This is the second part of a guest blog on double field theory (thanks again to Lubos for giving me this opportunity). I will introduce the extension of double field theory to `exceptional field theory', a subject developed in collaboration with Henning Samtleben, and explain how it allowed us to resolve open problems in basic Kaluza-Klein theory that could not be solved by standard techniques.

Exceptional field theory is the completion (in a sense I shall make precise below) of a research program that goes back to the early 80s and attempts to understand why maximal supergravity knows about exceptional groups, such as \(E_6\), \(E_7\) and \(E_8\). These groups emerge, miraculously, as global (continuous) symmetries upon compactifying maximal supergravity on tori. This looks like a miracle because exceptional groups had no role to play in the original construction of, say, 11-dimensional supergravity. Although these symmetries are now understood as the supergravity manifestations of the (discrete) U-dualities of string-/M-theory, they remained deeply mysterious from the point of view of conventional geometry. Exceptional field theory (EFT) makes these symmetries manifest prior to dimensional reduction, in the same sense that double field theory (DFT) makes the T-duality group \(O(d,d)\) manifest.

It should be emphasized that U-dualities are tied to toroidal backgrounds. Similarly, the continuous exceptional symmetries of supergravity only emerge for compactification on tori. For compactifcations on curved backgrounds, such as spheres, there is no exceptional symmetry. Understandably, this fact led various researchers to conclude that DFT and EFT are consistently defined only on toroidal backgrounds. This is not correct, however, despite the continuing claims by some people. In its most conservative interpretation, EFT (like DFT) is simply a reformulation of (maximal) supergravity that makes its duality properties manifest. In particular, it is background-independent, and so one may describe any desired compactification. The real question therefore is whether this formalism is useful for compactifications other than toroidal ones.

Since on curved backgrounds none of the exceptional symmetries are preserved, it is reasonable to expect that EFT is more awkward than useful for such compactifications. Remarkably, it turns out that, on the contrary, EFT allows one to describe such compactifications very efficiently as generalized Scherk-Schwarz compactifications, governed by `twist matrices' taking values in the duality group. For instance, the compactification of type IIB on \(AdS_5\times S^5\) can be described by a matrix valued in \(E_6\) (the U-duality group in \(D=5\)). Moreover, in this formulation one can solve problems that could not be addressed otherwise. Thus, although physically there is no \(E_6\) symmetry in any conventional sense, this group somehow still governs these spaces `behind the scenes'.

Before I explain this and EFT in more detail, let me first discuss what exactly the issues in conventional Kaluza-Klein theory are that we resolved recently. They are related to the `consistency of Kaluza-Klein truncations', a subject that unfortunately is not appreciated even by many experts. In Kaluza-Klein theory we start with some higher-dimensional theory and decompose fields and coordinates in a way that is appropriate for a lower-dimensional theory. For instance, the metric \(G\) is written as \[ G = \begin{pmatrix} g_{\mu\nu}+A_{\mu}{}^m A_{\nu}{}^n g_{mn} & A_{\mu}{}^{k}g_{kn}\\[0.5ex] A_{\nu}{}^{k} g_{km} & g_{mn} \end{pmatrix} \] with `external' indices \(\mu,\nu\) and `internal' indices \(m,n\). The resulting fields will eventually be interpreted as lower-dimensional metric, vectors and scalars. The question is how the fields depend on the internal coordinates \(y^m\), in other words, what the `Kaluza-Klein ansatz' is.

In one extreme we may declare the fields to be completely independent of the internal coordinates, which means we are effectively truncating to the massless modes of a torus compactification. In another extreme, we may keep the full \(y\)-dependence but expand the fields in a complete basis of harmonics (such a Fourier modes on a torus or spherical harmonics on a sphere), which means keeping the full tower of Kaluza-Klein modes. In both cases there is no danger of inconsistency. The interesting question is whether there is anything in between, i.e., a non-trivial truncation that is nevertheless consistent.

The standard lore is that for a compactification on a manifold with metric \(g_{mn}\) and isometry group \(G\) the appropriate ansatz is written in terms of the Killing vectors \(K_{\alpha}=K_{\alpha}{}^{m}\partial_m\) as \[ G_{\mu n}(x,y) = A_{\mu}{}^{\alpha}(x) K_{\alpha}{}^m(y) g_{mn}(y) \] and similarly for the other metric components. Working out the infinitesimal general coordinate transformations of the metric (using standard formulas from differential geometry that give these transformations in terms of Lie derivatives), one finds that the \(A_{\mu}{}^{\alpha}\) transform like Yang-Mills gauge fields with the gauge group given by the isometry group \(G\) of the internal manifold. Concretely, in order to verify this we have to use that the Killing vectors satisfy the following algebra \[ \big[K_{\alpha}, K_{\beta}\big] \ = \ f_{\alpha\beta}{}^{\gamma} K_{\gamma} \] where \(f_{\alpha\beta}{}^{\gamma}\) are the structure constants of \(G\). Moreover, if we take the gravity action and integrate over the internal manifold, we obtain a lower-dimensional Einstein-Yang-Mills theory. This is the famous `Kaluza-Klein miracle' in which an internal gauge symmetry (the Yang-Mills gauge group) is `geometrized' in terms of a higher-dimensional manifold and its spacetime (diffeomorphism) symmetry.

The trouble with this ansatz is that in general it is inconsistent! A Kaluza-Klein truncation is consistent if and only if any solution of the (truncated) lower-dimensional theory can be embedded into a solution of the (original) higher-dimensional theory. One way to see that Kaluza-Klein truncations on curved manifolds in general are inconsistent is to insert the Kaluza-Klein ansatz discussed above into the Einstein equations and to observe that the \(y\)-dependence does not factor out consistently: one may obtain equations in which the left-hand side depends only on \(x\), but the right-hand side depends on \(x\) and \(y\). (A nice discussion of this can be found in a classic 1984 PLB paper by Duff, Nilsson, Pope and Warner.) Consequently, a solution of the Einstein-Maxwell equations, following from the action obtained by simply integrating over the internal manifold, in general does not give rise to a solution of the original field equations. Consistency only holds for very specific theories and very special internal geometries and requires a suitable non-linear extension of the Kaluza-Klein ansatz.

The known consistent truncations include 11-dimensional supergravity on \(AdS_4\times S^7\), as established by de Wit and Nicolai in 1986, and \(AdS_7\times S^4\), shown to be consistent by Nastase, Vaman and van Nieuwenhuizen in hep-th/9911238. In contrast, until our recent paper, for the celebrated case of type IIB on \(AdS_5\times S^5\) there was no proof of consistency, except for certain truncations and sub-sectors. At this point let me stress that the size of the sphere is of the same order as the AdS scale. There is no low-energy sense which would justify to keep only the `massless' modes, and hence it is especially important to actually prove that the truncation is consistent.

What was known already since 1984 is 1) the complete Kaluza-Klein spectrum of type IIB on \(AdS_5\times S^5\), determined by Gunaydin and Marcus, which requires only the linearized theory, and 2) the complete \(SO(6)\) gauged supergravity in five dimensions, constructed directly in \(D=5\) by Gunaydin, Romans, and Warner, which was believed (and is now proven) to be a consistent truncation of type IIB. What was missing since 1984 is a way to uplift the \(D=5\) gauged supergravity to type IIB. This means that we didn't even know in principle how to obtain the \(D=5\) theory from the type IIB theory in \(D=10\), because we simply didn't have the Kaluza-Klein ansatz that needs to be plugged into the higher-dimensional action and field equations.

After this digression into the consistency issues of Kaluza-Klein theory, let me return to exceptional field theory (EFT) and explain how the above problems are resolved in a strikingly simple way. As in DFT, EFT makes the duality groups manifest by introducing extended/generalized spacetimes and organizing the fields into covariant tensors under these groups. In contrast to DFT, the coordinates are not simply doubled (or otherwise multiplied). Rather, the coordinates are split into `external' and `internal' coordinates as in Kaluza-Klein, but without any truncation, and the internal coordinates are extended to live in the fundamental representation.

EFT has been constructed for \(E_6\), \(E_7\) and \(E_8\) in the series of papers 1308.1673, 1312.0614, 1312.4542, 1406.3348, but for the present discussion I will focus on the \(E_6\) case. So let's first recall some basic facts about this group, which more precisely is here given by \(E_{6(6)}\). The extra 6 in parenthesis means that we are dealing with a non-compact version of \(E_6\), in which the number of non-compact and compact generators differs by 6. \(E_{6(6)}\) has two fundamental representations of dimension 27, denoted by \({\bf 27}\) and \(\bar{\bf 27}\), with corresponding lower and upper indices \(M,N=1,\ldots, 27\). There is no invariant metric to raise and lower indices, and so these two representations are inequivalent. \(E_{6(6)}\) admits two cubic fully symmetric invariant tensors \(d^{MNK}\) and \(d_{MNK}\).

The generalized spacetime of the \(E_{6(6)}\) EFT is given by `external' coordinates \(x^{\mu}\), \(\mu=0,\ldots,4\), and (extended) `internal' coordinates \(Y^M\) in the 27-dimensional fundamental representation. As for DFT, this does not mean that the theory is physically 32-dimensional. Rather, all functions on this extended space are subject to a section constraint, which is similar to the analogous constraint in DFT. In the present case it takes the manifestly \(E_{6(6)}\) covariant form \[ d^{MNK}\partial_N\partial_K A = 0 \qquad d^{MNK}\partial_NA\,\partial_K B = 0 \label{section0} \] with \(A,B\) denoting any fields or gauge parameters. Interestingly, this constraint allows for at least two inequivalent solutions: one preserves \(GL(6)\) and leaves six physical coordinates; the other preserves \(GL(5)\times SL(2)\) and leaves five physical coordinates. The first solution leads to a theory in \(5+6\) dimensions and turns out to be equivalent to 11-dimensional supergravity; the second solution leads to a theory in \(5+5\) dimensions and turns out to be equivalent to type IIB supergravity.

The field content of the theory comprises again a generalized metric, here denoted by \({\cal M}_{MN}\), which takes values in \(E_{6(6)}\) in the fundamental representation. Due to the splitting of coordinates, however, more fields are needed. The bosonic field content is given by \[ g_{\mu\nu}\;, \quad {\cal M}_{MN}\;, \quad {\cal A}_{\mu}{}^{M}\;, \quad {\cal B}_{\mu\nu M}\;. \] Here \(g_{\mu\nu}\) is the external, five-dimensional metric, while \({\cal A}_{\mu}{}^{M}\) and \({\cal B}_{\mu\nu M}\) are higher-form potentials needed for consistency. The fields depend on all \(5+27\) coordinates, subject to the above constraint.

The theory is uniquely determined by its invariance under the bosonic gauge symmetries, including internal and external generalized diffeomorphisms. Again, there is not enough space to explain this properly, but in order to give the reader at least a sense of the extended underlying geometry, let me display the generalized Lie derivative, which satisfies an algebra governed by the analogue of the `C-bracket' in DFT (which we call the `E-bracket'), and which encodes the internal generalized diffeomorphisms. Specifically, w.r.t. to vectors \(V^M\) and \(W^M\) in the fundamental representation it reads \[ \big(\mathbb{L}_{V}W\big)^M \ \equiv \ V^N\partial_NW^M-W^N\partial_N V^M+10\,d^{MNP}\,d_{KLP}\,\partial_NV^K\,W^L \] As for the C-bracket, the first two terms coincide with the Lie bracket between vector fields, but the new term, which explicitly requires the \(E_{6(6)}\) structure, shows that the full symmetry cannot be viewed as conventional diffeomorphisms on an extended space.

There is one more fascinating aspect of the symmetries of EFT that I can't resist mentioning. The vector fields \({\cal A}_{\mu}{}^{M}\) act as Yang-Mills-like gauge potentials for the internal diffeomorphisms. The novelty here is that the underlying algebraic structure is not a Lie algebra, because the E-bracket does not satisfy the Jacobi identity. The failure of the E-bracket to satisfy the Jacobi identity is, however, of a certain exact form. As a consequence, one can construct covariant objects like field strengths by introducing higher-form potentials, in this case the two-forms \({\cal B}_{\mu\nu M}\), and assigning suitable gauge transformations to them. This is referred to as the tensor hierarchy. One can then write a gauge invariant action, which structurally looks like five-dimensional gauged supergravity, except that it encodes, through its non-abelian gauge structure, the full dependence on the internal coordinates, as it should be in order to encode either 11-dimensional or type IIB supergravity. EFT explains the emergence of exceptional symmetries upon reduction, because the formulation is already fully covariant before reduction. (For more details see the recent review 1506.01065 with Henning Samtleben and Arnaud Baguet.)

[At this point let me stress that, as always in science, exceptional field theory did not originate out of thin air, but rather is the culmination of efforts by many researchers starting with the seminal work by Cremmer and Julia. The most important work for the present story is by de Wit and Nicolai in 1986, which made some symmetries, normally only visible upon reduction, manifest in the full \(D=11\) supergravity. It did not, however, make the exceptional symmetries manifest. These were discussed in more recent work dealing with the truncation to the internal sector governed by \({\cal M}_{MN}\). Notable work is due to West, Hillmann, Berman, Perry and many others.]

We are now ready to address the issue of consistent Kaluza-Klein truncations in EFT, following the two papers 1410.8145, 1506.01385. The Kaluza-Klein ansatz takes the form of a generalized Scherk-Schwarz reduction, governed by `twist matrices' \(U\in E_{6(6)}\). For technical reasons that I can't explain here we also need to introduce a scale factor \(\rho(Y)\). The ansatz for the bosonic fields collected above then reads \[ \begin{split} g_{\mu\nu}(x,Y) &= \rho^{-2}(Y)\,{g}_{\mu\nu}(x)\nonumber\\ {\cal M}_{MN}(x,Y) &= U_{M}{}^{{K}}(Y)\,U_{N}{}^{{L}}(Y)\,M_{{K}{L}}(x) \nonumber\\ {\cal A}_{\mu}{}^{M}(x,Y) &= \rho^{-1}(Y) A_{\mu}{}^{{N}}(x)(U^{-1})_{{N}}{}^{M}(Y) \nonumber\\ {\cal B}_{\mu\nu\,M}(x,Y) &= \,\rho^{-2}(Y) U_M{}^{{N}}(Y)\,B_{\mu\nu\,{N}}(x) \end{split} \] The \(x\)-dependent fields on the right-hand side are the fields of five-dimensional gauged supergravity.

We have to verify that the \(Y\)-dependence factors out consistently both in the action and equations of motion. This is the case provided some consistency conditions are satisfied, which have a very natural geometric interpretation within the extended geometry of EFT. To state these it is convenient to introduce the combination \[ {\cal E}_{{M}}{}^{N} \ \equiv \ \rho^{-1}(U^{-1})_{{M}}{}^{N} \] [Here I am deviating from the notation in the paper in order to simplify the presentation.] The consistency condition can then be written in terms of the \(E_{6(6)}\) generalized Lie derivative discussed above. It takes the form \[ \mathbb{L}_{\,{\cal E}_{{M}}}\,{\cal E}_{{N}} \ = \ -X_{{M}{N}}{}^{{K}}\, {\cal E}_{{K}} \] where the \(X_{{M}{N}}{}^{{K}}\) are the `structure constants' of gauged supergravity that encode the gauge group. This relation is the extended geometry version of the Lie bracket algebra of Killing vector fields given above. Thus, we can view the \({\cal E}_{M}\) as generalized Killing vectors on the extended space of EFT. An important and intriguing difference is that the \(X_{MN}{}^{K}\) in general are not the structure constants of a Lie group. In general they are not even antisymmetric in their lower two indices. They do, however, satisfy a quadratic Jacobi-type identity, leading to a structure that in the mathematics literature is referred to as a `Leibniz algebra'.

Due to these novel algebraic structures, there is no general procedure of how to solve the above consistency equations, i.e., of how, for given structure constants \(X_{MN}{}^{K}\), to find a twist matrix \(U\) satisfying the above equation. I think it is a mathematically fascinating open problem to understand systematically how to integrate the above Leibniz algebra to the corresponding `Leibniz group' (whatever that could mean). What we did in the paper instead is to solve the equations `by hand' for a few interesting cases, in particular spheres and their non-compact counterparts (inhomogeneous hyperboloidal spaces \(H^{p,q}\)).

These twist matrices take a surprisingly simple universal form, which then allows us to cover in one stroke the sphere compactifications of \(D=11\) supergravity (\(AdS_4\times S^7\) and \(AdS_7\times S^4\)) and of type IIB (\(AdS_5\times S^5\)). This, finally, settles the issue of consistency of the corresponding Kaluza-Klein truncation and also gives the explicit uplift formulas: they are given by the above generalized Scherk-Schwarz ansatz. Thus, for any solution of five-dimensional gauged supergravity, given by \(g_{\mu\nu}(x)\), \(M_{MN}(x)\), etc., we can directly read off the corresponding solution of EFT and, via the embedding discussed above, of type IIB. In particular, every stationary point and every holographic RG flow of the scalar potential directly lifts to a solution of type IIB.

This concludes my summary of exceptional field theory and its applications to Kaluza-Klein compactifications. Far from being impossible to describe in exceptional field theory, spheres and other curved spaces actually fit intriguingly well into these extended geometries, which allows us to resolve open problems. This is one example of a phenomenon I have seen again and again in the last couple of years: the application of this geometry to areas where duality symmetries are not present in any standard sense still leads to quite dramatic simplifications. I believe this points to a deeper significance of these extended geometries for our understanding of string theory more generally, but of course it remains to be seen which radically new geometry (if one may still call it that) we will eventually have to get used to.

First of all I would like to thank Luboš for giving me the opportunity to write a guest blog on double field theory. This is a subject that in some sense is rather old, almost as old as string theory, but that has seen a remarkable revival over the last five years or so and that, as a consequence, has reached a level of maturity comparable to that of many other sub-disciplines of string theory. In spite of this, double field theory is viewed by some as a somewhat esoteric theory in which unphysical higher-dimensional spacetimes are introduced in an ad-hoc manner for no reasons other than purely aesthetic ones and that, ultimately, does not give any results that might not as well be obtained with good old-fashioned supergravity. It is the purpose of this blog post to introduce double field theory (DFT) and to explain that, on the contrary, even in its most conservative form it allows us to attack problems several decades old that were beyond reach until recently.

Concretely, in the first part I will review work done in collaborations with Warren Siegel and Barton Zwiebach on a formulation of DFT that includes higher-derivative \(\alpha'\) corrections and that describes certain subsectors of string theory in a way that is exact to all orders in \(\alpha'\). This casts the old problem of determining and understanding these corrections into a radically new form that, we believe, provides a significant step forward in understanding the interplay of two of the main players of string theory: \(\alpha'\) and duality symmetries. In the second part, I will explain how an extension of DFT to exceptional groups, now commonly referred to as exceptional field theory, allows us to settle open problems in Kaluza-Klein truncations of supergravity that, although of conventional nature, were impossible to solve with standard techniques.

So let's start by explaining what DFT is. It is framework for the spacetime (target space) description of string theory that makes the T-duality properties manifest. T-duality implies that string theory on the torus \(T^d\) with background metric and B-field looks the same for any background obtained by an \(O(d,d;\ZZ)\) transformation. The discrete nature of the group is due to the torus identifications (periodicity conditions) which the transformations need to respect. In the supergravity approximation to string theory the dimensional reduction on the torus truncates the massive Kaluza-Klein modes (as it should be, since in the effective supergravity we have also truncated massive string modes), and so all memory of the torus is gone. Consequently, the duality symmetry visible in supergravity is actually the continuous \(O(d,d;\RR)\), and in the following I will exclusively consider this group. In contrast to what people sometimes suspect, the continuous symmetry is preserved by \(\alpha'\) corrections, which will be important below.

This implies that gravity in \(D=10\) or \(D=26\) dimensions, extended by the bosonic and fermionic fields predicted by string theory, yields an enhanced global symmetry upon reduction that cannot be understood in terms of the symmetries present in the standard formulation of gravity. Consider the minimal field content of all closed string theories, the metric \(g_{ij}\), the antisymmetric b-field \(b_{ij}\) and a scalar (dilaton) \(\phi\), with effective two-derivative action \[ S = \int d^Dx\,\sqrt{g}e^{-2\phi}\big[R+4(\partial\phi)^2-\tfrac{1}{12}H^2\big] \] This action is invariant under standard diffeomorphisms (general coordinate transformations) \(x^i\rightarrow x^i-\xi^i(x)\) and b-field gauge transformations \(b\rightarrow b+{\rm d}\tilde{\xi}\), with vector gauge parameter \(\xi^i\) and one-form parameter \(\tilde{\xi}_i\). The diffeomorphism symmetry explains the emergence of the \(GL(d,\RR)\) subgroup of \(O(d,d)\), representing global reparametrizations of the torus, while the b-field gauge symmetry permits a residual global shift symmetry \(b\rightarrow b+c\), with antisymmetric constant c. The full symmetry is larger, however: the complete \(O(d,d)\), as predicted by string theory. String theory is trying to teach us a lesson that we fail to understand by writing the spacetime actions as above. DFT is the framework that for the first time made the full symmetries manifest before dimensional reduction.

The idea behind DFT is to introduce a doubled space with coordinates \(X^M=(\tilde{x}_i, x^i)\), \(M=1,…,2D\), on which \(O(D,D)\) acts naturally in the fundamental representation. (Note that here, at least to begin with, we have doubled the number of all spacetime coordinates.) This idea is actually well motivated by string theory on toroidal backgrounds, where these coordinates are dual both to momentum and winding modes. In fact, in closed string field theory on such backgrounds, the doubled coordinates are a necessity and not a luxury. [Perhaps it's time now for some references: the idea of doubling coordinates in connection to T-duality is rather old, going back at least to the early 90's and work by Duff, Tseytlin, Kugo, Zwiebach and others, but the most important paper for the present story is hep-th/9305073 by Warren Siegel. The modern revival of these ideas was initiated in a paper by Chris Hull and Barton Zwiebach, 0904.4664, and then continued with myself in 1003.5027, 1006.4823. There is also a close relation to `generalized geometry', which I will comment on below. For more references see for instance the review 1309.2977 with Barton Zwiebach and Dieter Lust.]

In DFT we reorganize the fields into \(O(D,D)\) covariant variables as follows \[

\eq{
{\cal H}_{MN} &= \begin{pmatrix} g^{ij} & -g^{ik}b_{kj}\\[0.5ex] b_{ik}g^{kj} & g_{ij}-b_{ik}g^{kl}b_{lj}\end{pmatrix}, \\
e^{-2d} &= \sqrt{g}e^{-2\phi}
}

\] where the `generalized metric' \({\cal H}_{MN}\) transforms as a symmetric 2-tensor and \(e^{-2d}\) is taken to be an \(O(D,D)\) singlet. Moreover, \({\cal H}_{MN}\) can be thought of as an \(O(D,D)\) group element in the following way: Defining \[ {\cal H}^{MN} \equiv \eta^{MK}\eta^{NL}{\cal H}_{KL}\;, \qquad \eta_{MN} = \begin{pmatrix} 0 & 1\\[0.5ex] 1 & 0 \end{pmatrix} \] where \(\eta_{MN}\) is the metric left invariant by \(O(D,D)\), it satisfies \[ {\cal H}^{MK}{\cal H}_{KN} = \delta^{M}{}_{N}\;, \qquad \eta^{MN}{\cal H}_{MN} = 0 \] Conversely, the parametrization above is the most general solution of these constraints. Thus, we may forget about \(g\) and \(b\) and simply view \({\cal H}\) as the fundamental gravitational field, which is a constrained field.

If we forget about \(g\) and \(b\), how do we write an action for \({\cal H}\)? We can write an action in the Einstein-Hilbert-like form \[ S_{\rm DFT} = \int d^{2D}X\,e^{-2d}\,{\cal R}({\cal H},d) \] where the scalar \({\cal R}\), depending both on \({\cal H}\) and \(d\), denotes a generalization of the Ricci scalar in standard differential geometry. But how is it constructed? There is a beautiful story here, closely analogous to conventional Riemannian geometry with its notions of Levi-Civita connections, invariant curvatures, etc., but also with subtle differences. Most importantly, there is a notion of generalized diffeomorphisms, infinitesimally given by generalized Lie derivatives \({\cal L}_{\xi}\) parametrized by an \(O(D,D)\) vector \(\xi^{M}=(\tilde{\xi}_i,\xi^i)\) that unifies the diffeomorphism vector parameter with the one-form gauge parameter. These Lie derivatives form an interesting algebra, \([{\cal L}_{\xi_1},{\cal L}_{\xi_2}]={\cal L}_{[\xi_1,\xi_2]_C}\), defining the `C-bracket' \[

\eq{
\!\big[\,\xi_1\,,\;\xi_2\,\big]_{C}^M &= \xi_1^N\partial_N\xi_{2}^M \!-\xi_2^N\partial_N\xi_{1}^M\!-\\
&-\frac{1}{2}\xi_{1N}\partial^M \xi_{2}^N \!+ \frac{1}{2}\xi_{2N}\partial^M\xi_{1}^N
}

\] where indices are raised and lowered with the \(O(D,D)\) invariant metric. The first two terms look like the standard Lie bracket between vector fields, but the remaining two terms are new. Incidentally, this shows that these transformations are not diffeomorphisms on the doubled space, for these would close according to the Lie bracket, not the C-bracket. Due to lack of space I cannot review the geometry further, but suffice it to say that the generalized diffeomorphisms uniquely determine the Ricci scalar, and since I haven't introduced connections, etc., let me just give the explicit and manifestly \(O(D,D)\) invariant expression, written in terms of the derivatives \(\partial_M\) dual to the doubled coordinates, \[

\begin{split} {\cal R} \ \equiv &~~~4\,{\cal H}^{MN}\partial_{M}\partial_{N}d -\partial_{M}\partial_{N}{\cal H}^{MN}- \\[1.2ex]
~&-4\,{\cal H}^{MN}\partial_{M}d\,\partial_{N}d + 4 \partial_M {\cal H}^{MN} \,\partial_Nd+ \\[1.0ex] ~&+\frac{1}{8}\,{\cal H}^{MN}\partial_{M}{\cal H}^{KL}\, \partial_{N}{\cal H}_{KL}-\\
~&-\frac{1}{2}{\cal H}^{MN}\partial_{M}{\cal H}^{KL}\, \partial_{K}{\cal H}_{NL} \end{split}

\] With this form of the generalized Ricci scalar, the above DFT action reduces to the standard low-energy action upon truncating the extra coordinates by setting \(\tilde{\partial}=0\).

So far I have remained silent about the nature of the extended coordinates. Surely, we don't mean to imply that the theory is defined in 20 dimensions, right? Indeed, the gauge invariance of the theory actually requires a constraint, the `strong constraint' or `section constraint', \[ \eta^{MN}\partial_M\partial_N = 2\tilde{\partial}^{i}\partial_{i} = 0 \] This is supposed to mean that \(\partial^M\partial_MA=0\) for any field or gauge parameter A but also \(\partial^M\partial_M(AB)=0\) for any products of fields, which then requires \(\partial^MA\partial_MB=0\). One may convince oneself that the only way to satisfy these constraints is to set \(\tilde{\partial}=0\) (or \(\partial_i=0\) or any combination obtained thereof by an \(O(D,D)\) transformation). This constraint is therefore an \(O(D,D)\) invariant way of saying that the theory is only defined on any of the half-dimensional subspaces, on each of which it is equivalent to the original spacetime action. (This will change, however, once we go to type II theories or M-theory, in which case different theories emerge on different subspaces, but more about this later.)

Is it possible to relax this constraint? Indeed, some of the initial excitement about DFT was due to the prospect of having a framework to describe so-called non-geometric fluxes, which are relevant for gauged supergravities in lower dimensions that apparently cannot be embedded into higher-dimensional supergravity/string theory in any conventional `geometric' way. Those non-geometric compactifications most likely require a genuine dependence on both types of coordinates, and various proposals have been put forward of how to relax the above constraint. (I should also mention that in the full closed string field theory the constraint is relaxed, where one requires only the level-matching constraint, allowing for certain dependences on both \(x\) and \(\tilde{x}\).) While these preliminary results are intriguing and, in my opinion, capture part of the truth, I think it is fair to say that we still do not have a sufficiently rigorous framework. I will therefore assume the strong form of the constraint, and so if you don't like this constraint you may simply imagine at any step that \(\partial_M\) is really only a short-hand notation for \[ \partial_M = (0,\partial_i) \] In this way of thinking the extended coordinates play a purely auxiliary role. They are necessary in order to make certain symmetries manifest and thus analogous, for instance, to the fermionic coordinates in superspace (although technically they appear to be rather different). We then have a strict reformulation of supergravity; in particular, we are not tied to the torus. I want to emphasize that it is not only possible to use this theory for general curved backgrounds (spheres, for instance), but it is actually highly beneficial to do so, as will be explained in a follow-up post.

[Let me also point to a close connection with a beautiful field in pure mathematics called `generalized geometry', going back to work by Courant, Severa, Weinstein, Hitchin, Gualtieri and others, in which the generalized metric is also a central object. Subsequently, this was picked up by string theorists, suggesting that one should forget about \(g\) and \(b\) and view the generalized metric as the fundamental object. Curiously, however, before the advent of DFT, no physicist seemed to bother (or to be capable) to take the obvious next step and to formulate the spacetime theory in terms of this object, although this is technically straightforward once phrased in the right language. What is the reason for this omission? Of course I can only speculate, but the reason must be that the (auxiliary) additional coordinates, which are absent in generalized geometry, are really needed to get any idea of what kind of terms one could write to construct an action.]

Finally we are now ready to turn to higher-derivative \(\alpha'\) corrections. We constructed a particular subset of these corrections in the paper 1306.2970 with Warren Siegel and Barton Zwiebach by using a certain chiral CFT in the doubled space with a novel propagator and simplified OPEs, earlier introduced by Warren. This is a beautiful story, but slightly too technical to be properly explained here. Therefore, let me give the idea in an alternative but more pedagogical way. First, recall that the generalized metric satisfies a constraint, which we can simply write as \({\cal H}^2=1\) (leaving implicit the metric \(\eta\)). Is there a way to define the theory for an unconstrained metric? The trouble is that when checking gauge invariance of the action we use this constraint, and so simply replacing the constrained \({\cal H}\) by an unconstrained field, which in the paper we called the `double metric' \({\cal M}\), violates gauge invariance. However, by construction, the failure of gauge invariance must be proportional to \({\cal M}^2-1\), and therefore we can restore gauge invariance, at least to first order, by adding the following term to the action: \[

\eq{
S &= \int e^{-2d}\big[\tfrac{1}{2}{\eta}^{MN}({\cal M}-\tfrac{1}{3}{\cal M}^3)_{MN}+\\
&\qquad+{\cal R}'({\cal M},d) +\dots \big]
}

\] The variation of the first term is proportional to \({\cal M}^2-1\) and so whatever the `anomalous' transformation of the generalized Ricci scalar is, we can cancel it by assigning a suitable extra gauge variation to \({\cal M}\). (The scalar \({\cal R}'\) carries a prime here, because one actually has to augment the original Ricci scalar by terms that vanish when the metric is constrained.) Since \({\cal R}\) contains already two derivatives this means we have a higher-derivative (order \(\alpha'\)) deformation of the gauge transformations.

Now the problem is that we also have to use these \({\cal O}(\alpha')\) gauge transformations in the variation of \({\cal R}'\), which requires extra higher-derivative terms in the action, in turn necessitating yet higher order terms in the gauge transformations. One would think that this leads to an iterative procedure that never stops, thus giving at best a gauge and T-duality invariant action to some finite order in \(\alpha'\). Remarkably, however, we found an exact deformation, with gauge transformations carrying a finite number of higher derivatives. Moreover, the chiral CFT construction allowed as to define an explicit action with up to six derivatives, which is of the same structural form as above. Intriguingly, the action in terms of \({\cal M}\) is cubic. Since the \(O(D,D)\) metric \(\eta\) is used to raise and lower indices, we never need to use the inverse of \({\cal M}\) (in fact, the action it completely well-defined for singular \({\cal M}\)) and so we have a truly polynomial action for gravity.

How can this be, given the standard folklore that gravity must be non-polynomial? The resolution is actually quite simple: since \({\cal M}\) is now unconstrained it encodes more fields than the expected metric and b-field, and the extra field components act as auxiliary fields. Integrating them out leads to the non-polynomial form of gravity, but including infinitely many higher-derivative corrections. Actually, the fact that gravity can be made polynomial by introducing auxiliary fields is well known, but in all cases I am aware of this is achieved by using connection variables; the novelty here is that components of the metric itself (of the doubled metric, however) serve as auxiliary fields, in a way that does not simply reproduce Einstein gravity but also infinitetly many higher-derivative corrections!

What are these higher-derivative corrections in conventional language? This is actually a quite non-trivial question, since beyond zeroth order in \(\alpha'\) the conventional metric and b-field are not encoded in \({\cal M}\) in any simple manner, as to be expected, given the rather dramatic reorganization of the spacetime theory. Barton and I managed to show last year that to first-order in \(\alpha'\) the theory encodes in particular the deformation due to the Green-Schwarz mechanism in heterotic string theory. In this, the spacetime gauge symmetries (local Lorentz transformations or diffeomorphisms) are deformed in order to cancel anomalies, which in turn requires higher-derivative terms in the action in the form of Chern-Simons modifications of the three-form curvature. We are currently trying to figure out what exactly the higher derivative modifications are to yet higher order in \(\alpha'\). These are only a subsector of all possible \(\alpha'\) corrections. For instance, this theory does not describe the Riemann-square correction present in both bosonic and heterotic string theory. This is not an inconsistency, because T-duality is not supposed to completely constrain the \(\alpha'\) corrections; after all, there are different closed string theories with different corrections. This theory describes one particular \(O(d,d)\) invariant, and we are currently trying to extend this construction to other invariants.

I hope my brief description conveys some of the reasons why we are so excited about double field theory. In a follow-up blog post I will explain how the extension of double field theory to exceptional groups, exceptional field theory, allows us to solve problems that, although strictly in the realm of the two-derivative supergravity, were simply intractable before. So stay tuned.

Manifest unitarity (simplified template)

Guest blog by Prof Dejan Stojkovic, University of Buffalo

Dear Lubos,
First, I would like to thank you very much for his kind invitation for a guest post. I am certainly honored by this gesture.

We recently published a paper titled “Radiation from a Collapsing Object is Manifestly Unitary” in PRL. The title was carefully chosen (note the absence of the term “black hole”) because of its potential implications for a very touchy issue of the information loss paradox. I will use this opportunity to explain our points of view.

An evaporating landscape? Possible issues with the KKLT scenario

By Dr Thomas Van Riet, K.U. Leuven

What is this blog post about?

In 2003, in a seminal paper by Kachru, Kallosh, Linde and Trivedi (KKLT) (2000+ cites!), a scenario for constructing a landscape of de Sitter vacua in string theory with small cosmological constant was found. This paper was (and is) conceived as the first evidence that the string theory landscape contains a tremendous amount of de Sitter vacua (not just anti-de Sitter vacua) which could account for the observed dark energy.

The importance of this discovery cannot be underestimated since it fundamentally changed the way we think about how a fundamental, UV-complete theory of all interactions addresses apparent fine-tuning and naturalness problems we are faced with in high energy physics and cosmology. It changed the way we think string theory makes predictions about the low-energy world that we observe.

It is fair to say that, since the KKLT paper, the multiverse scenario and all of its related emotions have been discussed at full intensity, even been taken up by the media and it has sparked some (unsuccessful) attempts to classify string theory as non-scientific.

In this post I briefly outline the KKLT scenario and highlight certain aspects that are not often described in reviews but are crucial to the construction. Secondly I describe research done since 2009 that sheds doubts on the consistency of the KKLT scenario. I have tried to be as unbiased as possible. But near the end of this post I have taken the freedom to give a personal view on the matter.

The KKLT construction

The main problem of string phenomenology at the time of the KKLT paper was the so-called moduli-stabilisation problem. The string theory vacua that were constructed before the flux-revolution were vacua that, at the classical level, contained hundreds of massless scalars. Massless scalars are a problem for many reasons that I will not go into. Let us stick to the observed fact that they are not there. Obviously quantum corrections will induce a mass, but the expected masses would still be too low to be consistent with observations and various issues in cosmology. Hence we needed to get rid of the massless scalars. This is where fluxes come into the story since they provide a classical mass to many (but typically not all) moduli.

The above argument that masses due to quantum corrections are too low is not entirely solid. What is really the problem is that vacua supported solely by quantum corrections are not calculable. This is called the Dine-Seiberg problem and it roughly goes as follows: if quantum corrections are strong enough to create a meta-stable vacuum we necessarily are in the strong coupling regime and hence out of computational control. Fluxes evade the argument because they induce a classical piece of energy that can stabilize the coupling at a small value. Fluxes are used mainly as a tool for computational control, to stay within the supergravity approximation.

Step 1: fluxes and orientifolds

Step 1 in the KKLT scenario is to start from the classical IIB solution often referred to as GKP (1400+ cites), (see also this paper). What Giddings, Kachru and Polchinski did was to construct compactifications of IIB string theory (in the supergravity limit) down to 4-dimensional Minkowski space using fluxes and orientifolds. Orientifolds are specific boundary conditions for strings that are different from Dirichlet boundary conditions (which would be D-branes). The only thing that is required for understanding this post is to know that orientifolds are like D-branes but with negative tension and negative charge (anti D-brane charge). GKP understood that Minkowski solutions (SUSY and non-SUSY) can be build from balancing the negative energy of the orientifolds \(T_{{\rm O}p}\) against the positive energy of the 3-form fluxes \(F_3\) and \(H_3\):\[

V = H_3^2 + F_3^2 + T_{{\rm O}p} = 0

\] This scalar potential \(V\) is such that it does not depend on the sizes of the compact dimensions. Those sizes are then perceived as massless scalar fields in four dimensions. Many other moduli directions have gained a mass due to the fluxes and all those masses are positive such that the Minkowski space is classically stable.

The 3-form fluxes \(H_3\) and \(F_3\) carry D3 brane charges, as can be verified from the Bianchi identity for the five-form field strength \(F_5\)\[

\dd F_5 = H_3 \wedge F_3 + Q_3\delta

\] The delta-function on the right represent the D3/O3 branes that are really localised charge densities (points) in the internal dimensions, whereas the fluxes correspond to a smooth, spread out, charge distribution. Gauss' law tells us that a compact space cannot carry any charge and consequently the charges in the fluxes have opposite sign to the charges in the localised sources.

I want to stress the physics in equation the Bianchi identity. To a large extend one can think of the 3-form fluxes as a smeared configuration of actual D3 branes. Not only do they induce D3 charge, they also back-react on the metric because of their positive energy-momentum. We will see below that this is more than an analogy: the fluxes can even materialize into actual D3 branes.

This flux configuration is ‟BPS″, in the sense that various ingredients exert no force on each other: the orientifolds have negative tension such that the gravitational repulsion between fluxes and orientifolds exactly cancels the Coulomb attraction. This will become an issue once we insert SUSY-breaking anti-branes (see below).

Step 2: Quantum corrections

One of the major breakthroughs of the KKLT paper (which I am not criticizing here) is a rather explicit realization of how the aforementioned quantum corrections stabilize all scalar fields in a stable Anti-de Sitter minimum that is furthermore SUSY. As expected quantum corrections do give a mass to those scalar fields that were left massless at the classical level in the GKP solution. From that point of view it was not a surprise. The surprise was the simplicity, the level of explicitness, and most important, the fact that the quantum stabilization can be done in a regime where you can argue that other quantum corrections will not mess up the vacuum. Much of the original classical supergravity background is preserved by the quantum corrections since the stabilization occurs at weak coupling and large volume. Both coupling and volume are dynamical fields that need to be stabilized at self-consistent values, meaning small coupling and large (in string units) volume of the internal space. If this were not the case than one would be too far outside the classical regime for this quantum perturbation to be leading order.

So what KKLT showed is exactly how the Dine-Seiberg problem can be circumvented using fluxes. But, in my opinion, something even more important was done at this step in the KKLT paper. Prior to KKLT one could not have claimed on solid grounds that string theory allows solutions that are perceived to an observer as four-dimensional. Probably the most crude phenomenological demand on a string theory vacuum remained questionable. Of course flux compactifications were known, for example the celebrated Freund-Rubin vacua like \(AdS_5\times S^5\) which were crucial for developing holography. But such vacua are not lower-dimensional in any phenomenological way. If we were to throw you inside the \(AdS_5\times S^5\) you would not see a five-dimensional space, but you would observe all ten dimensions.

KKLT had thus found the first vacua with all moduli fixed that have a Kaluza-Klein scale that is hierarchically smaller than the length-scale of the AdS vacuum. In other words, the cosmological constant in KKLT is really tiny.

But the cosmological constant was negative and the vacuum of KKLT was SUSY. This is where KKLT came with the second, and most vulnerable, insight of their paper: the anti-brane uplifting.

Step 3: Uplifting with anti-D3 branes

Let us go back to the Bianchi identity equation and the physics it entails. If one adds D3 branes to the KKLT background the cosmological constant does not change and SUSY remains unbroken. The reason is that D3 branes are both BPS with respect to the fluxes and the orientifold planes. Intuitively this is again clear from the no-force condition. D3 branes repel orientifolds gravitationally as strong as they attract them "electromagnetically" and vice versa for the fluxes (recall that the fluxes can be seen as a smooth D3 distribution). This also implies that D3 branes can be put at any position of the manifold without changing the vacuum energy: the energy in the tension of the branes gets cancelled by the decrease in fluxes required to cancel the tadpole condition (Gauss' law).

Anti-D3 branes instead break SUSY. Heuristically that is straightforward since the no-force condition is violated. The anti-D3 branes can be drawn towards the non-dynamical O-planes without harm since they cannot annihilate with each other. The fluxes, however, are another story that I will get to shortly. The energy added by the anti-branes is twice the anti-brane tension \(T_{\overline{D3}}\): the gain in energy due to the addition of fluxes, required to cancel off the extra anti-D3 charges, equals the tension of the anti-brane. Hence we get\[

V_{\rm NEW} = V_{\rm SUSY} + 2 T_{\overline{D3}}

\] At first it seems that this new potential can never have a de Sitter critical point since \(T_{\overline{D3}}\) is of the order of the string scale (which is a huge amount of energy) whereas \(V_{\rm SUSY}\) was supposed to be a very tiny cosmological constant. One can verify that the potential has a runaway structure towards infinite volume. What comes to the rescue is space-time warping. Mathematically warping means that the space-time metric has the following form\[

\dd s_{10}^2 = e^{2A} \dd s_4^2 + \dd s_6^2

\] where \(\dd s_4^2\) is the metric of four-dimensional space, \(\dd s_6^2\) the metric on the compact dimensions (conformal Calabi-Yau, in case you care) and \(\exp(2A)\) is the warp-factor, a function that depends on the internal coordinates. A generic compactification contains warped throats, regions of space where the function \(\exp(A)\) can become exponentially small. This is often depicted using phallus-like pictures of warped Calabi-Yau spaces, such as the one below (taken from the KPV paper (I will come to KPV in a minute)):



Consider some localized object with some non-zero energy, then that energy is significantly red-shifted in regions of high warping. For anti-branes the tension gets the following redshift factor\[

\exp(4A) T_{\overline{D3}}.

\] This can bring a string scale energy all the way down to the lowest energy scales in nature. The beauty of this idea is that this redshift occurs dynamically; an anti-brane feels literally a force towards that region since that is where its energy is minimized. So this redshift effect seems completely natural, one just needs a warped throat.

The KKLT scenario then continues by observing that with a tunable warping, a new critical point in the potential arises that is a meta-stable de Sitter vacuum as shown in the picture below.



This was verified by KKLT explicitly using a Calabi-Yau with a single Kähler modulus .

The reason for the name uplifting then becomes obvious; near the critical point of the potential it indeed seems as if the potential is lifted with a constant value to a de Sitter value. This lifting did not happen with a constant value but the dependence of the uplift term on the Kähler modulus is practically constant when compared to the sharp SUSY part of the potential.

I am glossing over many issues, such as the stability of the other directions, but all of this seems under control (the arguments are based on a parametric separation between the complex structure moduli masses and the masses of the Kähler moduli).

The KKLT scrutiny

The issues with the KKLT scenario that have been discussed in the last five years have to do with back-reaction. As mentioned earlier, the no-force condition becomes violated once we insert the anti-D3 branes. Given the physical interpretation of the 3-form fluxes as a cloud of D3 branes, you can guess what the qualitative behavior of the back-reaction is: the fluxes are drawn gravitationally and electromagnetically towards the anti-branes, leading to a local increase of the 3-form flux density near the anti-brane.

Although the above interpretation was not given, this effect was first found in 2009 independently by Bena, Grana and Halmagyi in Saclay (France) and by McGuirk, Shiu and Sumitomo in Madison (Wisconsin, USA). These authors constructed the supergravity solution that describes a back-reacting anti-brane. Clearly this is an impossible job, were it not for three simplifying assumptions:

• They put the anti-brane inside the non-compact warped Klebanov-Strassler throat since that is the canonical example of a throat in which computations are doable. This geometry consists of a radial coordinate measuring the distance from the tip and five angles that span the manifold which is topologically \(S^2\times S^3\). The non-compactness implies that we can circumvent the use of the quantum corrections of KKLT to have a space-time solution in the first place. Non-compact geometries work differently from compact ones. For example, the energy of the space-time (ADM mass) does not need to effect the cosmological constant of the 4D part of the metric. Roughly, this is because there is no volume modulus that needs to be stabilized. In the end one should ‟glue″ the KS throat, at large distance from the tip, to a compact Calabi-Yau orientifold.
  
  
• The second simplification was to smear the anti-D3 branes over the tip of the throat. This means that the solution describes anti-D3's homogeneously distributed over the tip. In practice this implies that the supergravity equations of motion become a (large) set of coupled ODE's.
  
  
• These two papers solved the ODE's approximately: They treated the anti-brane SUSY breaking as small and expanded the solution in terms of a SUSY-breaking parameter, keeping the first terms in the expansion.

Regardless of these assumptions it was an impressive task to solve the ODE's. In this task the Saclay paper was the more careful one in connecting the solution at small radius to the solution at large radius. In any case these two papers found the same result, which was unexpected at the time: The 3-form flux density became divergent at the tip of the throat. More precisely, the following scalar quantity diverges at the tip:\[

H_3^2 \to \infty.

\] (I am ignoring the string coupling in all equations.) Diverging fluxes near brane sources are rather mundane (a classical electron has a diverging electric field near its position). But the real reason for the worry is that this singularity is not in the field sourced by the brane (since that should be the \(F_5\)-field strength and it indeed blows up as well).

In light of the physical picture I outlined above, this divergence is not that strange to understand. The D3 charges in the fluxes are being pulled towards the anti-D3 branes where they pile up. The sign of the divergence in the 3-form fluxes is indeed that of a D3 charge density and not anti-D3 charge density.

Whenever a supergravity solution has a singularity one has to accept that one is outside of the supergravity approximation and full-blown string theory might be necessary to understand it. And I agree with that. But still singularities can — and should — be interpreted and the interpretation might be sufficient to know or expect that stringy corrections will resolve it.

So what was the attitude of the community when these papers came out? As I recall it, the majority of string cosmologists are not easily woken up and the attitude of the majority of experts that took the time to form an opinion, believed that the three assumptions above (especially the last two) were the reason for this. To cut a long story short (and painfully not mention my own work on showing this was wrong) it is now proven that the same singularity is still there when the assumptions are undone. The full proof was presented in a paper that gets too little love.

So what was the reaction of the few experts that still cared to follow this? They turned to an earlier suggestion by Dymarsky and Maldacena that the real KKLT solution is not described by anti-D3 branes at the tip of the throat but by spherical 5-branes, that carry anti-D3 charges (a.k.a. the Myers effect). This then would resolve the singularity they argued (hoped?). In fact, a careful physicist could have predicted some singularity based on the analogy with other string theory models of 3 branes and 3-form fluxes. Such solutions often come with singularities that are only resolved when the 3-branes are being polarised. But such singularities can be of any form. The fact that it so nicely corresponds to a diverging D3 charge density should not be ignored — and it too often is.

So, again, I agree that the KKLT solution should really contain 5-branes instead of 3-branes and I will discuss this below. But before I do, let me mention a very solid argument of why also this seems not to help.

If indeed the anti-D3 branes "puff″ into fuzzy spherical 5-branes leading to a smooth supergravity solution then one should be able to "heat up″ the solution. Putting gravity solutions at finite temperature means adding an extra warp-factor in front of the time-component in the metric that creates an event horizon at a finite distance. In a well-known paper by Gubser it was argued that this provides us with a classification of acceptable singularities in supergravity. If a singularity can be cloaked by a horizon by adding sufficient temperature it has a chance of being resolved by string theory. The logic behind this is simple but really smart: if there is some stringy physics that resolves a sugra singularity one can still heat up the branes that live at the singularity. One can then add so much temperature that the horizon literally becomes parsecs in length such that the region at and outside the horizon become amendable to classical sugra and it should be smooth. Here is the suprise: that doesn't work. In a recent paper, the techniques of arXiv:1301.5647 were extended to include finite temperature and what happened is that the diverging flux density simply tracks the horizon, it does not want to fall inside. The metric Ansatz that was used to derive this no-go theorem is compatible with spherical 5-branes inside the horizon. So it seems difficult to evade this no-go theorem.

The reaction sofar on this from the community, apart from a confused referee report, is silence.

But still let us go back to zero temperature since there is some beautiful physics taking place. I said earlier that the true KKLT solution should include 5-branes instead of anti-D3 branes. This was described prior to KKLT in a beautiful paper by Kachru, Pearson and Verlinde, called KPV (again the same letter ‛K′). The KPV paper is both the seed and the backbone of the KKLT paper and the follow-up papers, like KKLMMT, but for some obscure reason is less cited. KPV investigated the ‟open-string″ stability of probe anti-D3 branes placed at the tip of the KS throat. They realised that the 3-form fluxes can materialize into actual D3 branes that annihilate the anti-D3 branes which implies a decay to the SUSY vacuum. But they found that this materialization of the fluxes occurs non-perturbatively if the anti-brane charge \(p\) is small enough\[

\frac{p}{M} \ll 1.

\] In the above equation \(M\) denotes a 3-form flux quantum that sets the size of the tip of the KS throat. The beauty of this paper resides in the fact that they understood how the brane-flux annihilation takes place, but I necessarily have to gloss over this such that you cannot really understand it if you do not already know this. In any case, here it comes: the anti-D3 brane polarizes into a spherical NS5 brane wrapping a finite contractible 2-sphere inside the 3-sphere at the tip of the KS throat as in the below picture:



One can show that this NS5 brane carries \(p\) anti-D3 charges at the South pole and \(M-p\) D3 charges at the North pole. So if it is able to move over the equator from the South to the North pole, the SUSY-breaking state decays into the SUSY vacuum: recall that the fluxes have materialized into \(M\) D3 branes that annihilate with the \(p\) anti-D3 branes leaving M-p D3 branes behind in the SUSY vacuum. But what pushes the NS5 to the other side? That is exactly the 3-form flux \(H_3\). This part is easy to understand: an NS5 brane is magnetically charged with respect to the \(H_3\) field strength. In the probe limit KPV found that this force is small enough to create a classical barrier if \(p\) is small enough. So we get a meta-stable state, nice and very beautiful. But what would they have thought if they could have looked into the future to see that the same 3-form flux that pushes the NS5 brane diverges in the back-reacted solution? Not sure, but I cannot resist from quoting a sentence out of their paper

‟One forseeable quantitative difference, for example, is that the inclusion of the back-reaction of the NS5 brane might trigger the classical instabilities for smaller values of \(p/M\) than found above.″

It should be clear that this brane-flux mechanism is suggesting a trivial way to resolve the singularity. The anti-brane is thrown into the throat and starts to attract the flux, which keeps on piling up until it becomes too strong causing the flux to annihilate with the anti-brane. Then the flux pile-up stops since there is no anti-brane anymore. At no point does this time-dependent process lead to a singular flux density. The singularity was just an artifact of forcing an intrinsically time-dependent process into a static Ansatz. This idea is explained in two papers: arXiv:1202.1132 and arXiv:1410.8476 .

I am often asked whether a probe computation can ever fail, apart from being slightly corrected? I am not sure, but what I do know is that KPV do not really have a trustworthy probe regime: for details explained in the KPV paper, they have to work in the strongly coupled regime and they furthermore have a spherical NS5 brane wrapping a cycle of stringy length scale, which is also worrisome.

Still one can argue that the NS5 brane back-reaction will be slightly different from the anti-D3 back-reaction exactly such as to resolve the divergence. I am sympathetic to this (if one ignores the trouble with the finite temperature, which one cannot ignore). However, again computations suggest this does not work. Here I will go even faster since this guest blog is getting lengthy.

This issue has been investigated in some papers such as arXiv:1212.4828, and there it was shown, under certain assumptions, that the polarisation does not occur in a way to resolve the divergence. Note that, like the finite temperature situation, the calculation could have worked in favor of the KKLT model, but it did not! At the moment I am working on brane models which have exactly the same 3-form singularity but are conceptually different since the 4D space is AdS and SUSY is not broken. In this circumstance the same singularity does get resolved that way. My point is that the intuition of how the singularity should get resolved does work in certain cases, but it does not work sofar for models relevant to KKLT.

What is the reaction of the community? Well they are cornered to say that it is the simplifications made in the derivation of the ‛no polarisation′ result that is causing troubles.

But wait a minute... could it perhaps be that at this point in time the burden of proof has shifted? Apparently not, and that, in my opinion, starts becoming very awkward.

It is true that there is still freedom for the singularity to be resolved through brane polarisation. There is just one issue with that: to be able to compute this in a supergravity regime requires to tune parameters out of the small \(p\) limit. Bena et. al. have pushed this idea recently in arXiv:1410.7776 and were so kind to assume the singularity gets resolved, but they found the vacuum is then necessarily tachyonic. It can be argued that this is obvious since they necessarily had to take the limit away from what KPV want for stability (remember \(p\ll M\)). But then again, the tachyon they find has nothing to do with a perturbative brane-flux annihilation. Once again a situation in which a honest-to-God computation could have turned into the favor of KKLT, it did not.

Here comes the bias of this post: were it not for a clear physical picture behind the singularity I might be finding myself in the position of being less surprised that there is a camp that is not too worried about the consistency of KKLT. But there is a clear picture with trivial intuition I already alluded to: the singularity, when left unresolved, indicates that the anti-brane is perturbatively unstable and once you realise that, the singularity is resolved by allowing the brane to decay. At least I hope the intuition behind this interpretation was clear. It simply uses that a higher charge density in fluxes (near the anti-D3) increases the probability for the fluxes to materialize into actual D3 branes that eat up the anti-branes. KPV told us exactly how this process occurs: the spherical NS5 brane should not feel a too strong force that pulls it towards the other side of the sphere. But that force is proportional to the density of the 3-form fluxes... and it diverges. End of story.

What now?

I guess that at some point these ‟anti-KKLT″ papers will stop being produced as their producers will run out of ideas for computations that probe the stability of the would-be KKLT vacuum. If the first evidence in favor of KKLT will be found in that endeavor, I can assure you that it will be published in that way. It just never happened thus far.

We are facing the following problem: to fully settle the discussion, computations outside the sugra regime have to be done (although I believe that the finite temperature argument suggests that this will not help). Where fluxes not invented to circumvent this? It seems that the anti-brane back-reaction brings us back to the Dine-Seiberg problem.

So we are left with a bunch of arguments against what is/was a beautiful idea for constructing dS vacua. The arguments against have an order of rigor higher than the original models. I guess we need an extra level of rigor on top from those that want to keep using the original KKLT model.

What about alternative de Sitter embeddings in string theory? Lots of hard work has been done there. Let me do injustice to it by summarizing it as follows: none of these models are convincing to me at least. They are borderline in the supergravity regime or we don't know whether it is trustworthy in supergravity (like with non-geometric fluxes). Very popular are F-term quantum corrections to the GKP vacuum which are used to stabilize the moduli in a dS vacuum. But none of this is from the full 10D point of view. Instead it is between 4D effective field theory and 10D. KKLT at least had a full 10-dimensional picture of uplifting and that is why it can be scrutinized.

It seems as if string theory is allergic to de Sitter vacua. Consider the following: any grad student can find an anti-de Sitter solution in string theory. Why not de Sitter? All claimed de Sitter solutions are always rather phenomenological in the sense that the cosmological constant is small compared with the KK scale. I guess we better first try to find unphysical dS vacua. Say a six-dimensional de Sitter solution with large cosmological constant. But we cannot, or nobody ever did this. Strange, right? Many say: "you just have to work harder". That ‛harder′ always implies ‛less explicit′ and then suddenly a landscape of de Sitter vacua opens up. I doubt that seriously, maybe it just means we are sweeping problems under the carpet of effective field theory?

I hope I have been able to convince you that the search for de Sitter vacua is tough if you want to do this truly top-down. The most popular construction method, the KKLT anti-brane uplifting, has a surprise: a singularity in the form of a diverging flux density. It sofar persistently survives all attempts to resolve it. This divergence is however resolved when you are willing to accept that the de Sitter vacuum is not meta-stable but instead a solution with decaying vacuum energy. Does string theory want to tell us something deep about quantum gravity?

Hinduism for physicists

Hinduism for Physicists! Or why Non-Abrahamic Eastern Religions do not have any conflict with Science. 

Kashyap Vasavada

            First of all, I thank Lubos for giving me an opportunity to express my views on this guest blog. I think, most of you, like me, read his blog to understand recent developments in physics. Lubos does an outstanding job in explaining these matters from a technical point of view and fills the gap between popular articles and original papers admirably.  Social and political issues are also discussed here often, but you may be wondering whether this blog belongs here. I am hoping that you will find it interesting. At the very outset, let me make it absolutely clear that there is no question about the great success of scientific method during the last few hundred years. As is well known, the scientific method consists of making observations with the sense organs (mainly eyes with the help of devices like telescopes, microscopes, electronics etc.); making models using our brains and checking if these models agree with the observations. This trivial statement will be important when we talk about the other method. By following the scientific method, we now know unbelievably large amount about the universe we live in. It will be foolish for anyone to suggest that scientists should abandon this method for investigating the universe. Thus I would be arguing for, not science or religion, but rather for science and religion.  In my opinion both science and religion have limitations and both are useful for the good of the mankind.

            The immediate reason for writing this blog is a recent survey that only about 21%of Americans believe in Big Bang theory, 27% in 4.5 Billion years old earth and 31% in theory of evolution. Assuming that the survey was correct, such a low rate of belief in science is probably based on literal interpretation by the participants, of their scriptures which were written thousands of years back. In the blog about that survey, I made a comment that eastern religions (in particular Hinduism and Buddhism) do not have any problem with science whatsoever. In the following I will try to explain this assertion. In fact you would have hard time
finding a single Indian (or someone from many other Asian countries) who believes in young earth creationism or is against Big Bang theory or theory of evolution.

            I will start with the concept of God in Hinduism. In some sense, Hinduism is the most misunderstood religion in the west, starting with its name! The real name is “Sanatana Dharma” meaning universal duty, responsibility or a prescribed way. The word “Hindu” was coined probably by Persians thousands of years back. They called people living near the river Sindhu (Indus) as Hindus! Unfortunately the word Hinduism is stuck like God particle for Higgs boson! So I will have to use it. The main scriptures are called “Vedas”, “Upanishads” and “Bhagvad Gita”. The religion did not originate from one prophet, but a number of sages contributed to the Vedic vision portrayed in the scriptures. Founding fathers of quantum theory like Schrodinger, Heisenberg and Bohr were familiar with these scriptures. These scriptures talk about (in Sanskrit) a formless, shapeless, omnipresent, omniscient God as Brahman. It is a universal, ultimate, super consciousness. It is supposed to be present in every particle of the universe. Thus the Hindu concept is: God is not like a king emperor who created the universe and rules arbitrarily from outside but it is an all pervading eternal entity. This consciousness may be present in every living and non-living matter i.e. in every single part of the universe and it would be synonymous with the laws of nature. The expression of consciousness in different systems may be different. So how do you realize it?  Well, the sages say that you have to prepare your mind to receive it.  Even in a scientific lab you have to prepare your system to observe any effect. In a sense this is similar to the fact that electromagnetic waves (including CMB) are present all around us and we do not realize their existence unless we have radio, TV, microwave detectors etc. Remember, Higgs field is all around us but we had to build a 10 Billion $ LHC to find Higgs particle! To realize this super consciousness, the prescription is to meditate, calm down your mind completely and only then you would realize it after a lot of hard effort. The scriptures say you do not have to believe it. If you are willing to put in effort you will verify it. This is a big hang up. It is not something which, like the result of a scientific experiment, can be seen by a crowd on a TV monitor. It is one on one. This is an experience outside our sense perception. How do I know it is real? Well, honestly, I do not have any personal experience. But I believe it is possible because I know some yogis have achieved it and they describe the experience.

            There is a beautiful interesting illustration in Vedas which Schrodinger mentions in his book on “what is life”. The question is why consciousness looks similar when our bodies look different. The answer in Vedas is that the source of consciousness is outside. We are merely reflecting it as multiple mirrors would reflect a single object!

            Then what about all these images and statues of hundreds of gods you see in a Hindu temple? The answer is:  everyone is not equipped to understand Brahman in the pure form. One needs some advancement. Suppose you are a physics teacher who is discussing electricity-magnetism. Your discussion  depends on whether the students are in  a primary school, high school, undergraduate program, graduate program or doing post-doctoral work. Similarly, everyone is not at the same spiritual level.  The suggestion is that, if it is too difficult for you to do meditation on the ultimate consciousness, it is OK to worship God in whatever deity form you are comfortable with. This gave rise to a concept of large number of different forms of God (Avatars or incarnations) in whom you can find solace and get guidance.  Over thousands of years a number of mythological stories also came up. Faithful believe that these deities are incarnations of God who came to earth in specific forms to achieve some purpose, typically to destroy evil and establish goodness. They may exist in a realm which is not directly accessible to our senses. Usually there are moral lessons also associated with Avatars. People may have their favorite deities to worship. The faith evolved over several thousand years as a kind of cafeteria system. You worship a deity depending on your comfort level and ability. Usually one’s upbringing also plays a role.  In the end it does not make any difference whom you worship. Everyone realizes what the icons represent symbolically. One strong point of Hinduism is the tolerance or (better) acceptance of different viewpoints. The belief is that there are thousand paths to realize God and everyone can choose his/her own path.  Some followers believe in miracles but not too strongly to challenge science! It is well known and even scientifically established that faith does work wonders and there is a close connection between body and mind.In my personal case, I find association with the temple activities very enriching. Not only that there is no anti-science talk, but also the prevalent view is that most or all activities are compatible with science. It does help that Hinduism is flexible and non-rigid. So I can continue my interest in learning and teaching physics and learn about metaphysics as part of religion too. One point I should emphasize to western readers: in Hinduism, philosophy, metaphysics and religious ritual practice are all mixed together. We think this is a strength rather than weakness. In a typical Hindu temple, you will find rituals going on all the time, together with philosophical talks. Rituals are prescribed for an average person because, at least for that time period he/she is engaged in thinking about God instead of mundane affairs of life and that would lead to cleansing  of mind  and get inspiration to lead a good life. Personally, I prefer philosophical, metaphysical discourses to rituals, but participating in rituals is also fine with me.

            As in other religions, Hinduism has commandments for leading a good life e.g., speaking truth, non-violence, love, compassion, ethics, morality etc. Hinduism believes that whatever one does, has consequences as a Karmic relationship. This would be similar to the law of action and reaction in physics. Admittedly, here, the mechanism for consequences remains unknown. But the belief is that one can wipe out bad Karmas with good Karmas. This may take several births. So Hinduism firmly believes in reincarnation, i.e. everyone has a soul (called Atman) which migrates from one body to another on death. Ultimately when one is completely free from Karmic bondage, he/she gets liberation called Moksha or Nirvana. Obviously, there is no scientific or material proof of these beliefs.

            Now let us consider two main issues in which some westerners see conflict with science: age of Universe and theory of evolution. On both of these issues, the Hindu sages got approximately correct ideas in agreement with science. Just by thought processes they realized that universe must be billions of years old, as noted by Carl Sagan in his book on cosmos. The other realization was that there must be some connection between animals and human beings. If human beings have souls, then animals too have souls. That gave rise to mythological stories that God came to earth in the form of first sea animals, then land animals and then human beings. These are ten Avatars of god Vishnu.

            How about origin of universe? Of course one cannot say that the ancient sages’ knowledge was anywhere comparable to the current knowledge.  But just see the astonishing description of origin from a scripture known as Vayupuran:

“In the beginning, there was nothing in the universe. The Brahman (the divine essence) alone was everywhere. The Brahman had neither color nor scent; it could not be felt or touched. It had no origin, no beginning or no end. The Brahman was constant and it was the origin of everything that was destined to be in the universe and the universe was shrouded in darkness.” Intersting! It was dark because visible light was not created yet!! In all Hindu scriptures, multiverse and cycles of creation and destruction of universe lasting billions and trillions of years are frequently mentioned.

            Another excerpt from Vedas: “The universe is brought about by the collapse of fullness in the transcendental field in which reside all the laws of nature responsible for the creation of the entire manifest universe. How is the transcendental level functioning? It is functioning from its unbounded nature to point to itself.  He who does not know that initial pure consciousness state, ultimate reality, what can the laws of nature accomplish for him? He who knows it, remains established in evenness, unity, wholeness of life”. Since Brahman was by itself, it is clear that it interacted with itself i.e. self-referral (like inflaton!!!) and eventually manifested in every particle of the universe. It is a very interesting parallel with modern cosmology. Strictly speaking the word “manifestation” rather than “creation” is used in Vedic cosmology with a subtle meaning.

            Now let us examine a frequent argument that science is based on logic and reason while religion is based strictly on faith. Well, our everyday logic and intuition are based on our life experiences with the world at the classical level. Modern physics has demolished this argument.  If you say that only thing physics should care for, is develop mathematically consistent models, (no matter how bizarre they appear to our intuition) and try to see if they are in agreement with experiments, then you are perfectly ok. In a sense, I agree with Lubos that as far as science is concerned, there is no need to look at the meaning of models. But the moment you look for meaning of the equations you get into mess (metaphysical if you will). As everyone on this blog knows, the world is made out of fuzzy wavelike dynamic stuff and not solid rigid objects we see around. The particles are in some sense both here and there at the same time and are described by a wave function, a superposition of seemingly contradictory properties. This closely parallels description of Brahman in Hindu scriptures “It moves and it moves not; it is far and it is near; it is within all this and it is also outside all this.”  The ultimate shock of quantum mechanics, for visualization in terms of our everyday life, came with Bell’s theorem and corresponding experiments on entanglements. Lubos has written about this topic because of its importance. One has to choose between locality and reality. I think most physicists choose to keep locality to save theory of relativity at the cost of reality i.e. the particles are believed to be in some kind of suspended state devoid of any specific properties until they are observed. It is well known that Einstein did not like this. Another basic finding of quantum theory is the involvement of the observer in the observed things. It is impossible to separate the effect of the measuring apparatus from the object measured. Detachment of the two is just not possible. As John Wheeler said “universe is indeed participatory!” Max Planck regarded consciousness as fundamental and matter as derivative from consciousness. At one time Wigner expressed his view that consciousness creates collapse of wave function. There have been debates for some 90 years about interpretation of quantum mechanics without any resolution in sight. Ideas about the entanglement of the observer and the object of observation are also emphasized in Upanishads. In the Hindu concept the observer (Brahman) is in the system itself and is time independent.

            There is a similar situation with theory of relativity, namely relativity of time for different observers, curving of space-time by matter in its neighborhood and possible singularity at the big bang. This theory has again challenged our intuition. Here also ancient Hindu sages did not have any problem. In certain scripture, you can find a statement that 100,000 human years is equivalent to 1 second of divine time!  They also talk about simultaneous visions of past, present and future and time travel!

            The readers of this blog already know about leading physicists discussing a multi-dimensional world of string theory. Hardly anyone would say that it is intuitive.  More recently, if BICEP2 experiment and its interpretation are verified, it would mean that our observable universe started with a size of smaller  than a proton  and grew by a factor of  e^80 or more by expansion of space. The conclusion is that our everyday logic just does not work in Modern Physics, although Mathematics works superbly.

            I am not saying that arguments of modern cosmology and modern physics are on the same footing as the metaphysical thoughts about universe and God in Hindu scriptures, but it sure should make one stop for a moment and think about the ultimate nature of reality in the two pictures. Again, all these intriguing developments are fine if you do not insist on visualizing by our poor little brains! I would say that one should not require higher standards of our intuitive understanding for religion than for science!

            My basic suggestion is that let us be modest. Although we can be proud of our achievement in understanding so much about the universe, just think for a moment. We are on a measly little planet bound to an average star in an average galaxy with more than 100 Billion stars. There are more than 100 Billion galaxies in our observable universe. There could be an infinite number of such universes. Our eyes and brains evolved in a specific manner on earth. Both of these have limitations. For example, our eyes are only sensitive to visible part of the electromagnetic spectrum only. Thus it would be height of arrogance and even stupidity to assume that what we can find with our sense organs and understand with our brains is all there is to it in the universe.  Although direct verification is hard at this point, it is not unreasonable to assume that there could be a world beyond our sensory perceptions.

            The reader might say that all this sounds like new age pundits talking mysticism! Well it is but there may be an underlying subtle reality! A number of authors have written books comparing modern physics with metaphysics of east. There are mystical ideas floating around about quantum consciousness and unified field of consciousness.  There is a Harvard trained theoretical physicist John Hagelin who is a professor of physics at Maharishi University in Fairfield, Iowa. He has written mathematical equations for such fields (using Lagrangian formalism like a respectable conventional theoretical physics). While these attempts are fine, my feeling is that they are somewhat premature for theoretical physics. It is not clear how one can verify the solutions of these equations which can satisfy scientists.   Then there are also serious scientific models by Penrose-Hameroff, Stapp and some others looking for quantum mechanical processes in our brain which could explain consciousness. As of now they are inconclusive and have not been widely accepted. The main issue is that, as yet, neuroscientists have not understood consciousness. It is not clear how far down some kind of primitive consciousness goes in the tree of life. Suggestion that there could be some presence of consciousness in non-living systems would be called metaphysical at best, although some argue that the quantum dynamics already resembles a kind of consciousness. So consciousness may be much more subtle than our brain functions, thought processes etc. But we have to travel a long way to theoretical physics if indeed these ideas work out. I just want the readers to be aware that such models could be part of reality whatever it turns out to be.

            Although I do not wish to belabor the point, question of existence of God is similar to some statements in Gödel’s theorems. He has proved that within a mathematical system, there would always be some statements which cannot be proved or disproved.

            To summarize: I am not saying that Hinduism is based on statements which have material, scientific proof.  But at this point there is no direct contradiction with any facts which science can establish. Hinduism goes beyond a point where science stops. It has concentrated on inner (non-sensory) understanding of reality through methodology known as Yoga. In terms of our concepts of reality, it seems that a previously assumed rigid line between physics and metaphysics may be rapidly disappearing! Normally I stay on the physics side of the line. But some time I am not so sure!

            I have described Hinduism in some detail but not Buddhism. Buddhism arose in India as an off-shoot of Hinduism but became a completely independent religion in several other Asian countries. It has many similarities with Hinduism and believes in Karmic consequences. The belief is that humans can achieve ultimate liberation (Nirvana) by following the rightful behavior recommended by Buddha. According to Buddha, explaining the concept of God would be very difficult; hence Buddha neither rejected nor accepted the existence of a creator deity.

            Now I will make few remarks about religions in general. Personally I am comfortable with most religions at the commandments level. Most religions maintain that morality, ethics, love, compassion etc. are integral parts of good life. Religious people believe that these commandments came as revelations from God. Whether one believes in God or not, we have to accept that these are good principles for the society. Unfortunately, many people (believers and non-believers) do not follows them. While it is possible to be virtuous and spiritual without being religious, believers would have special reasons to follow these. I can sense an immediate response from the readers about some religious zealots who maintain that if you do not believe in their God, they would kill you. While this is true unfortunately (in a small number of cases) this is not that wide spread and we have to work to eliminate these tendencies by education. In addition, there are social problems of drugs, high crime rates, murders etc. for which religious institutions can play an important role. I do not have to remind this scientific readership that dangerous weapons of mass destruction created by science, namely nuclear, chemical and biological weapons, can bring an end to all human life and perhaps the entire life on this planet. Thus we have to find some way of living with each other in a peaceful manner. My suggestion is that religion if properly followed can play a useful role, without which there would be nothing to hang on.

            Although the main focus of this blog is to inform the readers about Hinduism and its lack of conflict with science, I conclude with some general remarks about science and religion.  I would say this to the general readership: debate about science and religion is not all black or white. There are many scientists who are believers. Some surveys indicate 30 to 50% of all scientists are believers to a certain extent. Some of these are high profile scientists, at least one Nobel Laureate in physics. They realize that there are some finer, subtle points about their particular religions. At the same time, scientists should protest against ideas of young earth creationism, intelligent design and anti-evolution propaganda. In addition, they should insist that they are the ones who decide what to teach in science classes in schools and they should not let non-scientists dictate it. Europeans may find this funny but this simple issue is becoming important in U.S.! There are frequent court battles about whether teachers should teach creationism as part of a science class! I routinely write in our local newspaper on these issues. On the other hand, I do not care also for tirades against religion in which some prominent physicists and other scientists have engaged. I think science and religion can have a peaceful coexistence and can enrich human life. In a way I am calling for moderation and acceptance of importance of each other by both sides. Let us have a balanced view of science and religion.