• ## Brane dimers and quiver theories

This post reviews a few aspects of the article Brane Dimers and Quiver Gauge Theories, by Franco, Hanany, Kennaway, Vegh and Wecht. The illustrations are taken from this article. Here I will focus on an example, the general construction is pedagogically explained in the article linked above.

• ## Infrared physics is rich !

Today appeared on arxiv a set of lecture notes by Strominger, called Lectures on the Infrared Structure of Gravity and Gauge Theory. They review the progresses of the last few years in connecting different parts of physics, where the same mathematical object had been studies for decades but under very different guises. This is illustrated by the "infrared triangle", taken from the notes :

• ## Modular anomalies for E2 and eta are the same

In a previous post, I derived the modular anomaly using a straightforward computation (although a subtle one). Physicists may be more familiar with the transformation of the Dedekind eta function under S-duality, $$\label{Seta} \eta (-1/\tau) = \sqrt{- i \tau} \eta (\tau) \, .$$ This transformation is equivalent to the modular anomaly for the Eisenstein series $E_2$, thanks to the relation $$\label{relation} E_2 (\tau) = \frac{12}{\pi i} \frac{\mathrm{d}}{\mathrm{d} \tau} \log \eta (\tau) = \frac{12 \eta ' (\tau)}{\pi i \eta (\tau)} \, .$$ Indeed, taking the differential of (\ref{Seta}), we have $\eta ' (-1/\tau) = \frac{1}{2} \sqrt{-i} \tau^{3/2} \eta (\tau) + \sqrt{-i} \tau^{5/2} \eta ' (\tau)$, and the transformation rule for $E_2$ follows.

• ## M-theory to the rescue of Strings for understanding QTF

(fy)

#### M-theory

Let us first recall how M-theory emerges, from the superstring point of view. The tension of a $Dp$ brane is

\tau_p = g_s^{-1} (2 \pi)^{-p} \alpha ' ^{-(p+1)/2} \, .

Here $g_s$ is the string coupling constant, and $\alpha ' = l_s^2$ is the square of the string length. If we focus on the $D0$ brane, which is a particle, and for which the tension is just its mass, we find the mass $m_{D0} = \frac{1}{g_s \sqrt{\alpha '}}$. There also exist bound states of $n$ $D0$ branes, with mass $M = n m_{D0}$. Now when the string coupling becomes large, $g_s \rightarrow \infty$, those states become light particles, that can be interpreted as a Kaluza-Klein tower, with mass $M=n/R$ with a radius

R = g_s \sqrt{\alpha '} = g_s l_s \, .

This means that while at weak coupling $g_s \ll 1$ the strings length is much bigger that $R$ and the string theory is effectively ten-dimensional, but when the coupling becomes large, we can no longer ignore the additional compactified eleventh dimension. At low-energy, this theory is identified with eleven-dimensional supergravity, and at any energy scale, we have M-theory.

#### Branes in M-theory and string theory

The brane content of M-theory can be inferred from the low-energy supergravity. This theory has a three-form, which should be sourced by some three-dimensional object (counting time), that is the $M2$ brane. By eleven-dimensional Hodge duality[1. Recall that a $Dp$ brane in $d$ dimensions couples to a $p+1$ gauge field, which has a $p+2$-form field strength, whose dual is a $d-p-2$ field strength that derives from a $d-p-3$ gauge field that couples to a $D(d-p-4)$ brane. Applying to $d=11$ and $p=2$, we find that the dual of the $M2$ is the $M5$. ], we should also have an $M5$ brane. Somehow less well-known is the fact that there is a third kind of supersymmetric object, called the M-wave, or sometimes the $M0$ brane. Depending on whether these two types of branes wrap or not the compact cycle, they give various ten-dimensional objects :

• The M-wave can not wrap the circle (it is space-like). It gives the Kaluza-Klein states that are identified with bound states of $D0$ branes.
• If the $M2$ wraps the circle, it gives the fundamental type $IIA$ string, sometimes called $F1$.
• If the $M2$ does not wrap the circle, it gives the $D2$.
• If the $M5$ wraps the circle, it gives the $D4$.
• If the $M5$ does not wrap the circle, it gives the $NS5$.
• The $D6$ brane, magnetic dual in ten dimensions of the $D0$, corresponds to the Kaluza-Klein monopole, or Taub-NUT space.

All this is very nice, except that we didn't explain the $D8$ brane ! This is a more complicated story, maybe the subject of an following up note.

#### The fate of the M5 brane

In his seminal paper about solutions of four-dimensional theories via M-theory, Witten uses a construction with $D4$ and $NS5$ branes as follows.

\begin{array}{c|cccccccccc}
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
NS5 & - & - & - & - & - & - & \mathrm{Re} \, s & \cdot & \cdot & \cdot \\
D4 & - & - & - & - & \mathrm{Re} \, v & \mathrm{Im} \, v & - & \cdot & \cdot & \cdot
\end{array}

There are $n+1$ $NS5$ branes, and $k_\alpha$ $D4$ branes between the five-branes $\alpha -1$ and $\alpha$, for $\alpha=1,\dots,n$.

The four-dimensional interpretation is the following :

• There is a gauge group $\prod_{\alpha=1}^n SU(k_\alpha)$.
• The coupling constant is given by the positions of the $NS5$ in the direction $6$, and the theta angle is given by the eleventh dimension $10$: with $s=(x^6 + i x^{10})/R$ we have $-i \tau_{\alpha} = s_{\alpha} - s_{\alpha -1}$.
• There are hypermultiplets transforming in the representations $(\mathbf{k}_{\alpha} ,\bar{\mathbf{k}}_{\alpha+1})$
• The masses of the hypers are given by the positions $v = x^4 + i x^5$ of the $D4$ in the directions $4,5$.

This is very interesting, but the story is even more beautiful once one realizes that both the $NS5$ and the $D4$ come from the same object in M-theory, namely the $M5$! Not only is this beautiful, but it will also solve a problem of the type $IIA$ description, by smoothing the singular junctions where the $D4$ end on the $NS5$. We will therefore consider an $M5$ that lives in the four space-time dimensions, and in addition in a two-dimensional Riemann surface $\Sigma$ inside the $\mathbb{R}^3 \times S^1$ parametrized by $(v,s)$. If locally $\Sigma$ is given by the equation $v = \mathrm{cst}$, then the ten-dimensional interpretation is a $D4$, and of the equation is $s = \mathrm{cst}$, we have an $NS5$, as illustrated in the table below.

\begin{array}{c|ccccccccccc}
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
M5 & - & - & - & - & \Sigma & \Sigma & \Sigma & \cdot & \cdot & \cdot & \Sigma
\end{array}

Let us now add one last element to our construction, a few $D6$ branes.

\begin{array}{c|cccccccccc}
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
NS5 & - & - & - & - & - & - & \mathrm{Re} \, s & \cdot & \cdot & \cdot \\
D4 & - & - & - & - & \mathrm{Re} \, v & \mathrm{Im} \, v & - & \cdot & \cdot & \cdot \\
D6 & - & - & - & - & \mathrm{Re} \, v & \mathrm{Im} \, v & \mathrm{Re} \, s & - & - & -
\end{array}

More precisely, we put $d_{\alpha}$ $D6$ branes between the $NS5$ labeled $\alpha -1$ and $\alpha$. The gauge theory interpretation of the previous paragraph does not change, but we have now a different matter content, with additional hypermultiplets coming from the strings between the $D4$ and $D6$. Again, going to M-theory makes everything smooth, the singularity of the $D6$ core being replaced by the smooth and complete (multi-)Taub-NUT metric.

In the next article, I will explain how those constructions can be used to understand deep concepts of the field theories, both in four and higher dimensions.

#### References

• The classical book by C. Johnson, D-branes.
• For the M-wave, see this paper by Chu and Isono.
• The brane scan at nCatLab.
• E. Witten, Solutions Of Four-Dimensional Field Theories Via M Theory.
• ## Rigor and the Modular Anomaly

In this short note, I derive the modular transformation for the weight two Eisenstein series, and I emphasize the steps where problems that are sometimes neglected in the physical community, like absolute convergence or exchange of summation symbols and limits, are crucial. The Eisenstein $G_2$ function is defined by

G_2 (\tau) = \sum\limits_{c \in \mathbb{Z}} \sum\limits_{d \in \mathbb{Z}'_c} \frac{1}{(c \tau + d)^2} \, .

Here $\zeta$ is the Riemann zeta function, and we have $\zeta (2) = \pi^2 / 6$. In our notation $\mathbb{Z}'_c$ is just $\mathbb{Z}$ when $c \neq 0$ and is $\mathbb{Z} - \{0\}$ when $c=0$. In other words, we have

\label{defG2}
G_2 (\tau) = \sum\limits_{d \neq 0} \frac{1}{d^2} + \sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)^2} = 2 \zeta (2) + \sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)^2} \, .

Now what happens if we perform an $S$-duality, replacing $\tau \rightarrow -1/ \tau$ ? Well,

\frac{1}{\tau^2} G_2 \left(-\frac{1}{\tau} \right) = \sum\limits_{c \in \mathbb{Z}} \sum\limits_{d \in \mathbb{Z}'_c} \frac{1}{\tau ^2 (d- c / \tau )^2} \, ,

and if we change the names of the summation indices,

\label{G2dual}
\frac{1}{\tau^2} G_2 \left(-\frac{1}{\tau} \right) = \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \in \mathbb{Z}'_d} \frac{1}{ (c \tau + d )^2} = 2 \zeta (2) + \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)^2} \, .

Look at how similar equations (\ref{defG2}) and (\ref{G2dual}) are ! If we were not careful, we would conclude that they are equal... But the sum, although convergent, is not absolutely convergent, and more care is needed. We will see that the two expressions (\ref{defG2}) and (\ref{G2dual}) are not equal.

\begin{eqnarray}
G_2 (\tau) &=& \sum\limits_{c \in \mathbb{Z}} \sum\limits_{d \in \mathbb{Z}'_c} \frac{1}{(c \tau + d)^2} \\
&=& 2 \zeta (2) + \sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)^2} \\
&=& 2 \zeta (2) + \sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)^2} - \sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)(c \tau + d+1)} \\
&=& 2 \zeta (2) + \sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)^2(c \tau + d+1)} \\
&=& 2 \zeta (2) + \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)^2(c \tau + d+1)} \\
&=& 2 \zeta (2) + \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)^2} - \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)(c \tau + d+1)}\\
&=& 2 \zeta (2) + \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)^2} + \frac{2 \pi i}{\tau} \\
&=& \sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \in \mathbb{Z}'_d} \frac{1}{ (c \tau + d )^2} + \frac{2 \pi i}{\tau} \\
&=& \frac{1}{\tau^2} G_2 \left(-\frac{1}{\tau} \right)+ \frac{2 \pi i}{\tau} \, .
\end{eqnarray}
To go from the fourth to the fifth line, we have used the fact that the sum is absolutely convergent, and therefore we can exchange the two summation symbols. There are two crucial steps in this computation, where we need the astonishing results
\begin{eqnarray}
\sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)(c \tau + d+1)} &=& 0 \\
\sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)(c \tau + d+1)} &=& - \frac{2 \pi i}{\tau} \, .
\end{eqnarray}
Those two equalities are a wonderful example of why rigor is important in mathematics, and to quote Diamond and Shurman in their splendid book A First Course in Modular Forms, this is a calculation "that should leave the reader deeply appreciative of absolute convergence in the future". Let us prove the two equalities above:
\begin{eqnarray}
\sum\limits_{c \neq 0} \sum\limits_{d \in \mathbb{Z}} \frac{1}{(c \tau + d)(c \tau + d+1)} &=& \sum\limits_{c \neq 0} \lim\limits_{N \rightarrow \infty} \left( \sum\limits_{d=-N}^{N-1} \frac{1}{c \tau + d} - \frac{1}{c \tau + d+1} \right) \\
&=& \sum\limits_{c \neq 0} \lim\limits_{N \rightarrow \infty} \left( \frac{1}{c \tau -N} - \frac{1}{c \tau +N} \right) \\
&=& \sum\limits_{c \neq 0} 0\\
&=&0 \, .
\end{eqnarray}
\begin{eqnarray}
\sum\limits_{d \in \mathbb{Z}} \sum\limits_{c \neq 0} \frac{1}{(c \tau + d)(c \tau + d+1)} &=& \lim\limits_{N \rightarrow \infty} \sum\limits_{d=-N}^{N-1} \sum\limits_{c \neq 0} \left( \frac{1}{c \tau + d} - \frac{1}{c \tau + d+1} \right)\\
&=& \lim\limits_{N \rightarrow \infty} \sum\limits_{c \neq 0} \sum\limits_{d=-N}^{N-1} \left( \frac{1}{c \tau + d} - \frac{1}{c \tau + d+1} \right)\\
&=& \lim\limits_{N \rightarrow \infty} \sum\limits_{c \neq 0} \left( \frac{1}{c \tau -N} - \frac{1}{c \tau +N} \right) \\
&=& \lim\limits_{N \rightarrow \infty} \frac{-2}{\tau} \sum\limits_{c > 0} \left( \frac{1}{\frac{N}{\tau} + c} + \frac{1}{\frac{N}{\tau} - c} \right) \\
&=& \lim\limits_{N \rightarrow \infty} \frac{-2}{\tau} \left( \pi \, \mathrm{cot} \, \frac{\pi N}{\tau} - \frac{\tau}{N}\right)\\
&=& \frac{-2}{\tau} \left( \pi i - 0 \right)\\
&=& - \frac{2 \pi i}{\tau} \, .
\end{eqnarray}
In the process we have used the well-known identity

\frac{1}{\tau} + \sum\limits_{c > 0} \left( \frac{1}{\tau + c} + \frac{1}{\tau - c} \right) = \pi \, \mathrm{cot} \, \pi \tau \, .

As a final comment, note that in the literature a different normalization is often used, $E_2 = \frac{1}{2 \zeta (2)} G_2$. The transformation rule of this function under the modular $S$-duality then reads

\frac{1}{\tau^2} E_2 \left(-\frac{1}{\tau} \right) = E_2 (\tau) + \frac{6}{\pi i \tau} \, .

Note that the modularity can be restored if we agree to sacrifice holomorphy Define

E_2^{\ast} (\tau) = E_2 (\tau) + \frac{6}{\pi i (\tau - \bar{\tau})} \, .

\frac{1}{\tau^2} E_2^{\ast} \left(-\frac{1}{\tau} \right) = E_2^{\ast} (\tau)\, .

• ## A toy-model for instantons

Instantons are a central concept of modern theoretical physics, and they are by essence non-perturbative, which might make them somewhat frightening. Here I want to mention briefly one setup where they appear in such a simple guise that one can really study them explicitly at will. This is the $\mathbb{CP}^1$ model. Because I want to stress the simplicity of the description of the instantons, I will do that in the first section, and then in the second section I will explain the name $\mathbb{CP}^1$ given to the model. This is based on Shifman's book Advanced topics in Quantum Field Theory, section 29.

#### The model and its instantons

Consider a two-dimensional model with one complex scalar $\phi (x)$ with Lagrangian $$\mathcal{L} = \frac{2 \partial_{\mu} \bar{\phi} \partial_{\mu} \phi}{g^2 (1+ \bar{\phi} \phi)^2} \, ,$$ where $g$ is the coupling constant. In fact, we take $\phi$ to live on the Riemann sphere $\mathbb{CP}^1 = \mathbb{C} \cup \{\infty\}$, which is all right because of the form of the action. We also use the Euclidean metric $(+1,+1)$ and sum over repeated indices. One can check that $$\mathcal{L} = g^{-2} (1+ \bar{\phi} \phi)^{-2} \left[( \partial_{\mu} \bar{\phi} - i \epsilon_{\mu\nu} \partial_{\nu} \bar{\phi}) ( \partial_{\mu}\phi + i \epsilon_{\mu\rho} \partial_{\rho} \phi ) - 2i \epsilon_{\mu\nu} \partial_{\mu} \bar{\phi} \partial_{\mu} \phi \right] \, .$$ The last term in this equation is proportional to the derivative of $\epsilon_{\mu\nu} \bar{\phi} \partial_{\nu} \phi / (1+ \bar{\phi} \phi)$, so it reduces to a topological term. We deduce that the minimal action is achieved when $\partial_{\mu}\phi + i \epsilon_{\mu\rho} \partial_{\rho} \phi = 0$.

Define now the complex combination $z = x_1 + i x_2$. The above equation takes the very simple form $$\bar{\partial} \phi = 0 \, .$$ Since $\phi$ takes values on the Riemann sphere, it means that $\phi (z)$ minimizes the action if and only if it is a meromorphic function ! In fact, one can show that the variable $z$ should also be thought to live on the sphere, because as $|z| \rightarrow \infty$, $\phi (z)$ has to be constant for the action to remain finite. Therefore $\phi$ is a map $S^2 \rightarrow S^2$, and because $\pi_2 (S^2) = \mathbb{Z}$, each map belongs to a topological sector labeled by an integer.

Let us consider for instance $\phi (z) = \frac{a}{z-b}$. Plugging into the Lagrangian, only the topological term contributes, and the action evaluates to $$S = \frac{4 \pi}{g^2} \, .$$ The answer is independent of the constants $a$ and $b$, which are instanton moduli (the first one is related to the size of the instanton and the second one to its position). There are 4 real moduli for this instanton. Clearly, one can add as many terms as one wants to form a $k$-instanton solution with $4k$ moduli, and it is no more difficult to describe colliding instantons using poles of higher order.

#### The origin of the Lagrangian

Now that I have described the instantons, let us explain where does the Lagrangian written above come from. Using stereographic projection, we can send the plane parametrized by $\phi$ to the sphere $S^2$ -- incidentally, this explains why I insisted that $\phi$ live on the Riemann sphere. The equations for this projection are $$\begin{cases} X^1 = \frac{2 \mathrm{Re} \, \phi}{1+ \bar{\phi} \phi} \\ X^2 = \frac{2 \mathrm{Im} \, \phi}{1+ \bar{\phi} \phi} \\ X^3 = \frac{1 - \bar{\phi} \phi}{1+ \bar{\phi} \phi}\end{cases}$$ Using these new variables, that satisfy $||\vec{X}|| = 1$, we can rewrite the Lagrangian as $$\mathcal{L} = \frac{1}{2 g^2} ||\partial_{\mu} \vec{X}||^2 \, .$$ In other words, this is the theory of a free field on a sphere, otherwise known as the $O(3)$ model, for obvious invariance reasons. This is a very well-known model, that can be tracked back to Heisenberg's model of antiferromagnets formulated in the 1930s.

• ## Surface Defects and Resolvents

This is a short note based on the paper "Surface Defects and Resolvents" by Gaiotto, Gukov and Seiberg. I will illustrate how one can compute (twisted) chiral ring relations for a supersymmetric $2d$ theory coupled to a $4d$ theory. Here the focus is on the two-dimensional aspect, and a future post will shed another light on this problem, from the four-dimensional point of view.

#### $\mathcal{N}=(2,2)$ supersymmetry in two dimensions

Let us begin with a very quick review of some basic notions of two-dimensional theories with $\mathcal{N}=(2,2)$ supersymmetry. A good place to learn about $\mathcal{N}=(2,2)$ supersymmetry is the big book on mirror symmetry, available freely here. Let us first present the basic fields, and how we can construct supersymmetric actions.

It is useful to introduce the superspace formalism : the superfields depend on the time coordinate $x^0$, the space coordinate $x^1$ and the four Grassmann coordinates $\theta^{\pm}$ and $\bar{\theta}^{pm}$. The space-time coordinates are combined into $x^{\pm} = x^0 \pm x^1$. Then we can define eight operators $$D_{\pm} = \frac{\partial}{\partial \theta^{\pm}} - i \bar{\theta}^{\pm} \partial_{\pm} \, , \qquad Q_{\pm} = \frac{\partial}{\partial \theta^{\pm}} + i \bar{\theta}^{\pm} \partial_{\pm}$$ and similar definitions for $\bar{D}_{\pm}$ and $\bar{Q}_{\pm}$. We then say that a superfield $\Phi$ is

• Chiral if $\bar{D}_{\pm} \Phi = 0$ ;
• Anti-chiral if $D_{\pm} \Phi = 0$ ;
• Twisted chiral if $\bar{D}_+ \Phi = D_- \Phi = 0$ ;
• Twisted anti-chiral if $\bar{D}_- \Phi = D_+ \Phi = 0$.

Using these superfields, we can construct supersymmetric invariant actions by combining the following terms :

• The D-terms $\int \mathrm{d}^2 x \, \mathrm{d}^4 \theta \, K (\Phi_i)$ for any superfields $\Phi_i$, where $\mathrm{d}^4 \theta = \mathrm{d} \theta^+ \mathrm{d} \theta^- \mathrm{d} \bar{\theta}^+ \mathrm{d} \bar{\theta}^-$.
• The F-terms $\int \mathrm{d}^2 x \, \mathrm{d}^2 \theta \, W (\Phi_i)$ for any chiral superfields $\Phi_i$. Here $\mathrm{d}^2 \theta = \mathrm{d} \theta^+ \mathrm{d} \theta^-$.
• The twisted F-terms $\int \mathrm{d}^2 x \, \mathrm{d}^2 \tilde{\theta} \, \tilde{W} (U_i)$ for any twisted chiral superfields $U_i$. Here $\mathrm{d}^2 \tilde{\theta} = \mathrm{d} \bar{\theta}^- \mathrm{d} \theta^+$.

#### $\mathcal{N}=(2,2)$ gauge theories

If we want to construct gauge theories, we need another ingredient, namely the vector superfield $V$, which is a real superfield that transforms as usual under gauge transformations (see section 15.2.1 in the Mirror symmetry book for the details). We consider here a $U(1)$ gauge field. The important point is that the super-field-strength that we construct from $V$ is a twisted chiral superfield $$\Sigma = \bar{D}_+ D_- V \, .$$ The Lagrangian for the gauge part is then $$\mathcal{L}_{\mathrm{gauge}} = - \frac{1}{2e^2} \int \mathrm{d}^2 x \, \mathrm{d}^4 \theta \, \bar{\Sigma} \Sigma \, .$$ The kinetic terms for the charged chiral superfields $\Phi$ has the form $$\mathcal{L}_{\mathrm{kin}} = \int \mathrm{d}^2 x \, \mathrm{d}^4 \theta \, \bar{\Phi} e^V \Phi \, .$$

Now a point which is specific to two dimensions is that we can add a linear twisted superpotential $$\tilde{W}_{\mathrm{FI} , \theta} = -t \Sigma$$ where $t=r-i \theta$ with $r$ the Fayet-Iliopoulos parameter and $\theta$ the $theta$-angle. Finally, if it is possible to form a polynomial $W(\Phi)$ which is gauge invariant, we can of course add such a (non-twisted) superpotential term in the action.

Let us consider the supersymmetric gauged linear sigma-model (GLSM) Lagrangian $$\mathcal{L}_{\textrm{GLSM}} = \mathcal{L}_{\mathrm{gauge}} + \mathcal{L}_{\mathrm{kin}} + \frac{1}{2} \left( -t \int \mathrm{d}^2 x \, \mathrm{d}^2 \tilde{\theta} \, \Sigma + c.c. \right) \, .$$ We can compute the low-energy twisted effective superpotential as follows :

1. Integrate out the massive chiral matter fields. For a field of total mass $x$, one obtains a contribution $2 \pi i \tilde{W} = - x \log (x/e)$. The total mass can in general be written as $m+\sigma$ where $m$ is the bare mass and $\sigma$ is the scalar part of the vector multiplet.
2. Extremize the total effective $\tilde{W} [\sigma , m]$ with respect to $\sigma$. The equation $\partial_{\sigma} \tilde{W} [\sigma , m] = 0$ are the twisted chiral ring relations. When solved, the solutions plugged back into $\tilde{W} [\sigma , m]$ give the twisted effective superpotential.

As a final comment, note that a GLSM can be used to realize a non-linear sigma-model on certain target Kähler manifolds. For instance,

#### Four-dimensional theory with surface defects

We now show how the procedure of the previous paragraph can be applied to the computation of the twisted effective superpotential for a four-dimensional theory with surface defects. For simplicity we consider the example of pure $\mathcal{N}=2$ $SU(2)$ gauge theory in four dimensions, with a defect that breaks $SU(2) \rightarrow U(1)$. In the UV, this can be defined as a $CP(1)$ sigma-model coupled to the pure $SU(2)$ theory. This means that the $2d$ theory contains an $SU(2)$ doublet of fields  of unit charge under some $U(1)$ gauge field, and after integrating them out, we obtain an effective twisted superpotential $$2 \pi i \tilde{W} [\sigma , \Phi] = t \sigma - \mathrm{Tr} \, \left[ (\sigma + \Phi ) \log \frac{\sigma + \Phi }{e}\right] \, .$$ It is important to note that the $4d$ fields $\Phi$ play the role of $2d$ twisted masses, and we recall that the term $t \sigma$ was introduced in the previous paragraph as the original twisted superpotential.

Now the effects of the $4d$ gauge dynamics are introduced by noting that the second derivative gives $$- 2 \pi i \partial_{\sigma}^2 \tilde{W} [\sigma , \Phi] = \mathrm{Tr} \, \left[ \frac{1}{\sigma + \Phi }\right]$$ and that this quantity, which is usually denoted $T(\sigma)$, can be computed, for instance, using the Dijkgraaf-Vafa method. Naively, if we use the classical relation $\mathrm{Tr} \, \Phi^{2k} = 2u^k$, we would say that $T(x) = 2 \sigma / (\sigma^2 - u)$. But there are instanton corrections, and the result is $$T(x) = \frac{2 \sigma}{\sqrt{(x^2 - u)^2 - 4 \Lambda^4}} \, ,$$ where $\Lambda$ is as usual the scale factor. With this explicit expression at hand, we can integrate once to obtain $\partial_{\sigma} \tilde{W} [\sigma , \Phi]$, and therefore the twisted chiral ring relations. Here, this gives $$\sigma^2 = e^t + u + \Lambda^4 e^{-t} \, .$$

• ## Introduction to Matrix Models and Topological QFT

These are reading notes based on the book by Marcos Mariño, Chern-Simons Theory, Matrix Models and Topological Strings, Clarendon press, Oxford, 2005. It includes extensive quotations.

### Matrix Models

Matrix models are quantum gauge theories in zero dimensions. Consider an action $\frac{1}{g_s}W(M) = \frac{1}{2g_s} \mathrm{Tr} M^2 + \frac{1}{g_s} \sum\limits_{p \geq 3} \frac{g_p}{p} \mathrm{Tr} M^p$ for a Hermitian $N \times N$ matrix. The partition function $Z$ can be evaluated by perturbation theory around the Gaussian point as a power series in the $g_p$, using fatgraphs. The perturbative expansion of the free energy $F = \log Z$ will involve only connected vacuum bubbles and we can write

\label{eqFreeEnergyMatrix}
F (t) = \sum\limits_{g=0}^{\infty} \sum \limits_{h=1}^{\infty} F_{g,h} g_s^{2g-2} t^h = \sum\limits_{g=0}^{\infty} F_g(t) g_s^{2g-2} \, ,

where $g$ is the genus[1. A fatgraph is characterized by its number of edges $E$, of vertices $V$, and closed loops $h$. The genus is defined by $2g-2 = E-V-h$.] of the fatgraphs, $h$ is the number of holes and $t=Ng_s$ is the 't Hooft parameter. The right-hand side is the large $N$ expansion at fixed $t$.

How to compute $F_g(t)$ ? There is a clever trick to tackle this problem. The matrix model has a gauge symmetry $M \rightarrow U M U^{\dagger}$, which can be used to diagonalize $M$. Using the Fadeev-Popov technique we can rewrite $Z$ as an integral over the eigenvalues:

\begin{split}
Z & = \frac{1}{\mathrm{Vol} U(N)} \int \mathrm{d} M e^{- \frac{1}{g_s} W (M)}  \\
& = \frac{1}{N!} \int \prod\limits_{i=1}^N  \frac{\mathrm{d} \lambda_i}{2 \pi} e^{N^2 S_{\mathrm{eff}}(\lambda)} \, ,
\end{split}

where the effective action is

S_{\mathrm{eff}}(\lambda) = - \frac{1}{tN} \sum\limits_{i=1}^N W(\lambda_i) + \frac{2}{N^2} \sum\limits_{i<j} \log |\lambda_i - \lambda_j | \, .

In the large $N$ limit, the eigenvalues can be described by the density function $\rho (\lambda)$ that can be computed[1. One has to solve $\frac{1}{2t} W'(\lambda) = \mathrm{P} \int \frac{\rho(\lambda ') \mathrm{d} \lambda'}{\lambda - \lambda'}$, which can be done by introducing the resolvent -- there is a rich domain of research.] by variation of $S_{\mathrm{eff}}(\lambda)$. The effective action can be expressed in terms of $\rho$, and one can show that $F_0(t) = t^2 S_{\mathrm{eff}}(\rho)$. Higher-genus coefficients can also be obtained.

Note that a different strategy, involving orthogonal polynomials, can be used to compute the $F_g(t)$.

### Topological Sigma Models

#### Cohomological TQFT

A cohomological TQFT[1. The cohomological topological QFTs are also called topological theories of the Witten type. ] is a QFT defined on a manifold $M$ that has an underlying scalar symmetry $\delta$ (called topological symmetry) acting on the fields $\phi_i$ in such a way that the correlation functions don't depend on the background metric.

If the energy-momentum tensor $T_{\mu\nu}$ can be written as

\label{TensorEq}
T_{\mu\nu} = \delta G_{\mu\nu}

for some tensor $G_{\mu\nu}$, then by a standard calculation a correlator of $\delta$-invariant operators $\mathcal{O}$ doesn't depend on the metric.[1. We assume that $\delta$ is not anomalous, and we neglect boundary problems.] Here we will assume that $\delta^2 = 0$ and we restrict the observables to the cohomology of $\delta$. Another standard argument shows that in such cohomological theories, the semi-classical approximation for the computation of a correlation function is exact.

The descent equations are the equations $\mathrm{d} \phi^{(n)} = \delta \phi^{(n+1)}$ that, if solved for a scalar topological observable $\phi^{(0)}$, provide a family of topological non-local observables $\int_{\gamma_{i_n}} \phi^{(n)}$ for $i_n=1 , \cdots , b_n$ and $n=1 , \cdots , \mathrm{dim} \, M$.

#### Topological Twists

An $\mathcal{N}=2$ sigma model, defined on a Riemann surface $\Sigma_g$, has four supercharges $Q_{\pm \pm}$, in addition to the spacetime generators (the translations $P_\mu$ and the rotation $J$) and internal $U(1)$ currents $F_{L,R}$. We define the vectorial current $F_V = F_L + F_R$ and the axial current $F_A = F_L - F_R$. We consider $d$ chiral and $d$ anti-chiral superfields $\Phi^I=(x^I, \psi^I , F^I)$ and $\Phi^{\bar{I}}$ and the action

S = \int_{\Sigma_g} \mathrm{d}^2 z \int \mathrm{d}^4 \theta K (\Phi^I , \Phi^{\bar{I}}) \, .

This is a sigma model whose target is a Kähler manifold of complex dimension $d$ and metric $G_{I\bar{J}}= \partial_I \partial_{\bar{J}} K(x^I , x^{\bar{J}})$.

This sigma model can be twisted in two different ways, with a redefinition of the spin current:

• A-twist : $\tilde{J} = J - F_V$.
• B-twist : $\tilde{J} = J + F_A$.

Note that this amounts to gauging one of the two $U(1)$ global currents by coupling it to the spin connection. Since the axial current has an anomaly given by the first Chern class of $X$, the B-model makes sense only on a Calabi-Yau space, where $c_1(X)=0$. In each case, the four (fermionic) supercharges become two scalars (whose sum we call $\mathcal{Q}$) and one vector $G_\mu$ that satisfy

\mathcal{Q}^2 = 0 \quad \textrm{and} \quad \{ \mathcal{Q} , G_\mu \} = P_\mu \, .

One can prove that the two twisted theories are cohomological TQFTs, by taking $\delta = \mathcal{Q}$ and finding an appropriate tensor that satisfies (\ref{TensorEq}).

#### Correlation Functions

Let us focus on the A-model on a Calabi-Yau $X$. One finds that the $\mathcal{Q}$-cohomology is given by operators[1. We don't explain here how the operators are constructed.] $\mathcal{O}_{\phi}$ where $\phi \in H^p(X)$, so the $\mathcal{Q}$-cohomology is in one-to-one correspondence with the de Rham cohomology of the target $X$. Then one can prove that $\langle \mathcal{O}_{\phi_1} \cdots \mathcal{O}_{\phi_l} \rangle = 0$ unless

\sum\limits_{k=1}^l \mathrm{deg} \, \phi_k = 2d(1-g) \, .

This implies that for $g>1$ all correlation functions vanish. This problem will be addressed next, by coupling the theory with two-dimensional gravity.

### Topological Strings

Although the topological field theories described previously contain a lot of information in genus zero, they are trivial at higher genera due to selection rules, because a fixed metric was considered in the Riemann surface. In order to obtain a non-trivial theory at higher genus, we have to introduce the degrees of freedom of the two-dimensional metric. This means that we have to couple the TQFT to $2d$ gravity.

#### Closed strings

The twisted TQFTs of the previous section are very similar to the bosonic string, with $\mathcal{Q}$ playing the role of the BRST charge. This suggests the definition [1. See for instance equation (5.4.19) in Polchinski's book. ]

F_g = \int_{\bar{M}_g} \langle \prod\limits_{k=1}^{6g-6} \int_{\Sigma_g} \mathrm{d}^2 z \left( G_{zz}(\mu_k)_{\bar{z}} {}^z + G_{\bar{z} \bar{z}} (\bar{\mu}_k)_z {}^{\bar{z}} \right) \rangle \, ,

where $\mu_k$ are the Beltrami differentials and $\bar{M}_g$ is the moduli space of Riemann surfaces of genus $g$. We can decompose $F_g = \sum_{\beta \in H^2(X,\mathbb{Z})} N_{g,\beta} Q^\beta$, where $N_{g,\beta}$ are the Gromov-Witten invariants[1. These invariants are in general rational, and they can be written in terms of the integer Gopakumar-Vafa invariants.], with $Q^\beta = \exp \left( - \int_{\beta} \omega \right)$ and $\omega$ the complexified Kähler form on $X$.

There is a relation between topological string amplitudes and physical superstring amplitudes. For instance, type IIA/B compactified on $X$ is $\mathcal{N}=2$ supergravity in four dimensions. The low-energy effective action for the vector multiplets (up to two derivatives) is coded by the prepotential, which is $F_0$ of the A/B models of topological strings. The higher-genus $F_g$ corresponds to other couplings in the supergravity theory.

#### Open strings

The previous discussion can be extended to open strings if we replace the Riemann surface $\Sigma_g$ by $\Sigma_{g,h}$, with $h$ holes. It is then necessary to specify boundary conditions in $X$: for the A model it turns out that the relevant boundary conditions are Dirichlet and given by Lagrangian[1. A Lagrangian submanifold is a cycle on which the Kähler form vanishes.] submanifolds of $X$.

### String theories and Gauge theories

In equation (\ref{eqFreeEnergyMatrix}), the middle term involves coefficients $F_{g,h}$ that could be seen as open string amplitudes on $\Sigma_{g,h}$. Is there such a string theory? In some cases, the answer is yes, and involves open topological strings whose target is a Calabi-Yau with topological D-branes. The identification is obtained using string field theory.

Now the right-hand side of (\ref{eqFreeEnergyMatrix}) looks more like a closed string amplitude, which would be related to the open string theory by an open-closed duality. This kind of dualities are associated to geometric transitions that relate different geometric backgrounds.

• ## Groupe de Galois d'un polynôme : calcul concret

Je voudrais rassembler ici quelques techniques élémentaires permettant dans certains cas de calculer le groupe de Galois d'un polynôme. Les références que j'ai utilisées sont le cours que j'ai suivi à l'École Polytechnique, ainsi que ce TD pour l'exemple traité en détail.

Soit $k$ un corps parfait [1. Je ne veux pas entrer dans les détails de ce que cela signifie. Pour nous, il suffit de savoir que si un corps est fini, ou de caractéristique nulle, il est parfait. ] et soit $\Omega$ la clôture algébrique [2. Tout corps $k$ admet une clôture algébrique unique à $k$-isomorphisme près, c'est le théorème de Steinitz. Par définition, tout polynôme à coefficients dans $k$ est scindé dans la clôture algébrique, c'est-à-dire qu'il se factorise en produit de facteurs de degré 1 dans cette clôture. ] de $k$. Considérons un polynôme $P \in k[X]$ unitaire de degré $n$ et de racines distinctes [3. Que se passe-t-il si $P$ a des racines multiples ? Dans ce cas, on considère simplement le polynôme $Q = \prod (X-x_i)$ où le produit est sur toutes les racines distinctes de $P$, et le groupe de Galois de $Q$ sera le même que celui de $P$. Il n'y a donc pas à se soucier des racines multiples. ]$x_1 , \dots , x_n$. Le groupe de Galois de $P$ sur $k$ est le groupe de Galois de son corps de racines $K = k[x_1 , \dots , x_n]$, vu comme extension de $k$. Il est clair que l'on peut voir le groupe de Galois comme un sous-groupe de $\mathfrak{S}_n$, puisqu'un élément de $G$ laisse tous les coefficients du polynôme invariants, et qu'il n'est pas difficile de montrer que le morphisme de groupe est injectif.

Listons quelques propriétés du groupe de Galois :

1. $P$ est irréductible $\Leftrightarrow$ l'action de $G$ sur les racines est transitive (il n'y a qu'une seule orbite).
2. Si $k = \mathbb{F}_p$ est le corps à $p$ éléments ($p$ premier) et $P$ est irréductible de degré $n$, alors $G \approx \mathbb{Z}_n$ et est généré par un cycle de longueur $n$.
3. Le discriminant de $P$ est un carré de $k^{\ast}$ $\Leftrightarrow$ $G \subset \mathfrak{A}_n$ (pourvu que $k$ ne soit pas de caractéristique 2).
4. Si la réduction $\bar{P}$ modulo $p$ premier de $P \in \mathbb{Z}[X]$ unitaire est séparable, alors $P$ est séparable et $G = \mathrm{Gal}(P, \mathbb{Q})$ admet un sous-groupe isomorphe à $\mathrm{Gal}(\bar{P},\mathbb{F}_p)$.
5. Si $K$ et $L$ sont deux extensions de $k$, et[4. Si cette hypothèse n'est pas vérifiée, le groupe de Galois du produit n'est pas le produit usuel des deux groupes de Galois mais leur produit amalgamé. ] $K \cap L = k$,  alors $\mathrm{Gal}(KL/k) = \mathrm{Gal}(K/k) \times \mathrm{Gal}(L/k)$.

Dans certains cas, ces propriétés suffisent à calculer le groupe de Galois d'un polynôme à coefficients entiers. Donnons maintenant un exemple de calcul.

Exemple

Prenons $P = X^5 + 10 X^3 - 10 X^2 + 35 X - 18$. On calcule sans peine que le pgcd de $P$ et de $P'$ vaut $1$, ce qui prouve que le polynôme est séparable. Nous pouvons donc appliquer les propriétés ci-dessus. On peut commencer par calculer le discriminant, qui vaut $2^6 5^8 11^2$. C'est un carré, donc $G$ est un sous groupe de $\mathfrak{A}_5$. Ce groupe d'ordre $60$ a beaucoup de sous-groupes, donc il nous faut plus d'informations.

Pour utiliser la propriété 1, il faudrait déterminer si $P$ est réductible ou non. Un moyen efficace prouver l'irréductibilité d'un polynôme est le critère d'Eisenstein, éventuellement combiné avec un changement de variable $X \rightarrow X + a$, pour $a$ entier. Ici, on calcule avec un logiciel de calcul formel le pgcd des coefficients (excepté le coefficient dominant qui vaut $1$) de $P(X+a)$ pour plusieurs entiers $a$, et on trouve qu'il vaut $5$ pour $a=3$. Regardons cela plus en détail : $$P(X+3) = X^5+15 X^4+100 X^3+350 X^2+650 X+510 \, .$$ Comme $5^2$ ne divise pas $510$, on en déduit que $P$ est irréductible. Son groupe de Galois est donc transitif.

Il faut maintenant en savoir plus sur les éléments du groupe de Galois.

• Modulo $2$, on n'apprend pas grand chose, car le polynôme s'écrit $X(X+1)^4$, et le groupe de Galois est trivial.
• Regardons la réduction modulo $3$. On voit que $0$ et $1$ sont racines, donc on peut factoriser par $X(X-1)$. Le facteur restant est $X^3+X^2-X+1$, qui n'a clairement pas de racines dans $\mathbb{F}_3$. Comme ce polynôme est de degré $3$, il est en conséquence irréductible, et son groupe de Galois est $\mathbb{Z}_3$ d'après la propriété 2. D'après la propriété 5, le groupe de Galois de la réduction de $P$ modulo $3$ est donc $\mathbb{Z}_3$, et par la propriété 4, $G$ contient $\mathbb{Z}_3$ comme sous-groupe.
• Modulo $5$, $\bar{P}$ vaut simplement $(X+2)^5$, on n'apprend rien.
• Modulo $7$, on trouve que $\bar{P}$ n'a pas de racine ! C'est intéressant : soit $\bar{P}$ se factorise en $P_2 P_3$ (les indices indiquent le degré des polynômes), soit il est irréductible. Si $\bar{P}$ se factorise, alors le groupe de Galois de $P$ contiendrait comme sous-groupe un facteur $\mathbb{Z}_2$, autrement dit une transposition, [5. Cet argument est-il correct ?] ce qui est exclus puisque $G \subset \mathfrak{A}_5$. Donc $\bar{P}$ est irréductible, et les propriétés 2 et 4 garantissent que $G$ contient un cinq-cycle.

Nous allons maintenant pouvoir déterminer $G$. Son cardinal est divisible par $3$ et $5$, donc par $15$, et il divise $60$ qui est le cardinal de $\mathfrak{A}_5$. Il y a donc trois possibilités, $15$, $30$ ou $60$. On peut alors regarder sur les tables des sous-groupes de $\mathfrak{A}_5$, par exemple ici, et on constate qu'il n'y a pas de sous-groupe d'ordre $15$ ou $30$. On en déduit $G = \mathfrak{A}_5$.

Pour aller plus loin

On pourra me reprocher d'avoir utilisé des outils peu aisément généralisables, et d'avoir eu de la chance de trouver des informations utiles en observant les réductions de $P$ modulo $3$ et $7$. En ce qui concerne l'élimination du groupe d'ordre $15$, on pouvait aussi utiliser un résultat général sur les groupes d'ordre le produit de deux nombres premiers. Pour l'élimination du groupe d'ordre $30$, on peut se servir d'un résultat qui affirme grosso modo qu'un groupe simple non-abélien n'a pas de sous groupe d'indice petit (précisément, l'ordre du gros groupe simple doit diviser la factorielle de l'indice).

En dépit de toutes ces techniques, déterminer un groupe de Galois reste un problème en général difficile. Il est d'autant plus remarquable qu'un algorithme existe bel et bien (pour des corps raisonnables comme $\mathbb{Q}$ ou $\mathbb{F}_p$), comme expliqué ici et ! Malheureusement, cet algorithme n'est pas très utile en pratique, et il faut recourir au genre d'astuces présentées ici. Je ne peux pas ne pas citer en conclusion le très bon résumé présenté dans cette conversation.

• ## Le point météo à Oviedo

Depuis que je suis à Oviedo, on me parle des pluies torrentielles qui m'attendent, et de mois sans Soleil. Pour que chacun puisse se faire une idée de ce que peut être une année à Oviedo, j'ai pris les données de cette station météo pour l'année 2016, et j'ai tracé quelques courbes. Voici les plus significatives. Il faut cependant noter que ces données sont celles de l'aéroport d'Oviedo, qui se situe à côté de la mer, à une quarantaine de kilomètres. Le temps y est en fait assez différent de celui de la ville d'Oviedo.

Tout d'abord, les températures. La courbe rouge montre la température maximale et la bleue la température minimale (toutes deux en degrés Celsius, évidemment) pour chacun des 366 jours de l'année 2016 (ces jours apparaissent en abscisse). Puis, le second graphe montre les précipitations, en millimètres d'eau, pour chaque jour de l'année 2016.