I’ve been sick for a couple of weeks now. It’s that time of year, ‘freshers flu’ abound. At the beginning, when I knew I was coming down with an infection, I made the choice that it was also a good time for a full reset. Often when I am sick I still work on my physics, if capable, or in the very least dabble and then continue to studying new papers. But this time has been different: a complete and total sick break, the meaning of which has also been extended to include a complete and total break from everything in life. That is, no maths problems, no Polchinski textbook questions or random integrals to keep me occupied (ok, I admit, I worked on a few integrals and I am growing increasingly eager to work on some string problems), no Asperger’s appointments or random errands or administration. Just a proper shutdown, slowing life to a halt. Whilst dealing with being ill, I have taken advantage of the time to simply reflect on a busy year or more and to play some Magic (or other games). I also plan to catch up on some films this weekend in addition to the new Swamp Thing series on Netflix.
To be honest, it’s probably the first proper break I’ve had in 24+ months. Before contracting an infection and becoming ill, I was starting to experience mental burnout – the same hard crash and periodic fatigue that is a definite pattern in my life. I could objectively observe – like a narrator – silly errors that I was starting to make and I could see my computation time increasing.
Thankfully, though, I am starting to feel better and I am eager to get back to my studies and to working on a number of projects. For the first month of my MRES, I have spent a large chuck of my time learning the braneworld formalism and picking up some bits in advanced gravity theory that I previously missed. I also spent time starting to think more deeply about string geometry, classical vacua / moduli stabilisation, and to also start digging into the world of non-geometric backgrounds. There are still a lot of interesting questions and problems in these areas that one day I would like to be able to explore, but it means I first need to rebuild the picture for myself from first principles which takes time. This is how my brain best functions.
For the second month of my MRes, I imagine more time will be spent on an ongoing braneworld project alongside my professor and a PhD colleague, Cesc. It has been a lot to cram in a short amount of time, but I am very much looking forward to exploring some stringy questions in this area. I’ve also been learning more about black strings and an entirely new, wacky world of physics that includes black shells and black holes as bubble nucleation sites.
A short blog series on braneworlds is an idea that I have been playing with, but I still need to think about some of the logistics. Do I begin the series by building the geometry completely from first principles, or do I dive straight in to Randall-Sundrum? On the theme of the short-term future of this blog, I plan on uploading more string notes from my notebooks and also spending some time working on how the blog is organised (for easier reader navigation). I saw on someone else’s website that they have a section specifically for study notes, so I may do something similar. After creating such a page, I could make a table of contents for my string notes (linking to each entry) and I could also link to other documents I upload, like my notes on pure spinors or on Hodge theory or whatever. It seems a sensible way to organise things.
In the meantime, thanks for reading.
in front of the kinetic term,
(remember: a moduli space is a space parameterised by the value of some scalar fields). We will make it so 
, and we should note that 
, where a point in Moduli space is given by the expectation value for the scalar fields 
such that, in this set-up, the geodesic proper distance (i.e., the distance is equivalent to the vacuum expectation value in field space) between 
and 
may be denoted as 
(note: we measure the distance using the field space metric in front of the kinetic terms). Crucially, the first statement of the DC says this geodesic distance is infinite, which is to say the scalar field obtains an infinite vacuum expectation value. The second statement describes the behaviour at this infinity. That is to say, the second state describes that there exists an infinite tower of states with mass scale 
, such that 
as 
and where 
. 
, then all that we get is the familiar 
. In more complicated situations, such as when the scalar field is not canonically normalised, the refined DC tells us that we can apply it also to such completely general situations.
supergravity (SUGRA) in 4-dimensions. Due to limited space, we are not going to establish the supergravity formalism in these notes. The reader is instead directed to ref. [1, 3-5] for an introduction, where, for these notes, we are of course following Palti in ref. [1] quite strictly. Another very important paper, which we will cover in some depth is ref. [6] on infinite distances in field space. In fact, majority of what follows is based on this paper.
vector multiplets with bosonic content of a complex scalar field. Similar in a sense to past discussion about the presence of scalar fields with regards to the radius of the circle, in the present case the scalar fields we are interested in studying are complex structure moduli, 
, where 
. These complex structure moduli parameterise the geometry of the CY.
. These gauge fields are quite interesting, as we will elaborate. For now, note that there is a gravity multiplet which contains a (bosonic fields) graviton and graviphoton, 
. All of the gauge fields can be combined such that 
for 
.
. This means that for the field space in the effective field theory we find a space with 
complex dimensions (and so we have a 200 dimensional field space in total).
wrapping 3-cycles in the CY. Schematically, the moduli describe the size of the 3-cycle and then they describe the mass of the D3-branes that are wrapping the 3-cycles, behaving like particles in the external dimensions.![\displaystyle S_{\mathcal{N} = 2} = \int d^{4}x \sqrt{-g} [\frac{R}{2} - g_{ij} \partial_{\mu} t^{i} \partial^{\mu} \bar{t}^{j} -h_{\sigma \lambda} \partial_{\mu} l^{\sigma} \partial^{\mu} l^{\lambda} + \mathcal{I}_{IJ}\mathcal{F}^{I}_{\mu \nu}\mathcal{F}^{J, \mu \nu} + \mathcal{R}_{IJ}\mathcal{F}^{I}_{\mu \nu} (\star \mathcal{F})^{J, \mu \nu}] \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++S_%7B%5Cmathcal%7BN%7D+%3D+2%7D+%3D+%5Cint+d%5E%7B4%7Dx+%5Csqrt%7B-g%7D+%5B%5Cfrac%7BR%7D%7B2%7D+-+g_%7Bij%7D+%5Cpartial_%7B%5Cmu%7D+t%5E%7Bi%7D+%5Cpartial%5E%7B%5Cmu%7D+%5Cbar%7Bt%7D%5E%7Bj%7D+-h_%7B%5Csigma+%5Clambda%7D+%5Cpartial_%7B%5Cmu%7D+l%5E%7B%5Csigma%7D+%5Cpartial%5E%7B%5Cmu%7D+l%5E%7B%5Clambda%7D+%2B+%5Cmathcal%7BI%7D_%7BIJ%7D%5Cmathcal%7BF%7D%5E%7BI%7D_%7B%5Cmu+%5Cnu%7D%5Cmathcal%7BF%7D%5E%7BJ%2C+%5Cmu+%5Cnu%7D+%2B+%5Cmathcal%7BR%7D_%7BIJ%7D%5Cmathcal%7BF%7D%5E%7BI%7D_%7B%5Cmu+%5Cnu%7D+%28%5Cstar+%5Cmathcal%7BF%7D%29%5E%7BJ%2C+%5Cmu+%5Cnu%7D%5D++%5C+%5C+%283%29+&bg=ffffff&fg=000000&s=0 "\displaystyle S_{\mathcal{N} = 2} = \int d^{4}x \sqrt{-g} [\frac{R}{2} - g_{ij} \partial_{\mu} t^{i} \partial^{\mu} \bar{t}^{j} -h_{\sigma \lambda} \partial_{\mu} l^{\sigma} \partial^{\mu} l^{\lambda} + \mathcal{I}_{IJ}\mathcal{F}^{I}_{\mu \nu}\mathcal{F}^{J, \mu \nu} + \mathcal{R}_{IJ}\mathcal{F}^{I}_{\mu \nu} (\star \mathcal{F})^{J, \mu \nu}] \ \ (3)")
and 
. In totality, the moduli space is split into vector multiplets and hypermultiplets, 
. And so, as one would expect even notationally, these two metrics describe two separate manifolds. We are going to focus on the vector multiplets which span a special Kähler manifold, from which we can generalise for the hypermultiplets on the quaternionic Kähler manifold. What is important to note is the periodicity 
for the multiplet field space, in which we are dealing with holomorphic functions of 
and 
. These both contain real and imaginary parts of a complex matrix.
(of 
). This charge vector is in fact a 100-dimensional vector, where each entry is some holomorphic function of the 100’s of scalar fields in our theory. The basic idea, to give some more intuition, is that once we know the charge vector we know the mass of the BPS state, which, again, are the charged states under the gauge fields. Study (4) below,![\displaystyle m(\underline{q}) = \mid z(\underline{q}) \mid = \mid \frac{\underline{q \eta \underline{\prod}}(t)}{[i \underline{\prod}^{T}(t) \eta \bar{\prod}(t)]^{1/2}} \mid \ \ (4)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++m%28%5Cunderline%7Bq%7D%29+%3D+%5Cmid+z%28%5Cunderline%7Bq%7D%29+%5Cmid+%3D+%5Cmid+%5Cfrac%7B%5Cunderline%7Bq+%5Ceta+%5Cunderline%7B%5Cprod%7D%7D%28t%29%7D%7B%5Bi+%5Cunderline%7B%5Cprod%7D%5E%7BT%7D%28t%29+%5Ceta+%5Cbar%7B%5Cprod%7D%28t%29%5D%5E%7B1%2F2%7D%7D+%5Cmid+%5C+%5C+%284%29+&bg=ffffff&fg=000000&s=0 "\displaystyle m(\underline{q}) = \mid z(\underline{q}) \mid = \mid \frac{\underline{q \eta \underline{\prod}}(t)}{[i \underline{\prod}^{T}(t) \eta \bar{\prod}(t)]^{1/2}} \mid \ \ (4)")
is the period vector. Notice that in the denominator we have complex conjugation as the object 
must be real. Furthermore, all of the geometry of the CY is captured in the period vector 
. One can see that it is a function of 
and 
term in (4) is the natural symplectic form of this multiplet vector space, and so we may indeed construct the appropriate symplectic inner products.
is the Kähler potential, ![{K = -log [i \prod^{T} \eta \bar{\prod}]}](https://s0.wp.com/latex.php?latex=%7BK+%3D+-log+%5Bi+%5Cprod%5E%7BT%7D+%5Ceta+%5Cbar%7B%5Cprod%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{K = -log [i \prod^{T} \eta \bar{\prod}]}")
. In other words, we have the log of the period vector. This potential is actually very interesting, and one can derive it by considering in general a Kähler potential for some CY manifold, 
, of complex dimension 
where the complex structure moduli is given by a 
-dimensional Kähler manifold. The potential is then generally written as ![K = -log [-i^{D} \int_{Y_{D}} \omega \wedge \bar{\omega}]](https://s0.wp.com/latex.php?latex=K+%3D+-log+%5B-i%5E%7BD%7D+%5Cint_%7BY_%7BD%7D%7D+%5Comega+%5Cwedge+%5Cbar%7B%5Comega%7D%5D+&bg=ffffff&fg=000000&s=0 "K = -log [-i^{D} \int_{Y_{D}} \omega \wedge \bar{\omega}]")
in which one finds metric components of the form above. Once one finds the appropriate integral basis, the potential above is found.
going to 
. We now invoke the theorem that tells us that for such a point the period vector has a monodromy around it. In other words, if we send the real part of 
, which, in a sense, is like encircling the point at infinity, we have a transformation of the period vector. In fact, we see that the period vector transforms by the action of a monodromy matrix, 
. Then, due to properties studied in [6], we see that each 
can be decomposed and, with the monodromy matrix massaged in a way that it only gives its infinite order part, we can define the log of this 
is nilpotent. This means that if we take a high enough power we will get zero, 
. Moreover, remember that we have sent  + \mathcal{O}(E^{2\pi i t})) \ \ (8)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cprod+%28t%29+%3D+%5Cexp+%5Bt+N%5D%28a_%7B0%7D+%28S%29+%2B+%5Cmathcal%7BO%7D%28E%5E%7B2%5Cpi+i+t%7D%29%29+%5C+%5C+%288%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \prod (t) = \exp [t N](a_{0} (S) + \mathcal{O}(E^{2\pi i t})) \ \ (8)")
we get exponentially small corrections. In other words, because 
depends on the other moduli, but not ![{[t N]}](https://s0.wp.com/latex.php?latex=%7B%5Bt+N%5D%7D&bg=ffffff&fg=000000&s=0 "{[t N]}")
does not annihilate the vector 
. And so what we have, to be terse, is the following,
, quadratic 
it is found that the proper field distance is logarithmic when we send 
, as implied in (11), the effective theory at two points (P, Q) in the moduli space approach singularity. The mass of the BPS states decreases exponentially fast in the proper distance. And so, in a schematic way in these notes, we may describe this in the form of 
and 
, which is just the Distance Conjecture and the Weak Gravity Conjecture at work.
. For our current purposes, the result is notable it shows that the DC and WGC hold for any CY compactification for Type IIB string theory. And this result should not in any way be understated. Altgough we are dealing with a very complicated 100-dimensional field space, the fact the it can be proven mathematically that both of these first-class Swampland conjectures hold for any CY compactification – and that very powerful mathematical theorems tell us this is necessarily true – we are driven directly toward the suggestion of some deeply general physics.
, which is the volume of the CY and another field, 
, which is the string coupling times the volume of the CY. As a result of the flux being switched on, these two fields will have some potential. 
are positive components of the potential and so the following statement is made that, 
. But what does this mean? Well, if we write this in terms of the canonically normalised fields,
. Moreover, it says in the weakly coupled regime there is a non-interacting theory, and so any fluxes etc. vanish. This is a very generic statement; it applies to any point in the Hilbert space where many possible light tower of states may dominate. Consider, for example, a potential subject to the above statement regarding the string coupling in some expansion,
. If the potential looks like 
then one can quickly work out,
for which it is possible to then have these fields in minimum balance against each other. But then what of the C coefficient? One must ensure that this doesn’t takeover, so we could say 
. But what the Dine-Seiberg argument says that if 
then we will never find the minimum to the potential, because 
must be the leading term and we end up with a runaway direction in the field space. That is why for flux compactifications a general approach is to balance the coefficients by playing with the fluxes so that we can get a minimum for the potential.
, where 
is the infinitesimal parameter, ![\displaystyle \int [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = \int [d\phi]e^{-S[\phi]} \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint+%5Bd%5Cphi%5E%7B%5Cprime%7D%5De%5E%7B-S%5B%5Cphi%5E%7B%5Cprime%7D%5D%7D+%3D+%5Cint+%5Bd%5Cphi%5De%5E%7B-S%5B%5Cphi%5D%7D+%5C+%5C+%281%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = \int [d\phi]e^{-S[\phi]} \ \ (1)")
, where 
is an arbitrary function. Consider the following comments for clarity: in this change of variables what we are doing is basically promoting 
. In that this transformation is not a symmetry of the theory, because one will notice that the action and the measure are no longer invariant, what we find is that to leading order of 
. Notice, ![\displaystyle \int [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = \int [d\phi]e^{-S[\phi]}[1 + \frac{i\epsilon}{2\pi} \int d^{d}\sigma \sqrt{-g} J^{a}(\sigma) \partial_{a}\rho(\sigma) + \mathcal{O}(\epsilon^2)] \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint+%5Bd%5Cphi%5E%7B%5Cprime%7D%5De%5E%7B-S%5B%5Cphi%5E%7B%5Cprime%7D%5D%7D+%3D+%5Cint+%5Bd%5Cphi%5De%5E%7B-S%5B%5Cphi%5D%7D%5B1+%2B+%5Cfrac%7Bi%5Cepsilon%7D%7B2%5Cpi%7D+%5Cint+d%5E%7Bd%7D%5Csigma+%5Csqrt%7B-g%7D+J%5E%7Ba%7D%28%5Csigma%29+%5Cpartial_%7Ba%7D%5Crho%28%5Csigma%29+%2B+%5Cmathcal%7BO%7D%28%5Cepsilon%5E2%29%5D+%5C+%5C+%282%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = \int [d\phi]e^{-S[\phi]}[1 + \frac{i\epsilon}{2\pi} \int d^{d}\sigma \sqrt{-g} J^{a}(\sigma) \partial_{a}\rho(\sigma) + \mathcal{O}(\epsilon^2)] \ \ (2)")
is a local function that comes from the variation of the measure and the action. Indeed, it should be emphasised, both the measure and the action are local (p.41). The picture one should have in their mind is the same we have been building for some time: namely, we are working in some localised region within which all the operators we’re considering reside. This is one of the big ideas at this point in our study of CFTs. 
. This invariance of the path integral under change of variables gives the quantum version of Noether’s theorem 
, where `…’ are arbitrary additional insertions outside of the small local region in which 
is taken to be zero. This is precisely why Polchinski comments that, when we take `the function 
as an operator statement (p.42), we want to proceed to derive the Ward identity. It follows that, as motivated by Polchinski, given (2) we now want to insert new operators into the path integral, noting ![\displaystyle \int [d\phi^{\prime}] e^{-S[\phi^{\prime}]} A^{\prime}(\sigma_{0}) = \int [d\phi]e^{-S}[A(\sigma_{0}) + \delta A + \frac{i\epsilon}{2\pi} \int d^2\sigma\sqrt{-g} J^{a}(\sigma)A(\sigma_{0})\partial_{a}\rho + \mathcal{O}(\epsilon)^2] \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint+%5Bd%5Cphi%5E%7B%5Cprime%7D%5D+e%5E%7B-S%5B%5Cphi%5E%7B%5Cprime%7D%5D%7D+A%5E%7B%5Cprime%7D%28%5Csigma_%7B0%7D%29+%3D+%5Cint+%5Bd%5Cphi%5De%5E%7B-S%7D%5BA%28%5Csigma_%7B0%7D%29+%2B+%5Cdelta+A+%2B+%5Cfrac%7Bi%5Cepsilon%7D%7B2%5Cpi%7D+%5Cint+d%5E2%5Csigma%5Csqrt%7B-g%7D+J%5E%7Ba%7D%28%5Csigma%29A%28%5Csigma_%7B0%7D%29%5Cpartial_%7Ba%7D%5Crho+%2B+%5Cmathcal%7BO%7D%28%5Cepsilon%29%5E2%5D+%5C+%5C+%283%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int [d\phi^{\prime}] e^{-S[\phi^{\prime}]} A^{\prime}(\sigma_{0}) = \int [d\phi]e^{-S}[A(\sigma_{0}) + \delta A + \frac{i\epsilon}{2\pi} \int d^2\sigma\sqrt{-g} J^{a}(\sigma)A(\sigma_{0})\partial_{a}\rho + \mathcal{O}(\epsilon)^2] \ \ (3)")
to show, 
-derivative. But this statement is equivalent to, more generally, 
in some region R and 
outside that region. In the context of our present theory, the divergence theorem may then be invoked to give, ![\displaystyle \int_{R} d^2 \sigma \sqrt{-g} \nabla_{a}[J^{a}A(\sigma_{0})] = \int_{\partial R}dA n_{a}J^{a} A(\sigma_{0}) = \frac{2\pi}{i \epsilon} \delta A(\sigma_{0}) \ \ (8)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7BR%7D+d%5E2+%5Csigma+%5Csqrt%7B-g%7D+%5Cnabla_%7Ba%7D%5BJ%5E%7Ba%7DA%28%5Csigma_%7B0%7D%29%5D+%3D+%5Cint_%7B%5Cpartial+R%7DdA+n_%7Ba%7DJ%5E%7Ba%7D+A%28%5Csigma_%7B0%7D%29+%3D+%5Cfrac%7B2%5Cpi%7D%7Bi+%5Cepsilon%7D+%5Cdelta+A%28%5Csigma_%7B0%7D%29+%5C+%5C+%288%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{R} d^2 \sigma \sqrt{-g} \nabla_{a}[J^{a}A(\sigma_{0})] = \int_{\partial R}dA n_{a}J^{a} A(\sigma_{0}) = \frac{2\pi}{i \epsilon} \delta A(\sigma_{0}) \ \ (8)")
and 
the outward normal. As Polchinski explains, what we have is a relation between the integral of the current around the operator and the variation of that same operator (p.42). We can see this in the structure of the above equation. 
coordinates. To do this we may rewrite (8) in flat 2-dimensions as, 
is holomorphic, meaning 
. Therefore, also, 
is antiholomorphic, meaning 
. In these cases we use residue theorem, ![\displaystyle 2\pi i [Res J_{z}A(z_{0}, \bar{z}_{0}) + Res J_{\bar{z}}A(z_{0}, \bar{z}_{0})] = \frac{2\pi}{\epsilon}\delta A(z_{0}, \bar{z}_{0}) \ \ (10)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2%5Cpi+i+%5BRes+J_%7Bz%7DA%28z_%7B0%7D%2C+%5Cbar%7Bz%7D_%7B0%7D%29+%2B+Res+J_%7B%5Cbar%7Bz%7D%7DA%28z_%7B0%7D%2C+%5Cbar%7Bz%7D_%7B0%7D%29%5D+%3D+%5Cfrac%7B2%5Cpi%7D%7B%5Cepsilon%7D%5Cdelta+A%28z_%7B0%7D%2C+%5Cbar%7Bz%7D_%7B0%7D%29+%5C+%5C+%2810%29+&bg=ffffff&fg=000000&s=0 "\displaystyle 2\pi i [Res J_{z}A(z_{0}, \bar{z}_{0}) + Res J_{\bar{z}}A(z_{0}, \bar{z}_{0})] = \frac{2\pi}{\epsilon}\delta A(z_{0}, \bar{z}_{0}) \ \ (10)")
and 
are the coefficients of 
and 
. 
like dependence and 
like dependence. And in this way we will learn how operators transform. 
Heterotic duality. This will then lead us directly into the fourth and second-last entry of the series, where we will begin to review more advanced tests of the DC and WGC, using for instance arbitrary Calabi-Yau manifolds.
, we are going to instead consider a more general solution for the metric. The reason for this is because we want to study in particular the case of compactification with gauge fields. The metric may be written as follows,
in the d-dimensional theory, as opposed to a scalar field, which will also have a coupling 
. Furthermore, in that we have added another component to the metric, namely the 9-dimensional ![\displaystyle \int d^{D}X \sqrt{-G}R^{D} = \int d^{d}X\sqrt{-g} [R^{d} - \frac{1}{2}(\partial \phi)^{2} - \frac{1}{4}e^{-2(d - 1)\alpha \phi}F_{(A), \mu \nu} F^{\\mu \nu}_{(A)}] \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint+d%5E%7BD%7DX+%5Csqrt%7B-G%7DR%5E%7BD%7D+%3D+%5Cint+d%5E%7Bd%7DX%5Csqrt%7B-g%7D+%5BR%5E%7Bd%7D+-+%5Cfrac%7B1%7D%7B2%7D%28%5Cpartial+%5Cphi%29%5E%7B2%7D+-+%5Cfrac%7B1%7D%7B4%7De%5E%7B-2%28d+-+1%29%5Calpha+%5Cphi%7DF_%7B%28A%29%2C+%5Cmu+%5Cnu%7D+F%5E%7B%5C%5Cmu+%5Cnu%7D_%7B%28A%29%7D%5D+%5C+%5C+%282%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int d^{D}X \sqrt{-G}R^{D} = \int d^{d}X\sqrt{-g} [R^{d} - \frac{1}{2}(\partial \phi)^{2} - \frac{1}{4}e^{-2(d - 1)\alpha \phi}F_{(A), \mu \nu} F^{\\mu \nu}_{(A)}] \ \ (2)")
![{F_{(A), \mu \nu} =\frac{1}{2} \partial_{[\mu}A_{\nu]}}](https://s0.wp.com/latex.php?latex=%7BF_%7B%28A%29%2C+%5Cmu+%5Cnu%7D+%3D%5Cfrac%7B1%7D%7B2%7D+%5Cpartial_%7B%5B%5Cmu%7DA_%7B%5Cnu%5D%7D%7D&bg=ffffff&fg=000000&s=0 "{F_{(A), \mu \nu} =\frac{1}{2} \partial_{[\mu}A_{\nu]}}")
is the gauge field kinetic term or, in other words, the field strength of the gauge field. Recall, also, from before that the 
is a local gauge parameter. Notice that the metric remains invariant, and from this we can indeed see how lower d-dimensional theory has a U(1) gauge field with the above gauge coupling.
, notice that the gauge transformation (4) reveals that the KK modes 
obtain a charge under the U(1) gauge field. This charge is quantised, as anticipated, and for the nth KK mode it may be given as, 
, is related to the charge. More pointedly, we are already familiar with how, for the KK modes, there is an infinite tower of states. We see that the mass increases along this tower, and so too does the charge. In other words, it is argued that we have a charge-mass relation for the infinite tower of states. Here it is for arbitrary 
,
, which comes from the Kalb-Ramond B-field with a single index in the 
direction. For this Kalb-Ramond field we may write,![\displaystyle V_{\mu} \equiv B_{[\mu d]} \ \ (8)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++V_%7B%5Cmu%7D+%5Cequiv+B_%7B%5B%5Cmu+d%5D%7D+%5C+%5C+%288%29+&bg=ffffff&fg=000000&s=0 "\displaystyle V_{\mu} \equiv B_{[\mu d]} \ \ (8)")
![{B_{[mn]}}](https://s0.wp.com/latex.php?latex=%7BB_%7B%5Bmn%5D%7D%7D&bg=ffffff&fg=000000&s=0 "{B_{[mn]}}")
is an antisymmetric 2-form. If we also reduce ![\displaystyle \int d^{d}X \sqrt{-g} [R^{d} - \frac{1}{4}e^{-2(\alpha + \beta)\phi}F_{(V), \mu \nu}F^{\mu \nu}_{(V)}] \ \ (9)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint+d%5E%7Bd%7DX+%5Csqrt%7B-g%7D+%5BR%5E%7Bd%7D+-+%5Cfrac%7B1%7D%7B4%7De%5E%7B-2%28%5Calpha+%2B+%5Cbeta%29%5Cphi%7DF_%7B%28V%29%2C+%5Cmu+%5Cnu%7DF%5E%7B%5Cmu+%5Cnu%7D_%7B%28V%29%7D%5D+%5C+%5C+%289%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int d^{d}X \sqrt{-g} [R^{d} - \frac{1}{4}e^{-2(\alpha + \beta)\phi}F_{(V), \mu \nu}F^{\mu \nu}_{(V)}] \ \ (9)")
. From (9) one can again read off the gauge coupling,
very large, we obtain a strongly coupled theory. So, in taking from what we reviewed in the last entry, we know that charges under this gauge field are the winding modes of the string. That is, we have stringy or indeed quantum gravity states. Moreover, think about how if we take the basic Polyakov action for a string wrapping in the 
times in the Einstein frame, which means that we can set 
, then notice we have ![\displaystyle S_{P} = -\frac{T}{2} \int_{\sum} d\tau d\sigma [2i V_{\mu} \partial_{\tau} X^{\mu} \partial_{\sigma} (\frac{w\sigma}{2 \pi})]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+S_%7BP%7D+%3D+-%5Cfrac%7BT%7D%7B2%7D+%5Cint_%7B%5Csum%7D+d%5Ctau+d%5Csigma+%5B2i+V_%7B%5Cmu%7D+%5Cpartial_%7B%5Ctau%7D+X%5E%7B%5Cmu%7D+%5Cpartial_%7B%5Csigma%7D+%28%5Cfrac%7Bw%5Csigma%7D%7B2+%5Cpi%7D%29%5D+&bg=ffffff&fg=000000&s=0 "\displaystyle S_{P} = -\frac{T}{2} \int_{\sum} d\tau d\sigma [2i V_{\mu} \partial_{\tau} X^{\mu} \partial_{\sigma} (\frac{w\sigma}{2 \pi})]")
![\displaystyle \int_{\sum = C \times S^{1}} B_{[\mu d]} dX^{\mu} \wedge dX^{d} \sim \int_{C} V_{\mu} \ \ (13)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5Csum+%3D+C+%5Ctimes+S%5E%7B1%7D%7D+B_%7B%5B%5Cmu+d%5D%7D+dX%5E%7B%5Cmu%7D+%5Cwedge+dX%5E%7Bd%7D+%5Csim+%5Cint_%7BC%7D+V_%7B%5Cmu%7D+%5C+%5C+%2813%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\sum = C \times S^{1}} B_{[\mu d]} dX^{\mu} \wedge dX^{d} \sim \int_{C} V_{\mu} \ \ (13)")
, we obtain an infinite tower of light states. But this is just a special case of the DC, emphasising again the relation between the DC and the WGC. Furthermore, notice that the gauge coupling depends on the scalar field. So should we want to go to weak coupling, we must give the scalar a large expectation value that directly implies an infinite tower of states. 
, which is just a gauge coupling that is the graviphoton. The gauge group is U(1) and, it follows,
![\displaystyle S = \int d^{d}X \sqrt{-g} [] (\frac{M_{p}^{d}}{2})^{d-2}R^{d} - \frac{1}{4g_{s}^{2}} F^{2} + ... ] \ \ (17)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++S+%3D+%5Cint+d%5E%7Bd%7DX+%5Csqrt%7B-g%7D+%5B%5D+%28%5Cfrac%7BM_%7Bp%7D%5E%7Bd%7D%7D%7B2%7D%29%5E%7Bd-2%7DR%5E%7Bd%7D+-+%5Cfrac%7B1%7D%7B4g_%7Bs%7D%5E%7B2%7D%7D+F%5E%7B2%7D+%2B+...+%5D+%5C+%5C+%2817%29+&bg=ffffff&fg=000000&s=0 "\displaystyle S = \int d^{d}X \sqrt{-g} [] (\frac{M_{p}^{d}}{2})^{d-2}R^{d} - \frac{1}{4g_{s}^{2}} F^{2} + ... ] \ \ (17)")
and charge 
satisfying,
[3]. Again, much of the following discussion also echoes [1], where a summary with additional pedagogical references can be found. 
. This is important to note because compactifying on a 
. To understand why there are 28 U(1) gauge fields, simply remember that a 
.
from which we once again obtain an infinite tower of states. The first massive excitation is the 
depends on the expectation value of the dilaton. To put it explicitly, we have a dilatonic coupling. And, so, in terms of the bound set by the WGC for the mass the following inequality is satisfied,
and 
. In that the charges should be integer, they must satisfy the lattice condition 
.
factor is just 
, and 
. And so,
, where there is equal radius for the torus. The D-brane can be thought of as filling the non-compact spacetime. In considering string theory on this background, take in particular a Type IIB on a 
, and what we want to do is specifically put a D3-brane with its 4-dimensional world-volume completely in the 
external spacetime. The D3-brane of course carries U(1), so we therefore now have a U(1) gauge symmetry in our 4-dimensional theory.
.
, the relation between the string scale and the Planck scale can be found as (from dimensional reduction and re-writing everything in Planck units),
. So, in fact, the WGC is satisfied. That is,
we obtain a weakly coupled theory.
. We considered the case for 
when we compactified on a 
and especially 
, particularly due to backreaction on the space. In all cases, it can be seen that the Electric WGC holds for open string U(1)s [1].
at the base of a Calabi-Yau 3-fold. In these notes, we have not yet considered such complex extended objects. But the idea is that we then consider a D7-brane filling the 6 external dimensions and wrapping a holomorphic curves on the Kähler surface in the 4-dimensional space. In the uncompactified 6 dimensions, the D3-brane wrapping the 2-cycle produces a solitonic ring. Associated strings on the curve 
contained in 
, where the tower of states become light according the WGC, then the 2-cycle must become very large. But, if the 2-cycle becomes big, the volume of the 4-dimensional manifold changes and, impliedly, the values of 
and the string scale also change. So one approach is to keep the volume fixed. However, fixing the volume while making the 2-cycle big means that another 2-cycle needs to be small! 
goes to zero, 
. Moreover, as the tension of the string is actually the size of the cycle, the string itself asymptotically describes an open Heterotic string. And so we observe, 
fields. 
. This statement already offers a hint of what is to come both in this section and other future parts of our study on CFTs, particularly when we more explicitly discuss Wick’s theorem and mode expansions in relation to computing OPEs. For now, we may maintain an introductory tone and say that this statement leads us to an important insight early in Polchinski’s discussion in Section 2.2 of his textbook: notably that from the theory of complex variables a harmonic function may be decomposed locally as the sum of holomorphic and antiholomorphic functions. To begin to explain what this means, and to explain Polchinski’s discussion on pp.37-38 let us consider more deeply (17) from the last post. We can think of it this way,![\displaystyle \bar{\partial}_{1} [\partial_{1} :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbar%7B%5Cpartial%7D_%7B1%7D+%5B%5Cpartial_%7B1%7D+%3AX%5E%7B%5Cmu%7D%28z_%7B1%7D%2C+%5Cbar%7Bz%7D_%7B1%7D%29X%5E%7B%5Cnu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29%3A%5D+%3D+0+&bg=ffffff&fg=000000&s=0 "\displaystyle \bar{\partial}_{1} [\partial_{1} :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0")
![\displaystyle \bar{\partial}_{1} [:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0 \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbar%7B%5Cpartial%7D_%7B1%7D+%5B%3A%5Cpartial_%7B1%7D+X%5E%7B%5Cmu%7D%28z_%7B1%7D%2C+%5Cbar%7Bz%7D_%7B1%7D%29X%5E%7B%5Cnu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29%3A%5D+%3D+0+%5C+%5C+%282%29+&bg=ffffff&fg=000000&s=0 "\displaystyle \bar{\partial}_{1} [:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0 \ \ (2)")
derivatives vanish. This is telling us something we may perhaps already know or suspect, namely as we continue to think in terms of operators 
is in fact a holomorphic function. Now, as Polchinski explains, from the theory of complex analysis it is within the rules that we can Taylor expand such holomorphic (and antiholomorphic) functions. This use of Taylor expansion may be considered one of the first tools in understanding how to compute OPEs. Consider, for example, only the holormorphic case. When we proceed with Taylor expansion in 
it is implied that we have nonsingularity as 
and we obtain the following infinite series,
. We can rewrite (3) as follows, including also the antiholomorphic series,![\displaystyle = \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}(z_{2}, \bar{z}_{2}): + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}(z_{2}, \bar{z}_{2}):] \ \ (4)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D+%5B%5Cfrac%7Bz_%7B12%7D%5E%7Bk%7D%7D%7Bk%21%7D+%3AX%5E%7B%5Cnu%7D%5Cpartial%5E%7Bk%7DX%5E%7B%5Cmu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29%3A+%2B+%5Cfrac%7B%5Cbar%7Bz%7D_%7B12%7D%5E%7Bk%7D%7D%7Bk%21%7D+%3AX%5E%7B%5Cnu%7D%5Cbar%7B%5Cpartial%7D%5E%7Bk%7DX%5E%7B%5Cmu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29%3A%5D+%5C+%5C+%284%29+&bg=ffffff&fg=000000&s=0 "\displaystyle = \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}(z_{2}, \bar{z}_{2}): + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}(z_{2}, \bar{z}_{2}):] \ \ (4)")
. What this is telling us is that if we have some normal ordered product, we may write more generally for this product,
![\displaystyle = :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}): + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}:] \ \ (5)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+%3AX%5E%7B%5Cmu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29X%5E%7B%5Cnu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29%3A+%2B+%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D+%5B%5Cfrac%7Bz_%7B12%7D%5E%7Bk%7D%7D%7Bk%21%7D+%3AX%5E%7B%5Cnu%7D%5Cpartial%5E%7Bk%7DX%5E%7B%5Cmu%7D%3A+%2B+%5Cfrac%7B%5Cbar%7Bz%7D_%7B12%7D%5E%7Bk%7D%7D%7Bk%21%7D+%3AX%5E%7B%5Cnu%7D%5Cbar%7B%5Cpartial%7D%5E%7Bk%7DX%5E%7B%5Cmu%7D%3A%5D+%5C+%5C+%285%29+&bg=ffffff&fg=000000&s=0 "\displaystyle = :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}): + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}:] \ \ (5)")
![\displaystyle = - \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} + :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}) + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}:] \ \ (6)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+-+%5Cfrac%7B%5Calpha%5E%7B%5Cprime%7D%7D%7B2%7D%5Ceta%5E%7B%5Cmu+%5Cnu%7D+%5Cln+%5Cmid+z_%7B12%7D+%5Cmid%5E%7B2%7D+%2B+%3AX%5E%7B%5Cmu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29X%5E%7B%5Cnu%7D%28z_%7B2%7D%2C+%5Cbar%7Bz%7D_%7B2%7D%29+%2B+%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D+%5B%5Cfrac%7Bz_%7B12%7D%5E%7Bk%7D%7D%7Bk%21%7D+%3AX%5E%7B%5Cnu%7D%5Cpartial%5E%7Bk%7DX%5E%7B%5Cmu%7D%3A+%2B+%5Cfrac%7B%5Cbar%7Bz%7D_%7B12%7D%5E%7Bk%7D%7D%7Bk%21%7D+%3AX%5E%7B%5Cnu%7D%5Cpartial%5E%7Bk%7DX%5E%7B%5Cmu%7D%3A%5D+%5C+%5C+%286%29+&bg=ffffff&fg=000000&s=0 "\displaystyle = - \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} + :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}) + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}:] \ \ (6)")
is the regular part of the OPE that one may remember from the two-point function 
. Again, this is something we will become more familiar with as we progress. Furthermore, notice in general that (6) looks very much like an OPE as given in (1). In fact, it will become increasingly clear, especially toward the end of our present study, that we may think of this as the free field OPE hence the inclusion of the regular piece. Later, we will show explicitly the computation to achieve this result. In the meantime, since it is simply given in Polchinski’s textbook, it has also been stated here with addition of a few more comments as follows.
as above, note that one will obtain terms of the form 
in which the derivative is outside the normal ordering as opposed to inside the normal ordering. But differentiation and normal ordering commute, which can be proven using some basic identities of functional derivatives, hence the structure of the normal ordering in the OPE (6). Also, for any arbitrary expectation value that involves some product 
multiplied by a number of fields at other points, we have been building (and will continue to build) the intuition to understand exactly why the OPE describes the behaviour for when 
of some basis operators 
. As asymptotic expansions, we will come to write OPEs up to nonsingular terms.
![{\mathcal{F} = \mathcal{F}[\partial X^{\mu_{1} ... \mu_{n}}]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathcal%7BF%7D+%3D+%5Cmathcal%7BF%7D%5B%5Cpartial+X%5E%7B%5Cmu_%7B1%7D+...+%5Cmu_%7Bn%7D%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{\mathcal{F} = \mathcal{F}[\partial X^{\mu_{1} ... \mu_{n}}]}")
. The terms in brackets represent a combination of an arbitrary number of fields. If, as before, we Taylor expand and make this expression an expansion of polynomials of 
, it follows that we may then write the normal ordering for each monomial. This leads directly to the equation (2.2.7) in Polchinski,
is any functional of 
which is, of course, the subtraction. Now, reversely, if we act with the inverse exponential, we obtain the opposite of a sum of subtractions in the form of a sum of contractions,
instead of 
. It follows that for any pair of operators, we can generate the respective OPE 
, where, again, 
. 