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3c91ecf553b2bd370fbbbe732a2594883ebe8ed9 | subsection | 10 | 80 | Filtered Floer homology in symplectic disc bundles | The Floer boundary operator should enumerate solutions u\colon \,\mathbb {R}\times (\mathbb {R}/\mathbb {Z})\rightarrow E(R) to the equation\frac{\partial u}{\partial s}+J_t(u(s,t))\left(\frac{\partial u}{\partial t}-X_H(t,u(s,t))\right)=0which connect two generators of CF^{[a,b]}(H) (J_t is a t-parametrized family of ... | {
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468c423c1cbf872133ca6b03767d71283e8d4e7f | subsection | 11 | 80 | A radius-energy estimate | Choose a number \alpha \le R/4. The Hamiltonians that we consider will have the formH(t,x)=B+2\pi L(x)+f(t,x)+\delta h(\pi (x))where B is a constant; supp(f)\subset (\mathbb {R}/\mathbb {Z})\times E(\alpha ); \delta is a small positive number, which in particular should satisfy \Vert \delta h\circ \pi \Vert _{C^2}\le \... | {
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5ee474a867e40e7605dcf30669cbf0069f680fd1 | subsection | 12 | 80 | A radius-energy estimate | So, using the Schwarz inequality, we get&\left(\int _{0}^{1}|\dot{\gamma }(t)-X_H(\gamma (t))|^{2}dt\right)^{1/2}\ge \int _{0}^{1}|\dot{\gamma }(t)-X_H(\gamma (t))|dt\\&\ge \int _{0}^{1}|\dot{\gamma }(t)-X_{2\pi L}(\gamma (t))|-\int _{0}^{1}|X_{2\pi L}(\gamma (t))-X_H(\gamma (t))|dt\ge 7\alpha /4,since the hypothesis o... | {
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78f984a6d3b5171f7119eb68d7f66a4da872a893 | subsection | 13 | 80 | A radius-energy estimate | So (since |dL|\le \frac{1}{2}(2\alpha )=\alpha on E(\alpha ;2\alpha ))\frac{3\alpha ^{2}}{4}=\int _{t_0}^{t_1}dL(\dot{\gamma }(t))dt=\int _{t_0}^{t_1}dL(\dot{\gamma }(t)-X_H(\gamma (t)))dt \le \alpha \int _{t_0}^{t_1}|\dot{\gamma }(t)-X_H(\gamma (t))|dt.So the Schwarz inequality gives\int _{0}^{1}|\dot{\gamma }(t)-X_H(... | {
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cce0562196b07a5000ef6f15d770cd650fddab34 | subsection | 14 | 80 | A radius-energy estimate | ThenArea(S)+\int _{S}\left|\frac{\partial u}{\partial s}\right|^{2}_{J_t}dsdt\ge C\alpha ^2.Let \tilde{E}=[-1,2]\times (\mathbb {R}/\mathbb {Z})\times E(R), and define an almost complex structure \tilde{J} on \tilde{E} by, on each [-1,2]\times \lbrace t\rbrace \times E(R), setting \tilde{J}|_{TE(R)}=J_t, \tilde{J}\part... | {
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} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
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0f3ad91371d76ead0321d5fb46c7a9324e888a52 | subsection | 15 | 80 | A radius-energy estimate | So Proposition 4.3.1(ii) of gives a constant C^{\prime } such that \int _{S^{\prime }}\tilde{u}^{*}\tilde{\omega }\ge C^{\prime }\eta ^2\ge \frac{C^{\prime }}{16}\alpha ^2, from which the lemma immediately follows.Theorem 2.4 There are constants D and \alpha _0, depending only on J and the function \delta h\colon \,M\r... | {
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... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
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d34cd0f190cb0774cddca8c5276938ce1b1e0f9e | subsection | 16 | 80 | A radius-energy estimate | So (since (REF ) shows that \left|\frac{\partial u}{\partial s}\right|_{J_t}=\left|\frac{\partial u}{\partial t}-X_H(u(s,t))\right|_{J_t}) we have\int _{-\infty }^{\infty }\int _{0}^{1}\left|\frac{\partial u}{\partial s}\right|_{J_t}^{2}dsdt\ge \Vert J_t\Vert ^{-2}\int _{-\infty }^{\infty }\int _{0}^{1}\left|\frac{\par... | {
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c7484e51ab0165c10102230b96695af8e7ea46de | subsection | 17 | 80 | A radius-energy estimate | (For instance, a path in \mathbb {R}\times (\mathbb {R}/\mathbb {Z}) from \zeta ^{-1}(\lbrace 2\alpha +\epsilon \rbrace ) to \zeta ^{-1}(\lbrace 3\alpha -\epsilon ^{\prime }\rbrace ) can easily be seen to have a segment from \zeta ^{-1}(\lbrace 2\alpha +\epsilon \rbrace ) to \zeta ^{-1}(\lbrace 3\alpha -\epsilon ^{\pri... | {
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97309a058b9131faf951539bdea0985da0ff10d0 | subsection | 18 | 80 | A radius-energy estimate | Then Lemma REF gives that\int _{S}\left|\frac{\partial u}{\partial s}\right|_{J_t}^{2}dsdt\ge C\alpha ^2-Area(S).But since S\subset I\times (\mathbb {R}/\mathbb {Z})\subset \mathcal {Z}\times (\mathbb {R}/\mathbb {Z}) we have Area(S)\le C\alpha ^2/2, and so\int _{-\infty }^{\infty }\int _{0}^{1}\left|\frac{\partial u}{... | {
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bbc90e240dc318ab0444b79ee0f533fde135daa2 | subsection | 19 | 80 | Floer homology | With the above C^0-estimate established, the definition of our Floer groups becomes an application of standard machinery. Let \hbar be equal to one-half of the minimal energy of a nonconstant \bar{J}-holomorphic sphere in E(R); Gromov compactness of course implies that this is a positive number and that, if J_t is a t-... | {
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b1182840026273432a93ed7806230bfd6a320a45 | subsection | 20 | 80 | Floer homology | For any integer k, defineCF^{[a,b]}_{k}(H)=\left\lbrace \left.\sum _{i=1}^{l}a_i[\gamma _i,w_i]\right|l\in \mathbb {N}, a_i\in \mathbb {Z}_2, a\le \mathcal {A}_H([\gamma _i,w_i])\le b, \mu _{CZ}([\gamma _i,w_i])=k\right\rbraceandCF_{*}^{[a,b]}(H)=\bigoplus _{k\in \mathbb {Z}}CF^{[a,b]}_{k}(H).In the usual way, one then... | {
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} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
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af967adae908df8ab31219e1c6c8459a532a93f1 | subsection | 21 | 80 | Floer homology | (When [a,b] is understood from the context we will often write \partial _{H,J_t} for \partial _{H,J_t}^{[a,b]}.)We introduce the following standard definition:Definition 2.6 Let
H^- and H^+ be two nondegenerate Hamiltonians on E(R) such thatH^- and H^+ have the form (REF ),
H^-\le H^+ everywhere, and
(H^+-H^-)|_{E(\a... | {
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7b3f7de54ad26c2a452050d2276573814340de6e | subsection | 22 | 80 | Floer homology | In particular, since Energy(u)\ge 0 we have \mathcal {A}_{H^-}([\gamma _-,w_-])\ge \mathcal {A}_{H^+}([\gamma _+,w_+]), while since [\gamma _{\pm },w_{\pm }] are required to be generators of CF^{[a,b]}_{*}(H^{\pm }) and so have actions differing by at most b-a<D\alpha ^4<\hbar , we have Energy(u)<\min \lbrace \hbar ,D\... | {
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"Michael Usher"
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9978ecce61cc0ee1572c38f3d3b81d59f5e9de28 | subsection | 23 | 80 | Floer homology | The usual arguments (dating back to Theorem 4 of ) involving a homotopy of homotopies and the Floer gluing theorem show that, whenever (H^s,J_{s,t}) is a regular monotone homotopy from (H^-,J_{t}^{-}) to (H^+,J_{t}^{+}),\partial _{H^+,J_{t}^{+}}\circ \Phi _{H^s,J_{s,t}}=\Phi _{H^s,J_{s,t}}\circ \partial _{H^-,J_{t}^{-}... | {
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87b049d196d56bf47af4d454ba66e68891f88223 | subsection | 24 | 80 | Floer homology | Then, for all w\colon \,D^2\rightarrow E(R) such that \mathcal {A}_{H^-}([\gamma ,w])\in [a,b], we have\Phi _{H^{s},J_{s,t}}([\gamma ,w])=[\gamma ,w]+\sum c_{[\gamma ^{\prime },w^{\prime }]}[\gamma ^{\prime },w^{\prime }]where\mathcal {A}_{H^+}([\gamma ^{\prime },w^{\prime }])<\mathcal {A}_{H^+}([\gamma ,w])\mbox{ when... | {
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9c106fee557c57719374300265512a05df1ba3ef | subsection | 25 | 80 | Floer homology | Meanwhile, if u is any solution to (REF ) which is not of the form u(s,t)=\gamma (t), asymptotic say to [\gamma ^{\prime },w^{\prime }] as s\rightarrow \infty and to [\gamma ,w] as s\rightarrow -\infty , then we must have \frac{\partial u}{\partial s}(s,t)\ne 0 for some (s,t)\in \mathbb {R}\times (\mathbb {R}/\mathbb {... | {
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8979d377d07d55397db18481dce166fd6af56a44 | subsection | 26 | 80 | Special features of the Floer complexes of certain Hamiltonians | The proof of Theorem REF requires us to understand certain properties of the Floer complexes of Hamiltonians on E(R) having a particular form.As a first step, we prove the following elementary fact.Proposition 3.1 Let V be a C^1 vector field on a closed Riemannian manifold (M,g_0), 0\le \rho <1, and U\subset M an open ... | {
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b4d3dd878fbb65c350fb1954394d2e736cb8b415 | subsection | 27 | 80 | Special features of the Floer complexes of certain Hamiltonians | Where W\subset \mathbb {R}^{\dim M} is a neighborhood of the origin containing a ball of radius injrad(M,g_0), let \phi \colon \,W\rightarrow M be a normal coordinate chart with \phi (\vec{0})=x(0). Let B_0=\phi (W). Let V_0 denote the vector field on B_0 obtained by the parallel transport of V(x(0)) along geodesics in... | {
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9d197a02333041ebd1bd65525ca1067e4858e39a | subsection | 28 | 80 | Special features of the Floer complexes of certain Hamiltonians | \phi ^{-1}(x(t)) (0\le t\le 1) is thus a path contained entirely within a neighborhood W of the origin in \mathbb {R}^{\dim M} whose velocity vector has strictly positive inner product with the nonzero constant vector field \bar{V}_1 on W, and this precludes the possibility that x(1)=x(0), contrary to the hypothesis of... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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b5ba4f9f60cb50d412255bfabf5fb173d8863029 | subsection | 29 | 80 | Special features of the Floer complexes of certain Hamiltonians | Let Y_h denote the Hamiltonian vector field of h\colon \,M\rightarrow \mathbb {R} (using the symplectic form \omega _0 on M).We find, for w\in T^{hor}_{x}E,\omega _x(Y_{h}^{\#},w)=(\omega _0)_{\pi (x)}(Y_h,\pi _*w)+d\theta (Y_{h}^{\#},w)=d(h\circ \pi )_{x}(w)+d\theta (Y_{h}^{\#},w);thus\iota _{Y_{h}^{\#}-X_{h\circ \pi ... | {
"cite_spans": []
} | 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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800a86166a7cb081af50b555d5fcf6e504a8ff12 | subsection | 30 | 80 | Special features of the Floer complexes of certain Hamiltonians | This, together with the hypothesis on R (and the fact that \langle x(t),x(t)\rangle \le R^2), show that the above coefficient \sqrt{3} \Vert F_A(y(t))\Vert \langle x(t),x(t)\rangle is at most \sqrt{3}/{2}, and vanishes near the zeros of Y_h. Hence we may apply Proposition REF with \rho =\sqrt{3}/2 to deduce the result. | {
"cite_spans": []
} | 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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31fd20c3972cebe1d43f47c4ac266b7dd33119d4 | subsection | 31 | 80 | The perturbations and their periodic orbits | Suppose that H=f\circ L+\delta h\circ \pi is a \delta h-Hamiltonian, with \delta <\delta _0. We have just established that the 1-periodic orbits of X_H all lie in the fibers E_p for p some critical point of h\colon \,M\rightarrow \mathbb {R}. To specifically identify these orbits, note that our calculations have shown ... | {
"cite_spans": []
} | 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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941eb29ce706320fa86d5107a9731f4451c8e276 | subsection | 32 | 80 | The perturbations and their periodic orbits | Let (x_1+iy_1,\ldots ,x_r+iy_r) be fiberwise complex coordinates for E|_U, and define g_f\colon \,E(R)\rightarrow \mathbb {R} byg_f(u,x_1+iy_1,\ldots ,x_r+iy_r)=\chi (u)\beta _f(\sum (x_{j}^{2}+y_{j}^{2})/4)y_1for (u,x_1+iy_1,\ldots ,x_r+iy_r)\in E|_U\cong U\times \mathbb {C}^r and g_f|_{E(R)\setminus E|_U}=0.
Let \psi... | {
"cite_spans": []
} | 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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5c8ae1568348606ad4ba2a297cdba9239ff11183 | subsection | 33 | 80 | The perturbations and their periodic orbits | Now since the connection is trivial over V\subset M, we have \pi (\phi _H(z))=\eta _{\delta h}(\pi (z)) for z\in \pi ^{-1}(V), where \eta _{\delta h}\colon \,M\rightarrow M is the time-one map induced by the Hamiltonian \delta h on (M,\omega _0). So since (for \delta <\delta _0) the only fixed points of \eta _{\delta h... | {
"cite_spans": []
} | 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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1e119121d597271050e3240777b2c53787288422 | subsection | 34 | 80 | The perturbations and their periodic orbits | Since we assume that f^{\prime \prime }(\ell )\ne 0 for all \ell \in \mathcal {L}_f\setminus \lbrace 0\rbrace , for sufficiently small \epsilon (REF ) has precisely two solutions y_{+}^{\ell } and y_{-}^{\ell } with L close to \ell , of which the former has y_1<0 and f^{\prime }(\frac{\epsilon ^2+4y_{1}^{2}}{16}) sligh... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
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] | 2,008 | en | Mathematics | [
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858aa3a4e869749313ee1b56f2a6c3d46b724cef | subsection | 35 | 80 | The perturbations and their periodic orbits | Recall that the generators of the Floer complex of H^{\epsilon } are equivalence classes [\gamma _{j,0},w] or [\gamma _{j,l}^{\pm },w] where w\colon \,D^2\rightarrow E(R) is a nullhomotopy (or a “capping”) of the orbit \gamma _{j,0} or \gamma _{j,l}^{\pm }. The fact that the orbits are all contained in single fibers al... | {
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{
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"doi": "",
"end": 1367,
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"raw": "D. Salamon. Lectures on Floer homology. In Symplectic geometry and topology (Park City, Utah, 1997). AMS, Providence, 1999, 143–229. MR1702944.",
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097685eab28cb14323568e9536b6959379554216 | subsection | 36 | 80 | The perturbations and their periodic orbits | Then the Maslov indices for the periodic orbits of H^{\epsilon } with their fiberwise cappings w_0 are, for \epsilon sufficiently small, given by\mu _{H^{\epsilon }}([\gamma _{j,0},w_0])=2n-ind_{h}p_j;
For \ell \in \mathcal {L}_f\setminus \lbrace 0\rbrace ,
\mu _{H^{\epsilon }}([\gamma _{j,\ell }^{+},w_0])=\left\lbra... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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ba1ab5e6bdbdf3dcd5bea25608a392a381fac141 | subsection | 37 | 80 | Restrictions on Floer trajectories | We will be needing some information about the Floer complexes CF^{[a,b]}(H^{\epsilon }) of perturbations H^{\epsilon } of particular \delta h-Hamiltonians H=f\circ L+\delta h\circ \pi ; in our application the length of the interval [a,b] will be rather small. Lemma REF below will be a considerable help in this directio... | {
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{
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"doi": "",
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"raw": "D. McDuff and D. Salamon. J-holomorphic curves and symplectic topology. AMS Colloquium Publications 52, AMS, Providence, 2004. MR2045629.",
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] | [
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b41de0e41f67f06dfdbfc081117e997bdb24c048 | subsection | 38 | 80 | Restrictions on Floer trajectories | If the result were false, there would be a sequence w_n\colon \,(-1,2)\times \mathbb {R}/\mathbb {Z}\rightarrow M with
\left\Vert \frac{\partial w_n}{\partial s}+J_0\frac{\partial w_n}{\partial t}\right\Vert _{C^0}\rightarrow 0 such that \int _{(-1,2)\times (\mathbb {R}/\mathbb {Z})}\left|\frac{\partial w_n}{\partial s... | {
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"raw": "S. Ivashkovich and V. Shevchishin. Gromov compactness theorem for J-complex curves with boundary. Internat. Math. Res. Notices 2000, no. 22, 1167–1206. MR1807156.",
"source_ref_id": "0ec3fe33... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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3278084fe5f07935bb75ec1e812a1da79d198656 | subsection | 39 | 80 | Restrictions on Floer trajectories | Now the fact that the W_n have form (s,t)\mapsto (s,t,w_n(s,t)) implies that the limiting bubble tree has a principal component of form (s,t)\mapsto (s,t,w_{\infty }(s,t)) (having energy, as measured by the symplectic form ds\wedge dt+\omega _0 and the almost complex structure i\times J_0 on (-1,2)\times (\mathbb {R}/\... | {
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"source_ref_id": "0ec3fe33... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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7def1096894fb4c36bbe58f6496988457240ef38 | subsection | 40 | 80 | Restrictions on Floer trajectories | Let (H^s,\bar{J}) be a monotone homotopy from (H^-,\bar{J}) to (H^{+},\bar{J}), such that there are functions f_s\colon \,\mathbb {R}\rightarrow \mathbb {R} and \rho _s\colon \,(\mathbb {R}/\mathbb {Z})\times E(R)\rightarrow \mathbb {R}, with\Vert \rho _s\Vert _{C^1}<\eta _1 \mbox{ and } H^s(t,x)=f_s(L(x))+\rho _s(t,x)... | {
"cite_spans": []
} | 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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3dc436dea3a126f7a2051bb1824f8c8d998dc7a6 | subsection | 41 | 80 | Restrictions on Floer trajectories | Then\mbox{ either }A=0\mbox{ or }\int _{\mathbb {R}\times (\mathbb {R}/\mathbb {Z})}\left|\frac{\partial u}{\partial s}\right|^2>e_1.Remark 3.9 We emphasize that the constants \eta _1,e_1 are independent of the functions f_s.Applying the linearization \pi _* of the bundle map \pi \colon \,E(R)\rightarrow M to (REF ) an... | {
"cite_spans": []
} | 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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164ffcfe3b6d0544385f115e73524264cde6772f | subsection | 42 | 80 | Restrictions on Floer trajectories | So identifying S^2 with\frac{[-\infty ,\infty ]\times (\mathbb {R}/\mathbb {Z})}{(\pm \infty ,t)\sim (\pm \infty ,0)},\pi \circ u extends to a map \overline{\pi \circ u}\colon \,S^2\rightarrow M; this sphere is the image under \pi of the sphere obtained by attaching the fiberwise cappings of \gamma _{\pm } to the image... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
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] | 2,008 | en | Mathematics | [
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41b178cca064e10a53c2ebda1621d92c8ec4eb13 | subsection | 43 | 80 | Restrictions on Floer trajectories | Suppose also that a,b\in \mathbb {R} witha<b<a+e_1.ThenFor generic families of almost complex structures J_t sufficiently close to \bar{J}, the boundary operator \partial _{H,J_t}\colon \,CF^{[a,b]}_{*}(H^{\epsilon })\rightarrow CF^{[a,b]}_{*}(H^{\epsilon }) has the form
\partial _{H,J_t}[\gamma ,(w_0)_{\gamma }\#A]=\... | {
"cite_spans": []
} | 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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da17216c1c6e402f5c7f8390bffc2a61f869a530 | subsection | 44 | 80 | Restrictions on Floer trajectories | Since our restriction to the action window [a,b] forces the u being considered to have energy less than e_1, this is essentially an immediate consequence of Lemma REF (with the H^s of Lemma REF set equal to H^{\epsilon } independently of s for the first part and to H^{s,\epsilon } for the second part), except that Lemm... | {
"cite_spans": []
} | 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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92accd940e14f4bd7b87bea93464b07f506cea85 | subsection | 45 | 80 | Detecting periodic orbits | We now turn to the proof of Theorem REF . Throughout this section, we assume that either c_1(TE(R))=0 mod torsion or else that h has no critical points of index 2m-1. Our strategy is to some extent modeled on that used in Section 6 of to prove the result in the spherically rational case, though of course the possible i... | {
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{
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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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a2fee84d33e4187242f43f8c684a431e2c01e0c6 | subsection | 46 | 80 | Detecting periodic orbits | One property that the still-to-be-specified parameter \eta will have is that X_K has no nonconstant periodic orbits of period at most 2\eta (such an \eta , depending only on the C^2-norm of K, exists by the Yorke estimate ), so this means that any nonconstant 1-periodic orbit of X_{H_0} will correspond to a periodic or... | {
"cite_spans": [
{
"arxiv_id": "",
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"raw": "J. Yorke. Periods of periodic solutions and the Lipschitz constant. Proc. Amer. Math. Soc. 22, no. 2 (1969), 509–512. MR0245916.",
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804f54f7926cd51b1c6fe26845e7b77847aaa399 | subsection | 47 | 80 | Detecting periodic orbits | The horizontal axis, which is not to scale, is labeled in units of \rho .]The defining properties of the constants C_1 and C_2, together with the above properties of f_a,f_b, and g_{\rho ,\epsilon }, imply in particular thatf_b(L(x))\le g_{\rho ,\epsilon }(K(x))\le f_a(L(x))\mbox{ when }L(x)\le 10\rho /C_1.Choose a smo... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
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] | 2,008 | en | Mathematics | [
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0b6444143386ae7e237ca22c763647b661917187 | subsection | 48 | 80 | Detecting periodic orbits | Consider perturbations F_{b}^{\epsilon },F_{a}^{\epsilon } of F_a,F_b as constructed in the previous section, for sufficiently small \epsilon . Since F_{b}^{\epsilon } differs from F_b only where 4\rho /C_1\le L(x)\le 4\rho /C_1+\rho /C_2, and likewise F_{a}^{\epsilon } differs from F_a only where \rho /C_2\le L(x)\le ... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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] | 2,008 | en | Mathematics | [
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c1461a2ba297767e36d0716774893027b64699d4 | subsection | 49 | 80 | Detecting periodic orbits | Also (for sufficiently small \delta ), all 1-periodic orbits of H are nondegenerate except for possibly those contained in the region where 3\rho -\beta \le K\le 3\rho +\beta (which in turn is contained in the region where \frac{5\rho }{2C_2}\le L\le \frac{7\rho }{2C_1}), so there are arbitrarily small nondegenerate pe... | {
"cite_spans": []
} | 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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a24c924863fd0b10779b644335fbe3c94f9000e0 | subsection | 50 | 80 | Detecting periodic orbits | Adapting the notation of the previous section, F_{a}^{\epsilon } and F_{b}^{\epsilon } then have periodic orbits \gamma _{1,\ell _a}^{-}, \gamma _{1,\ell _b}^{-} lying in the fiber over p_1. Proposition REF then gives that, where as usual w_0 denotes the fiberwise capping,\mu _{F_{a}^{\epsilon }}([\gamma _{1,\ell _a}^{... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
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7b7fa668adbb27dedfa0ab18e22c3beda815309a | subsection | 51 | 80 | Detecting periodic orbits | The main result of this section is:Proposition 4.1 There is a 1-periodic orbit \gamma of X_H which is contained in the region \lbrace 3\rho -\beta \le K\le 3\rho +\beta \rbrace and a map w\colon \,D^2\rightarrow E(R) with w|_{\partial D^2}=\gamma such that the Salamon–Zehnder index \Delta ([\gamma ,w],H) satisfies -2r\... | {
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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": "9ea37cf5191a6... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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] | 2,008 | en | Mathematics | [
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44f7dc7c1dea897256ce9fb836e5d4c5ef5e453c | subsection | 52 | 80 | Detecting periodic orbits | In fact, if J_t is a generic path of almost complex structures sufficiently close to \bar{J}, then we have\partial _{F_{a}^{\epsilon },J_t} [\gamma _{1,\ell _a}^{-},w_0]=\partial _{F_{b}^{\epsilon },J_t} [\gamma _{1,\ell _b}^{-},w_0]=0and, for every generator [\gamma ,w] of CF^{[c(\rho ),d(\rho )]}_{2r+1}(F_{a}^{\epsil... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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] | 2,008 | en | Mathematics | [
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6ade0ac812bd644051457e9357c98bc8df545661 | subsection | 53 | 80 | Proof of Proposition | We prove the result for the generator [\gamma _{1,\ell _a},w_0] of CF^{[c(\rho ),d(\rho )]}(F_{a}^{\epsilon }); the proof when `a' is replaced by `b' is identical. (The proof will also apply equally well to the Hamiltonians F_{s}^{\epsilon } introduced in the proof of Proposition REF .)Where \partial =\partial _{F_{a}^... | {
"cite_spans": []
} | 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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81c3e8028cae9234b2966b65ea9eb2490c7f1e74 | subsection | 54 | 80 | Proof of Proposition | Also\mathcal {A}_{F_{a}^{\epsilon }}([\gamma _{1,\ell ^{\prime }_a}^{-},w_0])=4\pi \ell ^{\prime }_a-f_a(\ell ^{\prime }_a)+O(\delta ,\epsilon )>0since the construction of f_a ensures that f^{\prime }_a(\ell )<4\pi for \ell <\ell ^{\prime }_a, so that
\mathcal {A}_{F_{a}^{\epsilon }}([\gamma _{1,\ell ^{\prime }_a}^{-}... | {
"cite_spans": []
} | 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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87105849dbce1c99b0d756e37facea22c17f682a | subsection | 55 | 80 | Proof of Proposition | We then see that any of the [\gamma ,w_0] has action differing from that of [\gamma _{1,\ell _a}^{-},w_0] by an amount tending to zero with \delta and \epsilon , so that any cylinder u\colon \,\mathbb {R}\times (\mathbb {R}/\mathbb {Z})\rightarrow E(R) contributing to the matrix element\langle \partial [\gamma ,w_0],[\... | {
"cite_spans": []
} | 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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939b39a526b393ad0cc031f437aa4c60915dfd83 | subsection | 56 | 80 | Proof of Proposition | So in order for HF^{loc}_{*}(f_a\circ L,\mathcal {U}) to have \mathbb {Z}_2-summands in gradings separated by \dim N, it must be that some element of CF^{loc}_{*}(F_{a}^{\epsilon },f_a\circ L,\mathcal {U}) having the same relative grading as \gamma _{1,\ell _a}^{-} represents a nontrivial class in HF^{loc}_{*}(f_a\circ... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
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] | 2,008 | en | Mathematics | [
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7c298074e7434992bd0ecf66e7cc80a3d83b0392 | subsection | 57 | 80 | Proof of Proposition | Hence their periodic orbits also coincide, while for any [\gamma ,w]\in \tilde{\mathcal {L}} we have \mathcal {A}_{G+\mu }([\gamma ,w])=\mathcal {A}_G([\gamma ,w])-\mu . Assume that G has the form (REF ) and that a<b with b-a small enough as to allow for the construction of CF_{*}^{[a,b]}(G) as in Section 2.2. Then the... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
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] | 2,008 | en | Mathematics | [
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00b016505892a9044a4f4ba50e23e37e9d824759 | subsection | 58 | 80 | Proof of Proposition | Then, with respect to the identification (REF ), the map\Phi _{H^{s}_{2},J^{\prime }_{s,t}}\circ \Phi _{H^{s}_{1},J_{s,t}}\colon \,CF_{*}^{[a,b]}(G)\rightarrow CF_{*}^{[a+\mu ,b+\mu ]}(G)is chain homotopic to the homomorphism\Pi _{[a+\mu ,b]}^{[a,b]}\colon \,CF_{*}^{[a,b]}(G)\rightarrow CF_{*}^{[a+\mu ,b+\mu ]}(G)defin... | {
"cite_spans": []
} | 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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2b7d1b95e9ab35d84c7dbe3d6a90a7ebb246d6b9 | subsection | 59 | 80 | Proof of Proposition | Since the Hamiltonian vector field of G+\mu \chi (s) is independent of s, the map \Phi _{G+\mu \chi (s),J_t} induced by this monotone homotopy counts index 0 solutions u\colon \,\mathbb {R}\times (\mathbb {R}/\mathbb {Z})\rightarrow E(R) to the usual Floer equation \frac{\partial u}{\partial s}+J_t\left(\frac{\partial ... | {
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{
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"raw": "D. Salamon and E. Zehnder. Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure. Appl. Math. 45 (1992), no. 10, 1303–1360. MR1181727.",
"source_ref_id":... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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f3814298dce612550a587632c3da4ef62789b10d | subsection | 60 | 80 | Proof of Proposition | Let \chi \colon \,\mathbb {R}\rightarrow [0,1] be a smooth, monotone increasing function with \chi (s)=0 for s\le -1 and \chi (s)=1 for s\ge 1, and define \phi _s=\chi (s)\phi _a+(1-\chi (s))\phi _b; for any s this is a strictly increasing function from [0,\infty ) onto itself, so let f_s be the inverse of \phi _s. We ... | {
"cite_spans": []
} | 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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3e073b5f8b8c6e5321a5828c436d63abfb5fb14d | subsection | 61 | 80 | Proof of Proposition | Therefore, for each s and for any J^{\prime }_t\in \mathcal {J}^{reg}(F^{\epsilon }_{s}), we have \partial _{F^{\epsilon }_{s},J^{\prime }_t}^{[c(\rho ),d(\rho )]}[\gamma _{1,\ell (s)}^{-},w_0]=0 and, for any [\gamma ^{\prime },w^{\prime }]\in CF^{[c(\rho ),d(\rho )]}_{2r+1}(F^{\epsilon }_{s}),\langle \partial _{F^{\ep... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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] | 2,008 | en | Mathematics | [
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cf6b4a3729c749fc250bffc6f6c0fbb4f46a1289 | subsection | 62 | 80 | Proof of Proposition | We then obtain chain maps\Phi _{G_{j,1}^{s},J_{j,s,t}}\colon \,CF^{[c(\rho ),d(\rho )]}_{*}(F_{s_j}^{\epsilon })\rightarrow CF^{[c(\rho ),d(\rho )]}_{*}(F_{s_{j+1}}^{\epsilon }),\Phi _{G_{j,2}^{s},J^{\prime }_{j,s,t}}\colon \,CF^{[c(\rho ),d(\rho )]}_{*}(F_{s_{j+1}}^{\epsilon })\rightarrow CF^{[c(\rho ),d(\rho )]}_{*}(... | {
"cite_spans": []
} | 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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2501be64a7b41ca2dfe665039e5c4d0c1a0b5cae | subsection | 63 | 80 | Proof of Proposition | Thus (REF ) shows that\langle \Phi _{G_{j,2}^{s},J_t}\circ \Phi _{G_{j,1}^{s},J_t}([\gamma _{1,\ell (s_j)}^{-},w_0]),[\gamma _{1,\ell (s_j)}^{-},w_0]\rangle =1.But since for i=j,j+1, \gamma _{1,\ell (s_i)}^{-} is the only 1-periodic orbit \gamma of X_{F_{s_i}^{\epsilon }} having both \mu _{F_{s_i}^{\epsilon }}([\gamma ... | {
"cite_spans": []
} | 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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f28c78989a2709e15f2d16f3f755b9c0fb09f381 | subsection | 64 | 80 | Proof of Proposition | From this it follows that, for some c_j, c^{\prime }_j with c_jc^{\prime }_j=1, we have\Phi _{G_{j,1}^{s},J_t}[\gamma _{1,\ell (s_j)},w_0]=c_j[\gamma _{1,\ell (s_{j+1})},w_0], \quad \Phi _{G_{j,2}^{s},J_t}[\gamma _{1,\ell (s_{j+1})},w_0]=c^{\prime }_j[\gamma _{1,\ell (s_{j})},w_0].Since we are working over \mathbb {Z}_... | {
"cite_spans": []
} | 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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614bd43fe42d33166f8751d33815669ac3ec9a01 | subsection | 65 | 80 | Proof of Proposition | Since [\gamma _{1,\ell _b}^{-},w_0] is a cycle and
[\gamma _{1,\ell _a}^{-},w_0] is homologically nontrivial this is possible only if x=1, completing the proof of the proposition.The foregoing implies that the image of \Psi _{F_{b}^{\epsilon }}^{H^{\prime }}\colon \,HF^{[c(\rho ),d(\rho )]}_{2r}(F^{\epsilon }_{b})\righ... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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129a4cbcdda93e17882ebc8caa3e837fab5e00b8 | subsection | 66 | 80 | Proof of Proposition | Take for H^{\prime } a sufficiently small nondegenerate perturbation of H, with H^{\prime }-H supported in \lbrace 3\rho -\beta \le K\le 3\rho +\beta \rbrace (since the construction of H shows that its constant 1-periodic orbits are all nondegenerate, while all of its nonconstant 1-periodic orbits are (for small \delta... | {
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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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11237902a1c359ba344e000309346c52a6f6f437 | subsection | 67 | 80 | Proof of Proposition | In particular, in degrees *=2r-1,2r, we have CF_{*}^{[c(r),d(r)]}(H^{\prime })=CF_{*}^{[c(r),d(r)]}(\bar{F}) as \mathbb {Z}_2-modules.We have \bar{F}\le F_{b}^{\epsilon }\le H^{\prime }, so for generic J_t close to \bar{J} let (H^{s}_{1},J_{s,t}) be a regular monotone homotopy from (\bar{F},J_t) to (F_{b}^{\epsilon },J... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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bcf11e38b83cd7e4ceb5cd548cbb0280ae9da715 | subsection | 68 | 80 | Proof of Proposition | But \oplus _{k=2r-1}^{2r}CF_{k}^{[c(r),d(r)]}(H^{\prime }) and \oplus _{k=2r-1}^{2r}CF_{k}^{[c(r),d(r)]}(\bar{F}) are both just equal to the span over \mathbb {Z}_2 of the various generators [\gamma _{j,0},w] having appropriate action and grading. Therefore, Proposition REF shows that
the map\Phi _{H^{s}_{2},J^{\prime ... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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6017db41b71b13c62d240ca26fe06154d91233a0 | subsection | 69 | 80 | Proof of Proposition | Note that if (G^s,\bar{J}_{s,t}) is a regular monotone homotopy from (G^-,J^{-}_{t}) to (G^+,J^{+}_{t}), we have \mathcal {L}_{G^+}(\Phi _{G^s,\bar{J}_{s,t}}(x))\le \mathcal {L}_{G^-}(x) for all x\in CF_{*}^{[c(\rho ),d(\rho )]}(G^-). Note also that the map A=\sum _{j=0}^{\infty }(-D)^j of the previous paragraph satisf... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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706c0eac0a74fec508622e25eccc37a9d54dd5ee | subsection | 70 | 80 | Proof of Proposition | In view of this, (REF ) implies that\mathcal {L}_{H^{\prime }}(c)<\mathcal {L}_{F_{b}^{\epsilon }}([\gamma _{1,\ell _b}^{-},w_0]).Hence also\mathcal {L}_{F_{b}^{\epsilon }}(c^{\prime })<\mathcal {L}_{F_{b}^{\epsilon }}([\gamma _{1,\ell _b}^{-},w_0]).But then since [\gamma _{1,\ell _b}^{-},w_0] is the only generator of ... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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3c1979dcb027fc9e76995eb8df6f8e10614e0160 | subsection | 71 | 80 | Proof of Proposition | Now if the chain on the right hand side above were nullhomologous, there would need to be a nonzero matrix element \langle \partial _{F_{a}^{\epsilon },J_t}[\gamma ,w], [\gamma _{1,\ell _a}^{-},w_0]\rangle , whereas we have shown in Proposition REF that all such matrix elements are zero. So \Phi _{F_{s}^{\epsilon },\ba... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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546ab4b19b606d3d3edc78790b03307b7cea1d64 | subsection | 72 | 80 | Appendix: Background on local Floer homology for clean intersections | Let (P,\Omega ) be an arbitrary symplectic manifold, and suppose that H_0\colon \,P\rightarrow \mathbb {R} is an autonomous Hamiltonian on P inducing a flow \phi _{H_0}^{t} which has the property that fixed point set of \phi _{H_0}^{1} has a connected component N\subset Fix(\phi _{H_0}) such thatN is a compact submanif... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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838e02abe3ba56cfc17c8e6aca9dbdcb7fa0dd43 | subsection | 73 | 80 | Appendix: Background on local Floer homology for clean intersections | 581], for a (\mathbb {R}/\mathbb {Z})-parametrized family of \Omega -tame almost complex structures J_t and a Hamiltonian H\colon \,(\mathbb {R}/\mathbb {Z})\times P\rightarrow \mathbb {R} define\mathcal {M}_{J_t,H}(\mathcal {U})=\left\lbrace u\colon \,\mathbb {R}\times S^1\rightarrow P\left|\begin{array}{l}\int _{\mat... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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187c484bff9487c0e8258ebe77225425eccc4334 | subsection | 74 | 80 | Appendix: Background on local Floer homology for clean intersections | The extended map \bar{u}\colon \,[-\infty ,\infty ]\times (\mathbb {R}/\mathbb {Z})\rightarrow U is then homotopic rel boundary to \tau \circ \bar{u}\colon \,[-\infty ,\infty ]\times (\mathbb {R}/\mathbb {Z})\rightarrow N, and our definition of \mathcal {U} then ensures that, using the exponential map of the metric on ... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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2f2c920753faab9571533b778e50e0e21b4a0562 | subsection | 75 | 80 | Appendix: Background on local Floer homology for clean intersections | But we have \frac{\partial \bar{v}}{\partial t}=X_{H_0} everywhere, and the image of \bar{v} is contained in N, on which H_0 is constant, so that\Omega \left(\frac{\partial \bar{v}}{\partial s},\frac{\partial \bar{v}}{\partial t}\right)=-dH_0\left(\frac{\partial \bar{v}}{\partial s}\right)=0,and thus\int _{\mathbb {R}\... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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c0a85d52cd588e09b0ff97b05bb634c1a33ae686 | subsection | 76 | 80 | Appendix: Background on local Floer homology for clean intersections | Since the only such orbits belonging to \mathcal {U} are the elements of \mathcal {N}, this proves the proposition.If \epsilon >0, define\mathcal {M}_{J_t,H}^{\epsilon }(\mathcal {U})=\lbrace u\in \mathcal {M}_{J_t,H}(\mathcal {U})|\int _{\mathbb {R}\times S^1}\left|\frac{\partial u}{\partial s}\right|^{2}_{J_t}<\epsil... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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b8cccbf90731833c306dc728afc7e36631abbc2a | subsection | 77 | 80 | Appendix: Background on local Floer homology for clean intersections | Gromov compactness applied to the maps (s,t)\mapsto u_n(s-s_n,t) then implies that the u_n(s_n,\cdot ) converge to some element of \mathcal {S}_{J_t,H_0}(\mathcal {U}) (bubbling is precluded because the u_n have energy less than the minimal energy of any J_t-holomorphic sphere in U). But we have shown that \mathcal {S}... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
"math.SG",
"math.DG",
"math.DS"
] | 2,008 | en | Mathematics | [
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fbe4d048597220fc80357c4504de906f44d18997 | subsection | 78 | 80 | Appendix: Background on local Floer homology for clean intersections | Hence\left| \int _{\mathbb {R}\times (\mathbb {R}/\mathbb {Z})}u^*\Omega \right|\le \left|\int _{[0,1]\times (\mathbb {R}/\mathbb {Z})}u_{\gamma _-}^{*}\Omega \right|+
\left|\int _{[0,1]\times (\mathbb {R}/\mathbb {Z})}u_{\gamma _-}^{*}\Omega \right|<\delta /2whenever u\in \mathcal {M}_{J^{\prime }_t,H}^{\epsilon }(\ma... | {
"cite_spans": []
} | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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"math.DG",
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] | 2,008 | en | Mathematics | [
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08f7f15f6a8e2af2b1ab139e4a61bde793448ed1 | subsection | 79 | 80 | Appendix: Background on local Floer homology for clean intersections | The boundary operator \partial ^{loc}_{H} may be a priori defined to count elements of \mathcal {M}^{2\delta }_{J^{\prime }_t,H}(\mathcal {U}) for generic J^{\prime }_t in the usual fashion; the fact that any such element in fact belongs to \mathcal {M}^{\delta }_{J^{\prime }_t,H}(\mathcal {U}) implies that one has (\p... | {
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"raw": "M. Poźniak. Floer homology, Novikov rings, and clean intersections. In Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl., Ser. 2, 196, AMS, Providence, 1999, 119–181. MR1736... | 0807.1694 | Floer homology in disc bundles and symplectically twisted geodesic flows | [
"Michael Usher"
] | [
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] | 2,008 | en | Mathematics | [
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12aa578f8dff4fc2c1cbe8fc205ebb94501bbf13 | abstract | 0 | 38 | Abstract | The coupling of spin 0 and spin 1 external fields to Dirac fermions defines a
theory which displays gauge chiral symmetry. Quantum mechanically, functional
integration of the fermions yields the determinant of the Dirac operator, known
as the chiral determinant. Its modulus is chiral invariant but not so its
phase, whi... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
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535763d1f4144a726fa7d6a767bbb2d71f36219e | subsection | 1 | 38 | Introduction | In this work we consider even dimensional Dirac fermions which move in
the presence of external fields of the type scalar, pseudo-scalar,
vector and axial-vector and with general non abelian internal degrees
of freedom. At the classical level the theory is invariant under
chiral gauge transformations. Quantum-mechanica... | {
"cite_spans": [
{
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"doi": "",
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"raw": "R. D. Ball, Phys. Rept. 182, 1 (1989).",
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... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
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] | 2,008 | en | Physics | [
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a2ce34ef678610e23528d44be5e0bcab0f971eb8 | subsection | 2 | 38 | Introduction | This is the case of
regularizations such as the heat kernel of the squared Dirac operator,
{\bf D}^2 (a Klein-Gordon like operator) or direct
\zeta -function regularization of the {\bf D}
, or {\bf D}^2. This is
because chiral symmetry does not act as a similarity transformation on
the Dirac operator, and so powers of ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 205,
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"raw": "A. Dhar, R. Shankar and S. R. Wadia, Phys. Rev. D31, 3256 (1985).",
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"doi"... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
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e1e1ac6930420dc5c26c1b9721150e08f074df1a | subsection | 3 | 38 | The chiral determinant | In this section we summarize some theoretical results in
the literature regarding chiral fermions and their effective action. | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
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] | 2,008 | en | Physics | [
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5e5ae744b4f27a884f8eca32ca438e1fedc798fb | subsection | 4 | 38 | The Dirac operator | We consider a Dirac operator {\bf D} describing Dirac fermions
coupled to spin 0 and spin 1 external fields with non abelian degrees
of freedom:{\bf D}= \gamma _\mu (\partial _\mu +{\cal V}_\mu )
+\gamma _\mu \gamma _5 {\cal A}_\mu +{\cal S}
+\gamma _5 {\cal P} .The fermions live in a d-dimensional Euclidean space-time... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
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a578c550a1e74d5f0606c537256ea7caab4f72bc | subsection | 5 | 38 | The effective action | The fermionic effective action W is introduced through standard
functional integration of the fermionic fieldse^{-W}=\int {\cal D}\bar{\psi }{\cal D}\psi \,e^{-\int d^dx \,\bar{\psi }{\bf D}\psi }
= {\rm Det\,}{\bf D}so formallyW= -{\rm Tr}\,\log {\bf D}modulo ultraviolet (UV) ambiguities. {\rm Tr} denotes the function... | {
"cite_spans": [
{
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"raw": "J. Bijnens and J. Prades, Phys. Lett. B320, 130 (1994), [hep-ph/9310355].",
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0... |
a95fd7c7126368fb2651fa4f569065990a43fcff | subsection | 6 | 38 | The effective action | Because s=0 is a regular
point, the effective action can be defined asW({\bf D})=-\frac{d}{ds}{\rm Tr}({\bf D}^s) \Big |_{s=0} \,.Note that no further renormalization is needed, as the right-hand side
is already UV finite. An interesting property of this renormalization
is that it only depends on the spectrum of {\bf D... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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a8770ad47cf0f0edb2f483eeec1ab60a4341fb5d | subsection | 7 | 38 | The effective action | So in W^- the leading order (LO) is the only term
affected by UV ambiguities. | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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55061a0ebcac3cf94bcd9b13281de7ce374522ee | subsection | 8 | 38 | Chiral symmetry | The class of operators described in (REF ) is invariant under
the group of local chiral transformations. Let \Omega _R(x) and
\Omega _L(x) be matrices in internal space, assumed to be nowhere
singular. (In fact unitary, in order to preserve the hermiticity
properties of the external fields.) Then, the chirally rotated ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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6aec5f33c39036ab87c90c0a5a664303f5188a31 | subsection | 9 | 38 | Chiral symmetry | Likewise{\bf D}^\Omega =
\left(
\begin{matrix}
\Omega _L^{-1} & 0\cr 0 & \Omega _R^{-1}
\end{matrix}
\right)
\left(
\begin{matrix}
m_{LR}& \mathrel {\mathop {\!\!\!/}}_L \cr \mathrel {\mathop {\!\!\!/}}_R & m_{RL}\end{matrix}
\right)
\left(
\begin{matrix}
\Omega _R & 0\cr 0 & \Omega _L
\end{matrix}
\right) .Then if \ps... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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8dace6b6590356f0369c6b7d7a9e551d5624b438 | subsection | 10 | 38 | Chiral symmetry | Because chiral
rotations do not act as similarity transformations of {\bf D},
cf. (REF ), the spectrum is not preserved and an anomaly is
introduced. Nevertheless, the anomaly can be restricted to the
abnormal parity sector. Indeed, the adjoint Dirac operator{\bf D}^\dagger =
\left(
\begin{matrix}
m_{RL}& -\mathrel {\m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1148,
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"raw": "W. A. Bardeen, Phys. Rev. 184, 1848 (1969).",
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"end": ... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
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] | 2,008 | en | Physics | [
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3defe2af3a845c400793019f6fd3dcbbff69e204 | subsection | 11 | 38 | WZW term and invariant remainder | The variation of the effective action under a finite chiral
transformation can be obtained by integration of the infinitesimal
variation (the consistent anomaly). More specifically,
let (m_{LR},m_{RL},v_R,v_L) be the field configuration obtained by applying the
chiral rotation (\Omega _R,\Omega _L) to the configuration... | {
"cite_spans": [
{
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"doi": "",
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"raw": "L. L. Salcedo, Eur. Phys. J. C20, 161 (2001), [hep-th/0012174].",
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{
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"doi... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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2e93f0f6cfde48997f44a27326dbc30a7818809b | subsection | 12 | 38 | WZW term and invariant remainder | (Note that although the
WZW term depends on U, this dependence cancels in the anomaly.) For
instance, in two dimensions\Gamma _{\text{WZW}}(v_R,v_L,U) &=&
-\frac{i}{12\pi }\int \epsilon _{\mu \nu \alpha }
{\rm tr \,}\big (
U^{-1}\partial _\mu U\,
U^{-1}\partial _\nu U \,
U^{-1}\partial _\alpha U
\big )
\, d^3x
\\
&&
+\... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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1d4a042b1b7befa122fdb1e80960b5b3854e3df6 | subsection | 13 | 38 | WZW term and invariant remainder | That is,\Gamma _{\text{gWZW}}(m,v)-\Gamma _{\text{gWZW}}(\overline{m},\overline{v})
=
\Gamma (v_R,\Omega _R)-\Gamma (v_L,\Omega _L)
,
\qquad (v,m)=(\overline{m},\overline{v})^\Omega .Explicitly, in
two dimensions,\Gamma _{\text{gWZW}} &=&
-\frac{i}{24\pi }\int \epsilon _{\mu \nu \alpha }
{\rm tr \,}\big (
m_{LR}^{-1}\p... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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6ffff70c9df654ba6f44839495c1819bba010092 | subsection | 14 | 38 | WZW term and invariant remainder | At leading order in a
derivative expansion, the remainder W^-_c vanishes identically when
the scalar and pseudo-scalar fields satisfy a generalized chiral
circle constraint (namely, when m_{RL}m_{LR} is a c-number) but W^-_c
is a non trivial functional outside the chiral circle
or beyond LO . | {
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{
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{
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"doi": ... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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5c8195baa0ee4aa7ab2c11f7095117a72185b33f | subsection | 15 | 38 | Computation of | The operator {\bf D}^\dagger {\bf D} is of the Klein-Gordon type
therefore there are several techniques to address the computation of
W^+. This is further simplified by the fact that chiral symmetry is
preserved in the normal parity sector. This helps to reduce the number
of allowed structures.The situation in the abno... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 680,
"openalex_id": "",
"raw": "R. D. Ball, Phys. Rept. 182, 1 (1989).",
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{
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... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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8b2bc60224ba97ab540ecef6c57204b1e4be06a3 | subsection | 16 | 38 | Computation of | For instance, for d=2, one
finds P^{R,L}_\mu = i\epsilon _{\mu \nu }v^{R,L}_\nu /4\pi . This term is
just the polynomial non covariant part of the current derived from
\Gamma _{\text{gWZW}} in (REF ).Let us remark that the current
coming from W^-_c is both covariant and consistent, and does not
coincide with the covari... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 641,
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... | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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017cfb6aee6dbee7bf590743f6f5a39bc437f2cc | subsection | 17 | 38 | The invariant part of the effective action | In this section the main result of the paper is presented, namely, we
show how, upon separation of the anomalous WZW contribution, the
effective action can also be expressed as the {\rm Tr}\log of a local
operator of the Klein-Gordon type which, in addition, is manifestly
chiral covariant. | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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-... |
66d02d1ff1daaaaadc963b304eb387f26be9f5da | subsection | 18 | 38 | Covariant Klein-Gordon operator | In order to construct such a covariant operator of the Klein-Gordon
type, we will use the convenient matrix notation of (REF ).
That is, provided that the operators A and D commute with
\gamma _5 and B and C anticommute with \gamma _5, we can use
\left(
\begin{matrix}
A & B \cr C & D
\end{matrix}
\right) to represent P... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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8fc8eb895514783f8fe6391905954bbea53cfd6d | subsection | 19 | 38 | Covariant Klein-Gordon operator | The only thing to be noted is
the relation{\rm Tr}\left(
\begin{matrix}
A & B \cr C & D
\end{matrix}
\right)
=\frac{1}{2}{\rm Tr}(A+D)+\frac{1}{2}{\rm Tr}(\gamma _5(A-D))
.In this notation{\bf D}=
\left(
\begin{matrix}
m_{LR}& \mathrel {\mathop {\!\!\!/}}_L \cr \mathrel {\mathop {\!\!\!/}}_R & m_{RL}\end{matrix}
\right... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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905c9f0803ea7100d84b720e0bcdb482758623b2 | subsection | 20 | 38 | Covariant Klein-Gordon operator | The natural
assignments would be achieved by using instead
{\bf D}^m{}^\prime =
\left(
\begin{matrix}
m_{RL}& 0 \cr 0 & m_{LR}\end{matrix}
\right)
{\bf D}\left(
\begin{matrix}
m_{LR}^{-1} & 0 \cr 0 & m_{RL}^{-1}
\end{matrix}
\right)
and {\bf K}^\prime ={\bf D}^m{}^\prime {\bf D}^\prime =
\left(
\begin{matrix}
K_R^\da... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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41be17a859c75709435286654e5283d76d482ea5 | subsection | 21 | 38 | Covariant Klein-Gordon operator | (In other way, in even dimensions the
representations \gamma _\mu and -\gamma _\mu are equivalent.)
Therefore, we can symmetrize with respect to \mathrel {\mathop {\!\!\!/}}_{R,L}\rightarrow -\mathrel {\mathop {\!\!\!/}}_{R,L}. In the third and fourth equalities we make use of
the formal identity{\rm Tr}\log (AB)={\rm ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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a4935124f5f6455f7a985230a72b01b1499f1a40 | subsection | 22 | 38 | Covariant Klein-Gordon operator | In fact we would
expect it to hold to all orders, included the LO one. The reason is
that in most expansions of interest (such as perturbation theory or
inverse mass expansions) higher orders are increasingly UV convergent
and so (REF ) should be fulfilled to all UV convergent orders
of all those expansions. This cover... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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7081f486ec5343caba8e34cf3291852395479ffc | subsection | 23 | 38 | Proof of the main result | In order to prove the identification in (REF ) we should make
the statement precise. | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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e9cf66f560c50259f22afa3dd54d5543b4a34ded | subsection | 24 | 38 | The standard LR effective action of | The functional W^-({\bf D}) is perfectly well defined once the two
following conditions are met, first, the LR version of the anomaly is
chosen, and second, W^-({\bf D}) depends only on {\bf D} (no new
field absent from {\bf D} is introduced in the functional). This is
unique because the allowed ambiguity would be an a... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0.00304335053078... |
153bd95b38bf22cb77b462bef1288e55851d0098 | subsection | 25 | 38 | Natural and standard effective action from | Let us now consider the functionalW({\bf K})= -\frac{1}{2}{\rm Tr}\log {\bf K}\,.As noted before, given a differential operator, such as {\bf D} or
{\bf K}, the logarithm of its determinant is unique modulo a
counter-term action which is polynomial regarding its dependence on
the fields present in the operator and thei... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
-0.017868654802441597,
-0.006759875454008579,
-0.05023365840315819,
0.008568104356527328,
-0.01902836374938488,
-0.027817727997899055,
0.032838042825460434,
0.01609857566654682,
0.008125584572553635,
0.03299063444137573,
0.01311537902802229,
-0.01017796155065298,
-0.006962061394006014,
0.0... |
79af52c5a2d2daeabd55b15eb18effb11b4d9626 | subsection | 26 | 38 | Natural and standard effective action from | We will take the
\zeta -function determination as the standard one, to be denoted
W_s({\bf K}):W_s({\bf K})= -\frac{1}{2}\frac{d}{ds}{\rm Tr}({\bf K}^s)\Big |_{s=0}
.It order to analyze this point further, let us
introduce the chiral covariant quantitiesM_R=m_{RL}m_{LR}\,,\quad M_L=m_{LR}m_{RL}\,,
\\
Q^R_\mu = m_{LR}^{... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
-0.029165828600525856,
-0.012629368342459202,
-0.03809414431452751,
-0.03778890147805214,
-0.0359574519097805,
-0.01732245646417141,
0.02744121477007866,
0.013110123574733734,
0.02034434862434864,
0.019810175523161888,
0.01108026783913374,
-0.023442549630999565,
-0.003453044919297099,
0.00... |
33fb221aa8728047e0443672e7c7ac411cde3ac7 | subsection | 27 | 38 | Natural and standard effective action from | Such polynomials are absent in two dimensions in the abnormal
parity sector.The possible candidates, \int d^2x\epsilon _{\mu \nu }{\rm tr \,}(Q^R_\mu Q^R_\nu -Q^L_\mu Q^L_\nu ) , and
\int d^2x\epsilon _{\mu \nu }{\rm tr \,}([D^R_\mu ,Q^R_\nu ]-[D^L_\mu ,Q^L_\nu ])
, vanish. But they exists in four or more dimensions. T... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
-0.056870609521865845,
0.034201692789793015,
-0.01099122129380703,
-0.03142528235912323,
-0.03813748061656952,
0.028160715475678444,
0.015102442353963852,
-0.025613131001591682,
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0.005316364578902721,
-0.011189536191523075,
0.0005806431290693581,
-0.014255790039896965,... |
a27ba4a32adee289326d19a6b131f2672617589e | subsection | 28 | 38 | Proof of the statement | The statement to be proven is then that for a certain natural
determination of W({\bf K}) the relation (REF ) holds. Or
equivalently,W_s({\bf D})-\Gamma _{\text{gWZW}}({\bf D})
-W_s({\bf K})
& &
\\
&&
\hspace{-99.58464pt}= \text{``Chiral invariant polynomial of $m_{LR},m_{RL},D^{L,R}_\mu $, and
$Q^{L,R}_\mu $''}
.Note ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
-0.03396761417388916,
0.0422992929816246,
-0.028748871758580208,
-0.054842591285705566,
-0.04571741819381714,
0.01215418055653572,
0.0011311094276607037,
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0.01594616286456585,
0.03277737647294998,
-0.00927776750177145,
0.02548334188759327,
-0.0020504933781921864,
0.03... |
a1c1f1d9f46ce3613aacf0c9df1329dec1de6aa2 | subsection | 29 | 38 | Proof of the statement | Therefore,
using (REF ) and (REF ),W_s({\bf D}^m)
=
W_s({\bf D})-\Gamma (v_R,m_{RL}^{-1})+\Gamma (v_L,m_{LR}^{-1})
.Substituting in (REF ) we findW_s({\bf D})
-
W_s({\bf K})
-\frac{1}{2}(
\Gamma (v_R,m_{RL}^{-1})-\Gamma (v_L,m_{LR}^{-1})
)
&&
\\
&&
\hspace{-99.58464pt}= \text{``Polynomial of $m_{LR},m_{RL},\partial _\m... | {
"cite_spans": []
} | 10.1140/epjc/s10052-008-0782-4 | 0807.1696 | The invariant factor of the chiral determinant | [
"L. L. Salcedo"
] | [
"hep-th"
] | 2,008 | en | Physics | [
-0.022402597591280937,
0.0403185710310936,
-0.03586247190833092,
-0.058051418513059616,
-0.048406705260276794,
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0.039525020867586136,
0.0009676190675236285,
0.0019571753218770027,
0.01037722546607256,
-0.022478900849819183,
0.017259767279028893,
-0.028277939185500145,
... |
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