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4e304b9739d267859e8e5c64b2026d12e2257b40 | subsection | 8 | 58 | Dual graphs and Cremona transformation | It is nonsingular for n=2,3, but singular for n\ge 4.The open set {\mathcal {U}} as above is the complement of the divisor \Sigma _n of (REF ).
The inverse image of \Sigma _n in {\mathcal {G}}({\mathcal {C}}) can be described easily.
It consists of the points((t_1:\cdots :t_n),(s_1:\cdots :s_n))such that\lbrace i \,| \... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
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] | 2,008 | en | Physics | [
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944196ec154de964fe372d155e6b63cdd32847f0 | subsection | 9 | 58 | Dual graphs and Cremona transformation | \pi _1:{\mathcal {G}}({\mathcal {C}})\rightarrow {\mathbb {P}}^{n-1} is the blow-up along {\mathcal {S}}_n.
{\mathcal {L}} intersects every component of {\mathcal {S}}_n transversely.
\Sigma _n cuts out a divisor with simple normal crossings on {\mathcal {L}}.(1) Notice that the definition (REF ) of the Cremona
trans... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
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df9989b755034799aadc16f87bdf35affdebf9b8 | subsection | 10 | 58 | Dual graphs and Cremona transformation | Figure REF .We recall here a well known result (see e.g. , Proposition
8.3), which will be very useful in the following.Lemma 1.3
Suppose given a planar graph \Gamma with \#E(\Gamma )=n, with dual
graph \Gamma ^\vee . Then the graph polynomials satisfy\Psi _\Gamma (t_1,\ldots ,t_n)=
(\prod _{e\in E(\Gamma )} t_e)\,\, ... | {
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"start": 12
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} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
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9544012ef86b61f9c9454f773e6fcfac5c0c604b | subsection | 11 | 58 | Dual graphs and Cremona transformation | \end{array}The third equality uses the fact that \# E(\Gamma )=\#E(\Gamma ^\vee )
and \# V(\Gamma ^\vee )=b_0(S^2\setminus \Gamma ), so that
\deg \Psi _\Gamma + \deg \Psi _{\Gamma ^\vee } = \# E(\Gamma ), and the
fact that there is a bijection between complements of spanning tree
T in \Gamma and spanning trees T^{\prim... | {
"cite_spans": [
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"raw": "A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., Vol.210 (2000) 249–27... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
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] | 2,008 | en | Physics | [
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f658f23b1eb88768e4c02fda2aed09fe17d4d65c | subsection | 12 | 58 | Dual graphs and Cremona transformation | An explicit example of combinatorially inequivalent graphs with the
same graph hypersurface, obtained as dual graphs of different planar
embeddings of the same graph, is given in Figure REF .We see a direct application of this general result for planar graphs
in §REF below, where we derive a relation between the
classe... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
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9b3ac06b7624f1f3a0558b6cc1d0e25f0af29158 | subsection | 13 | 58 | An example: the banana graphs | In this paper we concentrate on a particular example, for which
we can carry out complete and explicit calculations. We consider
an infinite family of graphs called the “banana graphs”.
The n-th term \Gamma _n in this family is a vacuum bubble Feynman
graph for a scalar field theory with an interaction term of the
form... | {
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"raw": "D.R. Grayson, M.E. Stillman, Macaulay 2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/",
"source_ref_id": "6e32dc48b4f94d2f9224a35aaa87... | 0807.1690 | Feynman motives of banana graphs | [
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65af887b6252fd4ad7e4be5ac722f136c7ca1005 | subsection | 14 | 58 | An example: the banana graphs | At present we do not have a
conceptual explanation for this positivity phenomenon, but we can
state the following tentative guess, based on the sparse
numerical and theoretical evidence gathered so far.Conjecture 1.5
The coefficients of all the powers H^k in the CSM class of an
arbitrary graph hypersurface X_\Gamma ar... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
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ffe4a13b2ec3ffaff99fafd616ac0c5d57d8e176 | subsection | 15 | 58 | An example: the banana graphs | In fact, from (REF ) one
sees that the determinant satisfies\det M_{\Gamma _n} (t)= (t_{n-1}+t_n)\, \det M_{\Gamma _{n-1}}(t) \, -
t_{n-1}^2\, \det M_{\Gamma _{n-2}}(t) .It then follows by induction that the determinant satisfies the
recursive relation\det M_{\Gamma _n}(t)= t_n \, \det M_{\Gamma _{n-1}}(t) \, + t_1 \cd... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
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] | 2,008 | en | Physics | [
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... | |
5235cc3449c32643834ca0762ad0cd1df010510b | subsection | 16 | 58 | An example: the banana graphs | This will depend on which
scalar field theory the graph belongs to, but the resulting integral
is unaffected by this, as long as we have nonzero external momenta
flowing through the graph.Lemma 1.8
The Feynman integral (REF ) for
the banana graphs \Gamma _n is of the formU(\Gamma ,p)= \frac{\Gamma ((1-D/2)(n-1) +1) C(... | {
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"raw": "S. Bloch, E. Esnault, D. Kreimer, On motives associated to graph polynomials, Commun. Math. Phys., Vol.267 (2006) 181–225.",
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... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
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a8bd9feb700fea31b532fb8887dfce84faf831d2 | subsection | 17 | 58 | An example: the banana graphs | In general, this type of
regularization procedures requires a detailed knowledge of the
singularities of the hypersurface X_\Gamma to be carried out,
and that is one of the reasons for introducing invariants of singular
varieties in the study of graph hypersurfaces. | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
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c7331809d56b6b1ba7da07ec8694b1da47b657c9 | subsection | 18 | 58 | Characteristic classes and the Grothendieck ring | In order to understand the nature of the part of the cohomology of the
graph hypersurface complement that supports the period corresponding
to the Feynman integral (ignoring divergence issues momentarily), one
would like to decompose {\mathbb {P}}^{n-1}\setminus X_{\Gamma } into
simpler building blocks. As in §8 of , t... | {
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"Paolo Aluffi",
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a69310eb373074964a1d345a00d6eba3ea0b957d | subsection | 19 | 58 | The Grothendieck ring | Let {\mathcal {V}}_K denote the category of algebraic varieties over a field K.
The Grothendieck ring K_0({\mathcal {V}}_K) is the abelian group generated by
isomorphism classes [X] of varieties, with the relation[X]=[Y]+[X\setminus Y],for Y\subset X closed. It is made into a ring by the product
[X\times Y]=[X][Y].An a... | {
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"raw": "H. Gillet, C.Soulé, Descent, motives and K-theory. J. Reine Angew. Math. 478 (1996), 127–176.",
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} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0112e0b89fc469a3c1a10dcb5e484d601f0f9dcc | subsection | 20 | 58 | The Grothendieck ring | Since the Lefschetz motive is
invertible in K_0({\mathcal {M}}_K), its inverse being the Tate motive {\mathbb {Q}}(1),
the ring homomorphism (REF ) induces a ring homomorphism\chi : K_0({\mathcal {V}}_K)[{\mathbb {L}}^{-1}] \rightarrow K_0({\mathcal {M}}_K).Thus, in the following we can either regard the classes [X_\Ga... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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a7abc87eaf8fad4407fc263d76d072a824981088 | subsection | 21 | 58 | CSM classes as a measure of
singularities | The Chern class of a nonsingular complete variety V is the
`total homology Chern class' of its tangent bundle. We write
c(V):=c(TV)\cap [V]_* to indicate the result of applying the Chern
class of the tangent bundle of V to the fundamental class [V]_*
of V. (We use the notation [V]_* rather than the more common [V]
in o... | {
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{
"arxiv_id": "",
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"Paolo Aluffi",
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] | [
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8f750b9bb8fc1f2637789eb297aacdb0a9514637 | subsection | 22 | 58 | CSM classes as a measure of
singularities | First of all, c(X) must agree with its namesake when X
is a complete nonsingular variety: that is, c(X)=c(TX)\cap [X]_*
in this case. Secondly, associate with every variety X an abelian
group F(X) of `constructible functions': elements of F(X) are
finite integer linear combinations of functions {\bf 1}_Z (defined
by {\... | {
"cite_spans": [
{
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}... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
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ba7c652ca625096a577baf4db8bb9d8cb29dcf6a | subsection | 23 | 58 | CSM classes as a measure of
singularities | This pushforward is understood
in the table in §REF , and will be often understood in the
explicit computations of §. | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
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ff8d31cdd66c2d4ea2b5ea0960e3e32d89a7ac84 | subsection | 24 | 58 | CSM classes versus classes in the
Grothendieck ring | CSM classes are defined in by relating them to
a different class, called `Chern-Mather class', by means of a local
invariant of singularities known as the `local Euler obstruction'.
As noted above, once the existence of the classes has been established,
then their computation may be performed by systematic use of
resol... | {
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"Paolo Aluffi",
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f17dcac4373b8f38660f1287120b81fd15c46e0a | subsection | 25 | 58 | CSM classes versus classes in the
Grothendieck ring | This parallel cannot be taken too far, since
the `embedded' Chern-Schwartz-MacPherson treated here
is not an invariant of isomorphism classes.Example 2.1 Let Z_1 and Z_2 be, respectively, a linearly embedded {\mathbb {P}}^1 and a
nonsingular conic in {\mathbb {P}}^2. Denoting by H the hyperplane
class in {\mathbb {P}}^... | {
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} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
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] | [
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"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.05076302960515022,
-0.0055746231228113174,
0.0029723690822720528,
0.0026881054509431124,
0.025763075798749924,
-0.024557340890169144,
0.040445588529109955,
0.016865046694874763,
0.013362306170165539,
0.009287223219871521,
0.000474329455755651,
-0.012942587956786156,
-0.020070169121026993,... | |
768b48ed45cad6f691ae2d765b5d761850fdf8ba | subsection | 26 | 58 | CSM classes versus classes in the
Grothendieck ring | With notation as above, assumec(X)=\sum a_i H^{-i}\quad .Then the class of X in the Grothendieck group of varieties
equals[X]= \sum a_i {\mathbb {T}}^i ,where {\mathbb {T}}=[{\mathbb {G}}_m] is the class of the multiplicative group, see §.Thus, adopting a variable T=H^{-1} in the CSM environment,
and T={\mathbb {T}} in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1635,
"openalex_id": "",
"raw": "P. Aluffi, Modification systems and integration in their Chow groups. Selecta Math. (N.S.) 11 (2005), no. 2, 155–202.",
"source_ref_id": "bdd07b2e67a3b9f2469ba3083bf244d67773ad6c",
"sta... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.04390357434749603,
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0.009206325747072697,
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0.010251286439597607,
0.0022462825290858746,
0.02019747532904148,
0.0029022428207099438,
-0.006483328528702259,
0.007524474523961544,
-0.007734229322522879,
... | |
c492b6fe8619d37299951ebdea128b763f9023f1 | subsection | 27 | 58 | Banana graphs and their motives | In this section we give an explicit formula for the classes
[X_{\Gamma _n}] of the banana graph hypersurfaces X_{\Gamma _n}
in the Grothendieck ring. The procedure we adopt to carry out
the computation is the following. We use the Cremona transformation of
(REF ). Consider the algebraic simplex \Sigma _n
placed in the ... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.045893315225839615,
0.013861062936484814,
-0.030300574377179146,
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0.011839499697089195,
-0.03646443784236908,
0.05080609768629074,
0.009474650956690311,
0.006552918814122677,
0.005561207886785269,
-0.002033006865531206,
0.03454204276204109,
0.008933024480938911,
-0... | |
aaf86abea4b7770365e3fa895ad809c75623e748 | subsection | 28 | 58 | Banana graphs and their motives | In fact, the variables P^i satisfy a different
multiplication rule, which we denote by \bullet and which is given byP^i \bullet P^j = P^{i+j}+P^{i+j-1}+\cdots +P^j-P^{i-1}-\cdots -1and which recovers in this way the class [{\mathbb {P}}^i\times {\mathbb {P}}^j]. This follows from Lemma REF , by converting each of the
t... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.04159864783287048,
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0.00998001266270876,
0.012070626951754093,
0.005054861307144165,
0.017472652718424797,
-0.03750897943973541,
0.0005703409551642835,
-0.04135448858141899,
-0.0... | |
975f7073c53a6b43385c10392604370b47003bf2 | subsection | 29 | 58 | Banana graphs and their motives | In terms of classes [{\mathbb {P}}^r] it corresponds to mapping
[{\mathbb {P}}^0] to zero and [{\mathbb {P}}^r] to [{\mathbb {P}}^{r-1}] for r\ge 1.One can see that, for g({\mathbb {T}})=[{\mathbb {P}}^r]=\frac{(1+{\mathbb {T}})^{r+1}-1}{{\mathbb {T}}},
we have\frac{g({\mathbb {T}})-g(-1)}{{\mathbb {T}}+1}=\frac{\frac{... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.05279020592570305,
-0.009604460559785366,
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0.04833507910370827,
-0.004676357842981815,
0.06548427045345306,
0.01742381975054741,
0.00788038782775402,
0.03335242345929146,
-0.014257933013141155,
0.009985893033444881,
-0.023145301267504692,
0.... | |
97b23d2a4784c756c6f44a74d1be146c73a9e661 | subsection | 30 | 58 | Banana graphs and their motives | One then
applies the transformation {\mathcal {H}} of (REF ) to obtain\begin{array}{ll}
[{\mathcal {L}}\cap \Sigma _{r+1}] &=\left(\frac{(1+{\mathbb {T}})^{r+1}-1-{\mathbb {T}}^{r+1}}{{\mathbb {T}}}-\frac{-1-(-1)^{r+1}}{-1}\right)/({\mathbb {T}}+1) \\[3mm]
&=\frac{(1+{\mathbb {T}})^r-1}{{\mathbb {T}}}- \frac{{\mathbb {... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.0325591042637825,
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0.025388168171048164,
0.03101811558008194,
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0.00021348317386582494,
0.004874711390584707,
0.022123102098703384,
0.004592450801283121,
0.03... | |
481a2480fbb7bef84d8e02dbce436a768f28e387 | subsection | 31 | 58 | Banana graphs and their motives | This hyperplane intersects \Sigma _n transversely, so
that (REF ) applies and gives[{\mathcal {L}}\setminus \Sigma _n]=[{\mathcal {L}}]-[{\mathcal {L}}\cap \Sigma _n]=\frac{{\mathbb {T}}^{n-1}-(-1)^{n-1}}{{\mathbb {T}}+1}.Next we examine how the graph hypersurface X_{\Gamma _n} intersects
the algebraic simplex \Sigma _... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.07665888965129852,
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0.009277190081775188,
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0.0029143805149942636,
-0.01545689720660448,
0.01... | |
b7d36faeef65cbd1d26d4446b8fe4a61f98d90eb | subsection | 32 | 58 | Banana graphs and their motives | We then have[{\mathcal {S}}_{r+1}] = [\Sigma _{r+1}]- (r+1){\mathbb {T}}^{r-1}
=\frac{(1+{\mathbb {T}})^{r+1}-1-(r+1){\mathbb {T}}^r-{\mathbb {T}}^{r+1}}{{\mathbb {T}}}.This gives the formula (REF ).We then have the following result.Theorem 3.10
The class in the Grothendieck ring of the graph hypersurface
X_{\Gamma _n... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.04146862402558327,
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0.02885105460882187,
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0.014463669620454311,
0.008521054871380329,
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0.030437786132097244,
0.005450576078146696,
0.014189043082296848,
-0.004920394159853458,
... | |
f05c8b94198faf0b3885dd061f27eee262fe0f3a | subsection | 33 | 58 | Banana graphs and their motives | The class in the Grothendieck ring
is[X_{\Gamma _4}]= {\mathbb {T}}^2 + 5{\mathbb {T}}+ 5 .In terms of the Lefschetz motive {\mathbb {L}}, the formula (REF )
reads equivalently as[X_{\Gamma _n}]= \frac{{\mathbb {L}}^n-1}{{\mathbb {L}}-1}-\frac{({\mathbb {L}}-1)^n-(-1)^n}{{\mathbb {L}}}-n\,
({\mathbb {L}}-1)^{n-2} .In t... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.08954776078462601,
0.011765733361244202,
0.01796143688261509,
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0.014115828089416027,
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0.02765176258981228,
0.049687713384628296,
0.0019666780717670918,
0.00333820260129869,
-0.01698477379977703,
0.03201622515916824,
-0.023714590817689896,
-0.0... | |
2fc0b92b455c0fd28476b8f7fe8b9b3eae30ea7e | subsection | 34 | 58 | Banana graphs and their motives | Then the formula
(REF ) for the class of the hypersurface complement shows
that\chi ({\mathbb {P}}^{n-1}\setminus X_{\Gamma _n})= \chi ({\mathbb {T}}^{n-1})+(n-1)
\chi ({\mathbb {T}}^{n-2})+\chi ({\mathbb {T}}^{n-3})-\cdots \pm 1 = (-1)^{n-1}.Since \chi ({\mathbb {P}}^{n-1})=n we obtain\chi (X_{\Gamma _n})= \chi ({\mat... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.049714360386133194,
-0.0067292917519807816,
-0.01576271653175354,
-0.01579323597252369,
-0.009239424020051956,
-0.019546989351511,
0.03714080527424812,
0.06689617782831192,
0.00836202222853899,
-0.005966333206743002,
-0.009536977857351303,
0.02348385378718376,
-0.01172666810452938,
0.00... | |
5044a1f939a23f960d3c557e8469e69e186c8ba6 | subsection | 35 | 58 | Banana graphs and their motives | However, how geometrically one can
associate a {\mathbb {P}}^n to a graph hypersurface X_{\Gamma _n}\subset {\mathbb {P}}^{n-1} is unclear, so that a satisfactory conceptual explanation of
the occurrence of (REF ) in (REF ) is still
missing.For completeness we also give the explicit formula of the class
(REF ) written ... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.06647378206253052,
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0.0223410502076149,
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0.02331770956516266,
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0.04031766206026077,
-0.034518755972385406,
0.0... | |
bdf10ea46b1f31a2a22be51d9e3a1c2793f657fb | subsection | 36 | 58 | Classes of dual graphs | In the result obtained above, we used essentially the relation between
the graph hypersurface X_{\Gamma _n} and the hypersurface of the dual
graph, which is, in this case, a hyperplane. More generally, although
one cannot obtain an explicit formula, one can observe that for any
given planar graph the relation between t... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0.013774468563497066,
0.02902940660715103,
0.010431965813040733,
0.015812020748853683,
-0.0038385360967367887,
0.014896267093718052,
-0.02170337364077568,
0... | |
1ee108aa2d5a9357bc4456d5ac7f47e97227a5e1 | subsection | 37 | 58 | CSM classes for banana graphs | We now give an explicit formula for the Chern–Schwartz–MacPherson
class of the hypersurfaces of the banana graphs, for an arbitrary number
of edges.The computation of the CSM class is substantially more involved than
the computation of the class in the Grothendieck ring we obtained in
the previous section, although the... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.03800401836633682,
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0.015125294215977192,
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0.021276146173477173,
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0.033974677324295044,
0.017124703153967857,
-... | |
a2883c0dda6404effc439dae8207058f1bf2a69f | subsection | 38 | 58 | Characteristic classes of blowups | Let D be a divisor with simple normal
crossings and nonsingular components D_i, i=1,\dots ,r, in a
nonsingular variety M. Then TM(-\log (D)) denotes the sheaf of
vector fields with logarithmic zeros (i.e. the dual of the
sheaf \Omega ^1_M(\log D) of 1-forms with logarithmic poles).
In terms of Chern classes one has (cf... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 380,
"openalex_id": "",
"raw": "P. Aluffi, Modification systems and integration in their Chow groups. Selecta Math. (N.S.) 11 (2005), no. 2, 155–202.",
"source_ref_id": "bdd07b2e67a3b9f2469ba3083bf244d67773ad6c",
"star... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.07002920657396317,
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0.017400456592440605,
0.... | |
6dfe2305afd09ac0b4dfeb159849aa16d4ad1b47 | subsection | 39 | 58 | Characteristic classes of blowups | Thus, we obtain (REF ).One derives directly from this result the following formula for Chern
classes.Corollary 4.2
Under the same hypothesis as Theorem REF , the Chern classes
satisfy\frac{c(TW)}{(1+F)\prod _{j\in J}(1+F_j)} \cap [W]
=\pi ^*\left(\frac{c(TV)}{\prod _{j\in J}(1+E_j)}\cap [V]\right) .In other words, if ... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.037132203578948975,
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0.02561420388519764,
0.021724017336964607,
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0.041922472417354584,
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-0.0026163405273109674,
-0.01280710194259882,... | |
c14da6ae98b8fd71c7f42bed2a49f6a62fc6a8f5 | subsection | 40 | 58 | Characteristic classes of blowups | By Corollary REF , we have the identityc(T\tilde{V}(-\log \tilde{E}))\cap [\tilde{V}]=\tilde{\pi }^*(c(TV(-\log E))\cap [V])
=\alpha ^* \pi ^* (c(TV(-\log E))\cap [V]) .As in (REF ) of §REF , this is saying thatc({\bf 1}_{\tilde{\pi }^{-1}(U)})=\alpha ^* \pi ^*( c({\bf 1}_U)).The statement then follows by pushing forwa... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 712,
"openalex_id": "",
"raw": "R.D. MacPherson, Chern classes for singular algebraic varieties. Ann. of Math. (2) 100 (1974), 423–432.",
"source_ref_id": "e54088067d3abbdb0da711f39183bde1525e144f",
"start": 569
}
... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0.04671580716967583,
0.021969549357891083,
0.030421724542975426,
0.006854041945189238,
0.006445927079766989,
0.0012939531588926911,... | |
e372f7f50f7375639cd99c8f9e520867b1d5b0fb | subsection | 41 | 58 | Computing the characteristic classes | In this section we give the explicit formula for the CSM class of the
graph hypersurface X_{\Gamma _n} of the banana graph \Gamma _n. The
procedure is somewhat similar conceptually to the one we used in the
computation of the class in the Grothendieck ring, namely we will
use the inclusion–exclusion property of the Che... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1303,
"openalex_id": "",
"raw": "P. Aluffi, Classes de Chern des variétés singulières, revisitées. C. R. Math. Acad. Sci. Paris 342 (2006), no. 6, 405–410.",
"source_ref_id": "9f2c98f33c535e85d9de9775f65982083dd7229b",
... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.05406610667705536,
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0.019603541120886803,
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0.007177794352173805,
0.0026869054418057203,
0.026605894789099693,
0.02468368038535118,
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0.03713231161236763,
0.03877992182970047,
-0... | |
a3ce5f93d5fb7bd3feb97672c9fb109523b7e009 | subsection | 42 | 58 | Computing the characteristic classes | By (2) and (3) of Lemma REF , we know that \pi _1^{-1}({\mathcal {L}}) is
the blowup of {\mathcal {L}} along {\mathcal {L}}\cap {\mathcal {S}}_n, that is, the singularity subscheme
of {\mathcal {L}}\cap \Sigma _n. The blowup of a variety along the singularity
subscheme of a divisor with simple normal crossings is
domin... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.047938570380210876,
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0.03500034660100937,
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0.04650438204407692,
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0.0006808589096181095,
-0.04113379865884781,
0.022367265075445175,
0.010359736159443855,
0... | |
b8175c28d7d8d94b5bde6c5d41da8a226355b559 | subsection | 43 | 58 | Computing the characteristic classes | Since the CSM class of a divisor with normal
crossings is computed by the Chern class of the bundle of vector fields with
logarithmic zeros along the components of the divisor, we findc({\mathcal {L}}\setminus \Sigma _n)= \frac{c(T{\mathcal {L}})\cap [{\mathcal {L}}]_*}{(1+h)^n}
=\frac{(1+h)^{n-1}}{(1+h)^n}\,h\cdot [{\... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
-0.046089958399534225,
-0.022709224373102188,
-0.03943591192364693,
-0.012132951989769936,
0.026372002437710762,
-0.010812826454639435,
0.003330838866531849,
0.02469322830438614,
-0.0016625578282400966,
0.009248514659702778,
-0.009874239563941956,
0.019473770633339882,
0.022617654874920845,
... | |
8d25efd0bbb1ff893eae773dc925afe9a7145a19 | subsection | 44 | 58 | Computing the characteristic classes | Thus, we obtainc(X_{\Gamma _n}\setminus \Sigma _n)={\pi _2}_*
\left(\sum _{i=1}^{n-1} (-1)^{i-1}h^i (h+H)^{n-1}\cdot [{\mathbb {P}}^{n-1}\times {\mathbb {P}}^{n-1}]_* \right).Finally, we have to evaluate the pushforward via \pi _2. We can writec(X_{\Gamma _n}\setminus \Sigma _n) = \sum _{i=1}^{n-1} a_i
H^i\cdot [{\math... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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-... | |
79c82807fe02f30c5488dbefdfb4dee32698837a | subsection | 45 | 58 | Computing the characteristic classes | Thus, banana graphs provide
an infinite family of graphs for which Conjecture REF holds.Remark 4.11 As pointed out in §REF , CSM classes are defined (as classes
in the Chow group of an ambient variety) for locally closed subsets.
It follows from
Theorem REF that the CSM class of the
complement of X_{\Gamma _n} in {\mat... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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... | |
ec404dfe7075655556b2009fa9b3a50a620db205 | subsection | 46 | 58 | The CSM class and the class in the Grothendieck ring | We discuss here the formal similarity, as well as the discrepancy,
between the expression for the CSM class and the formula for the
class in the Grothendieck ring of the graph hypersurface
X_{\Gamma _n}.As noted in Propostion REF , the CSM class and the
class in the Grothendieck group carry the same information for
sub... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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... | |
df3e27c19ff44857dc7fed5bc994051ebe707ff7 | subsection | 47 | 58 | Classes of cones | We make here a general observation which may be useful in other
computations of CSM classes and classes in the Grothendieck ring
for graph hypersurfaces. One can observe
that often the graph hypersurfaces X_\Gamma happen to be cones over
hypersurfaces in smaller projective spaces.There are simple operations one can per... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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16626013697e77fedb666cb51217532406fa4ed7 | subsection | 48 | 58 | Classes of cones | Then, by Claim 1 of we havef(H)\cap [{\mathbb {P}}^m]_* =i_* c(X)=H\cdot (1+H)^{-1} \cap j_* c(C(X))
=H(1+H)^{-1} g(H) \cap [{\mathbb {P}}^{m+1}]_* .This implies(1+H) f(H) \cap [{\mathbb {P}}^m]_* = g(H)\cap [{\mathbb {P}}^m]_* .This determines all the coefficients in g(H) with
the exception of the coefficient of H^{m+... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 150,
"openalex_id": "",
"raw": "P. Aluffi, MacPherson's and Fulton's Chern classes of hypersurfaces, Internat. Math. Res. Notices, Vol.11 (1994) 455–465.",
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"... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
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] | 2,008 | en | Physics | [
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... | |
19a9592c8d91973f57c634f5cabcf67f4f7f8759 | subsection | 49 | 58 | Classes of cones | Doubling a disconnecting edge in a non-1PI graph.In these cases the resulting graph hypersurface is obtained by first
taking a cone over the original hypersurface in one extra dimension
and then taking the union with a transversal hyperplane, respectively
given by the vanishing of the coordinate corresponding to the lo... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
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"math.AG",
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] | 2,008 | en | Physics | [
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0.... | |
2acb69ab9da9051f0a46ea978918df650ac193c1 | subsection | 50 | 58 | Classes of cones | Given a pair of 1PI graphs \Gamma _1 and \Gamma _2 and
two additional edges joining them as in Figure REF , the
graph polynomial becomes of the form\Psi _\Gamma (t)=(t_1+t_2)\Psi _{\Gamma _1}(t_3,\ldots ,t_{n_1+2})\Psi _{\Gamma _2}(t_{n_1+3},
\ldots , t_{n_1+n_2+2}) +\Psi _{\Gamma _1,\Gamma _2}(t_3,\ldots ,t_{n_1+n_2+2... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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-0... | |
0eafa751083e5f2329e3eff14377846fe7e3cbc9 | subsection | 51 | 58 | Classes of cones | Then\begin{array}{rl}
c({\mathbb {P}}^n\setminus X_{\hat{\Gamma }}) = & (1+H)g(H)\cap [{\mathbb {P}}^n]_* \\[2mm]
c({\mathbb {P}}^n\setminus X_{\Gamma ^{\prime }}) = & g(H)\cap [{\mathbb {P}}^n]_*
\end{array}Similarly, the classes in the Grothendieck group satisfy\begin{array}{rl}
[{\mathbb {P}}^n\setminus X_{\hat{\Gam... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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-0... | |
621b4c17a1498e32b6285052b89616f78eb0510c | subsection | 52 | 58 | Classes of cones | The class for the
`banana split' graph is{\mathbb {T}}^n+n{\mathbb {T}}^{n-1}+n{\mathbb {T}}^{n-2}-(-1)^n .
[Figure: Banana split graph.] | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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... | |
35cad4b252ee71bac1430c106e234503c7886b07 | subsection | 53 | 58 | Banana graphs in Noncommutative QFT | Recently there has been growing interest in investigating the
renormalization properties and the perturbative theory for certain
quantum field theories on noncommutative spacetimes. These arise,
for instance, as effective limits of string theory ,
. In particular, in dimension D=4, when the underlying
{\mathbb {R}}^4 i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 249,
"openalex_id": "",
"raw": "A.Connes, M.Douglas, A.Schwarz, Noncommutative geometry and matrix theory: compactification on tori. JHEP 9802 (1998) 3–43.",
"source_ref_id": "d9d8412798ea32acb6a2b5ed8a4ae7c06de093a1",
... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0.017927609384059906,
0.0028398095164448023,
0.00... | |
bbe3340a0af37e15e749e998c7c0799ca9d233ca | subsection | 54 | 58 | Banana graphs in Noncommutative QFT | The hypersurface corresponding
to the value s = 0 has a singularity at the origin 0\in {\mathbb {A}}^n whose
tangent cone is the (affine) graph hypersurface X_\Gamma .This follows directly from the relation between the graph
polynomial for the ribbon graph \tilde{\Gamma } given in
and the Kirchhoff polynomial \Psi _\G... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 326,
"openalex_id": "",
"raw": "R. Gurau, V. Rivasseau, Parametric Representation of Noncommutative Field Theory, Commun. Math. Phys. Vol. 272 (2007) N.3, 811–835",
"source_ref_id": "ab54b3a86acf3a0d5a821c23c8edf6ed225e347d"... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0.0... | |
b12babf95b79e70560a66cbffd0bd756f90c83e9 | subsection | 55 | 58 | Banana graphs in Noncommutative QFT | Notice also how, at the limit value s= 0 of the parameter, the
two polynomials for the two different ribbon graphs corresponding to
the third banana graph \Gamma _3 agree.For each value of the parameter s=(4\theta \Omega )^{-1} one obtains
in this way an affine hypersurface, which is a curve in {\mathbb {A}}^2 or a
sur... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1671,
"openalex_id": "",
"raw": "P. Aluffi, Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes. Pure Appl. Math. Q. 2 (2006), no. 4, 915–941.",
"source_ref_id": "8827baa619e454a1d9a2b3189661cc... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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dcfc167654f8a67c7707f8041b9e65a48308c6e3 | subsection | 56 | 58 | Banana graphs in Noncommutative QFT | For general values of the
parameter s=(4\theta \Omega )^{-1} the CSM class is given byc(\bar{X}_{\tilde{\Gamma }_3})= 14H^3 + 4H.For the special value s=1/2 of the parameter, the CSM class becomes
of the formc(\bar{X}_{\tilde{\Gamma }_3})|_{s=1/2}= 5H^3 + 5H^2 + 4H,while in the limit s\rightarrow 0 one hasc(\bar{X}_{\t... | {
"cite_spans": [
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"doi": "",
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"raw": "D. Broadhurst, D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, Vol.393 (1997) 403–412.",
"source_ref_id": "d29d0ef6dc447... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0864616f8769265e3abfcf31fe6108e115e17024 | subsection | 57 | 58 | Banana graphs in Noncommutative QFT | It would be interesting to
see whether it also has an effect on invariants such as the CSM class. | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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8c558ae26b22a5bc3e8f0b5bfa0cbb95be1a4fe7 | abstract | 0 | 16 | Abstract | In this paper, we investigate the atom-molecule conversion dynamics of a
generalized many-body model that includes the atom-atom, atom-molecule, and
molecule-molecule interactions, emphasizing the efficiency of the Feshbach
molecular formation. We show that the picture of two-body molecular production
depicted by the L... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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4f266cebc0cbdfe31927c71e06e5f0786ff31f89 | subsection | 1 | 16 | introduction | The production of ultracold diatomic molecules in bosonic systems is an
exciting area of research with important applications ranging from the
production of molecular Bose-Einstein condensates (BECs) to the
study of chemical reaction dynamics. A widely used production
technique involves the association of ultracold ato... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 243,
"openalex_id": "",
"raw": "S. Inouye, M. R. Andrews, J. Stenger, H. -J. Miesner, D. M. Stamper-Kum, and W. Ketterle, Nature (London) 392, 151 (1998).",
"source_ref_id": "8ee1d60ed3fcee226efa2eafe822284fa37365bd",
... | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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... |
7d90c8bc6a929bb3609991cbc6b9513f335b47b2 | subsection | 2 | 16 | Model | A widely used molecular production technique involves the association of
ultracold atom pairs (open channel) into very weakly bound diatomic
molecules (close channel) by applying a time varying magnetic field in the
vicinity of a Feshbach resonance, . Precisely speaking,
atoms are converted into molecules at an avoided... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 251,
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"raw": "E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999).",
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"start": 0
},
{
"arxiv_id... | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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010e93180a7bee38cdee8ffa69193290436f3c75 | subsection | 3 | 16 | Model | Here, N_{a}=\hat{a}^{\dagger }\hat{a} and N_{b}=\hat{b}^{\dagger }\hat{b} are the atom number and molecule number, respectively.
We introduce the parameter V to denote the quantized volume of the trapped
particles, therefore n=N/V is the mean density of the initial bosonic
atoms.Choosing the Fock states as the basis, t... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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6fdcf75f65f1106f1efa02a9f633a7db6bcd275c | subsection | 4 | 16 | Model | When the
external magnetic field is linearly swept across the Feshbach resonance at \Delta \simeq 0, a fraction of atoms will be converted to molecules at the
avoided-crossing of energy levels. The conversion efficiency as a function
of the sweeping rate (i.e., \alpha =\dot{\Delta }=\mu _{co}\dot{B}/4 ) and coupling st... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 407,
"openalex_id": "",
"raw": "L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); G. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).",
"source_ref_id": "d13ef9e517982105e494d5437b869ad5bb85fa5a",
"start": 194
}... | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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f577259d78bb866034996e7358fe867a8698a16d | subsection | 5 | 16 | Many-body effects ( | As the total particle number N increases, Eq.(REF ) is no longer
analytically solvable. In addition, the computational demand increases
dramatically as N becomes very large. In the mean-field limit where N\rightarrow \infty , the quantum fluctuation is negligible. It is
appropriate to replace all the quantum operators ... | {
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"raw": "Jie Liu, Li-Bin Fu, Bi-Yiao Ou, Shi-Gang Chen, and Qian Niu, arXiv:quant-ph/0105140v1; Jie Liu, Libin Fu, Bi-Yiao Ou, Shi-Gang Chen, Dae-Il Choi, Biao Wu and Qian Niu, Phys. Rev. A 66, 023404 (2002... | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
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] | 2,008 | en | Physics | [
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17884f3390607ad1fe1cfe9cebf72307677060c4 | subsection | 6 | 16 | Many-body effects ( | In addition, the absence of hermicity as
well as the lack of U(1) invariance of the Hamiltonian restrict the
mean-field motion to a “tear-drop” shaped equal-single-pair-entropy surface, , rather than the surface of a Bloch sphere. As we will
show latter, the interplay of these new features leads to a very different
ene... | {
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{
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"end": 230,
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"raw": "I. Tikhonenkov, E. Pazy, Y. B. Band, M. Fleischhauer, and A. Vardi, Phys. Rev. A 73, 043605 (2006).",
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},
{... | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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... |
8aaa27bb218a61b3181d9bc056bc0668c8ea19ee | subsection | 7 | 16 | General properties | We first show how the nonlinear interactions lead to the deformation of the
eigen-energy levels. The eigen-states of the system satisfy thatH\left(
\begin{array}{c}
a \\
b\end{array}\right) =\mu \left(
\begin{array}{c}
a \\
2b\end{array}\right) .Solving the above nonlinear equations together with total particle
conserv... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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... |
0199e6a0364423f6ddfd61b61f36b5d44139d9ef | subsection | 8 | 16 | General properties | The conversion efficiency as a function of the
sweeping rate \alpha is plotted against the nonlinear parameters ranging
from weak nonlinearity to strong nonlinearity. Fig.REF shows that
(1) the nonlinear interaction always suppresses the conversion from atoms to
molecules. For example, in the case of strong nonlinearit... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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0290bfadf5651b636c2a18ff9f990b4281a3c9ff | subsection | 9 | 16 | Adiabatic limit | In the adiabatic limit where the external field varies slowly compared with
the intrinsic motion of the system, the conversion dynamics are entirely
determined by the phase-space structure evolution of the classical
Hamiltonian (REF ). The fixed points (i.e., the energy extrema of
the classical Hamiltonian) on the phas... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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bde18345aad97b2cc6a40dc8647b01f7e9f87697 | subsection | 10 | 16 | Adiabatic limit | \frac{\partial \dot{\theta }}{\partial S}\right|_{S_{c}}=-4U+\Omega \frac{5-3S_{c}}{2(1-S_{c})^{3/2}}=0.The critical energy bias \Delta _{c} and obit energy \mathcal {H}_{c} can
be obtained through Eq.() and Eq.(REF ), respectively\Delta _{c} &=&2US_{c}+\Omega \frac{1-3S_{c}}{2\sqrt{1-S_{c}}}, \\
\mathcal {H}_{c} &=&-2... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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69f0aadf96789ad2506391e4e421e477f01f176a | subsection | 11 | 16 | Adiabatic limit | For this purpose, we introduce a
small variable \delta =U/\Omega -\sqrt{2}/4, and calculate the critical
values with perturbation theory,S_{c} &=&-1+\frac{16\sqrt{2}}{3}\delta -\frac{256}{9}\delta ^{2}+\frac{5888\sqrt{2}}{81}\delta ^{3}, \\
\Delta _{c}/\Omega &=&\frac{\sqrt{2}}{2}-2\delta +\frac{16\sqrt{2}}{3}\delta ^{... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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e642454fc41c19ba68672a5fffeda6347ec30704 | subsection | 12 | 16 | Adiabatic limit | After integrating Eq.(REF ), we
finally come to a power law of the conversion efficiency,\chi \sim 1.2\lambda ^{2/3}=1.2\left( \frac{U}{\Omega }\right) ^{-2/3},
\frac{U}{\Omega }>>1.We would like to mention that, in the nonlinear Landau-Zener model
describing the tunneling dynamics of BEC atoms in a double-well
potenti... | {
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molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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bbd6792bd30d3e12535ae2a1e20888ba0493a125 | subsection | 13 | 16 | Sudden limit | The sudden limit corresponds to nonadiabatic conversion. The conversion
efficiency is not strongly related to the structure of the energy levels. In
this limit, we can derive the analytical expression of the conversion
efficiency using the stationary phase approximation (SPA). Because of the
large sweeping rate \alpha ... | {
"cite_spans": []
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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8e0e540715d9bcffb01f0c6bcc92b3b4205b0e4d | subsection | 14 | 16 | Sudden limit | The dominant contribution comes
from the stationary point t_{0} of the phase around which we have4\Delta +8U(4\left|b\right|^{2}-1)=\bar{\alpha }(t-t_{0}),with\bar{\alpha }=4\alpha +32U\left[ \frac{d\left|b\right|^{2}}{dt}\right] _{t_{0}}.Since \left|b\right|^{2}=|b^{^{\prime }}|^{2}, then we have\left|b\right|^{2}=\le... | {
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]
} | 10.1103/PhysRevA.79.025602 | 0807.1691 | The role of particle interactions in a many-body model of Feshbach
molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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87ddec6b36ddb18b146c69ef9dba87d7b4118f01 | subsection | 15 | 16 | Conclusion and discussion | In conclusion, we have both numerically and analytically investigated a
generalized many-body model that includes the atom-atom, atom-molecule, and
molecule-molecule background scattering interactions, emphasizing the
dynamics of Feshbach molecular formation. Compared to the simple two-body
molecular production picture... | {
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molecular formation in bosonic systems | [
"Jing Li",
"Di-Fa Ye",
"Chao Ma",
"Li-Bin Fu",
"Jie Liu"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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5e40aa1aa183327d7dbc9fc4efbaafca70f33dc1 | abstract | 0 | 14 | Abstract | In 0.9 fb$^{-1}$ of $p \bar p$ collisions, D0 has observed an excess of
events with an isolated lepton, missing transve rse momentum, and two to four
jets. This excess is consistent with single top quark production. We examine
these data to study the Lorentz structure of the $Wtb$ coupling. The standard
model predicts ... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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1e2f68f55c4fb0850b151c81afb5eb1b5befc4b5 | subsection | 1 | 14 | Body | Search for anomalous {Wtb} couplings in single top quark production
V.M. Abazov^{36}
B. Abbott^{75}
M. Abolins^{65}
B.S. Acharya^{29}
M. Adams^{51}
T. Adams^{49}
E. Aguilo^{6}
M. Ahsan^{59}
G.D. Alexeev^{36}
G. Alkhazov^{40}
A. Alton^{64,a}
G. Alverson^{63}
G.A. Alves^{2}
M. Anastasoaie^{35}
L.S. Ancu^{35}
T. Andeen^{5... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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... |
222c02bc1d36cbaf7df09ebc6e19b98cf1fdb853 | subsection | 2 | 14 | Body | Chan^{55}
A. Chandra^{48}
E. Cheu^{45}
F. Chevallier^{14}
D.K. Cho^{62}
S. Choi^{32}
B. Choudhary^{28}
L. Christofek^{77}
T. Christoudias^{43}
S. Cihangir^{50}
D. Claes^{67}
J. Clutter^{58}
M. Cooke^{50}
W.E. Cooper^{50}
M. Corcoran^{80}
F. Couderc^{18}
M.-C. Cousinou^{15}
S. Crépé-Renaudin^{14}
V. Cuplov^{59}
D. Cutts... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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ce81023476331cf10bf9f8aa043afd5688da29d7 | subsection | 3 | 14 | Body | Hadley^{61}
P. Haefner^{25}
S. Hagopian^{49}
J. Haley^{68}
I. Hall^{65}
R.E. Hall^{47}
L. Han^{7}
K. Harder^{44}
A. Harel^{71}
J.M. Hauptman^{57}
R. Hauser^{65}
J. Hays^{43}
T. Hebbeker^{21}
D. Hedin^{52}
J.G. Hegeman^{34}
A.P. Heinson^{48}
U. Heintz^{62}
C. Hensel^{22,d}
K. Herner^{72}
G. Hesketh^{63}
M.D. Hildreth^{5... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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ab175772d7f0c9292112f80e222f822b9a4dadec | subsection | 4 | 14 | Body | Maciel^{2}
D. Mackin^{80}
R.J. Madaras^{46}
P. Mättig^{26}
C. Magass^{21}
A. Magerkurth^{64}
P.K. Mal^{82}
H.B. Malbouisson^{3}
S. Malik^{67}
V.L. Malyshev^{36}
H.S. Mao^{50}
Y. Maravin^{59}
B. Martin^{14}
R. McCarthy^{72}
A. Melnitchouk^{66}
L. Mendoza^{8}
P.G. Mercadante^{5}
M. Merkin^{38}
K.W. Merritt^{50}
A. Meyer^... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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a2fa8f9b3810b20487f339b37c3afb888e2a6226 | subsection | 5 | 14 | Body | Rizatdinova^{76}
S. Robinson^{43}
R.F. Rodrigues^{3}
M. Rominsky^{75}
C. Royon^{18}
P. Rubinov^{50}
R. Ruchti^{55}
G. Safronov^{37}
G. Sajot^{14}
A. Sánchez-Hernández^{33}
M.P. Sanders^{17}
B. Sanghi^{50}
G. Savage^{50}
L. Sawyer^{60}
T. Scanlon^{43}
D. Schaile^{25}
R.D. Schamberger^{72}
Y. Scheglov^{40}
H. Schellman^{... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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e6ece710f851f6ea878bc0b05b6e1878a5ec6845 | subsection | 6 | 14 | Body | Wang^{50}
J. Warchol^{55}
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M. Wayne^{55}
G. Weber^{24}
M. Weber^{50}
L. Welty-Rieger^{54}
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W.-C. Yang^{44}... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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7f1ea15f342d3df5673121aedb6129b3767baef0 | subsection | 7 | 14 | Body | Physikalisches Institut A, RWTH Aachen University,
Aachen, Germany
^{22}Physikalisches Institut, Universität Bonn,
Bonn, Germany
^{23}Physikalisches Institut, Universität Freiburg,
Freiburg, Germany
^{24}Institut für Physik, Universität Mainz,
Mainz, Germany
^{25}Ludwig-Maximilians-Universität München,
München, Germany... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
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38394542c0393310c1fb9331964f51dc79598b83 | subsection | 8 | 14 | Body | Petersburg, Russia
^{41}Lund University, Lund, Sweden,
Royal Institute of Technology and
Stockholm University, Stockholm, Sweden, and
Uppsala University, Uppsala, Sweden
^{42}Lancaster University, Lancaster, United Kingdom
^{43}Imperial College, London, United Kingdom
^{44}University of Manchester, Manchester, United K... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
] | [
"hep-ex"
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5d472f75efe317bb8d3bb9fd8b988d960b2b2b68 | subsection | 9 | 14 | Body | The most general lowest dimension, CP-conserving Lagrangian admits right-handed vector and left- or right-handed tensor couplings as well. We find that the data prefer the left-handed vector coupling and set upper limits on the anomalous couplings. These are the first direct constraints on a general Wtb interaction and... | {
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"D0 Collaboration",
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c38eca1bbfe36cc219f3c2a2f293aa159295987b | subsection | 10 | 14 | Body | In this case the predicted cross section for single top quark production is 2.9\pm 0.3 pb .
[Figure: Feynman diagrams for (a) s-channeland (b) t-channel single top quark production.]The presence of anomalous couplings can change angular distributions and event kinematics as demonstrated by the p_T spectrum of the charg... | {
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"D0 Collaboration",
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57e8248605002d568064f820cb557cc2900774c5 | subsection | 11 | 14 | Body | The W+jets background is normalized such that the number of events predicted by the simulation agrees with the number of events observed in each analysis channel (defined by lepton flavor and jet multiplicity) before b tagging is applied.After all cuts we select 1,398 b tagged lepton+jets events, which we expect to con... | {
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production | [
"D0 Collaboration",
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c2ceeb502f470271374c706392fd3296521dc150 | subsection | 12 | 14 | Body | Figure REF shows representative output distributions for the data and the sum of SM signal and backgrounds for the electron channel with two jets and one b-tagged jet in each of the three anomalous coupling scenarios.
[Figure: Boosted decision tree output distributions for data and sum of SM signal and backgrounds for ... | {
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} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
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852934d57bf60d65f7284fa644e9299c92a19946 | subsection | 13 | 14 | Body | These are the first direct constraints on a general Wtb interaction and the first direct limits on left- and right-handed tensor couplings.
[Table: Measured values of the total cross section for single top production and one-dimensional limits on Wtb couplings in the three scenarios.]We thank the staffs at Fermilab and... | {
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} | 10.1103/PhysRevLett.101.221801 | 0807.1692 | Search for anomalous $\boldmath{Wtb}$ couplings in single top quark
production | [
"D0 Collaboration",
"V. M. Abazov"
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9ea9d7075f73d48b0e3bcb78823bca8bafdc772a | abstract | 0 | 10 | Abstract | Large volume cosmological simulations succeed in reproducing the large-scale
structure of the Universe. However, they lack resolution and may not take into
account all relevant physical processes to test if the detail properties of
galaxies can be explained by the CDM paradigm. On the other hand, galaxy-scale
simulatio... | {
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} | 10.1017/S1743921308027920 | 0807.1693 | Numerical simulations of galaxy evolution in cosmological context | [
"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
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480e6ac83993c8a41baca81fedfbea61483fb309 | subsection | 1 | 10 | Introduction | The morphology of galaxies in the Local Universe is well constrained by observations, but is still largely unexplained. Indeed, large volume cosmological simulations fail to reproduce realistic galaxies. For instance, the disks formed are often too concentrated : it is the “angular momentum problem”, well known since t... | {
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c47071af53709de7f6e945d7b3ef08ad3e14a0c3 | subsection | 2 | 10 | Analysis of the cosmological simulation | Merger histories and accretion data are extracted from a dark matter only cosmological simulation performed with the AMR code RAMSES (). This simulation has an effective resolution of 512^3 and a comoving box length of 20 h^{-1} Mpc. The mass resolution is 6.9\times 10^6 M_{\odot }, so that a Milky Way type halo is mad... | {
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"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
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799dd93c1a8ef6efd8ab77c6a3246095a179fdea | subsection | 3 | 10 | The PM code | The history that has been extracted from the cosmological simulation is re-simulated with a particle-mesh code ().
Gas dynamics is modeled with a sticky-particle scheme with \beta _r=0.8 and \beta _t=0.7, and star formation is computed according to a Kennicutt law with an exponent 1.5.The maximum spatial resolution is ... | {
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"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
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056119ef9beadddaa7c0139763628afa98c3c1d6 | subsection | 4 | 10 | Model galaxies | Each halo of the cosmological simulation (i.e. the main halo as well as all the interacting satellites) is replaced with a realistic galaxy, having a disk, a bulge and of course a dark matter halo. The total mass of the galaxy is divided in 20% of baryons and 80% of dark matter (the mass of dark matter being given by t... | {
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"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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f8faddb3b4a0a40b8c7c92b2d5433842bc9649d1 | subsection | 5 | 10 | Diffuse accretion | Each dark matter particle that is considered as diffuse accretion in the cosmological simulation is replaced with a small blob of particles, containing in mass 20% of gas and 80% of dark matter.The right side of figure REF shows an example of simulation where the main galaxy (edge-on) is surrounded by accreted gas (cle... | {
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} | 10.1017/S1743921308027920 | 0807.1693 | Numerical simulations of galaxy evolution in cosmological context | [
"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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262f2e5c2749ae0a7307a1c4ae48e9f7bf1dfa8b | subsection | 6 | 10 | Two examples | We present here the first results concerning two simulations, that have been chosen to have a mass at z=0 of the order of magnitude of the mass of the Milky Way. They have very different histories.In the the first one, the mass growth of the galaxy is dominated by diffuse accretion (at a mean rate of \sim 5 M_{\odot } ... | {
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} | 10.1017/S1743921308027920 | 0807.1693 | Numerical simulations of galaxy evolution in cosmological context | [
"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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1c2ac831ad9454faa43850cb2ddf358489a06d7b | subsection | 7 | 10 | The calm case | The evolution of the distribution of gas and stars is shown in figure REF . Gas is smoothly accreted around the galaxy and falls onto the disk. Minor mergers are not strong and frequent enough to destroy the stellar disk. They only slightly heat it, and a thin stellar disk is rebuilt thanks to gas from diffuse accretio... | {
"cite_spans": []
} | 10.1017/S1743921308027920 | 0807.1693 | Numerical simulations of galaxy evolution in cosmological context | [
"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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81a8bc345477f2a8680707d9c8db180a1181b81f | subsection | 8 | 10 | The violent case | In this case, the evolution of the morphology of the galaxy is totally different (see figure REF ). The disk is destroyed early by the first series of mergers. In fact, after the first of these mergers (which has a mass ratio of 8:1) the disk is already very perturbed, and the following mergers contribute to the transf... | {
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} | 10.1017/S1743921308027920 | 0807.1693 | Numerical simulations of galaxy evolution in cosmological context | [
"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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e734609450c3a5c569fbfa312e8d96f997005e1e | subsection | 9 | 10 | Conclusion | In order to study galaxy evolution in cosmological context, we have successfully developed a technique that allows us to perform high resolution simulations taking into account realistic merger and gas accretion histories.The first two simulations shown here do not allow us to draw any general conclusion on galaxy evol... | {
"cite_spans": []
} | 10.1017/S1743921308027920 | 0807.1693 | Numerical simulations of galaxy evolution in cosmological context | [
"Marie Martig",
"Frederic Bournaud",
"Romain Teyssier"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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b665907bc6470b483335487aa8de844e923d5387 | abstract | 0 | 80 | Abstract | We show that if K: P \to R is an autonomous Hamiltonian on a symplectic
manifold (P,\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along
a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then
the Hamiltonian flow of K has contractible periodic orbits with bounded period
on all su... | {
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} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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7468218397d3b40901fd0e0e32a231b2222e51b8 | subsection | 1 | 80 | Introduction | Consider a symplectic manifold (P,\Omega ) containing a closed, connected symplectic submanifold M, with 2m=\dim M, 2n=\dim P, and r=n-m. In recent years, there has been significant interest in the following question:Question 1.1 If K\colon \,P\rightarrow [0,\infty ) is a proper smooth function with K^{-1}(\lbrace 0\rb... | {
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"raw": "V. L. Ginzburg and B. Z. Gürel. Relative Hofer–Zehnder capacity and periodic orbits in twisted cotangent bundles. Duke Math. J. 123 (2004), no. 1, 1–47. MR2060021.",
"source_ref_id": "0166aad2... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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9153dda4fffdcf5dde54ad37db8d3e7f194bb1ca | subsection | 2 | 80 | Introduction | Of course, the case \sigma =0 just corresponds to the geodesic flow on M; accordingly the Hamiltonian flow of K on (T^*M,\Omega _{\sigma }) is sometimes called the \sigma -twisted geodesic flow. The search for periodic orbits of twisted geodesic flows was initiated by V. Arnol'd (for the case M=T^2) in , and has contin... | {
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{
"arxiv_id": "",
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"raw": "V. I. Arnol'd. On some problems in symplectic topology. In Topology and geometry–Rohlin Seminar. Lect. Notes Math. 1346, Springer, Berlin, 1988, 1–5. MR0970068.",
"source_ref_id": "def6c9309eb... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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c5f088167e9b2538c6b9e1816c8f56f64a358332 | subsection | 3 | 80 | Introduction | An important ingredient in the proof of the corresponding theorem in is a result (Propositions 3.1 and 3.2 of ) which, in certain situations which include the case where c_1(TP)|_{\nu M} is torsion, allows one to use grading information in Floer homology to bound the period of a periodic orbit. As such, in order to obt... | {
"cite_spans": [
{
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"raw": "V. L. Ginzburg and B. Z. Gürel. Periodic orbits of twisted geodesic flows and the Weinstein-Moser theorem. arXiv:0705.1818, to appear in Comment. Math. Helv.",
"source_ref_id": "9ea37cf5191a6d... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
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"math.SG",
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0ce123e84641d151dcd08a8c17592501a2d3168a | subsection | 4 | 80 | Introduction | We should point out that since we define X_K by \iota _{X_K}\omega =dK instead of \iota _{X_K}\omega =-dK as is done in , our Hamiltonian vector fields have periodic orbits which are related to those in by time reversal, as a result of which the Salamon–Zehnder indices \Delta ([\gamma ,w],K) of these orbits have opposi... | {
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"raw": "V. L. Ginzburg and B. Z. Gürel. Periodic orbits of twisted geodesic flows and the Weinstein-Moser theorem. arXiv:0705.1818, to appear in Comment. Math. Helv.",
"source_ref_id": "9ea37cf5191a6d... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
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] | 2,008 | en | Mathematics | [
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80fea5d2fc3e853d1a0da0d577e30ada79418bef | subsection | 5 | 80 | Introduction | The Arzelà-Ascoli theorem applied to the orbits so obtained from a sequence \epsilon _k\searrow 0 then shows that X_K has a contractible-in-\nu M periodic orbit of period at most T_0 in K^{-1}(\lbrace \rho \rbrace ).Now Theorems REF and REF depend only on the behavior of K in a (sufficiently small) tubular neighborhood... | {
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"raw": "D. McDuff and D. Salamon. Introduction to symplectic topology. Oxford Mathematical Monographs. Oxford University Press, Oxford, 1998. MR1698616.",
"source_ref_id": "2378df4359598566bc14c83d5ef... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
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0fda5109f101e2dd0f778a2d14f68262f005a905 | subsection | 6 | 80 | Introduction | (The main subtlety here is that the Hamiltonians can behave quite wildly very close to the zero section, so we need our constants to depend only on the behavior of the Hamiltonians away from the zero section.) This enables us to define Floer groups HF^{[a,b]}(H) in a fairly standard way. Similar constructions are carri... | {
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"source_ref_id": "9ea37cf5191a6d... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
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5a885bb916ca4371aae888980a9272270a9e705c | subsection | 7 | 80 | Introduction | For a judiciously chosen action interval [a,b] (equal to [c(\rho ),d(\rho )] in the notation of Section 4), in grading 2r one has, for i=0,1, CF_{2r}^{[a,b]}(F_i)=\mathbb {Z}_2\oplus N_i, where the \mathbb {Z}_2 is generated by a “fiberwise-capped” (in the terminology of Section 3) periodic orbit x_i, while the (typica... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
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525f688affd2a0132d31d95e31183bb530172bbb | subsection | 8 | 80 | Filtered Floer homology in symplectic disc bundles | As our input we take:a closed connected symplectic manifold (M,\omega _0) of (real) dimension 2m;
a Morse function h\colon \,M\rightarrow \mathbb {R} having just one local maximum (i.e., just one critical point of index 2m; Theorem 8.1 of shows how to construct such an h). In the situation of case (ii) of Theorem REF ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 275,
"openalex_id": "",
"raw": "J. Milnor. Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, NJ, 1965. MR0190942.",
"source_ref_id": "c052e5331b1cbc6c678e5eac13... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
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2e7bdc07b370c4f16ec1638fc7276ebd77c1951b | subsection | 9 | 80 | Filtered Floer homology in symplectic disc bundles | Thus provided that R^2\le \frac{1}{2}\Vert F_A\Vert _{\infty }^{-1}, \omega :=\pi ^{*}\omega _0+d\theta defines a symplectic form on E(R), which tames the almost complex structure \bar{J} on E obtained by lifting J_0 to T^{hor}E and using the complex vector bundle structure on T^{vt}E. Choose an R such that this is the... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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