K\"ahler moduli space has a cone structure since if $J$ satises the positivity condition, so does $sJ$ for any positive number $s$.</p> <p>In fact, in the study of string theory, the K\"ahler form of a Calabi-Yau 3-fold arises as one of the moduli fields of the associated 2d conformal field theory (Landau-Ginzburg model). Investigation of 2d CFT moduli space reveals that the corresponding geometrical description necessarily involves configurations in which the (supposed) K\"ahler form lies outside of the K\"ahler cone of the particular Calabi-Yau being studied. This may be thought of as residing in a new K\"ahler cone which shares a common wall with the original one. It was first shown by Witten in the <a href="http://arxiv.org/abs/hep-th/9301042" rel="nofollow">seminal paper</a> that this is a flop transition as you move from the one cone to the next cone, which is a famous example of topology change in string theory.</p> <p>In string theory, there is NS-NS two form $B$ which is an element of $H^{1,1}$. It is natural to combine the $B$-field with the K\"ahler form $J$, providing the complexied K\"ahler form $B+iJ$.</p> <p>As for the stability condition in the context of string theory, the <a href="http://arxiv.org/abs/hep-th/0403166v1" rel="nofollow">paper</a> by Aspinwall is a good point to start.</p>