A Mind for Madness

Musings on art, philosophy, mathematics, and physics


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Music 2008-2010

In preparation for the music 2011 list I thought I’d look back at my 2008-2010 lists and see what I still listen to and what has fallen by the wayside as impressive at first but not lasting. It is kind of funny to do this, because really these lists tend to be way off in terms of long-term listening. They reflect how excited I was about certain albums at first, but when I want something solid to fall back on when I’m bored with the current I tend to pick things in the middle or lower of the list.

What do I continually go back to in 2008? Definitely NIN’s The Slip. Honestly, I still listen to this on a regular basis. It is fantastic. I sometimes have cravings for it. Out of everything I post this is probably the thing I return to the most. Lightspeed Champion’s Crossing the Lavender Bridge has not faded as a fantastic album either. The third place by a significant gap is The Helio Sequence’s Keep Your Eyes Ahead. It is quite good and I return to it frequently, but not the same as the other two. Another really good one that I periodically come back to (but not that frequently, more for nostalgia’s sake) is Land of Talk’s Some Are Lakes. I recommend this above all other things they’ve put out. The others are good as well, but this one I come back to the most.

How about 2009? I return to almost nothing from this year. What happened guys? Sometimes I come back to Loney, Dear’s Dear John or Regina Spektor’s Far. I still think that Phoenix’s Wolfgang Amadeus Phoenix is probably the best thing to come out that year. The Dirty Projector’s Bitte Orcha was amazing, but honestly I rarely return to it.

What about last year? Has it been long enough? Well, Joanna Newsom’s Have One on Me is still so good…so, so good. Really amazing. It is my ring tone for goodness sake. OK, I have several ring tones depending on who calls me, but all of them are off this album. Very rarely do I come back to anything else. I don’t think I gave The Tallest Man on Earth enough credit last year. I continually come back to that. It is really good. Other than that I sometimes come back to Moonface, Interpol, or … I hate to admit it Kanye West (come on, admit it, that was a pretty awesome album). I unfortunately missed Lightspeed Champion’s new album last year Life Is Sweet! Nice to Meet You which has been consistently amazing for me this year.

That is my summary of what in retrospect was good in 2008-2010. In a few weeks I’ll post my new best of 2011 series.


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Mirror Symmetry A-branes

I started writing this post this past weekend, but got stuck really quickly and then kept putting it off. I don’t want to leave anyone following this hanging with no idea what the A-model is. This is harder for me to describe than the A-model for some reason. Mostly I’m running into the problem of either just saying what the A-side is without explanation or I’m getting too bogged down in details. Both seem bad. In conclusion, I think I’ll err on the side of too few details, and then hopefully make sense of what is going on by completely describing mirror symmetry in the easiest case possible: the one dimensional case, i.e. for an elliptic curve.

I’m going to semi-cheat right off and refer to posts over a year old. Recall what a symplectic form is on a smooth manifold is. It is just a closed non-degenerate 2-form. A smooth manifold plus symplectic form is called a symplectic manifold. The cotangent bundle always has a canonical symplectic form on it. An example that may be less well-known is that any smooth complex projective variety is symplectic because the Fubini-Study Kähler form on {\mathbb{P}^n} restricts to a symplectic form.

If we just think about vector spaces for a second, then given a symplectic form, we say that a subspace {S} is isotropic if {S\subset S^\perp} and coisotropic if {S^\perp \subset S}. The subspace is Lagrangian if it is both isotropic and coisotropic. This extends to manifold language easily by saying an embedded submanifold {S\subset M} is Lagrangian if the tangent subspace {T_sS\subset T_sM} is Lagrangian for every point of {S}. If you want to get used to these definitions, a quick exercise would be to check that the zero section of the cotangent bundle is Lagrangian with respect to the canonical symplectic structure.

My second semi-cheat is to ask you to recall the definition of an almost complex structure from close to two years ago. The way to think about it is that it is a bundle map {J: TM \rightarrow TM} that behaves similarly to “multiplication by {i}“. The condition is that {J^2=-Id}, and indeed multiplication by {i} when identifying {\mathbb{R}^2\simeq \mathbb{C}} gives an example of an almost complex structure. In fact, since we’ll always work over {\mathbb{C}}, any complex manifold does have multiplication by {i} as a natural almost complex structure.

It is possible that all these things are related by the following. Suppose {(M, \omega)} is a symplectic manifold, {J} an almost complex structre, and {g} a Riemannian metric. These three structures are called compatible if {\omega(J(-), -)=\langle - , -\rangle_g}. I am far out of my depth here, but I’m pretty sure such a manifold is called Kähler if this happens, but maybe some slight more conditions are needed (e.g. does this automatically imply that {g} is Hermitian? If so, then this is definitely what people call Kähler).

Now for the definition of the A-model. Let {(M, \omega)} be a Kähler (in the sense of the previous paragraph) manifold. We define the Fukaya category {Fuk(M)} to have as objects the Lagrangian submanifolds. The morphisms require a bit of technicality to define, but essentially are a way to intersect the submanifolds. It involves all the structures above and is called Floer cohomology. Recall that we’re merely sketching an idea here! Somehow this should be an {A_\infty} or dg-category if you remember from last time, and this just comes from the fact that the morphisms have to do with cohomology classes of intersections.

If you’ve been following this at all, then you should be in utter amazement. We can state mirror symmetry now as an equivalence of {A_\infty} categories {D^b(X)\rightarrow Fuk(\widehat{X})} where {X} is a Calabi-Yau. Why is this amazing (for those not following along)? Look at the left side of this equivalence. The bounded derived category of coherent sheaves (in the Zariski topology!!) on {X} is something that has to do purely with the algebraic data of {X}. I mean, the Zariski topology is algebraic, the definition of coherent is very algebraic, the construction of the derived category is algebraic, etc.

The right hand side seems to have forgotten all of the algebraic data. You forget that it is a variety and instead think of it as a smooth manifold. You consider a bunch of structure that helps you study the smooth structure. You consider Lagrangian submanifolds. The Fukaya category is almost entirely analytic in nature. But now the conjecture of Kontsevich mirror symmetry is that the two are always equivalent. That’s it for today. There should be one more post in this series in which I try to sketch the conjecture in the case of an elliptic curve.


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Side Note

When I told people I was typing up notes on Deligne’s proof (written by Illusie) that all K3 surfaces lift from positive characteristic to characteristic zero they were really excited and wanted a copy. I’m not sure if any of those people read my blog (in fact I’m pretty sure they don’t!), but just in case other people are interested I just put up a rough copy under Expository/Notes at my department webpage.


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The B-model

Sorry for the delay, but I’ve been incredibly busy and this topic is basically lowest priority right now. Today we will finish describing the “B-side” of mirror symmetry. The typical way this is done is with {A_\infty}-categories. I’m going to do it using dg-categories, because I’m much more comfortable with the language. Since we are working over {\mathbb{C}} it turns out these two things are exactly the same. I won’t go into why that is the case (probably because I don’t have an understanding why they are). The reason it is called the “B-side” is that we are constructing what is called the B-model. The statement of mirror symmetry will be something along the following lines: the B-model on {X} is equivalent to the A-model on {\widehat{X}} where {\widehat{X}} is that mirror pair we looked at here.

The two different “models” have something to do with string theory that I definitely don’t understand. As I said above we will need the notion of a dg-category. Without being incredibly careful with this definition (we’ll see later that this essentially adds no information to our derived category) a dg-category, {T}, (over {\mathbb{C}}) is a collection of objects {Ob(T)} such that for any pair of objects and any integer we have {T(x,y)^n} a {\mathbb{C}}-module. This should be thought of as morphisms in degree {n}. There is a composition of morphisms, and it should act like multiplication in a graded ring, so {T(x,y)^n\times T(y,z)^m\rightarrow T(x,z)^{n+m}} is bilinear and associative.

There needs to be an identity map in degree zero for any object which we notate {e_x\in T(x,x)^0}. So far nothing should feel strange to you, but our final condition is that there is a differential {d: T(x,y)^n\rightarrow T(x,y)^{n+1}} with the property that {d^2=0} (i.e. {Hom(x,y)} is a complex with respect to the differential) satisfying a graded Leibniz rule {d(fg)=d(f)g+(-1)^mfd(g)}.

If you want something to help you wrap your head around this here is an analogy. A groupoid is a category in which every morphism is invertible, so you can think of any group {G} as a groupoid by taking the category to have only one object and {Hom(*,*)=G}. In a similar way, a dg-category or “differential graded” category can be formed by taking any differential graded algebra {A=\bigoplus A^n} and then forming the category with exactly one object and {T(*,*)^n=A^n}. All those definitions are just basically turning the definition of a differential graded algebra into a category in a consistent way.

Let {X} be a smooth projective variety over {\mathbb{C}}. Then we have already constructed a triangulated category {D^b_{Coh}(X)}, the derived category. This is not a dg-category, but we can ask if there is a dg-enhancement. It is a bit technical to describe what this means, but roughly an enhancement is asking for a dg-category {T} where the homotopy category {H^0(T)} is equivalent to {D^b(X)}. Here {H^0(T)} means to take the same objects and define the morphisms to just be the {0}-th cohomology {H^0(Hom(x,y))}.

There are varying levels of strength of what it means to be a unique enhancement mostly stemming from the fact that there is a choice of equivalence {H^0(T)\rightarrow D^b(X)} that may or may not be respected. Again, we’ll skip over these technicalities because the hope is to take away from these posts a general flavor of what mirror symmetry says rather than trying to describe all the technical details (the Clay Math book called “Mirror Symmetry” is 929 pages for goodness sake!). Thanks to Lunts and Orlov when our variety is projective we get that there always exists a dg-enhancement and it is unique in the strongest sense.

A priori a dg-category is big with lots of information and behaves nicely (is relatively easy to work with). A triangulated category has very little structure and behaves rather poorly (is difficult to work with). This enhancement theorem says that it doesn’t really matter which one we work with if we care about {D^b(X)} which should be surprising. Somehow {D^b(X)} is just the {0}-th part of the big dg-category so for instance having an equivalence of the triangulated categories doesn’t seem strong enough to be able to extend it to an equivalence of the whole dg-structure, but it is.

To conclude this half of the series, suppose we have a Calabi-Yau threefold {X}, then for our purpose we will call the B-model of {X} denoted by {B(X)} to be the unique dg-enhancement of {D^b(X)}. Mirror symmetry will eventually be an equivalence (as dg-categories) with {A(\widehat{X})}. This means we need to move on to the {A}-model.

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