 Welcome, everyone, to the second day of our workshop. And we are happy to have Mayuka Yamashita. And she will tell us about differential push forwards under some duality and invertible field theories. Thank you for the introduction. And thank you for this opportunity. Today, I'm going to talk about differential homology theory and its relation with invertible theories. This is a joint work with Kazeya Yonekura, who gave a talk yesterday. And this talk is based on the following two sequential papers. The first one is a collaboration. And the second one, this is a new article which I put last week. And also, we made a replacement of the first article last week. We changed a lot and improved a lot. So today, I'm going to talk about the new ingredients added last week. So many of us are familiar with the idea that physical objects are classified by generalized homology theory in mathematics. This gives anthropological classification. And in mathematics, there is a notion called the differential homology theory. This is a refinement of generalized homology theory. And this refines the original theory or manifold with some differential geometric data. And it has been also known that using such a differential homology theory, we get a differential classification, so more refined classification of physical objects. For example, P-form gauge field in physics can be understood in terms of ordinary differential homology theory. And Raman-Raman fields can be understood in terms of differential k-theory or differential ko-theory. And actually, in the joint work with Kiyonori-Gomi, we constructed a model of differential extension of k-theory or ko-theory in terms of thermionic mastons. Actually, this paper, I put this paper today. And I'm going to mention a little bit about this work later. But the main player is the following example. We constructed a model of differential extension of the Anderson-Dale theory. And using this model, this element here can be understood as partition function of invertible theories or manifold with differential structure. Here, in the yesterday's talk, Kazeya explained the physical motivation for this work. And I'm going to review the mathematical aspect of the construction in the first part of the talk. Now using the correspondence between differential homology and physics, some operations in physics can be understood in terms of transformations of differential homology theories in mathematics. Today, I'm going to talk about such examples. I explained that many interesting examples are related to something called the push forward in differential homology theory. This is something like fiber integration in differential theory. I'm going to talk about two results. The first one constructs a transformation like this, of this form. So for example, this, an example of such a transformation is the following one, the Anderson-Dale theory and the Atiyabot-Shakira orientation in the Kazeori. So actually this corresponds to the question which was asked yesterday. And physical meaning of such a transformation is a theory on G-manifold with connection with background field living in this differential homology group. And secondly, the second result is something like fiber integration in the Anderson-Dale itself. And this has the meaning of as compactification of QFTs in physical interpretation. So let's start. First time I'm going to talk about the differential model of the Anderson-Dale. So we constructed a differential extension of a generalized homology theory called the Anderson-Dale to G-Voldism theory. Here mathematically speaking, for any spectrum or generalized homology theory, we can take the Anderson-Dale to produce another spectrum. But the mathematical definition is, I don't explain it today, but we are interested in this theory because it is related to QFTs. In the conjecture by Fredon Hopkins, they conjecture that this group classifies invertible theory on the dimensional manifolds with G structure with connection, so gene diagram manifold. Here, D means the dimension of the spacetime and the degree appearing here is D plus one. And we consider them up to the formation equivalence. Motivated by this conjecture, we constructed a model of the differential extension of the Anderson-Dale theory in terms of QFTs. Now, before explaining the construction, I will first review the differential homology theory, general theory. So given a generalized homology theory, a differential extension or itself called the differential homology theory is defined on manifolds and refined the original theory with some differential geometric data. So there is a set of actions and I don't explain the detail, but it says that we have structure homomorphism like this. And this I is a forgetful map, we recover the topological information. And I have this map, R, this map is called a curvature homomorphism. And this map says that we can get differential form, differential closed form out of a differential homology element. And in general, the coefficient is a non-trivial vector space. This is specified by this generalized homology theory. So let's look at some examples. The first example is ordinary homology. So second ordinary homology classifies vector line bundle after isomorphism. And refining this, the differential second homology classifies line bundle with connection up to isomorphism. So this group remembers something like autonomy and curvature. Indeed, in this case, this homomorphism, curvature homomorphism is just taking the curvature. This is a two form. Next we consider the case theory and I use the model by Fried and Lott. The topological case theory classifies complex vector bundle up to homotopy and refining this, the differential theory basically classifies Hermitian vector bundle with connection. And the curvature homomorphism in this case is given by taking the Chan character form of the connection. So Chan character form is an even degree differential form. And indeed, in the case theory case, the coefficient is like this. And this is naturally identified with the differential, even degree differential forms. So the commutativity of the square means that this refines the topological Chan character homomorphism. Now I want to construct such a differential extension of the Anderson-Dierle theory. And here G specifies the tangential structure group. In general, G is a sequence of compact D groups with mapped orthogonal groups. Here I'm allowing it to non-oriented. Given such a sequence G, a G nabla structure on a manifold is a G structure on a stable tangent bundle with connection. So we consider examples like this. The first example is the case where G is trivial. This is no physical, but in this case, we are considering stable reframed manifold. And also we consider the case of SO or spin, as usual. And we can also consider the internal symmetry group. We fix a compact D group H and consider GD, something like SO times H. And mapped to SO is given by crushing H. Then we are considering oriented Riemannian manifold with principal H bundle, which is not related to its tangent bundle. So given such G, I construct this group so that it models invertible theory on D dimension with G nabla structure with mapped to X. So this X is a target space. So we are based on the following picture or empirical fact about invertible theories. This was explained in the yesterday's talk. If you have such a theory, we get a partition function. This assigns some number to this triple. This consists of D dimensional closed manifold with G nabla structure with mapped to X. And first empirical fact says that up to the deformation, we can make it into U1 for invertible theory. And the next empirical fact is the important one. The partition function varies by local integration. This means that if we have a closed manifold and vary the structure in a smooth way like this, then the partition function varies accordingly. And the variation can be measured by integration of some differential form constructed on this cylinder. And generalizing this to non-trivial codeism, we interpret in this way. So we have a local procedure, some local procedure to produce a top form to D plus one dimensional manifold. One dimensional higher. Here, more precisely, in the case of non-oriented theory, we cannot integrate the differential form, but we need to consider density. So orientation bundle coefficient. And they satisfy the compatibility condition. So G and omega satisfies the following compatibility condition. If you have a manifold with boundary D plus one dimension and the structure and map to X, then we can consider the partition function on the boundary. And also we get the differential form on the bulk. Then they are related by this integration formula. So this is the empirical fact and we start from this picture. Now let's look at some easy examples of such a compatible pair. The first example corresponds to the polynomial theory. This is the case of oriented manifold with one dimension. And the target space, for target space, we fix a line bundle with connection. Then the partition function for this theory is given by the polynomial. Indeed, in this case, the closed manifold of one dimension with orientation mapping to X is specifying a closed curve in X. Then this partition function assigns a horonomy around this curve. In this case, what is omega here? Omega assigns a differential two-form to two-dimensional manifold, oriented manifold with map to X. Then you easily see that this can be taken as a curvature of the target manifold or first jump form and pullback by this map. Then this satisfies the compatibility condition because of the relation between horonomy and curvature. So let's look at another example corresponding to the classical Chan-Simon theory. Here, classical means that we are considering a connection as a background field. In this case, the structure is a so-times, so orientation with principle H-bundle with connection for a fixed compact V-group. And the partition function is given by Chan-Simon's invariant for H-connection on a three-manifold. If the bundle is trivial and connection is written like this, the Chan-Simon's invariant is an integration of the Chan-Simon's form. Then in this case, the four-form assigned to four-dimensional manifold is the second-channel character form for the H-connection, background connection. Locally written like this. Then we know that they satisfy the bulk and its correspondence compatibility condition. Now, in general, I want to make this local procedure omega into more mathematical object. And first, for simplicity, I explain the case where G is oriented. Then we can consider two types of such a procedure. The first one is coming from invariant polynomial on the real algebra of the structure groups. And the second one comes from closed differential forms on the target space. This corresponds to the example I gave. So the first example, if you have invariant polynomial on the real algebra like this, then by the Chan-Vey construction, if you have G connection, then applying the Chan-Vey construction, we get the characteristic form. In the second case, if you have a closed differential form on the target space, then we can just pull back along this map. Then combining these two types, we regard this omega, local procedure omega as an element here. A closed differential form on the target space is co-efficient in the invariant polynomial on the real algebra. Then indeed, if you have such an element of total degree n, then we get this procedure for G nabla manifold with map to x, then we get the differential n form, closed form, by first pull back this differential form to my manifold and then apply the Chan-Vey construction on the co-efficient. Now in the case of non-oriented theory, we can make it if you... So I want to get an orientation bundle co-efficient differential form instead of just differential form in order to integrate. So in order to do that, I replace the invariant polynomial by polynomials which changes signs according to the action of G. So this means that we consider the one-dimensional G module G acting by the plus minus one and consider the invariant part here. Then by the same procedure, we produce the orientation bundle co-efficient differential form. But if you are not familiar with this idea, you can just concentrate on the oriented case. Then after this preparation, I define the differential group like this. If you have a smooth manifold, then this group is defined to be the group consisting of this pair. Here, omega is the differential form with co-efficient in the invariant polynomial and total degree D plus one. And H is just the partition function applying this identification of R over G and U1. This H assigns some R over G value to this triple consisting of D-dimensional closed manifold with G nubra structure with map to X as before. And this is monoid by disjoint union and I require H to be the monoids homomorphism. And I require the compatibility condition as explained. So if you have a compact manifold with boundary with structure and map to X, then the partition function H evaluated at the boundary is given by the integration of the form constructed on the bulk by the channel construction. So you see that this definition is very much direct from the empirical fact about QFTs. Now to define the topological group, I divide it by some relation corresponding to deformation equivalence of theories. Mathematically, we take a differential form of one degree lower, then consider this relation. This pair of the alpha and this pair satisfies the compatibility condition because of the Stokes lemma. So you can divide it by this relation to get a smaller group. Then the main result says that this construction actually gives a model for the Anderson-Dale theory or manifolds. And moreover, this group with hat is a differential extension of the Anderson-Dale theory. Now I explained some examples of elements of our group. So recall the polynomial theory. In this case, the structure is orientation with dimension is one, and we fix a line band with connection on the target. Then the partition function is given by the polynomial. It is regarded as an element in this differential group of the target by this pair of the first channel form of the connection and polynomial function. Then you can also consider the deformation equivalence in this example. Suppose that you have two connections on the same line bundle on the target space. Then the resulting theories are different because they have different partition functions. But they are expected to be deformation equivalent because any connections can be connected by linear homotopy. Indeed, in the topological group, we have this equality because you can make the difference between connection by differential forms. So this reflects the deformation equivalence of the theory. Now let's look at the case of classical Chan-Simon theory. So in this case, the partition function was given by the Chan-Simon invariant, and Omega was second Chan character form of the background connection. This is regarded as an element in this group like this. So consisting of, this is regarded as invariant polynomial on the real zero of H, and the partition function is a Chan-Simon's invariant. Now let's introduce another example. This corresponds to the case of complex fermions. In this case, the structure is spincy structure with connection. And on the target space, we fix Hermitian vector bundle with connection. Then the partition function for this theory is given in terms of the Dirac operator twisted by the pullback of this connection. Here the reduced data invariant of Dirac operator is defined like this using the data invariant. And the data invariant is some spectral invariant for self-adjoint operator. And the Omega, the differential form in this case is given by this formula. This is the Chan character form for the connection. And this is the pullback. And also we multiply by this characteristic form of the background spincy connection. This is a taught form. Then the compatibility condition for this pair is checked by the Atyapadodi-Singer index theorem. So this theorem says that if you have a compact manifold with boundary with spincy structure with connection, then the APS index are the Fredholm index of the twisted Dirac operator with APS boundary condition is given by this formula. The integration of this form on the bulk and the reduced data invariant on the boundary. So this is a Fredholm index for integer. So this means that this pair satisfies the compatibility condition. Now I move on to the main parts of the talk. So I'm going to explain about differential push forwards. So we start from the following question. We explained the model of the Anderson dual in terms of QFTs. And I explained some examples arising from examples in physics. And a natural question is that, can we have some nice characterization or unified understanding of such an examples? And I want to say that the answer is yes. Actually, in many cases, they are related to differential push forwards in differential homology theory. So I'm going to first introduce the notion of push forwards, also called integrations. First I explained the topological push forward, which is very well known subject. So this is a push forward mapping eco-homology theory and push forward map is the map in the covariant direction. This is opposite to the usual pullback, which is contravariant. So this basically says that if you have something like a fiber bundle with nice structure on the relative tangent bundle, then we can do the fiber integration in eco-homology theory, reducing the degree by the relative dimension. So dimension of the fiber. So the setting is that first we assume that E is multiplicative homology theory and structure group G is also multiplicative. For example, SO in the case of SO or spin or spin C, they are multiplicative. And the multiplicative basically says that if you have two G manifold, then the product also becomes G manifold. Then we also assume that we are given a homomorphism of ring spectrum from the Madsen-Tillman spectrum to the eco-homology. Here this Madsen-Tillman spectrum is classified as the tangential body groups, a variant of tom spectrum. And such a multiplicative homomorphism is called a multiplicative genus. Given such data for in general for proper map, proper smooth map with relative tangential topological G structure, we get a topological push forward like this. So the typical example comes from fiber bundle and the requirement in this case is that the fibers are closed manifold with a G structure, topological G structures. So let's look at some examples. And for simplicity, I only explained the basic case of push forward along a map to point. So in this case, the requirement is that M is a closed manifold with topological G structure. So in the case of ordinary homology, we know that we can integrate along the oriented manifolds like this. So this corresponds to the map of spectrum, multiplicative genus like this. In the case of case theory or KO theory, we need more than orientation. In the case of case theory, we need spin C structures. And in the case of KO theory, we need spin structure. And they are called Attia-Bott Shapiro maps. In both cases, the push forward map is given by taking the index of twisted Dirac operator. Here the zero degree element can be represented by vector bundles. In the case of K, this is complex and in this KO case, the real. In the case of case theory, if you have a closed manifold with spin C structure, then we can take the index of the twisted Dirac operator G value. In the case of spin manifold, we can take the KO value index. And this map goes to the KO value index. Now I want to explain the differential refinement of push forward. So basically the differential push forward map is the corresponding map between differential homology theories. Here, in order to have such an integration map, I first assumed that this is a proper sub-margin, which is the same as fiber bundle. And assume that the fibers are equipped with G-nabra structures. So this is the fiber-wise tangent form. G-nabra structure, not just that topological G structure. And technically speaking, in order to have such a nice theory, I want to assume that E is rationally even. This means that the coefficient is torsion in the odd degree, but this condition is satisfying interesting examples. In particular, for closed manifold with G-nabra structure, we get the push forward map, differential push forward map to point. So this produces a differential homology element of the point. Now let's look at some examples. In the case of differential homology, ordinary homology, the differential push forward is actually horonomy. As this group classifies line bundle with connection, and interesting degree of the push forward is one-dimensional. So let's take a one-dimensional closed manifold with orientation. Then actually, in this case, we don't need the data of connection. But anyway, in this case, the push forward map is like this, producing the element here, the first differential homology of point. And this actually is isomorphic to R over G. Indeed, this map is given by taking the horonomy along this one-dimensional manifold. You see that the compatibility of horonomy and curvature can be stated in terms of differential homology. Indeed, if you have two-dimensional line for this orientation, like this, and given a line bundle with connection, specifying an element in differential homology, then we restrict this element to the boundary and differential push forward. Then we get the horonomy. On the other hand, we apply the curvature homomorphism on the bulk. Then we get the curvature of this connection and they satisfy the compatibility condition. Actually, this is a general feature about the differential push forward. So let's look at the case of case theory. This group classifies Hermitian vector bundle with connection. And in this case, the differential push forward is given by reduced data invariant. In this case, the structure is spin C nabra structure, spin C structure with connection. So let's take a odd dimensional closed manifold with this structure. Then the push forward is like this. And odd degree differential homology, differential case theory of point is isomorphic to R over G. And this push forward map is indeed given by taking the reduced data invariant of the Dirac operator twisted by this connection. You see that the compatibility condition is satisfied. So if you have compact manifold with boundary and differential case theory element like this, then we restrict it to the boundary and push forward. Then we get the reduced data invariant. And on the bulk, we take the curvature homomorphism like this and recall that this curvature homomorphism is given by the Chan character. Then we get differential form. And actually we need to multiply something. So this is a characteristic form for the background spin C connection, total form. And they satisfy the compatibility condition. Actually, this is also the feature, general feature about the differential push forward. Now from this explanation, it is obvious that those are related to the element in differential under some deal. Indeed, in the case of ordinary homology corresponds to the homomy theory. We fixed the line bandwidth connection on the target. This fixed differential homology element on the target space. Then the partition function is given by the push forward and the omega differential form is given by the curvature. In the case of differential case theory corresponds to the case of fermions. In this case, we fixed the differential case theory element in the target space. And the partition function is given by the differential push forward. And omega in this case is given by the curvature of the dis-element and also the characteristic form. Now a general property about the differential push forward includes the following. The first property is the commutativity of this diagram. The left square just says that this is refinement of the topological push forward. And the right square says that this is a refinement of push forward in differential forms. Here, the push forward is not just the fiber integration but we multiply the fiber wise characteristic form. In general, multiplicative genus specifies a characteristic polynomial called the Chandler's character. In the case of usual orientation, this is trivial. So we don't need any differential, so characteristic form. In the case of K-theory, ABS orientation, this is taught polynomial. And in the case of K-theory, this is a hard polynomial of the spincy connection. And another important property for us is the following borders formula. This generalizes the compatibility condition. In general, we assume that we have a compact manifold with boundary and having a structure, a G-nabla structure. And let's take a differential element here. Then we have the following equality in this group. Here, this equality is the same as the commutativity of this diagram. So let's look at this. Here, we have this element, E hat. And we can restrict it to the boundary to get this element. And we can push forward by this G-nabla structure to get this element. On the other hand, on the bulk, we apply the Carbacher homomorphism to produce a differential form here. This is appearing here. And we multiply by the characteristic form and integration. Then in general, we get the element in this vector space. And in the case of interesting example, like ordinary cohomology and case theory in this degree, this is isomorphic to R. And this A is in general the structure of homomorphism. I explained in the first part. This is like this. But in the case of ordinary cohomology or case theory, this group in this degree is isomorphic to R over G. And this map A is just taking the modularity. So you see that this is something very much like the compatibility condition in the definition of the Anderson-Dale differential model. But you see that this is an equality in E hat group. So E hat of the point. So we need to work a little bit to get R over G value. But it is natural to expect that using this autism formula, we can produce a lot of elements of the differential Anderson-Dale in terms of differential push forward. So our main result sets us doing something like that. So we construct the natural transformation of differential cohomology like this. So this sets that if you insert this target space here, then it sets that if you have a target space cohomology element, differential cohomology element, then we get a corresponding theory with target X. And the partition function is given by, basically given by the push forward. And in order to translate this element in the differential cohomology of point to R over G, we need to fix additional data. This is an element in the Anderson-Dale of the cohomology theory E. And typically there is a preferred choice of such an element. I will see that in the next slides. And but in general, it works. The construction works for any beta in this group, assuming that this degree is taken so that this condition is satisfied. But actually, for example, if you take, no, sorry, sorry. Your screen. Everything okay? Okay, no? No, we see it here. Sorry. So let's continue. So if you have, so this is very, for example, if you take N to be an even integer, this condition is automatically satisfied because we are assuming that is a rational even. Anyway, if you fix such an element, we can show that we can produce R over G value from this differential cohomology group. So after this preparation, we can define this transformation. This is the, so the resulting theory, the partition function of the resulting theory is given by the following composition. So first we have the differential cohomology element in the target space. Then pull back by this map to get an element in my manifold. Then use the G structure, to push forward to get this element. And finally, we replace it by R over G value by this, using this beta. So mathematically speaking, the main result concerns the topological characterization of this transformation. So I defined the differential cohomology transformation using the push forward. And the topological characterization means that we get some element in the topological group and characterize the topological element by some map of spectrum. Actually, on the topological level, there is some expected choice of such a transformation using the multiplicative genus and this Anderson-Dale element. And basically this theorem says that indeed this transformation refines such a expected topological transformation. Here, we are using the mathematical definition of the Anderson-Dale as a map of spectrum like this. And we are using the element E here and multiplicative genus G here and the Anderson-Dale element here. Now let's look at some examples. The first example, Holognomy theory one, Holognomy theory. So in this case, there is a natural choice of the element called the Anderson-Dale element for the ordinary cohomology. Then applying the construction to this element, we get the following assignment. So this transformation assigns a line band with connection to the corresponding Holognomy theory. And we can characterize this resulting element by this composition using the main theorem. Now let's do... So we can generalize this to the higher degree. So in general, the differential cohomology element can be regarded as deformed gauge field and the differential push forward in higher degree is called the higher Holognomy function, which appears in the definition of Chan-Simon's invariant. So for the dimensional closed manifold with orientation, the push forward map is like this. Then the transformation applied to this degree produces the following theory. So you have the differential cohomology element in the target space, then just pull back and take the higher Holognomy. Then we get the R over the value and we get the same topological characterization of the theory. Now using this, I can consider the classical Chan-Simon theory of any level. So in general, the level of Chan-Simon theory is specified by integer cohomology element of the classifying space. Then the definition of the Chan-Simon's invariant is the following. So in order to define it, I first fix some nice approximation, manifold approximation of classifying space and take a differential lift of this element. Then basically the Chan-Simon's invariant for H connection is given by first pulling back by the classifying map of this element, then taking the higher Holognomy function. So you see that this indeed coincides with the natural transformation I gave. So this means that the natural transformation applied to this element topologically coincides with the corresponding Chan-Simon's theories. Now in the case of complex fermions, the stories are parallel. So in this case, also we have the Anderson-Self duality element. And applying this element to the general construction, this transformation means that if you have a target vector bundle with connection, then it maps to the corresponding eta invariant theory. And also we get applying the main theorem, we get the characterization, following characterization of the transformation. So this basically says that this is under the sondiality of the Adiabot-Shapiro orientation. And this is one answer to the question we, which was asked yesterday, at least in the case of K theory. Finally, I wanted to explain that this another, there is another viewpoint of this transformation as defining a theory on Ginevra Manifold with element in background feeling like this. But actually I want to skip this. And I briefly mentioned the work by Gomis and me, which was put on archive today. This actually corresponds to exactly to the question raised yesterday and also the one conjecture appearing in the Frida Hopkins paper, Reflection Positivity Paper. So we constructed a model of differential chaos theory in terms of fermionic mastants. So I don't go into the detail, but basically this group, actually this is level by PQ. And maybe it looks strange, but this is a refinement of differential chaos theory of this degree, Q minus P minus two. Anyway, this group is defined in a way that and triple like this define the group in this, defines an element in this group. Here S is a Creeford module, CLPQ module. And M0 and M1 is a pair of invertible mastants on this module. A mastants are defined, formulated in terms of some scalar joint operators on Creeford modules. And we constructed this model, but we didn't, so we have not yet described the differential push forward around differential spin manifold. But I explained in the introduction of that article that this push forward map is expected to be assigning this value, this expression, we see in the physics literature. So perfume of the massive dilacobrida, the ratio of the perfumes. So if you can justify this expectation, then applying the general theory constructed in my work explained today, this means that this transformation is a theory with partition function regarding this expression. And also we get the topological characterization by using the main theorem. So this basically says that the undersondarity of the Atyabot-Shapiro map. So maybe this is a very interesting point. So finally, I explained the main results as second example of differential push forward. Actually, this is something like differential push forward in undersonderal theory itself. But I note that this is not a special case of the differential push forward for multiplicative genus because this is not associated with multiplicative genus. And this transformation corresponds to taking the compactification of theories in physics. So basically it says that you can get integration along a proper sub-margin fiber bundle with fiber-wise structure, differential structure with appropriate G. The setting is that in general, we assume that we have three sequences of tangential structure groups and we have this multiplication map, a homomorphism of tangential structure groups. Then we claim that for a proper sub-margin fiber bundle with fiber-wise G2 structure with connection, we can define the push forward map in this way on the from G3, the undersonderal to G3-bodism of the total space. We produce an element in the undersonderal of G1 on the base space and degree decreases. And actually on the topological counterpart, the topological counterpart is given by an easy generalization of the user's notion of the push forward in multiplicative genus. I will explain it later. First, I want to look at what kind of G1, G2, and G3 we are interested in. So in the first, the first example is given by any multiplicative structure group. Then in this case, we can take G1, G2, and G3, all of them equal to G. And this includes these examples and multiplication map is equipped in G. But we can also consider examples which is not multiplicative. So in this case, G1 and G3 is pin plus and G2 is spin. This means that product of pin plus manifold and spin manifold becomes pin plus manifold. And also for any G, we can consider this example and this is just a trivial group. So we can multiply in a clinical way. Actually, the construction applied to fiber integration applied to this example recovers the S1 integration which refines the suspension, the suspension map in the differential homology level. And given such a structure groups, the compactification of theory is the following procedure. So if you have a fiber bundle with fiber-wise G2 structure with connection, then from a d-dimensional theory on G3 manifold with connection, basically by taking the fiber product, you can produce a lower dimensional theory and structure becomes G1 manifold with connection. And the target space becomes N from N to X. So the fiber product is a pullback of the fiber bundle. So if you have such a fiber bundle with fiber-wise G2 structure with connection and assume that we have a map to X from some manifold, then we can pull back this fiber bundle and this total space is equipped with G3 structure if M is equipped with a G1 structure. And the dimension is related like this. So in this way, using this construction, we can define this transformation in a very straightforward way. So if you have a compatible pair in this group, then this means that we get some R over Z value to d-dimensional manifold with G3 structure with connection with map to N. Then we map it to the following theory. Here the resulting theory, the partition function of the resulting theory is given by evaluating the partition function on the fiber product constructed like this. So this is very natural construction and corresponding to so compactification of theories. And indeed, you can check that this is a compatible pair and that defines the well-defined group homomorphism. Now mathematically speaking, the topological push forward in this case is actually an easy generalization of the ones associated with multiplicative genus. The point is that you notice that in the definition of push forward in the case of multiplicative genus, we only use the associated MTG module structure of this homology theory and we produce this map. And we can generalize this to the situation like this. We start from a morphism of spectrum like this. I assume, I don't assume anything for E and F and G. Then by the same procedure, usual procedure to produce the push forward map, we can define the push forward in this way. So eco-homology element of the total space becomes F-co-homology element in the base space. So applying this to our situation, we have the multiplication map like this and correspondingly, we have the multiplication map between Anderson-Diard of the Mada-Sentilman spectrum to Mada-Sentilman spectrum. Then as applying the general construction, this transformation becomes like this and indeed this is the same form as the construction, differential push forward constructed above. And indeed mathematically speaking, the main theorem says that the differential push forward constructed by the making the fiber product coincide, defines the topological push forward I described in the last page. So this is the end of the talk. Thank you for the listening. Thank you for the talk. Are there any questions?