Periodic Reporting for period 3 - Symmetries-Cosmology (Duality Symmetries, Higher Derivatives, and their Applications in Cosmology)
Okres sprawozdawczy: 2021-09-01 do 2023-02-28
The problems that are addressed concern the potential use of novel
symmetries, dualities and new mathematical techniques in the cosmology
of the early Universe close to the big bang. The evolution of the Universe
is very well understood in terms of accepted physical theories like
Einstein's theory of general relativity and the standard model of particle physics.
These theories allow us to conclude, with a high degree of confidence, that roughly 14 billion years ago
the entire Universe was in a very dense and hot state,
from which our current Universe evolved through expansion and subsequent cooling.
This is known as the big bang model. While the history of the Universe
starting very shortly after the big bang must be considered well established science,
we also know that our current physical theories break down when trying to approach
the big bang itself. More precisely, Einstein's general relativity
predicts a singularity, a state of infinite density and spacetime curvature in which the
theory itself becomes meaningless. We thus know that the framework of Einstein's theory
can only be an approximation to a more general theory that must take over close to the
big bang (and similarly close to the center of a black hole). My research program explores
the framework of string theory in order to find such a more general theoretical description
that could be relevant close to the big bang.
Technically, there are three important respects in which any string theory introduces new
features compared to general relativity (to which it reduces in an appropriate low-energy limit):
First, it features certain so-called dualities, which means that two seemingly different
solutions of the equations should actually be viewed as physically equivalent, despite
having very different properties from the viewpoint of standard geometry and physics.
The prime example is T-duality, the phenomenon that string theory on a space of size R
(like the radius of a circle) is equivalent to string theory on a space of size 1/R.
Second, string theory features so-called higher-derivative corrections. While almost all
of physics since Newton is governed by differential equations of second order, which means
that two initial conditions are needed in order to find a solution (we may think of the initial position
and velocity of a baseball, from which its trajectory can be computed in principle),
in string theory, at least as currently understood, there must be an infinite number of
higher-derivative corrections, controlled by a fundamental parameter of string theory,
the tension of the string (or its inverse denoted by $\alpha'$).
Third, string theory features so-called higher string modes, massive states that are not
expected to be excitable in current accelerators but which may become relevant in
the very early Universe.
My goals in this research program are to illuminate all three aspects of string theory
by focusing on certain sub-sectors of the full string theory (technically, consistent truncations
that are duality invariant) and to explore their possible uses in cosmology.
The first feature, duality invariance, indeed immediately suggest promising scenarios for cosmology,
since a Universe of very small size could equivalently be thought of as being very large.
There have indeed been numerous proposals in the string theory literature along such lines,
but unfortunately they all fall short of producing a full-fledged cosmological model with potentially
falsifiable predictions. One reason is the presence of the second and third feature mentioned above,
the higher-derivatives corrections and massive string modes, whose explicit formulation
is only poorly understood.
The more detailed research objectives are thus: 1) to find the higher-derivative corrections
possibly to all orders by exploiting the duality properties in various schemes, 2)
to write explicit field equations that govern those massive string modes that are needed
for a duality invariant formulation and 3) to apply these findings to cosmology,
for instance by determining whether massive string modes may leave imprints in the
cosmic microwave background.
This line of research is important for society because it aims to satisfy humanity's most
basic curiosity: where does the world come from? The fact that already now,
equipped with the physical theories that are extremely well tested, we are able to go back
in time (figuratively speaking) and reconstruct the entire history of the Universe almost all the way
to the big bang is truly astonishing. There is every reason to believe that we can go even further
once we find the next theory transcending Einstein's theory.
symmetries, dualities and new mathematical techniques in the cosmology
of the early Universe close to the big bang. The evolution of the Universe
is very well understood in terms of accepted physical theories like
Einstein's theory of general relativity and the standard model of particle physics.
These theories allow us to conclude, with a high degree of confidence, that roughly 14 billion years ago
the entire Universe was in a very dense and hot state,
from which our current Universe evolved through expansion and subsequent cooling.
This is known as the big bang model. While the history of the Universe
starting very shortly after the big bang must be considered well established science,
we also know that our current physical theories break down when trying to approach
the big bang itself. More precisely, Einstein's general relativity
predicts a singularity, a state of infinite density and spacetime curvature in which the
theory itself becomes meaningless. We thus know that the framework of Einstein's theory
can only be an approximation to a more general theory that must take over close to the
big bang (and similarly close to the center of a black hole). My research program explores
the framework of string theory in order to find such a more general theoretical description
that could be relevant close to the big bang.
Technically, there are three important respects in which any string theory introduces new
features compared to general relativity (to which it reduces in an appropriate low-energy limit):
First, it features certain so-called dualities, which means that two seemingly different
solutions of the equations should actually be viewed as physically equivalent, despite
having very different properties from the viewpoint of standard geometry and physics.
The prime example is T-duality, the phenomenon that string theory on a space of size R
(like the radius of a circle) is equivalent to string theory on a space of size 1/R.
Second, string theory features so-called higher-derivative corrections. While almost all
of physics since Newton is governed by differential equations of second order, which means
that two initial conditions are needed in order to find a solution (we may think of the initial position
and velocity of a baseball, from which its trajectory can be computed in principle),
in string theory, at least as currently understood, there must be an infinite number of
higher-derivative corrections, controlled by a fundamental parameter of string theory,
the tension of the string (or its inverse denoted by $\alpha'$).
Third, string theory features so-called higher string modes, massive states that are not
expected to be excitable in current accelerators but which may become relevant in
the very early Universe.
My goals in this research program are to illuminate all three aspects of string theory
by focusing on certain sub-sectors of the full string theory (technically, consistent truncations
that are duality invariant) and to explore their possible uses in cosmology.
The first feature, duality invariance, indeed immediately suggest promising scenarios for cosmology,
since a Universe of very small size could equivalently be thought of as being very large.
There have indeed been numerous proposals in the string theory literature along such lines,
but unfortunately they all fall short of producing a full-fledged cosmological model with potentially
falsifiable predictions. One reason is the presence of the second and third feature mentioned above,
the higher-derivatives corrections and massive string modes, whose explicit formulation
is only poorly understood.
The more detailed research objectives are thus: 1) to find the higher-derivative corrections
possibly to all orders by exploiting the duality properties in various schemes, 2)
to write explicit field equations that govern those massive string modes that are needed
for a duality invariant formulation and 3) to apply these findings to cosmology,
for instance by determining whether massive string modes may leave imprints in the
cosmic microwave background.
This line of research is important for society because it aims to satisfy humanity's most
basic curiosity: where does the world come from? The fact that already now,
equipped with the physical theories that are extremely well tested, we are able to go back
in time (figuratively speaking) and reconstruct the entire history of the Universe almost all the way
to the big bang is truly astonishing. There is every reason to believe that we can go even further
once we find the next theory transcending Einstein's theory.
The main scientific results achieved since the beginning of the project can be grouped into three categories,
along the lines
of the three core research programs outlined in the proposal: 1) duality invariant
formulations of higher derivative $\alpha'$ corrections; 2) formulations of higher (Lie-infinity-type)
algebras in gauge theory, with the central goal of formulating what is known as weakly constrained double field theory;
and 3) investigation of string cosmologies that include all $\alpha'$ corrections,
and/or that encode cosmological perturbation with massive string modes.
Under research program 1), we have shown, in collaboration with Eloy and Samtleben, that in generic dimensional reductions a novel Green-Schwarz type mechanism
is needed in order to realize the duality symmetries mentioned above when going beyond lowest order in $\alpha'$.
Under 2) I have developed, in collaboration with Roberto Bonezzi, a mathematical theory for dealing with the
gauge theories arising in duality covariant formulations of string theory (known as tensor hierarchies).
Most importantly, we have shown that the dynamics of such theories can be encoded in a tower of so-called
duality relations, using the mathematical structures of differential graded Lie algebras.
Under program 3) I have shown, in collaboration with Zwiebach, how to develop an ``$\alpha'$ complete cosmology",
which classifies all $\alpha'$ corrections and thereby allows one to characterize the ``space of string cosmologies"
explicitly to all orders in $\alpha'$. Moreover, in collaboration with Codina and Marques, we used these methods in order
to determine the first two non-trivial coefficients arising in type II string theory, which turn out to be proportional to the transcendental number $\zeta(3)$.
Finally, in collaboration with Chiaffrino and Pinto, we have shown how to use results from higher L-infinity algebras,
notably what is known as the homotopy transfer theorem, in order to effectively construct gauge invariant variables to all orders in
cosmological perturbation theory, thereby providing cross disciplinary results from pure mathematics (notably algebraic topology)
and cosmology.
along the lines
of the three core research programs outlined in the proposal: 1) duality invariant
formulations of higher derivative $\alpha'$ corrections; 2) formulations of higher (Lie-infinity-type)
algebras in gauge theory, with the central goal of formulating what is known as weakly constrained double field theory;
and 3) investigation of string cosmologies that include all $\alpha'$ corrections,
and/or that encode cosmological perturbation with massive string modes.
Under research program 1), we have shown, in collaboration with Eloy and Samtleben, that in generic dimensional reductions a novel Green-Schwarz type mechanism
is needed in order to realize the duality symmetries mentioned above when going beyond lowest order in $\alpha'$.
Under 2) I have developed, in collaboration with Roberto Bonezzi, a mathematical theory for dealing with the
gauge theories arising in duality covariant formulations of string theory (known as tensor hierarchies).
Most importantly, we have shown that the dynamics of such theories can be encoded in a tower of so-called
duality relations, using the mathematical structures of differential graded Lie algebras.
Under program 3) I have shown, in collaboration with Zwiebach, how to develop an ``$\alpha'$ complete cosmology",
which classifies all $\alpha'$ corrections and thereby allows one to characterize the ``space of string cosmologies"
explicitly to all orders in $\alpha'$. Moreover, in collaboration with Codina and Marques, we used these methods in order
to determine the first two non-trivial coefficients arising in type II string theory, which turn out to be proportional to the transcendental number $\zeta(3)$.
Finally, in collaboration with Chiaffrino and Pinto, we have shown how to use results from higher L-infinity algebras,
notably what is known as the homotopy transfer theorem, in order to effectively construct gauge invariant variables to all orders in
cosmological perturbation theory, thereby providing cross disciplinary results from pure mathematics (notably algebraic topology)
and cosmology.
The main results that went beyond the state of the art, also as far as methodology is concerned,
are the formulation of the ``$\alpha'$ complete cosmology" and the use of homotopy transfer in
gauge invariant perturbation theory. First, the formulation of string cosmology to all orders in $\alpha'$ was not expected
to be possible until a full-fledged double field theory was available to all orders in $\alpha'$, at best toward
the very end of the project. However, upon closer examination it turned out that the large symmetries of cosmology
allow one, when used in combination with field redefinitions, to directly classify $\alpha'$ corrections to all orders.
Second, the formulation of gauge invariant variables was facilitated by the surprising insight, in part due to my postdoc
Chiaffrino, that this can be interpreted in terms of homotopy transfer. This in turn allowed us to use the powerful
homotopy transfer theorem (or the related perturbation lemma) to give an effective procedure to define gauge invariant
variables to all orders.
The expected (or at least desired) results until the end of the project are: 1) The determination of a double field theory
(duality invariant formulation of string theory) that include exact or $\alpha'$-complete higher-derivative deformations,
going beyond the cosmological setting. Such a theory exists for a non-standard
string theory, constructed in collaboration with Siegel and Zwiebach, but the goal here is to achieve something similar
for the much more involved and more ``realistic" string theories in ten dimensions.
2) Formulation of a ``weakly constrained" double field theory, which means that it encodes
massive string modes (winding and momentum modes). 3) Applications in string inspired cosmology, notably
the computation of cosmological correlation functions including such massive string modes.
are the formulation of the ``$\alpha'$ complete cosmology" and the use of homotopy transfer in
gauge invariant perturbation theory. First, the formulation of string cosmology to all orders in $\alpha'$ was not expected
to be possible until a full-fledged double field theory was available to all orders in $\alpha'$, at best toward
the very end of the project. However, upon closer examination it turned out that the large symmetries of cosmology
allow one, when used in combination with field redefinitions, to directly classify $\alpha'$ corrections to all orders.
Second, the formulation of gauge invariant variables was facilitated by the surprising insight, in part due to my postdoc
Chiaffrino, that this can be interpreted in terms of homotopy transfer. This in turn allowed us to use the powerful
homotopy transfer theorem (or the related perturbation lemma) to give an effective procedure to define gauge invariant
variables to all orders.
The expected (or at least desired) results until the end of the project are: 1) The determination of a double field theory
(duality invariant formulation of string theory) that include exact or $\alpha'$-complete higher-derivative deformations,
going beyond the cosmological setting. Such a theory exists for a non-standard
string theory, constructed in collaboration with Siegel and Zwiebach, but the goal here is to achieve something similar
for the much more involved and more ``realistic" string theories in ten dimensions.
2) Formulation of a ``weakly constrained" double field theory, which means that it encodes
massive string modes (winding and momentum modes). 3) Applications in string inspired cosmology, notably
the computation of cosmological correlation functions including such massive string modes.