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Toroidal compactifications of 11-D supergravity naturally induce exceptional symmetries in  that can be realized as U-duality symmetries of M-theory upon Z-discretization and without Betti-truncations. Hence, exceptional field theory based on the modular group  uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory. By mirror symmetry, there ought to be a deep internal symmetry induced between M-theory and F-theory upon KK-reduction to Type-IIB SUGRA. In the formalism taking the 6-8 limit, the content of the theory is given by the action:

with:

and:

where the Chern-Simons-topological Lagrangian has covariant variational form:

with:

and the Yang-Mills field equation for the covariant field strength form is:

Thus, we can derive the Chern-Simons-type topological action:

with:

and:

and the covariant curvature form  and holomorphic curvature form are, respectively:

and:

where the Ramond-Ramond gauge-coupling sector is given by the action:

and the Ramond-Ramond term being:

thus giving us the Type-IIB Calabi-Yau three-fold superpotential:

Before we can see the duality relations between M-theory and F-theory elliptic fibrational Standard-Model constructions, note that the topologically mixed Yang-Mills action:

where the corresponding Chern-Simons action is:

with the Ramond-Ramond coupling-term:

has variational action:

with:

Now, since 11-D SUGRA on a torus is equivalent to Type-IIB string-theory on a circle, the action of the modular group on the Type-IIB axio-dilaton allows us to take the zero limit of:

and by mirror symmetry, we get a Type-IIA dimensional uplift to M-theory, given that in the Einstein frame, the Type-IIB bosonic SUGRA action is:

with:

This is one aspect of appreciating, via the exceptional field theory (EFT) modular group action, the F-theory/M-theory duality. The essence of EFT is thus a deeper double duality relating M-theory/Type-IIA and F-theory/Type-IIB. Key is the role of U-duality in the modular holomorphic action on the Neveu-Schwarz sector of Type-IIB. Our generalized diffeomorphisms, generated by a vector , act fully locally on  yielding the Lie derivative  that differs from the classic Lie derivative by a Calabi-Yau induced -tensor and is implicitly defined by the transformation rules for a generalized vector:

The associated diffeomorphism algebra has an exceptional field bracket:

with closure condition:

The action diffeomorphism-symmetries are parametrized by vector bundles over the metaplectic space and take the form:

with:

where the gauge vector transforms as:

The corresponding generalized exceptional scalar metric  has the following property:

which decomposes in light of the orbifold blow-up:

as:

thus allowing us to define the crucial exceptional metric:

Since the full Type-IIB Calabi-Yau superpotential is given by:

where the Kähler Type-IIB orientifold moduli is:

with:

and the volume of the divisor, , is:

with:

thus we now have the ingredients to write the modular exceptional field theory action as:

with the exceptional Ricci scalar:

the kinetic part:

and the gauge term:

and the 10+3-D Chern-Simons topological term:

where the potential has the form:

This is a theory dynamically equivalent to 11-D SUGRA and Type-IIB under covariantized U-duality group-action. However, the gauged kinetic terms corresponding to the gauge form  appears only topologically in:

Hence, the EoM for the field  is given by:

Since exceptional field theory based on the modular group  uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory, homological mirror symmetry entails there ought to be a deep internal symmetry induced between M-theory and F-theory upon dimensional-reduction to Type-IIB SUGRA which in the formalism, taking the 6/8 Klebanov-Witten limit, is defined by the action:

We are now in a position to explore this EFT-duality between M-theory and F-theory, noting that it is a duality that is rich in F-theory phenomenology, and by the Type-IIB duality, such phenomenology is inherited by 11-D SUGRA under U-duality. -EFT is equivalent to both, 11-D SUGRA and Type-IIB SUGRA. The field content for EFT consists of:

the field content for M-theory consists of:

and that of Type-IIB being:

The Kaluza-Klein and gauge fields for M-theory are, respectively:

and:

For the split-Type-IIB theory, the Kaluza-Klein and gauge fields are respectively:

where we have:

We then parametrize  in terms of the familiar axio-dilaton  as:

The -EFT/11-D/Type-IIB duality can now be written in terms of the gauge field equations as:

Now the definition of F-theory can be summarized as a 12-D lift of Type-IIB with varying axio-dilaton and a 7-brane geometric backreaction where the  duality U-action yields a monodromy-group representation induced by elliptic toric fibration that admits a duality with M-theory via a KK-reduction. The -EFT diffeomorphism group action has two Sen singularity solutions, one corresponding to F-theory, the other to 11-D SUGRA. Hence, such a 12-D field theory dimensionally reduces in the Witten-Vafa limit, to 11-D SUGRA and 10-D Type-IIB theories.

Since the D7-brane backreaction is central to F-theory, let’s study it under the -EFT/11-D/Type-IIB duality. Solutions to the D7-brane worldvolume action in Type-IIB SUGRA possess non-trivial axio-dilaton and metric. Let a D7-brane extend along the six-dimensions of the compactifying manifold in the  and  directions, and we represent the time-direction transverse to the brane in polar coordinates . Thus, the harmonic functional of the D7-brane is  and the solutions to our system are given by:

Holomorphically circling the transverse dimension yields, by monodromy group action on , :

The D7-brane is derived via Type-IIB sectional dimensional reduction under the elliptic fibrational structure underlying the monodromy action on the toric geometry:

On the M-section, we have a monopole-smearing solution:

hence, by the M/Type-IIB duality, the smearing relates the first Chern class of the Witten-Vafa fibration to the co-dimension 2 monodromy hypercharges. By modularity, the D7-brane hyperdoublet and the D3-brane multiplet are smeared monopoles that pq-7-branes in Type-IIB with p-cycles and q-cycles holomorphically wrap. The metric of a pq-Type-IIB theory is given as:

Thus, the exceptionality property determines two holomorphic functions  and  where the toric modulus takes the form:

with the elliptic invariant Jacobian. The polynomial roots give rise to Type-IIB singularities localizing the 7-branes. Hence we get:

and for the M-section, we have:

Both maintain the elliptic fibrational base singularities under which the action of the M/Type-IIB duality symmetry gives rise to generalized Klebanov-Witten quiver gauge theories that yield the YM gauge theories of the SM. Exceptionality hence eliminates the need to go to the full-blown 12-D F-theory and thus eliminates both, Betti-truncations and the KK-conical blow-up sector that would have to be super-Higgsed away upon dimensional reduction to 11-D SUGRA. This is phenomenologically important as we shall see in upcoming posts.