For a given CFT one can consider its group of symmetries (i.e. automorphisms of the space $H=\oplus_{p,q}H^{p,q}$ preserving all the structures). It is expected that the group of symmetries is a compact Lie group of dimension less or equal than $dim\,H^{1,0}$.

Let us fix $c_{0}\geq 0$ and $E_{min}>0$, and consider the moduli space ${\cal M}_{c\leq c_{0}}^{E_{min}}$ of all irreducible CFTs with the central charge $c\leq c_{0}$ and

$$min\{p+q>0|H^{p,q}\neq 0\}\geq E_{min}$$

It is expected that ${\cal M}_{c\leq c_{0}}^{E_{min}}$ is a compact real analytic stack of finite local dimension. The dimension of the base of the minimal versal deformation of a given CFT is less or equal than $dim\,H^{1,1}$. We define ${\cal M}_{c\leq c_{0}}=\cup_{E_{min}>0}{\cal M}_{c\leq c_{0}}^{E_{min}}$. We would like to compactify this stack by adding boundary components corresponding to certain asymptotic descriptions of the theories with $E_{min}\to 0$. The compactified space is expected to be a compact stack $\overline{{\cal M}}_{c\leq c_{0}}$. In what follows we will loosely use the word “space” instead of the word “stack”.

In order to compactify ${\cal M}_{c\leq c_{0}}$ we consider degenerations of CFTs as $E_{min}\to 0$. A degeneration is given by a one-parameter (discrete or continuous) family $H_{\varepsilon},\varepsilon\to 0$ of bi-graded spaces as above, where $(p,q)=(p(\varepsilon),q(\varepsilon))$. These spaces are equipped with OPEs. The subspace of fields with conformal dimensions vanishing as $\varepsilon\to 0$ gives rise to a commutative algebra $H^{small}=\oplus_{p(\varepsilon)\ll 1}H_{\varepsilon}^{p(\varepsilon),p(\varepsilon)}$ (the algebra structure is given by the leading terms in OPEs). The spectrum $X$ of $H^{small}$ is expected to be a compact space (“manifold with singularities”) such that $dim\,X\leq c_{0}$. It follows from the conformal invariance and the OPE, that the grading of $H^{small}$ (rescaled as $\varepsilon\to 0$) is given by the eigenvalues of a second order differential operator defined on the smooth part of $X$. The operator has positive eigenvalues and is determined up to multiplication by a scalar. This implies that the smooth part of $X$ carries a metric $g_{X}$, which is also defined up to multiplication by a scalar. Other terms in OPEs give rise to additional differential-geometric structures on $X$.

Thus, as a first approximation to the real picture, we assume the following description of a “simple collapse” of a family of CFTs. The degeneration of the family is described by the point of the boundary of $\overline{{\cal M}}_{c\leq c_{0}}$ which is a triple $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X})$, where the metric $g_{X}$ is defined up to a positive scalar factor, and $\phi_{X}:X\to{\cal M}_{c\leq c_{0}-dim\,X}$ is a map. One can have some extra conditions on the data. For example, the metric $g_{X}$ can satisfy the Einstein equation.

Although the scalar factor for the metric is arbitrary, one should imagine that the curvature of $g_{X}$ is “small”, and the injectivity radius of $g_{X}$ is “large”. The map $\phi_{X}$ appears naturally from the point of view of the simple collapse of CFTs described above. Indeed, in the limit $\varepsilon\to 0$, the space $H_{\varepsilon}$ becomes an $H^{small}$-module. It can be thought of as a space of sections of an infinite-dimensional vector bundle $W\to X$. One can argue that fibers of $W$ generically are spaces of states of CFTs with central charges less or equal than $c_{0}-dim\,X$. This is encoded in the map $\phi_{X}$. In the case when CFTs from $\phi_{X}(X)$ have non-trivial symmetry groups, one expects a kind of a gauge theory on $X$ as well.

Purely bosonic sigma-models correspond the case when $c_{0}=c(\varepsilon)=dim\,X$ and the residual theories (CFTs in the image of $\phi_{X}$) are all trivial. The target space $X$ in this case should carry a Ricci flat metric. In the supersymmetric case the target space $X$ is a Calabi-Yau manifold, and the residual bundle of CFTs is a bundle of free fermion theories.

In terms of the Virasoro operator $L_{0}$ the collapse is described by a subset (cluster) $S_{1}$ in the set of eigenvalues of $L_{0}$ which approach to zero “with the same speed”, as $E_{min}\to 0$. The next level of the collapse is described by another subset $S_{2}$ of eigenvalues of $L_{0}$. Elements of $S_{2}$ approach to zero “modulo the first collapse” (i.e. at the same speed, but “much slower” than elements of $S_{1}$). One can continue to build a tower of degenerations. It leads to an hierarchy of boundary strata. Namely, if there are further degenerations of CFTs parametrized by $X$, one gets a fiber bundle over the space of triples $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X})$ with the fiber which is the space of triples of similar sort. Finally, we obtain the following qualitative geometric picture of the boundary $\partial\overline{{\cal M}}_{c\leq c_{0}}$.

A boundary point is given by the following data:

1) A finite tower of maps of compact topological spaces $p_{i}:\overline{X}_{i}\to\overline{X}_{i-1},0\leq i\leq k$, $\overline{X}_{0}=\{pt\}$.

2) A sequence of smooth manifolds $({X}_{i},g_{{X}_{i}}),0\leq i\leq k$, such that ${X}_{i}$ is a dense subspace of $\overline{X}_{i}$, and $dim\,X_{i}>dim\,X_{i-1}$, and $p_{i}$ defines a fiber bundle $p_{i}:X_{i}\to X_{i-1}$.

3) Riemannian metrics on the fibers of the restrictions of $p_{i}$ to $X_{i}$, such that the diameter of each fiber is finite. In particular the diameter of $X_{1}$ is finite, because it is the only fiber of the map $p_{1}:X_{1}\to\{pt\}$.

4) A map $X_{k}\to{\cal M}_{c\leq c_{0}-dim\,X_{k}}$.

The data above are considered up to the natural action of the group $({{\bf R}}_{+}^{\ast})^{k}$ (it rescales the metrics on fibers).

There are some additional data, like non-linear connections on the bundles $p_{i}:X_{i}\to X_{i-1}$. The set of data should satisfy some conditions, like differential equations on the metrics. We cannot formulate this portion of data more precisely in general case. It will be done below in the case of $N=2$ SCFTs corresponding to sigma models with Calabi-Yau target spaces.

Non-supersymmetric toroidal model is described by the so-called Narain lattice, endowed with some additional data. More precisely, let us fix the central charge $c=n$ which is a positive integer number. What physicists call the Narain lattice $\Gamma^{n,n}$ is a unique unimodular lattice of rank $2n$ and the signature $(n,n)$. It can be described as ${{\bf Z}}^{2n}$ equipped with the quadratic form $Q(x_{1},...,x_{n},y_{1},...,y_{n})=\sum_{i}x_{i}y_{i}$. The moduli space of toroidal CFTs is

$${\cal M}_{c=n}^{tor}=O(n,n,{{\bf Z}})\backslash O(n,n,{{\bf R}})/O(n,{{\bf R}})\times O(n,{{\bf R}}).$$

Equivalently, it is a quotient of the open part of the Grassmannian $\{V_{+}\subset{{\bf R}}^{n,n}|\,dim\,V_{+}=n,Q_{|V}>0\}$ by the action of $O(n,n,{{\bf Z}})=Aut(\Gamma^{n,n},Q)$. Let $V_{-}$ be the orthogonal complement to $V_{+}$. Then every vector of $\Gamma^{n,n}$ can be uniquely written as $\gamma=\gamma_{+}+\gamma_{-}$, where $\gamma_{\pm}\in V_{\pm}$. For the corresponding CFT one has

$$\sum_{p,q}dim(\,H^{p,q})z^{p}\bar{z}^{q}=\bigl|\prod_{k\geq 1}(1-z^{k})\bigr|^{-2n}\sum_{\gamma\in\Gamma^{n,n}}z^{Q(\gamma_{+})}\bar{z}^{\,-Q(\gamma_{-})}$$

Let us try to compactify the moduli space ${\cal M}_{c=n}^{tor}$. Suppose that we have a one-parameter family of toroidal theories such that $E_{min}(\varepsilon)$ approaches zero. Then for corresponding vectors in $H_{\varepsilon}$ one gets $p(\varepsilon)=q(\varepsilon)\to 0$. It implies that $Q(\gamma(\varepsilon))=0,Q(\gamma_{+}(\varepsilon))\ll 1$. It is easy to see that one can add vectors $\gamma(\varepsilon)$ satisfying these conditions. Thus one gets a (part of) lattice of the rank less or equal than $n$. In the case of “maximal” simple collapse the rank will be equal to $n$. One can see that the corresponding points of the boundary give rise to the following data: $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X}^{triv};B)$, where $(X,g_{X})$ is a flat $n$-dimensional torus, $B\in H^{2}(X,{{\bf R}}/{{\bf Z}})$ and $\phi_{X}^{triv}$ is the constant map form $X$ to the trivial theory point in the moduli space of CFTs. These data in turn give rise to a toroidal CFT, which can be realized as a sigma model with the target space $(X,g_{X})$ and given B-field $B$. The residual bundle of CFTs on $X$ is trivial.

Let us consider a $1$-parameter family of CFTs defined by the family $(X,\lambda g_{X},\phi_{X}^{triv};B=0)$, where $\lambda\in(0,+\infty)$. There are two degenerations of this family, which define two points of the boundary $\partial\overline{\cal M}_{c=n}^{tor}$. As $\lambda\to+\infty$, we get a toroidal CFT defined by $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X}^{triv};B=0)$. As $\lambda\to 0$ we get $(X^{\vee},{{\bf R}}_{+}^{\ast}\cdot g_{X^{\vee}},\phi_{X}^{triv};B=0)$, where $(X^{\vee},g_{X^{\vee}})$ is the dual flat torus.

There might be further degenerations of the lattice. Thus one obtains a stratification of the compactified moduli space of lattices (and hence CFTs). Points of the compactification are described by flags of vector spaces $0=V_{0}\subset V_{1}\subset V_{2}\subset...\subset V_{k}\subset{{\bf R}}^{n}$. In addition one has a lattice $\Gamma_{i+1}\subset V_{i+1}/V_{i}$, considered up to a scalar factor. These data give rise to a tower of torus bundles $X_{k}\to X_{k-1}\to...\to X_{1}\to\{pt\}$ over tori with fibers $(V_{i+1}/V_{i})/\Gamma_{i+1}$. If $V_{k}\simeq{{\bf R}}^{n-l},l\geq 1$, then one has also a map from the total space $X_{k}$ of the last torus bundle to the point $[H_{k}]$ in the moduli space of toroidal theories of smaller central charge: $\phi_{n}:X_{k}\to{\cal M}_{c=l}^{tor}$, $\phi_{k}(X_{k})=[H_{k}]$.

In this case we have a discrete family with $c={3k\over{k+2}}$, where $k\geq 1$ is an integer number called level. In the limit $k\to+\infty$ one gets $X=SU(2)=S^{3}$ equipped with the standard metric. The corresponding bundle is the trivial bundle of trivial CFTs (with $c=0$ and $H=H^{0,0}={{\bf C}}$). Analogous picture holds for an arbitrary compact simply connected simple group $G$.

The boundary of the compactified moduli space $\overline{{\cal M}}^{N=2}$ of $N=2$ SCFTs with a given central charge contains an open stratum given by sigma models with Calabi-Yau targets. Each stratum is parametrized by the classes of equivalence of quadruples $(X,J_{X},{{\bf R}}_{+}^{\ast}\cdot g_{X},B)$ where $X$ is a compact real manifold, $J_{X}$ a complex structure, $g_{X}$ is a Calabi-Yau metric, and $B\in H^{2}(X,i{\bf R}/{\bf Z})$ is a $B$-field. The residual bundle of CFTs is a bundle of free fermion theories.

As a consequence of supersymmetry, the moduli space ${\cal M}^{N=2}$ of superconformal field theories is a complex manifold which is locally isomorphic to the product of two complex manifolds. ¹¹ 1 Strictly speaking, one should exclude models with chiral fields of conformal dimension $(2,0)$, e.g. sigma models on hyperkähler manifolds, see [AM]. It is believed that this decomposition (up to certain corrections) is global. Also, there are two types of sigma models with Calabi-Yau targets: $A$-models and $B$-models. Hence, the traditional picture of the compactified moduli space looks as follows:

Here the boundary consists of two open strata (A-stratum and B-stratum) and a mysterious meeting point. This point corresponds, in general, to a submanifold of codimension one in the closure of A-stratum and of B-stratum.

We argue that this picture should be modified. There is another open stratum of $\partial\overline{{\cal M}}^{N=2}$ (we call it T-stratum). It consists of toroidal models (i.e. CFTs associated with Narain lattices), parametrized by a manifold $Y$ with a Riemannian metric defined up to a scalar factor. This subvariety meets both $A$ and $B$ strata along the codimension one stratum corresponding to the double collapse. Therefore the “true” picture is obtained from the traditional one by the real blow-up at the corner:

Mirror symmetry is related to the existence of two different strata of the boundary $\partial\overline{{\cal M}}^{N=2}$ which we called A-stratum and B-stratum. As a corollary, same quantities admit different geometric descriptions near different strata. In the traditional picture, one can introduce natural coordinates in a small neighborhood of a boundary point corresponding to $(X,J_{X},{{\bf R}}_{+}^{\ast}\cdot g_{X},B)$. Skipping $X$ from the notation, one can say that the coordinates are $(J,g,B)$ (complex structure, Calabi-Yau metric and the B-field). Geometrically, the pairs $(g,B)$ belong to the preimage of the Kähler cone under the natural map $Re:H^{2}(X,{{\bf C}})\to H^{2}(X,{{\bf R}})$ (more precisely, one should consider $B$ as an element of $H^{2}(X,i{{\bf R}}/{{\bf Z}})$). It is usually said, that one considers an open domain in the complexified Kähler cone with the property that with the class of metric $[g]$ it contains also the ray $t[g],t\gg 1$. The mirror symmetry gives rise to an identification of neighborhoods of $(X,J_{X},{{\bf R}}_{+}^{\ast}\cdot g_{X},B_{X})$ and $(X^{\vee},J_{X^{\vee}},{{\bf R}}_{+}^{\ast}\cdot g_{X^{\vee}},B_{X^{\vee}})$ such that $J_{X}$ is interchanged with $[g_{X^{\vee}}]+iB_{X^{\vee}})$ and vice versa.

We can describe this picture in a different way. Using the identification of complex and Kähler moduli, one can choose $([g_{X}],B_{X},[g_{X^{\vee}}],B_{X^{\vee}})$ as local coordinates near the meeting point of A-stratum and B-stratum. There is an action of the additive semigroup ${{\bf R}}_{\geq 0}\times{{\bf R}}_{\geq 0}$ in this neighborhood. It is given explicitly by the formula $([g_{X}],B_{X},[g_{X^{\vee}}],B_{X^{\vee}})\mapsto(e^{t_{1}}[g_{X}],B_{X},e^{t_{2}}[g_{X^{\vee}}],B_{X^{\vee}})$ where $(t_{1},t_{2})\in{{\bf R}}_{\geq 0}\times{{\bf R}}_{\geq 0}$. As $t_{1}\to+\infty$, a point of the moduli space approaches the B-stratum, where the metric is defined up to a positive scalar only. The action of the second semigroup ${{\bf R}}_{\geq 0}$ extends by continuity to the non-trivial action on the B-stratum. Similarly, in the limit $t_{2}\to+\infty$ the flow retracts the point to the A-stratum.

This picture should be modified, if one makes a real blow-up at the corner, as discussed before. Again, the action of the semigroup ${{\bf R}}_{\geq 0}\times{{\bf R}}_{\geq 0}$ extends continuously to the boundary. Contractions to A-stratum and B-stratum carry non-trivial actions of the corresponding semigroups isomorphic to ${{\bf R}}_{\geq 0}$. Now, let us choose a point in, say, A-stratum. Then the semigroup flow takes it along the boundary to the new stratum, corresponding to the double collapse. The semigroup ${{\bf R}}_{\geq 0}\times{{\bf R}}_{\geq 0}$ acts trivially on this stratum. A point of the double collapse is also a limiting point of a $1$-dimensional orbit of ${{\bf R}}_{\geq 0}\times{{\bf R}}_{\geq 0}$ acting on the T-stratum. Explicitly, the element $(t_{1},t_{2})$ changes the size of the tori defined by the Narain lattices, rescaling them with the coefficient $e^{t_{1}-t_{2}}$. This flow carries the point of T-stratum to another point of the double collapse, which can be moved then inside of the B-stratum. The whole path, which is the intersection of $\partial\overline{\cal M}^{N=2}$ and the ${{\bf R}}_{\geq 0}\times{{\bf R}}_{\geq 0}$-orbit, connects an A-model with the corresponding B-model through the stratum of toroidal models. We can depict it as follows:

The T-portion of the path (we call it T-path) connects dual torus fibrations over the same Riemannian base. This is mirror symmetry in our picture.

This description is inspired by [SYZ]. The reader notices however, that in our picture, the mirror symmetry phenomenon is explained entirely in terms of the boundary of the compactified moduli space. In order to explain the mirror symmetry phenomenon it is not necessary to build full SCFTs. It is sufficient to work with simple toroidal models on the boundary of the compactified moduli space $\overline{{\cal M}}^{N=2}$. Also, in contrast with [SYZ], we do not use supersymmetric cycles (D-branes) in our description.