# Scalars and special geometry

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1. Nov 1, 2015

### Emilie.Jung

Crossing over the following paragraph:
There are three types of special manifolds which we shall discuss, related to the real scalars
of gauge multiplets in D = 5, the complex scalars of D = 4 gauge multiplets and the
quaternionic scalars of hypermultiplets. Since there are no scalars in the gauge multiplets of D = 6, there is no geometry in that case.

On another occasion, I crossed over the following statement:
That special geometry is determined by scalars of vector multiplets in N=2 Supergravity theories.

I wonder how scalars determine geometry of a manifold. In other words, how would scalars of vector multiplets characterize the geometry (special geometry) for a manifold?

2. Nov 1, 2015

### JorisL

One obvious reason is that they influence the Stress-Energy tensor of the Einstein Field Equations.
So the properties of those scalars inherently influence the geometry.
Note that this already happens in regular GR. I'm not sure regarding technical details but at first sight I don't see any problems one could have.

Another remark is that scalars show up as (by-)product of compactification.
Stability considerations will depend on the scalars in general (I think).
The latter I'm not 100% certain about, only working on that for a little while.

@fzero Can you look at my answer here? Just to make sure I didn't give wrong information.

3. Nov 1, 2015

### fzero

There are a couple of ways to connect these scalar fields to geometry. The first is the one that Joris suggests, namely, the VEVs of the scalar fields appear in the classical field equations, so we will find that quantities like the space time metric are actually functions of the scalar VEVS: $G_{mn} = G_{mn}(\phi_i , \ldots)$. In this picture, we call the scalar fields moduli. Since their values control the spacetime metric, changes in the scalar field values correspond to changes in the size and shape of the spacetime. The term modulus is basically inherited from the study of Riemann surfaces, whose metrics can rather explicitly be described in terms of a collection of complex parameters also called moduli.

There is another type of geometry, which is the one being discussed in the reference http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/specgeom.pdf from your earlier thread. Namely the target space geometry directly associated to the scalar fields. In this picture, we have a space $B$ corresponding to the spacetime of the SUSY theory, with coordinates $x^\mu$. In the examples mentioned in your quotes this would probably be a 5 or 6d Minkowski or Euclidean space. We could call $B$ the base space, or the worldvolume if we had a brane theory: the context will usually suggest the terminology. The scalar fields in the theory can be viewed as maps from $B$ to some target space $\mathcal{M}$. The fields themselves are coordinates $\phi_i = \phi_(x^\mu)$ on $\mathcal{M}$.

The quantum theory of these fields will generate corrections to the free-field theory action, so we might expect that the kinetic term will become something like
$$S = \int d^Dx \frac{1}{2} g_{ij}(\phi) \partial_\mu \phi^i \partial^\mu \phi^j + \cdots.$$
For this to be a well-defined quantum theory we would require that $g_{ij}(\phi)$ be a positive-definite and non-degenerate function of the classical values, so that we don't have negative or zero-norm states. Furthermore, field redefenitions of the form $\phi^i \rightarrow \phi^i + \xi^i (\phi)$ can be interpreted as diffeomorphisms of the target space as long as the functions $g_{ij}$ transform as a rank 2 tensor. Therefore $g_{ij}$ can be interpreted as a metric on the target space. When $D=2$, theories of this type are known as nonlinear sigma models and have been studied since the 1960s.

Just as the first picture led to the nomenclature of moduli for the scalar fields, sometimes the geometry contained in the nonlinear sigma model interpretation of the action for scalar fields is called the geometry of moduli space or, equivalently, moduli space geometry, since it is the geometry of the scalar fields themselves.

To be brief, the constraints of various types of SUSY constrain the possible corrections to the scalar field action, which in turn places restrictions on the metric of the target space. These restricted geometries are what is being called special geometry. The notes that you are reading seem to systematically go through that material.

4. Nov 1, 2015

### mitchell porter

See the first paragraph of the paper that introduced the name "special geometry". The basic thing to understand is that the "target manifold" whose geometry is determined by the scalars, is not the space-time manifold that you start with, it's the "space" of possible values of the scalar fields. It's just another example in physics, of something relatively abstract being described geometrically, as when the set of possible states of a physical system are called a "state space" or "configuration space". In this case, you can apparently go quite far in giving a geometric interpretation to some of the algebraic relations between the scalars.

5. Nov 2, 2015

### Emilie.Jung

@fzero as always thank you for your rich answers. I have few questions on this

-1-
Do you mean by this sentence present in the summary of your answer what you presented by
If so, how? I didn't get how the latter idea is a SUSY constraint?
-2-
What kind of restrictions did the SUSY constraints place on the metric in your description above?

6. Nov 2, 2015

### fzero

No, the general form there isn't a SUSY constraint. Just think of this as an effective field theory, where in addition to the kinetic terms $(\partial \phi_i)^2$, we can have higher-order terms $\phi^k (\partial \phi)^2$, subject to whatever symmetries are supposed to be preserved. Even things like rotational symmetries can put constraints on these terms, like an $O(N)$ symmetry.

Let's consider $D=4$ and $N=1$ SUSY to be specific. Then we consider complex scalar fields $\phi_i$ that are components of chiral superfields $\Phi_i$. The most general SUSY Lagrangian can be written in the form
$$L = \int d^4\theta K(\Phi_i,\bar{\Phi}_\bar{i}) + \int d^2\theta W(\Phi_i) + \text{h.c.},$$
where $K(\Phi_i,\bar{\Phi_i})$ is a real function and $W(\Phi_i)$ is a function of only the left-chiral superfields. Some superspace manipulation will reveal that
$$\int d^4\theta K(\Phi_i,\bar{\Phi_i}) = g_{i\bar{i}} \partial_\mu \phi^i \partial^\mu \bar{\phi}^{\bar{i}} + \text{fermions, auxiliary fields},$$
where
$$g_{i\bar{i}} = \left. \frac{ \partial^2 K}{\partial \Phi_i \partial \bar{\Phi}_\bar{i}} \right|_{\Phi = \phi}.~~~(*)$$

Mathematically, the target space is a complex manifold with a Hermitian metric derivable from a potential function. Furthermore, (*) implies that the corresponding 2-form
$$\omega = g_{i\bar{i}} d\phi^i \wedge d \bar{\phi}^{\bar{i}}$$
is closed. In the theory of complex manifolds $\omega$ is known as a Kähler form and a manifold supporting such a form and associated Hermitian metric is known as a Kähler manifold.

Therefore $N=1$ SUSY has placed the constraint that the target/moduli space is a Kähler manifold.

7. Nov 3, 2015

### Emilie.Jung

Manyyy thanks for the nice illustration, @fzero!

8. Nov 10, 2015

### samuelphysics

@fzero , I have to say that your answers are quite impressive!

Kindly, may you explain how did you conclude this sentence:
Sounds as if you concluded it from the two sentences that preceded it, but I cannot relate why would positive-definite metric and the diffeomorphism map you pointed out lead to the fact that $g_{ij}$ is a metric?

9. Nov 11, 2015

### fzero

Let's talk about this geometrically. The fields $\phi^i(x^\mu)$ are maps from $B\rightarrow \mathcal{M}$. Then the objects $\partial_\mu\phi^i$ are the components of what is called the pushforward map. We don't need to dwell too much on the significance apart from the fact that this is a map from the tangent space of $B$ at the point $x$ to the tangent space of $\mathcal{M}$ at the point $\phi$. So $\partial_\mu\phi^i$ should be viewed as tangent vector-valued on $B$ and also on $\mathcal{M}$.

Now the metric is defined as a function that acts on a pair of tangent vectors that is bilinear, symmetric and nondegenerate and returns a real number. Assume for the sake of argument that we consider the $\phi^i$ as real (we could take the real and imaginary parts of the complex fields). Then we have an expression $g_{ij} \eta^{\mu\nu} \partial_\mu \phi^i \partial_\nu\phi^j$, where $\eta_{\mu\nu}$ is the metric that we assume exists on the base space. This is already written as a matrix equation, so we know that $g_{ij}$ is bilinear. Furthermore, the expression $\eta^{\mu\nu} \partial_\mu \phi^i \partial_\nu\phi^j$ is symmetric in $ij$, so only the symmetric part of $g_{ij}$ can ever appear in the equation. Last, we had to assume that $g_{ij}$ is nondegenerate for the quantum theory to make sense. So $g_{ij}$ satisfies the definition of a metric on $\mathcal{M}$ and we therefore interpret it as such.

10. Nov 11, 2015

### samuelphysics

Uh, I read you @fzero !! Thanks!!! Just few quick questions

The fields are themselves the maps? Or did you mean that they were mapped by some functions?

Sorry, how did you know that it was a bilinear? Not familiar with bilinears.

Speaking of that, does the process usually go that we seek a quantum theory of the fields of our SUSY theory? If so, why should we knock ont he doors of quantum world?

11. Nov 11, 2015

### MacRudi

Samuelphysics wrote:
"Sorry, how did you know that it was a bilinear? Not familiar with bilinears."

Yes can you @fzero explain it more detailed? There's the point, where it is very interesting for understanding

12. Nov 11, 2015

### fzero

More precisely, the vacuum expectation values of the fields are the maps. A nonzero VEV takes the coordinate $x^\mu$ as input and returns the coordinate $\phi^i$ as output.

A linear function of a vector $\mathbf{f(v)}$ satisfies the conditions that $\mathbf{f(v+w)}= \mathbf{f(v)+f(w)}$ and $\mathbf{f}(a \mathbf{v}) =a\mathbf{f(v)}$, when $a$ is a scalar. A bilinear function is a function of two vectors $F\mathbf{v,w})$ that is linear in both arguments, so for example, $F(\mathbf{v,w+x})=F(\mathbf{v,w})+F(\mathbf{v,x})$ and $F(\mathbf{v+x,w})=F(\mathbf{v,w})+F(\mathbf{x,w})$. Matrices are natural examples of bilinear maps, in the form $M(\mathbf{w,v}) = \mathbf{w}^T\mathbf{M} \mathbf{v}.$

From the physics perspective, we are typically studying SUSY theories as models of the real world, so we must understand the quantum theory. Either we have in mind the idea that nature might be supersymmetric at some high energy scale, or we wish to study some general property of QFT itself and we can view SUSY as a tool to make the QFT model a bit simpler than the nonSUSY models. Along the way to understanding the quantum theory we must also understand the classical version, so this is where most of this geometry discussion lies.

13. Nov 11, 2015

### MacRudi

And why do we set SUSY a priori as models of the real world? Because without we could not make predictions with the Lie group standard Model for our QT. The basics of Particle physics and our QT is grounded on Lie Group theory. Without we are lost in the whole particle physics and QT. The rest of chance to keep our QT as kind of theory, which has something to do with the real world lies in SUSY Models. Although we should have found long time ago an experimental prove for SUSY like Up sQuarks, we haven't found yet and I would bet, we will not find.

14. Nov 11, 2015

### samuelphysics

@fzero thank you sooo much

15. Nov 11, 2015

### fzero

There are a few motivations for considering SUSY extensions of the Standard Model. One is the hierarchy problem that asks why the Higgs mass could be small compared to the scale at which new physics occurs, be that a grand unification, or gravity, or something else. Another motivation is that coupling constant matching is better if SUSY exists at some scale lower than the GUT scale. SUSY, GUTs and many other ideas in particle physics could well be wrong, but we would not be able to realize that if the implications of these theories were not studied.

There is no reason to think that we absolutely should have seen experimental proof for SUSY by now. Rather, current observations indicate two things: 1. SUSY is certainly broken at some scale beyond the electroweak scale, though the precise limits on the scale of SUSY breaking are very dependent on the particular SUSY model, and 2. As the limits on this scale are pushed up away from the electroweak scale, the possibility that SUSY explains the hierarchy problem is diminished.

There is a vast window of over 10 orders of magnitude of energy scale up to the Planck scale that we cannot probe with current technology (and perhaps not ever in Earth-based experiments). We cannot completely rule out many possibilities for new physics, no matter how we feel about them.

16. Nov 12, 2015

### MacRudi

@fzero thank you very much for this complete resumee about the problems we have to solve, if we want to solve QT like the problems. This is a chalenge we need to solve more carefully with the understanding of mathematics. I tried to study it the last years with Grassmann Algebra and its geometry and found out and came to the opinion that we cannot really hope to let the underlying mathematics of QT be as true real world. It is only the first step as Newton made with Integral analysis mathematics and we need the Manifold of the Tensor mathematics to describe the whole world with locality. Similar to these first steps of Newton and the extension of Einstein I see it with our case here. Special Geometry is like the Tensor mathematic of einstein and Lagrangian is only the underlying Integral analysis of Newton. But if we understand the whole special geometry, then we will come to the conclusion, that we have a different world view like we had from Newton to Einstein. And in the next step we will see, that we have to think completely in special geometry to understand what the sun looks like when we are looking out of the cave directly (Plato).

Maybe we need a new generation of scientists, who have learned at school (from the grammar school) about different Algebra to think easily in a Grassmann World. It is very hard to think in other mathematics. The next generation could think in it as a native grassman "language thinker". It will take more time.