Untitled

1: Introduction to Homological Mirror Symmetry

Title: Backlandt: A gentle introduction to homological mirror symmetry

“What is the slant from algebra in this question?”

QFT:

1. Spacetime (Minkowski spacetime ${\mathbb{R}}^{n,1}$, $d{x}^{\mu }\otimes d{x}^{\nu }$) $\to$ $n$-D QFT ($M^n,g$)
2. Fields (Highly generalized: functions on $M^n$) $\to$ $\text{Map}(M^n,\mathbb{R})$, $\text{Map}(M^n,\mathbb{C})$, etc.
   • $\text{Map}(M^n,X)$ where $X$ is the target space (e.g., $Y/G$ - stack)
3. Equations of motions satisfied by fields

Mathematical principle: To study a space $M$, it’s very helpful to understand functions on that. Example: To understand $M^n\in n$-manifold, we have to understand all mapping spaces $\text{Map}(M^n,X)$, and their relations $X\to Y\Rightarrow \text{Map}(M^n,X)\to \text{Map}(M^n,Y)$ (morphism spaces).

Page 2: Linearization and Spectra

Thm: Yoneda embedding theorem

More concretely, given $M$, the task of “linearization” of $M$ is to find numbers, vector spaces, 1-categories, etc., out of $M$.

• $X(M)$, $H^{\ast }(M)$, $\text{Fuk}(M)$, $\text{Coh}(M)$, etc.

All of these linearizations only see part of the geometry ($M$). $n$-categories on $M$ is better and better at seeing ${\Pi }_k(M)$ for $k$ big as we increase.

Let $(M^n,g)$ be a Riemannian manifold, compact, oriented.

$L^2(M)\supset {\Delta }_g$ ask about the eigenvalue of this operator.

(Old question: How much does the spectrum know about $(M^n,g)$?)

Much easier variant: ${\Delta }_g\curvearrowright L^2$-differential forms on $M$.

$d\delta +\delta d={\Delta }_g$

where $d:p\text{-forms}\to (p+1)\text{-forms}$ is the de Rham differential, and $\delta :(p+1)\text{-forms}\to p\text{-forms}$.

$\ker {\Delta }_g$ is easier to find.

$H_{dR}^{\ast }(M)$ (topological) $\iff$ Atiyah-Singer index theorem.

Margin Note: TQFT generated into HDQFT. 1-D TQFT.

Page 3: 1D-QFT and Hamiltonian Formalism

1D-QFT on $M={\mathbb{R}}_t^1$

1. Fields: $\text{Map}({\mathbb{R}}_t^1,X)$ where $X$ is the target manifold (want to study index theory on $X$)
2. EOM (Equations of Motion)

Hamiltonian Formalism

${\Delta }_g=\Delta \curvearrowright \text{Hilb}=L^2\text{ diff form on }X$. ($\Delta$ is the Hamiltonian).

Goal: diagonalize ${\Delta }_g$.

Partition function $Z(\beta )=\text{tr}(e^{-\beta {\Delta }_g})$.

Example: $\text{Spec }{\Delta }_g=\{1,2,3\}\Rightarrow Z(\beta )=e^{-\beta }+e^{-2\beta }+e^{-3\beta }+\dots$

Dictionaries: 1D-QFT

1. Hilbert space $\mathcal{H}$

2. Hamilton op $\Delta \in \mathcal{H}$

• $Z(pt)=\mathcal{H}$

• $Z(\text{interval})=⟨v_1,\Delta v_2⟩\in \mathbb{C}\Leftarrow$ scattering amplitude

• $Z(S_{\beta }^1)=\sum ⟨v_i,1\cdot e^{-\beta \Delta }{v}_i⟩={\text{tr}}_{\mathcal{H}}{e}^{-\beta \Delta }$

Arrow pointing down: K-theory (dimension reduction).

Page 4: 2D-TQFTs and Mirror Symmetry

2-D TQFTs:

$Z(1\text{-mfd})=\mathcal{H}$, $Z(pt)=\text{category}(\mathcal{C})$

$Z(\text{interval})={\text{Hom}}_{\mathcal{C}}(C_1,C_2)$ for $C_1,C_2\in \mathcal{C}$ (no operads here as we consider simplified TQFT).

$Z(S^1)=\text{Hochschild homology of }\mathcal{C}$.

Yoneda $\Rightarrow ⟨\cdot ,\cdot ⟩$ is non-degenerate.

Homological Mirror Symmetry $\subseteq$ Mirror Symmetry

Kontsevich 1994 ICM. T-duality in String theory $\iff$ duality between 2D QFTs.

$M$ symplectic structure $\to$ Riemannian manifold and a compatible complex structure.

Mirror Symmetry: $M$ and $M^{\vee }$.

$Z_A^M(pt)=Z_B^{M^{\vee }}(pt)$

$\text{Fuk}(M)\simeq \text{Coh}(M^{\vee })\text{and vice versa ideally}$

• A-twist: $Z^A(pt)=\text{Fuk}(M)$ (symplectic invariant only)

• B-twist: $Z^B(pt)=D^b\text{Coh}(M^{\vee })$ (complex geometric invariant)

Page 1: Constructing the Dual Mirror (SYZ)

Q: How to construct $\check{M}$ (canonically)?

Strominger-Yau-Zaslow (Mirror Symmetry is T-duality)

$M^{2n}\stackrel{\pi }{\to }{B}^n\text{and}{\check{M}}^{2n}\stackrel{\check{\pi }}{\to }{B}^n$

1. Generically, both arrows represent a Lagrangian torus fibration $(S^1{)}^n$.

2. Generically, the fibers $M_b^{2n}$ and ${\check{M}}_b^{2n}$ need to be dual.

Example: $n=1,B^n={\mathbb{R}}^1$

• $M^{2n}={\mathbb{R}}^1\times S_R^1\to {\mathbb{R}}_B$

• ${\check{M}}^{2n}={\mathbb{R}}^1\times S_{1/R}^1\to {\mathbb{R}}_B$

$\begin{array}{ccc}{\mathbb{R}}^1\times S_R^1& & {\mathbb{R}}^1\times S_{1/R}^1\\ \text{(Cylinder)}& & \text{(Dual Cylinder)}\\ \searrow & & \swarrow \\ & {\mathbb{R}}_B& \end{array}$

• On $\mathbb{R}\times S^1$, we have a complex structure: ${\mathbb{C}}_z^{\ast }\cong \mathbb{R}\times S^1$

  $\begin{array}{ccc}\mathbb{R}\times S^1& \stackrel{\exp }{\to }& {\mathbb{C}}_z^{\ast }\\ \text{(Cylinder)}& & \text{(Punctured Plane)}\end{array}$

• On $\mathbb{R}\times S^1$, we have a symplectic structure: $T^{\ast }{S}^1\cong \mathbb{R}\times S^1$

(Def: $(M^{2n},\omega )$ is a symplectic manifold if $\omega$ is a closed ($d\omega =0$) nowhere degenerated 2-form. $M^{2n}=T^{\ast }{W}^n$ locally has $\omega =\sum d{p}_i\wedge d{q}^i$.)

Q1: If $M,\check{M}$ is a “good” mirror pair, then $M_{\text{complex}}(M)\simeq M_{\text{sympl str}}(\check{M})$.

• $M_{\text{complex}}(M)\supset {\mathbb{R}}_{>1}\cup \{+\infty \}$ (moduli of complex structures)

• $M_{\text{sympl str}}(\check{M})\supset \text{total symplectic area}\in {\mathbb{R}}_{>0}\cup \{+\infty \}$

Page 2: Torus Example and Complex/Symplectic Duality

Example: $n=1,B^n=S_{R_1}^1$

$M^2=S_{R_1}^1\times S_{R_2}^1\stackrel{B=S_{R_1}^1}{\to }{S}_{R_1}^1\times S_{1/R_2}^1={\check{M}}^2$

$\begin{array}{ccc}{S}_{R_1}^1\times S_{R_2}^1& & S_{R_1}^1\times S_{1/R_2}^1\\ \text{(Torus }{M}^2\text{)}& & \text{(Dual Torus }{\check{M}}^2\text{)}\\ \searrow & & \swarrow \\ & B^{\prime }=S_{R_1}^1& \end{array}$

Cross-Mapping of Structures:

• $M^2$ (complex structure $\sim \frac{R_1}{R_2}$) $⟷$ ${\check{M}}^2$ (Symplectic area $\sim R_1\cdot \frac{1}{R_2}=\frac{R_1}{R_2}$)

• $M^2$ (Symplectic area $\sim R_1{R}_2$) $⟷$ ${\check{M}}^2$ (complex structure $\sim \frac{R_1}{1/R_2}=R_1{R}_2$) (B-field)

(Note on the side: $\mathbb{H}/SL(2,\mathbb{Z})$ — many complex structures.)

Page 3: Singular Fibrations and the Fukaya Category

Example: Singular $S^1$ fibration

$\begin{array}{ccc}M\text{ (Pinched Cylinder)}& & \check{M}\text{ (Pair of Pants)}\\ \downarrow & & \downarrow \\ \times \in {\mathbb{R}}^1& & B\text{-graph (Y-shape)}\end{array}$

• B-graph $\to$ Y-shape. $⟵$ Tropical geometry.

Bocklandt: 90% of the examples deal with $\text{Fuk}(\text{open Riemann surfaces})$.

${\text{End}}_{\text{Fuk}}(\text{generating objects})\cong \text{gentle algebra (}{A}_{\infty }\text{ algebra)}$

How do we think about $\text{Fuk}(M^{2n},\omega )$?

• It is easier to consider exact symplectic manifolds, i.e., $\omega =d\lambda \Rightarrow$ (non-compact) + “conical boundary conditions” (Liouville domain).

Page 4: Objects in the Fukaya Category

• Obj of $\text{Fuk}(M^{2n},\omega )$: $L^n\subseteq M^{2n}$ Lagrangian submanifold + conditions ($\omega |{}_L=0$).

Prototype: $L=W^n\subseteq T^{\ast }{W}^n$ (zero section).

Weinstein neighborhood thm: Let $L\subseteq (M^{2n},\omega )$ be a compact smooth Lagrangian. Then $\exists$ a tubular neighborhood $N_L\supset L$ s.t. $N_L\cong T^{\ast }L$ as a symplectic manifold.

${\text{Hom}}_{\text{Fuk}}(L_1,L_2):=\mathbb{C}⟨L_1\cap L_2⟩\supset d_{\text{dis}}(d^2=0)$

(Hamiltonian perturbation)

Example: $L\subseteq T^{\ast }L$, $L$ compact. Want $\text{Hom}(L,L):=\text{Hom}(L(C),C)$.

(Morse theory) Pick a Morse function $f\in C^{\infty }(L)$. ${\Gamma }_{df}\subseteq T^{\ast }L$.

• Ex: ${\Gamma }_{df}$ is transversal to $L\iff f$ is a Morse function.

• Intuitively: $d(x^2)=2xdx$. (Graph intersecting x-axis).

Page 5: Homomorphisms and Homological Mirror Symmetry (HMS) of ${\mathbb{C}}^{\ast }$

$\text{Hom}(L,L)=\text{Hom}({\Gamma }_{df},C)=\mathbb{C}⟨df=0⟩+\text{differential }d$

$L=S^1\subset T^{\ast }{S}^1$. Two holomorphic strips $\leftrightarrow$ gradient flow lines. $df=\sin \theta d\theta$.

$\begin{array}{c}\text{Cylinder with holomorphic strips }{D}_1,D_2\text{ between }p,q\\ \updownarrow \\ \text{Gradient flow lines on circle }{S}^1\end{array}$

$\mathbb{C}⟨p,q⟩\supset d$.

$d(p)=\pm (\#\text{ of holomorphic strips running from }p\text{ to }q)\cdot e^{\int \omega }\cdot q$

$dp=\left(e^{\text{Area}(D_1)}-e^{\text{Area}(D_2)}\right)q=0\Rightarrow {\text{Hom}}_{\text{Fuk}(T^{\ast }{S}^1)}(S^1,S^1)\cong H^{\ast }(S^1)$

(The ring structure also matches).

HMS of ${\mathbb{C}}^{\ast }$:

$\text{Fuk}(T^{\ast }{S}^1,\omega )\simeq D^b\text{Coh}({\mathbb{C}}^{\ast })$

$\begin{array}{ccc}{T}^{\ast }{S}^1& \stackrel{\text{dual to}}{\to }& {\mathbb{C}}_z^{\ast }\\ \text{(Cylinder, }\mathbb{R}\text{-many objects)}& & \text{(Plane, }{\mathbb{C}}^{\ast }\text{-many objects)}\end{array}$

We have to consider “Lagrangian decorated by plane waves”.

More precisely: $(L,\mathcal{L})$, where $\mathcal{L}\in U(1)$ is a local system on $L$.

• Local system $\mathcal{L}$ on $S^1$ is a vector space of $\mathcal{L}|{}_{x\in S^1}\cong V$ + monodromy $T_{{\pi }_1(S^1)}\subset V\iff \text{an invertible matrix in }U(1)$.

$\text{Cylinder}⟶\text{Line Bundle }{\mathcal{O}}_{{\mathbb{P}}^1}\text{ (twisted/Mo¨bius strip)}$

Page 6: Finding Mirrors of Complicated Objects

(Example from the book): Find mirrors of some more complicated objects.

e.g., $D^b\text{Coh}({\mathbb{C}}_z^{\ast })\cong D^b(\mathbb{C}[z,z^{-1}]\text{-modules})$ as $\mathbb{R}$-modules and compute.

Any finitely generated $\mathbb{C}[z,z^{-1}]$-module is a finite direct sum ($\oplus$) of:

1. $\mathbb{C}[z,z^{-1}]$ free module

2. $\mathbb{C}[z,z^{-1}]/(z-\lambda {)}^n(\lambda \in {\mathbb{C}}^{\ast },n\in {\mathbb{N}}_{>0})$.

What are the mirrors of these? $⟵$ Jordan decomposition of an invertible matrix (is the monodromy matrix).

Example: Take $(L_r,{\mathcal{L}}_{e^{i\theta }})⟷\mathbb{C}[z,z^{-1}]/(z-r{e}^{i\theta })\in \text{Coh}({\mathbb{C}}_z^{\ast })$.

Fact: $\text{End}(L_r,{\mathcal{L}}_{e^{i\theta }})\cong \text{End}(\text{zero section }{S}^1)\cong H^{\ast }(S^1)\cong \mathbb{C}[ϵ]/⟨{ϵ}^2⟩,|ϵ|=1$ as a dg algebra.

We compute $R{\text{Hom}}_{\mathbb{C}[z,z^{-1}]}(\mathbb{C}[z,z^{-1}]/(z-\lambda ),\mathbb{C}[z,z^{-1}]/(z-\lambda ))$:

${\text{Hom}}^0\left(\mathbb{C}[z,z^{-1}]/(z-\lambda ),\mathbb{C}[z,z^{-1}]/(z-\lambda )\right)\cong \mathbb{C}$

(where $1\mapsto \alpha$)

Page 7: Ext and Projective Resolutions

${\text{Ext}}^1={\text{Hom}}^1(-,-)\cong \mathbb{C}$

Replace this by non-torsion through projective resolution:

$0\to \mathbb{C}[z,z^{-1}]\stackrel{z-\lambda }{\to }\mathbb{C}[z,z^{-1}]\to \mathbb{C}[z,z^{-1}]/(z-\lambda )\to 0$

Apply $\text{Hom}(-,\mathbb{C}[z,z^{-1}]/(z-\lambda ))$ to yield the following commutative diagram mapping the resolution:

$\begin{array}{ccc}\mathbb{C}[z,z^{-1}]& \stackrel{z-\lambda }{\to }& \mathbb{C}[z,z^{-1}]\\ \downarrow \text{Hom}& & \downarrow \text{Hom}\\ \text{Hom}\left(\mathbb{C}[z,z^{-1}],\mathbb{C}[z,z^{-1}]/(z-\lambda )\right)& \stackrel{z-\lambda }{\to }& \text{Hom}\left(\mathbb{C}[z,z^{-1}],\mathbb{C}[z,z^{-1}]/(z-\lambda )\right)\\ \Vert & & \Vert \\ \mathbb{C}[z,z^{-1}]/(z-\lambda )& \stackrel{0}{\to }& \mathbb{C}[z,z^{-1}]/(z-\lambda )\end{array}$

(Diagram indicates that the resulting mapped differential goes to 0).