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fb93024fe4977a0e01b359185a388e83c8ecccb1 | abstract | 0 | 6 | Abstract | We consider a supersymmetric completion of codimension-two branes with
nonzero tension in a 6D gauged supergravity. As a consequence, we obtain the
football solution with 4D Minkowski space as a new supersymmetric background
that preserves 4D N=1 SUSY. In the presence of brane multiplets, we derive the
4D effective sup... | {
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} | 10.1063/1.3052086 | 0807.1858 | U(1)_R mediation from the flux compactification in six dimensions | [
"Hyun Min Lee"
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a295fb0521752ce2725fa282610f2627b231f9d9 | subsection | 1 | 6 | Body | 8x11doubleU(1)_R mediation from the flux compactificationin six dimensions11.10.Kk,11.25.Mj,04.65.+e,12.60.Jv
Gauged supergravity, Codimension-two branes, Flux compactifications, Supersymmetry breakingHyun Min Lee
address=
Department of Physics,
Carnegie Mellon University, Pittsburgh, PA 15213, USAWe consider a supersy... | {
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93a0584d710c91c831d8b357b9f28544c23b0173 | subsection | 2 | 6 | Body | But, once the T modulus is stabilized by a T-dependent superpotential, the brane scalar soft mass does not vanish due to modulus mediation.In this talk, we consider a flux compactification in a 6D gauged supergravity with codimension-two branes.
The U(1)_R gauge flux in the internal dimensions is turned on
to stabilize... | {
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ff6d336433b457257b890ff546b3cb199456fbb7 | subsection | 3 | 6 | Body | In particular, for a constant dilaton and {\hat{F}}_{\rho \theta }=4g\epsilon _{\rho \theta }, the football solution is obtained aswith r^2_0=\frac{1}{2g^2}. Two codimension-two branes with equal tensions
T_1=T_2=4\pi (1-\lambda ) are located at the poles of the football.
The gauge field strength is modified due to the... | {
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3512aea6c040090269f94e712cf699231d18312d | subsection | 4 | 6 | Body | After plugging the above solutions into the 6D action together with e^f G_{\mu \nu \rho }=\epsilon _{\mu \nu \rho \tau }\partial ^\tau \sigma
and integrating over the extra dimensions,
we identify the Kähler potential aswhere \delta _{GS}=8g_R and \xi _R=2g_RM^2_P with g_R=g/\sqrt{V} and
the scalar components of the m... | {
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19eb9259ddbf59d7d0364102b0a1ebc2b14d9b21 | subsection | 5 | 6 | Body | For example, we assume that the uplifting potential is given solely by the F-term with
W=W(S)+\mu ^2 Q^{\prime }+W_v(Q), i.e. F_{Q^{\prime }}=\mu ^2.
Then, for the vanishing vacuum energy condition |F_{Q^{\prime }}|^2\simeq -{\rm Re}(S){\rm Re}(T)V_1={\rm Re}(S){\rm Re}(T)(2m^2_{3/2}M^2_P-|F_S|^2) with \alpha {\rm Re}(... | {
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} | 10.1063/1.3052086 | 0807.1858 | U(1)_R mediation from the flux compactification in six dimensions | [
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18e778616cbaa1ec288efa787e9af011a2c704b2 | abstract | 0 | 22 | Abstract | The dependence of the Casimir force on material properties is important for
both future applications and to gain further insight on its fundamental
aspects. Here we derive a general theory of the Casimir force for
low-conducting compounds, or poor metals. For distances in the micrometer
range, a large variety of such m... | {
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} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
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"cond-mat.mtrl-sci"
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bb78f95e38d269fac690ef983da85e144d5e45a9 | subsection | 1 | 22 | Introduction and motivation | The Casimir force has demonstrated the reality of
zero-point field fluctuations, which played a significant role in
the development of quantum field theory (see, e.g., the
monographs , and review
papers , , , ). The Casimir effect attracts
considerable attention because of its numerous applications in
quantum field th... | {
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628dff601f69b45e76263b482e897785bebb092f | subsection | 2 | 22 | Casimir force for good metals and dielectrics | The Casimir force between two macroscopic samples
is caused by a spatial redistribution of the fluctuations of the
electromagnetic field compared to that of free space because of
the presence of the samples. For the simplest case of two parallel
perfectly conducting thick metallic plates placed in vacuum
and separated ... | {
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"E. G. Galkina",
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"V. A. Yampol'skii",
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9faf0fd7fb1739d1e1332f7f301f7f17a73ed5c4 | subsection | 3 | 22 | Material aspects of the Casimir force | To study the Casimir force, different materials can be used.
Indeed, it is important to understand how this force is
affected by the choice of different materials. For example,
recent studies, using silicon with different degrees of
doping , or materials for sensors, like
Vanadium oxide , have shown numerous specific
f... | {
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bd0d74953d1ff891938da0494125d69f752a318b | subsection | 4 | 22 | Material aspects of the Casimir force | For metals, the
plasma frequency \omega _{\mathrm {p}} is the highest frequency
\omega _0 . Thus, the standard Casimir result (REF ) is
valid for large distances l > \lambda _p = c / \omega _p between
the plates, see Ref. 3. For the opposite limit
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04a2bdbe02c49f5fb3bb76d6dd996a782e534e09 | subsection | 5 | 22 | Caviats and limitations | It is worth noting here that, as far as we know, only one
experiment has been performed using the parallel-plate
configuration originally envisioned by Casimir. Most measurements
of the Casimir force have studied the interaction of a spherical
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transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
] | 2,008 | en | Physics | [
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2932085abd311d7d26dbc1b7359976f6c713592d | subsection | 6 | 22 | Casimir force for pure metals and compounds | Numerous compounds are known for which the carrier density and
plasma frequency \omega _p are abnormally small. The
investigation of such conducting systems, which can be called
“poor metals”, is of interest from the point of view of both
fundamental physics and applications. Examples include highly
doped silicon , , l... | {
"cite_spans": [
{
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"end": 535,
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"raw": "F. Chen, U. Mohideen, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. A 72, 020101(R) (2005); Phys. Rev. A 74, 022103 (2006); F. Chen, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohidee... | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
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3e6eefec6ddc567630ade2735a76c577cc46905e | subsection | 7 | 22 | Derivation of the Casimir force for poor metals | For general dispersive media, the Casimir force is determined by
the integral in Eq. (REF ). Keeping in mind the large
variety of poor-metal parameters discussed above, we now need to
develop an analytical approach to estimate the integral
(REF ) and to study the role of different parameters,
like \epsilon _\infty or \... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
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8c2274131ae4f8c3c7e9e4c6588b520497ea8e16 | subsection | 8 | 22 | Derivation of the Casimir force for poor metals | (REF ) for the first integral and omitted the Drude
multiplier for the second integral.Expanding the integration region over \zeta in both integrals
up to 0 \le \zeta < \infty , and subtracting the extra terms, we
present the Casimir force in the form,F = F^{(m)} + \Delta F,withF^{(m)}=\frac{\hbar }{2\pi ^2c^3}\int \li... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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139ce21731e555cd33e26b25bec64c793cf8fdd2 | subsection | 9 | 22 | Computing the Casimir force integrals | To proceed further, let us change the variable z by x=z/2p in
Eq. (REF ), and note, that the integral \int \limits _0^\infty { (Ae^x - 1)^{ - 1}x^3dx} , with A = A(x,p) being a smooth
function of x, can be approximated by 1 / A(x_0 ,p) with x_0
\sim 4. The problem is then reduced to calculating two
one-dimensional inte... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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b2e42762568c660bcba3bec2ff8b90d67a467efd | subsection | 10 | 22 | Computing the Casimir force integrals | (REF ),
is not realized for any \varepsilon _\infty \ne 1.Otherwise, in the limit case \tilde{l} \rightarrow \infty , the
integrals J_1=J_2=1, and the ideal Casimir limit (REF ) is
recovered. In contrast to the case of small values of \tilde{l}
, the dependence of \Pi on l for large, but finite values of
\tilde{l} show... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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49c239e39d754aaab33e21743fe76814e2c035f1 | subsection | 11 | 22 | Computing the Casimir force integrals | But for the case of interest, \tilde{l} \ge 3, the
integral J_1 can be well approximated with the simpler
expressionJ_{1, \, \mathrm {appr}}=\frac{\sqrt{(3+
\varepsilon _{\infty })(3+5\alpha \tilde{l}) +3
\varepsilon _{\infty }\tilde{l}^2 }-2\sqrt{3+5\alpha \tilde{l}}}{\sqrt{(3+ \varepsilon _{\infty })(3+5\alpha \tilde... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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61fc2f24fbda1494a4d4812ce5d3a17d75ac8b45 | subsection | 12 | 22 | Computing the Casimir force integrals | Indeed, all the curves with
0 \le \alpha < 0.3 merge together, and for describing of
J_{2} within an accuracy of 1.5%, the function J_{2}(\tilde{l}) found for \alpha =0.1 can be used. Even for small
\varepsilon _{\infty } = 1.5, the inaccuracy of this approximation
is less than 5%.
[Figure: The integral J_2 calculatedn... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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235ce5ba988fa865dead774d0e6787c775f522f7 | subsection | 13 | 22 | Change in the Casimir force near the metal-insulator
transition | The analytical formulae derived above give a good description of
the behavior of the Casimir force when the metal-insulator
transition occurs. Usually, the metal-insulator transition is
associated with an abrupt change, by a few orders of magnitude, of
the conductivity at a transition temperature T = T_c. Let us
start ... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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ca976e4a8b3ed5aba0260f6f967f6b3183dbaeb9 | subsection | 14 | 22 | Change in the Casimir force near the metal-insulator
transition | (REF ) is valid, and the formula for \Delta F
reads\Delta F = \frac{ \pi ^2 }{240}\frac{c\hbar }{l^4}
\cdot \left[1- \left( {\frac{\varepsilon _\infty -
1}{\varepsilon _\infty + 1}} \right)\varphi _1(\varepsilon _\infty ) - \sqrt{\frac{ c }{\omega _p \varepsilon _\infty \, l}} \right].Note that our results differ sign... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
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dd545077e4aef15e241a1849f225090f4d40e00f | subsection | 15 | 22 | Composite media and the intermediate region for
metal-insulator transition | The very abrupt (by a few orders of magnitude) change of the
conductivity at the metal-insulator transition occurs for the dc
case only. At finite frequencies, the behavior of the complex
permittivity of compounds near metal-insulator transition is more
complicated and the jump-like behavior, typical for the static
con... | {
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... | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
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029a26dc2bc478e552841d9811abf1b8cdd60930 | subsection | 16 | 22 | Composite media and the intermediate region for
metal-insulator transition | In the intermediate region, the effective
permittivity \varepsilon _{\mathrm {eff}} (i\zeta ) as a function
of the phase concentration f can be written as follows:2\frac{\varepsilon _{\mathrm {eff}} (i\zeta )}{\varepsilon _\infty } = (2f -
1)\frac{\omega _p^2 }{\zeta (\zeta + \gamma )} \\
+ \sqrt{4 + \frac{4\omega _p^2... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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248396cb891ebcac2e53f403d1721a77c7e5548a | subsection | 17 | 22 | Composite media and the intermediate region for
metal-insulator transition | REF .It is useful to note here that a linear temperature dependence of
\omega _p^2 was observed in
La_{0.7}Sr_{0.3}MnO_3 for T < T_c . Thus, we can describe
the Casimir force in the intermediate region as a series of
curves, with their shape only depending on l / \lambda _p, as
shown in Fig. REF . | {
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"raw": "K. H. Kim, J. H. Jung, and T. W. Noh, Phys. Rev. Lett. 81, 1517 (1998).",
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} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
] | 2,008 | en | Physics | [
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0319b2e721cfb06aab8bbe71c52b90bea1155dfa | subsection | 18 | 22 | Predictions for specific materials | Using the results obtained in the previous sections, here we
estimate numbers for different materials showing the
metal-insulator transition. To study the Casimir force in the
vicinity of the metal-insulator transition, we choose two typical
compounds: vanadium dioxide VO_{2} and the manganites exhibiting
colossal magn... | {
"cite_spans": []
} | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
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8beecf46dfdb4ee038c66a38eddda1f634cdb4e6 | subsection | 19 | 22 | Vanadium dioxide VO | Vanadium dioxide, VO_{2}, shows a jump in the static
conductivity (a metal-insulator transition) a little bit above
room temperature, at T = T_c \approx 68 \ ^\circ C. The pure
metallic phase of VO_{2} is realized at T > 88 \ ^\circ C, and
pure insulator phase (more exactly, semiconducting
phase with a gap of the orde... | {
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"end": 537,
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"raw": "N. F. Mott, Metal-Insulator Transitions, 2nd ed. (Taylor&Francis, London, 1990); M. Imada, A. Fujimori, Y Tokura, Rev. Mod. Phys., 70, 4 (1998).",
"source_ref_id": "01bbc0bb94f2bdad28b17f88b04... | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
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e0cd64d793edbe78d153bc60ce1b5231f3250c86 | subsection | 20 | 22 | Manganites. | Manganites (with antiferromagnetic insulators LaMnO_{3} or
NdMnO_{3} as parent compounds, after substitution of La by
divalent ion) show a metal-insulator transition at the dopant
concentration x\sim 0.3, with a ferromagnetic metallic phase in
the low temperature range. These systems are very
popular now in the contex... | {
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"raw": "Physics of Manganites, edited by T. A. Kaplan and S. D. Mahanti (Kluwer Academic/Plenum publishers, New York, 1999).",
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"star... | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
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7a451acf9ebd842140b92e7d78cb5621386f67df | subsection | 21 | 22 | Conclusions | The Casimir force depends on the materials used, and we have
studied some of these material-dependent aspects. The Casimir
force F_\mathrm {C} for a mechanical system containing compounds
with a metal-insulator transition shows an abrupt
temperature dependence in the transition region. The relative
change of the force ... | {
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"raw": "V. Petrov, M. Petrov and V. Bryksin, J. Petter and T. Tschudi, Opt. Lett. 31, 3167 (2006).",
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... | 10.1103/PhysRevB.80.125119 | 0807.1859 | Large temperature dependence of the Casimir force at the metal-insulator
transition | [
"E. G. Galkina",
"B. A. Ivanov",
"Sergey E. Savel'ev",
"V. A. Yampol'skii",
"Franco Nori"
] | [
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7f7a1eb428171c8c2fda85c43e00cd701cbb03c2 | abstract | 0 | 15 | Abstract | We give an elementary introduction to a recent diagrammatic extension of
dynamical mean field theory (DMFT) coined dynamical vertex approximation
(D$\Gamma$A). This approach contains the important local correlations of DMFT,
giving, among others, rise to quasiparticle renormalizations, Mott-Hubbard
transitions and magn... | {
"cite_spans": []
} | 0807.1860 | Dynamical vertex approximation -- an introduction | [
"K. Held",
"A. A. Katanin",
"A. Toschi"
] | [
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a29536e61c8c87a7bfaaad97aaa530efcd3d7624 | subsection | 1 | 15 | Introduction | Dynamical mean field theory (DMFT) , , ,
has been a success story
for the theoretical modeling of
strongly correlated electrons systems since it contains the major contribution
of electronic correlations: the local ones.
Diagrammatically, DMFT corresponds to
all topologically distinct Feynman diagrams, albeit only
for... | {
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"A. A. Katanin",
"A. Toschi"
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00fda8691cb9e7b79c69a0cc2a2b23a6505213f6 | subsection | 2 | 15 | Introduction | For the one-band Hubbard model, approximately 30-60 sites
are possible which means a correlation length of about 3 sites in two dimensions and 1 site in three dimensions.
In realistic multi-orbital calculations, the effort increases further with the number of orbitals.
Hence, hitherto only two sites
have been considere... | {
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6a629bdeda08ee3ff83e521c31a5df1015f65512 | subsection | 3 | 15 | Full (self-consistent) D | Taking the locality of the fully irreducibleThat is, cutting two Green function lines of the Feynman diagram for the vertex does not divide the vertex into two parts. vertex as a starting point,
we can formulate
the D\Gamma A approach as following, see Fig. REF for the algorithm:
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e36272effff6af2ba3cc015187ba40cdd7f9fbd3 | subsection | 4 | 15 | Full (self-consistent) D | (REF ),
the dispersion relation \epsilon _{\bf {k}} of the lattice and the
chemical potential \mu ,
we can now calculate a new local and non-local
Green function via the Dyson equationG_{\bf {k},\nu }=[i \nu _n-\epsilon _{\bf {k}}+\mu -\Sigma _{\bf {k},\nu }]^{-1}
.After this fourth step of the algorithm,
we can go bac... | {
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"K. Held",
"A. A. Katanin",
"A. Toschi"
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8b076d635573908418b3e2c70ce4bb93a2f16f4c | subsection | 5 | 15 | Restriction to the two-particle-hole channels | Instead of this full (self-consistent) D\Gamma A approach, we have, in a first step,
implemented a simplified scheme : Neglecting the particle-particle
channel of the Parquet diagrams, we
restricted ourselves
to the two particle-hole channels which are longitudinal and transversal, respectively,
in the spin indices; se... | {
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d887b34845c91a1c49ceab3efb10bfb1f7859755 | subsection | 6 | 15 | Restriction to the two-particle-hole channels | Here, \Gamma _{s(c),\text{loc}} is the local (\mathbf {q}-summed) counterpart of \Gamma _{s(c),\mathbf {q}}.
With the restriction to the two particle-hole channels, actually a non-self consistent calculation
is preferable so that the algorithm ends with a calculation of the D\Gamma A
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d25c5bb56f97913aae6010eebdfd29b1899eacee | subsection | 7 | 15 | Moriyaesque | As a numerically inexpensive alternative to the fully self-consistent D\Gamma A algorithm,
we recently introduced a Moriyaeque \lambda correction to the
algorithm of Section REF . In the Moriya theory of weak
itinerant magnets , the perturbatively obtained
susceptibility is corrected as follows\chi _{\mathbf {q}\omega ... | {
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e3049bc7d401049d1cfd8deca0193e1449481ece | subsection | 8 | 15 | Three-dimensional Hubbard model | Let us turn to the results, starting with the three-dimensional
Hubbard model on a cubic latticeH=-t\sum _{\langle ij\rangle \sigma }c_{i\sigma }^{\dagger }c_{j\sigma }+U\sum _{i}n_{i\uparrow }n_{i\downarrow }.Here, t denotes the hopping amplitude between nearest-neighbors, U the
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9601c5c0c6bfa2d7b572d6ed0253db36fdf32b4c | subsection | 9 | 15 | Three-dimensional Hubbard model | When increasing U to the crossover regime, this quasiparticle peak
is first strongly damped, i.e, smeared out
(second point considered at U=1.5 D, \beta =0.089D; for the DMFT spectrum see Fig. REF ).
Note, this change from a metal to a strongly damped quasi-metal
was confirmed recentely by photoemssion experiments for ... | {
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032ca61949916f0b59cb64853c1ee83c9f79cb8e | subsection | 10 | 15 | Three-dimensional Hubbard model | It also shows that the Mott-Hubbard transition is strongly affected
by antiferromagnetic fluctuations.The results shown here are close to the antiferromagnetic phase transition, where the antiferromagnetic fluctuations
are particularly strong. If we increase temperature, i.e., if we are deep inside the paramagnetic pha... | {
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bf1b08278033fb7e41dd0f5c02ab3406a6bfdf4e | subsection | 11 | 15 | Two-dimensional Hubbard model | Let us turn now to the two-dimensional Hubbard model on a square lattice where antiferromagnetic fluctuations are expected to be much more pronounced since the particle-hole ladder diagrams diverges logarithmically in two dimensions. Our insight into these antiferromagnetic fluctuations has been recently improved
not o... | {
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98fbd527782ebb5b360ca8ff201719aad804b4de | subsection | 12 | 15 | Two-dimensional Hubbard model | Note, however the strong anisotropy of the damping which is much stronger in the anti-nodal direction.
[Figure: Spectral function for the two-dimensional Hubbard model off half-fillingin the nodal (left) and anti-nodal (right)direction. Parameters are n\approx 0.8 electrons/site, a next-nearest neighbor hopping t^{\pri... | {
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d50b4f03e5c5bdd4e97d4e5b071a4409402e9101 | subsection | 13 | 15 | One-dimensional Hubbard model | One of the fascinating effects of electronic correlation is the separation
of spin and charge in one dimension. Slezak et al. addressed this
issue in their one-dimensional D\Gamma Aesque calculation. As a starting point the authors
considered an 8-site DCA cluster and supplemented it with additional longer range correl... | {
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362775c516176b0a676fb5822251554555a3a4a2 | subsection | 14 | 15 | Summary and Outlook | In summary, we have presented the D\Gamma A approach in an algorithmic
form, including the restriction to the two particle-hole channels and the
Moriyaesque \lambda -correction. Let us also emphasize here that D\Gamma A is a complementary
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33d46c3b59700bf4cd218b3afaa975afb310d55f | abstract | 0 | 15 | Abstract | We present an empirical assessment of the use of broadband optical colours as
age indicators for unresolved extragalactic clusters and investigate stochastic
sampling effects on integrated colours. We use the integrated properties of
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032ec0c78223a57d18a8aa45793db1c238299ecf | subsection | 1 | 15 | INTRODUCTION | Thousands of luminous young star clusters have been discovered in
external galaxies by the Hubble Space Telescope (HST) (e.g.,
, , , , , , , , , , ,
and references therein). Population synthesis analyses show that
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ee92cefcaa18374ce086492ea4606a396a400ad3 | subsection | 2 | 15 | INTRODUCTION | This suggests that the observed
integrated colours of low mass clusters may not be good indicators of
age when compared to the integrated colours of a stellar population
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d4663f662bb202557447c10d40a3ee6b7a6f8651 | subsection | 3 | 15 | THE DATA SAMPLE | We started with the set of Galactic open clusters
from the WEBDAhttp://www.univie.ac.at/webda/(Web Base
Données Amas) database operated at the Institute for Astronomy of
the University of Vienna . The WEBDA database contains
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710b8f767e9a8c06bb02fabe89d94dadc2695569 | subsection | 4 | 15 | THE MODEL CLUSTERS | We used a set of evolutionary synthesis models from the Starburst99 (SB99) code . We used the new v5.1 code,
which includes the Padova asymptotic giant branch (AGB) stellar models
. The new version accounts for all stellar phases that
contribute to the integrated light of a stellar population with
arbitrary age from ex... | {
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Clusters | [
"M. Hancock",
"B. J. Smith",
"M. L. Giroux",
"C. Struck"
] | [
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cf2df37d5ecd73276ecc65719c1efb270f472b66 | subsection | 5 | 15 | DATA ANALYSIS | This study is not intended as a test of the SB99 model's ability to
reproduce the colours of young clusters. This has already been
demonstrated (see for e.g., ). We intend to test how
well various colours and initial model assumptions can recover the
ages of well studied Galactic open clusters. In addition to using the... | {
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c1febad63f36acb90e70da47b86c38e3738739ca | subsection | 6 | 15 | Predicting Ages | In Figure 2, we plot the predicted ages against the published ages, if
only (U-B) and (B-V) are used to estimate the ages. We also provide
a histogram showing the distribution of the differences between the
predicted and published ages, a plot of the predicted E(B-V)s against the
published E(B-V)s, and a histogram show... | {
"cite_spans": []
} | 10.1111/j.1365-2966.2008.13705.x | 0807.1862 | An Assessment of Broadband Optical Colours as Age Indicators for Star
Clusters | [
"M. Hancock",
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"M. L. Giroux",
"C. Struck"
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7977e14d17aaa88d5daa2df8b70e03c2e426f9bb | subsection | 7 | 15 | Predicting Ages | This is because
we have assumed the maximum uncertainty in measured colours for the
OCs. Using the actual uncertainties in integrated colours for this
sample would not change the predicted ages but would result in better
age constraints. Unfortunately the uncertainties in integrated
colours of each of these OCs are not... | {
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e3bbbc91ce580c3124ffab28f22176376cf6b4de | subsection | 8 | 15 | Extinction and Metallicity | The extinction is predicted well by the same colours that predict the
age well. For example, using the (U-B) and (B-V) colours
we find that 62% of the E(B-V) estimates are within 0.2 mag of the
published values. The colour that is the most sensitive to age,
(U-B), is once again critical.We get similar numbers of good f... | {
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} | 10.1111/j.1365-2966.2008.13705.x | 0807.1862 | An Assessment of Broadband Optical Colours as Age Indicators for Star
Clusters | [
"M. Hancock",
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52e7537e5ab7670870057ef6ffd4b214328b0dd5 | subsection | 9 | 15 | The Effect of H | The SB99 model colours do not include nebular H\alpha emission.
This does not impact our analysis of the integrated colours of the OCs
because the stellar components are resolved and no H\alpha is
included in either the models or the data. However, this should be
considered when models are compared to young unresolved ... | {
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Clusters | [
"M. Hancock",
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80f6ea1251245abc09196863d42e6e0a972d6637 | subsection | 10 | 15 | The Effect of Stochastic Sampling of the IMF | It has been shown that traditional synthesis models usually return the
mean value of the integrated stellar population distribution
(e.g. , ). While this is correct on average, it
is not necessarily correct in individual cases because the mean may not be
representative of the actual values. Galactic open clusters are
t... | {
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"M. Hancock",
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bd3b26c0ab3cb2ce6d9ec937fefe5ceec50375bb | subsection | 11 | 15 | The Effect of Stochastic Sampling of the IMF | The scatter appears slightly larger for the fainter samples,
with the (B-V) difference being slightly skewed to negative
numbers (see Figure 4).Figure 5 shows the differences between the published and model (U-B)
and (B-V) colours plotted against the published ages using the same
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ad5076bab1af2735ba3b8688122b72913ed206f3 | subsection | 12 | 15 | The Effect of Stochastic Sampling of the IMF | In a follow up
paper (Hancock et al., in prep.) we will explore a more detailed
analysis of the stellar content of these resolved OCs to further
investigate the extent to which stochastic sampling of the IMF affects
the integrated broadband colours. If stochastic sampling effects are
observationally characterized in th... | {
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} | 10.1111/j.1365-2966.2008.13705.x | 0807.1862 | An Assessment of Broadband Optical Colours as Age Indicators for Star
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af6071572898be97a442a9d7a45747fde778ffd6 | subsection | 13 | 15 | The Effect of Cluster Dissolution | Current stellar population synthesis models do not take into account
the effects of cluster dissolution. Mass segregation causes the
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1c1bc825cb3e3617a95bf410142e0d772510b9d6 | subsection | 14 | 15 | SUMMARY | We present an empirical assessment of the use of broadband colours as
age indicators for unresolved extragalactic clusters and investigate
stochastic sampling effects on integrated colours. The population
synthesis code Starburst99 and four optical
colours were used to estimate how well we can recover the ages of 62
we... | {
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5c083eec4d8e858226f4723cae07dfe4e22affab | abstract | 0 | 153 | Abstract | We use the theory of pseudo-holomorphic quilts to establish a counterpart, in
symplectic Floer homology, to the Gysin sequence for the homology of a
sphere-bundle. In a motivating class of examples, this "symplectic Gysin
sequence" is precisely analogous to an exact sequence describing the behaviour
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} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
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94496fadca917918b9d2b6aebeefe73573b26f00 | subsection | 1 | 153 | Spherically-fibred coisotropic submanifolds | This paper is about the following situation. We are givencompact symplectic manifolds (M, \omega _M) of dimension 2n and (N,\omega _N) of dimension 2(n-k) , where 0\le k \le n;
a smooth fibre bundle \rho \colon V \rightarrow N whose typical fibre F is a k-dimensional manifold admitting a Morse function with precisely ... | {
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dc569bc92b0df43c6cd99ba0f3234dd41db03006 | subsection | 2 | 153 | Spherically-fibred coisotropic submanifolds | Modulo Hamiltonian diffeomorphisms, the infinitesimal deformations are parametrised by H^0(N; \mathcal {H}^1 ) where \mathcal {H}^1 is the local system on N associated with (V, \rho ) with fibres H^1 (\rho ^{-1}(x);).A simple property of fibred coisotropic submanifolds is the following:Proposition 1.1 The graph \wideha... | {
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7eae6b1424d11a342a7d16a10dbb99e968102eeb | subsection | 3 | 153 | Statement of results | To ensure that our Floer homology groups and the maps between them are well-defined we impose monotonicity hypotheses on the symplectic manifolds M and N and the Lagrangian correspondence \widehat{V}. A connected symplectic manifold (M,\omega ) is monotone if the the symplectic area homomorphism \pi _2(M)\rightarrow , ... | {
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aadb80c35c71196f74c9e4657df72f2e2f9c7c98 | subsection | 4 | 153 | Statement of results | All three terms in the triangle are taken with coefficients in the universal Novikov ring \Lambda _R of a commutative ring R of characteristic 2.Remarks (i) All three maps in the sequence have a definite degree, but since Floer-theoretic gradings are relative, only the horizontal arrow has a straightforward numerical d... | {
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6c356f47f219a2e5f53f4a8e23e36abf5d2ed898 | subsection | 5 | 153 | Statement of results | One has \operatorname{HF}_*(\mu ) =\operatorname{HF}^*(\mu ^{-1}) and \operatorname{HF}_*(\widehat{V}, (\mathrm {id}\times \mu )\widehat{V})=\operatorname{HF}^*(\widehat{V},\mathrm {id}\times \mu ^{-1}\widehat{V}) (`Poincaré duality'). Conjecture: under Poincaré duality, \rho ^! (for \mu ) corresponds to the adjoint of... | {
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f0e632b9d62472db9c992ce6f82fdb325093903e | subsection | 6 | 153 | Examples and corollaries | The basic example of a spherically-fibred coisotropic submanifold is the following.Example 1.10
Let S=S^{2n-1}(r) be a sphere of radius r in (standard symplectic) n. It is coisotropic, and the isotropic leaves are the orbits under the scaling S^1-action; thus the reduced manifold is \mathbb {CP}^{n-1} and the reductio... | {
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040dc28c7c57d23d91fd209299e854f95499233f | subsection | 7 | 153 | Examples and corollaries | For further details, see .Example 1.11 Let (M,\omega ) be a compact symplectic manifold, and fix p\in M. Let \widetilde{M\times be a symplectic blow-up of M\times near (p,0). Then the blow-down map \widetilde{M\times \rightarrow M\times followed by second projection defines a degeneration \widetilde{M\times \rightarrow... | {
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e2c6eca3c979845f6561ed730df6bf80003b7d14 | subsection | 8 | 153 | Examples and corollaries | On the other hand, if N is positive-dimensional, mas is a weaker assumption than Oh's. Our result implies that if the quantum cap product on N is undeformed, or if e(V)=0, one has \operatorname{HF}_*(\widehat{V},\widehat{V};\Lambda _R) \cong H_*(V;\Lambda _R).Example 1.12
Consider the spherical Lagrangian \widehat{S}=... | {
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ca30d325e043b35c0ebca91a6669f5dd690ce698 | subsection | 9 | 153 | Examples and corollaries | Their hypotheses are that m_L^{\min } \ge 2 and that the cohomology ring H^*(L;/2) is generated in degrees \le m_L^{\min } - 1. This, together with our vanishing criterion, has the following corollary.Theorem 1.14 Suppose that the hypotheses of Theorem REF hold, and that H^*(\widehat{V};/2) is generated in degrees \le ... | {
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258390a8428b0e8e2dca1c50948b2bcb438c8d1c | subsection | 10 | 153 | Remarks on the proof | One approach to Theorem REF , along the lines of a standard proof of its classical counterpart, would be to study the spectral sequence associated with a natural filtration on \operatorname{CF}_*(\widehat{V}, (\mathrm {id}_{M_-}\times \mu ) \widehat{V}). A generator for this complex projects to a fixed point of \mu , a... | {
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8694b178356583b075e300248bec2d04042cf251 | subsection | 11 | 153 | Remarks on the proof | Its natural habitat is the theory of `Lagrangian matching conditions' in the sense of , alias the `quilted' Floer theory of Wehrheim–Woodward . To prove that \Phi is an isomorphism we show that its low-action part defines a linear isomorphism and then fall back on the algebra of -graded complexes. We introduce an algeb... | {
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7a4edb687e29212c5fa14839088d69f21120c501 | subsection | 12 | 153 | Remarks on the proof | They focus on circle-bundles appearing as the boundary of a tubular neighbourhood of symplectic hypersurfaces, and proceed by a direct comparison of the `pearl complexes' computing Floer self-homology. The results appear to be similar to ours where both apply, but the motivations, methods and applications are quite dif... | {
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91fae3173c3b9631980e27b7799c2cc8f260c51e | subsection | 13 | 153 | Navigation | Section reviews the basic Floer-theoretic constructions, as well as some basic facts about symplectic fibrations. Section , which is largely to blame for the extravagant length of the paper, explains the framework of `Lagrangian matching conditions'. There is more generality here than is really needed, but this framewo... | {
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03144cdd24fca6b0620fa90883b4ebe99aaf7e1b | subsection | 14 | 153 | Locally Hamiltonian fibre bundles | Definition 2.1
A locally Hamiltonian fibration (LHF) is a triple (E,p,\Omega ) consisting of a smooth, proper fibre bundle p\colon E \rightarrow B equipped with
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6585f692d1bef928b1ff7566c693e1d4dbd5818e | subsection | 15 | 153 | Locally Hamiltonian fibre bundles | We can therefore understand F_\Omega through the map (X,Y)\mapsto \kappa _{X,Y}, which is a two-form on B
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86997b131a9cf741e426888c6f3e1ede6e92eadb | subsection | 16 | 153 | Locally Hamiltonian fibre bundles | The monodromies of null-homotopic loops are in fact Hamiltonian automorphisms .Example 2.2
A fundamental example is the mapping torus (\operatorname{T}_\mu ,p_\mu ,\omega _\mu ) of an automorphism \mu \in \operatorname{Aut}(M,\omega ).
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56651c23c079fc9f2ad9c4a8b576aa071ca89272 | subsection | 17 | 153 | Locally Hamiltonian fibre bundles | The techniques of readily yield the following.Lemma 2.3
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40dc9633dba869302e86a4594eaaf649d231a223 | subsection | 18 | 153 | Floer homology for fixed points and Lagrangian intersections | There are two variants of symplectic Floer homology:Floer homology for fixed points of symplectic automorphisms (not necessarily isotopic to the identity); and
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8312728b26fd93463675775c5c49044eb55eb4bf | subsection | 19 | 153 | Fixed-point Floer homology | Suppose that (T,p,\Omega ) is a LHF (REF ) over a closed oriented 1-manifold Z. We suppose the typical fibre N to be compact and either monotone or strongly negative (see (REF )).To (T,p,\Omega ) we attach its Floer homology \operatorname{HF}(T,\Omega ).
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2cfa4b5eb2426e4b8d4d5dc73f1613f385a5901f | subsection | 20 | 153 | Fixed-point Floer homology | When T is non-degenerate, any u\in (T ; J^v), thought of as a path of sections u(s,\cdot ) of T, converges uniformly as s\rightarrow \pm \infty to horizontal sections \nu _\pm \in \operatorname{{H}}(T,\Omega ) . Hence (T; J^v) is partitioned into components (T; \nu _-, \nu _+; J^v). The Conley–Zehnder index \operatorna... | {
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039974d5845b0a399513118c825c2d95c17c6620 | subsection | 21 | 153 | Fixed-point Floer homology | But when c_1^{\min }(N)\ge n - 2, this dimension is \le -1 for all \beta with \langle \beta ,[\omega ]\rangle \ge 0, and so {N}^s(T)^l is empty for all l. Since every sphere factors through a simple one, there are then no holomorphic spheres in the fibres of T.We shall need to consider the subset (T; J^v)_q^r \subset (... | {
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2df734375f059a0b6899a3a943783a933c8f3292 | subsection | 22 | 153 | Fixed-point Floer homology | We often write \operatorname{HF}(\mu ) instead of \operatorname{HF}(T,\Omega ).\operatorname{HF}(T,\Omega ) is initially defined when Z is connected and the monodromy \mu has non-degenerate fixed points. It is then the homology of a complex \operatorname{CF}(T,\Omega ;J^v), where the underlying \Lambda _{R}-module \ope... | {
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78a0ea27552a3b446967e5d3bb7d274efcf84415 | subsection | 23 | 153 | Fixed-point Floer homology | The Conley–Zehnder index \operatorname{ind}\colon (T; \nu _-, \nu _+; J^v)\rightarrow , which coincides with the Fredholm index of the linearisation of the Cauchy–Riemann equation \tilde{J^v} \circ Du = Du \circ i, further partitions (T; \nu _-, \nu _+; J^v) into subsets (T; \nu _-, \nu _+; J^v)^q:=\operatorname{ind}^{... | {
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f71ddb54f6c271bcfc47d6c27c41f56388404c18 | subsection | 24 | 153 | Fixed-point Floer homology | Since every sphere factors through a simple one, there are then no holomorphic spheres in the fibres of T.We shall need to consider the subset (T; J^v)_q^r \subset (T; J^v)^q of sections u whose action \operatorname{\mathsf {A}}(u) equals r. We shall also need to divide out by the -action, by translation, on the moduli... | {
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6b2524fcbb5d772b710b54e2c4f9c1afcb7f1173 | subsection | 25 | 153 | Lagrangian intersection Floer homology | Suppose that L_0 and L_1 are monotone Lagrangian submanifolds in an (M,\omega _M) which is closed or has convex contact-type boundary. Assume that \min ( m^{\min }_{L_0}, m^{\min }_{L_1})\ge 3. One then has a Lagrangian Floer homology module \operatorname{HF}_*(L_0,L_1)= H(\operatorname{CF}_*(L_0,L_1; J)) over \Lambda ... | {
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d1a41e04e38052a7e5242caea277e8bdf23f0188 | subsection | 26 | 153 | Lagrangian intersection Floer homology | This ensures that these moduli spaces are negative-dimensional manifolds, hence empty. The possibility of nowhere-injective discs is ruled out by a lemma of Kwon–Oh and Lazzarini , to the effect that the image of any pseudo-holomorphic disc attached to a Lagrangian contains that of a somewhere-injective disc.Suppose th... | {
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9be1f7d57329de24066b4a96740a4d96a7f76bcb | subsection | 27 | 153 | Lagrangian intersection Floer homology | The definition of {P}^{\mathrm {reg}}(M) is slightly more elaborate than in the monotone case: the additional requirements are that the moduli spaces of simple J_t-spheres, somewhere-injective J_0-discs attached to L_0, and somewhere-injective J_1-discs attached to L_1 are all cut out transversely. This ensures that th... | {
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8946d132a5c78898203b196cf71764627d9bae8d | subsection | 28 | 153 | Open manifolds | In Theorem REF we allow (M,\omega ) to have a convex contact-type boundary: thus there is a non-vanishing, outward-pointing vector field X defined near \partial M, satisfying the Liouville condition L_X\omega =\omega . For the purposes of pseudo-holomorphic theory on such manifolds, one imposes a convexity condition on... | {
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323dc7c0f35c834ef030f8468b71d601c5391711 | subsection | 29 | 153 | A cobordism category for Lagrangian correspondences | Floer homology groups for Lagrangian correspondences fit into a sort of TQFT, or sigma-model, of a non-standard kind, which we now outline. It could be more accurately described as a (1+1)-dimensional topological field theory coupled to locally Hamiltonian fibrations and subject to `Lagrangian matching conditions'. It ... | {
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6e389a5f28aaaba03824b85703faeb46d1ee428b | subsection | 30 | 153 | A cobordism category for Lagrangian correspondences | Define a cycle of LHFs in terms of the following data: an oriented 1-manifold Z \cong S^1 with a finite set of marked points enumerated in cyclic order as (z_1,\dots ,z_l) dividing Z into intervals [z_i,z_{i+1}]; and, for each i\in /l, an LHF (E_i,\pi _i ,\Omega _i) over [z_{i-1},z_i], together with a Lagrangian\Lambda... | {
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df62111453afc9e2571c0ddd6ceb25519a2d85e6 | subsection | 31 | 153 | A cobordism category for Lagrangian correspondences | The theory as we shall describe it differs from that of only in some rather minor aspects.We first describe the objects in a cobordism category, {C} (the morphisms will be described later).Definition 3.1 A cycle of Lagrangians, of length l\ge 1, is a sequence of symplectic manifolds \mathsf {M}= (M_1,\dots , M_{l}) tog... | {
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5af951f1fba9f0e8f2abad414421fbff68b82f39 | subsection | 32 | 153 | A cobordism category for Lagrangian correspondences | One has Lagrangian correspondences L_{i,i+1} = (\mathrm {id}\times m_i^{-1}) \Lambda _{i,i+1} \subset (M_i)_- \times \pi _i^{-1}(z_i). Thus one obtains a cycle of Lagrangians (M_i; L_{i,i+1})_{i\in /l}. Conversely, given a cycle of Lagrangians, one can build a cycle of (trivial) LHFs. These constructions are mutually i... | {
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56e9b4afb6d8f8209032cb5e2252896369621363 | subsection | 33 | 153 | Non-degeneracy | A cycle of Lagrangians L_{i,i+1}\subset (M_i)_-\times M_{i+1} (i\in /l) is called non-degenerate when the product \mathbb {L}:=\prod _{i\in /l }{L_{i,i+1} } (a Lagrangian in \prod _{i\in /l}{(M_i)_-\times M_{i+1}}) meets the multi-diagonal \Delta _l = \lbrace (x_1,x_2, x_2, \dots , x_l,x_l, x_1)\rbrace (another Lagrang... | {
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c71911cea9557115e0ef8d9a5591982325657961 | subsection | 34 | 153 | Non-degeneracy | A cycle of Hamiltonian chords (\sigma _1,\dots ,\sigma _l) is a sequence of \Omega _i-horizontal sections \sigma _i\colon [z_{i-1},z_i] \rightarrow E_i where (\sigma _{i-1}(z_i),\sigma _i(z_i))\in L_{i-1,i} for each i\in /l. Cycles of Hamiltonian chords are in natural bijection with intersections \mathbb {L}\cap \Delta... | {
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8e592ced480b4b95a825ca2c83ff632ff6d09704 | subsection | 35 | 153 | Non-degeneracy | Form the fibre productZ= \left( \prod _i{ C^\infty ([0,1]\times M_{i+1}) } \times \mathbb {L} \right)\times _q \Delta _l,and note that transversality of (\prod {\mathrm {id}_{M_i}\times \phi _{H_i}} ) (\mathbb {L}) with \Delta _l is equivalent to (H_1,\dots , H_l) being a regular value of the projection Z\rightarrow \p... | {
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b550461e08d0d510a9982008e22a7e4dcd0c22e1 | subsection | 36 | 153 | Floer homology from cycles of correspondences | Before turning to the morphisms in the category {C}, we explain how to attach homology groups to objects.Definition 3.4 A compact symplectic manifold (M,\omega ) is normalised monotone if \int _{S^2} {f^*\omega } = 2 \langle f^* c_1(TM,\omega ) , [S^2]\rangle for all smooth maps f\colon S^2\rightarrow M. A compact Lagr... | {
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794ee607198032f0373ccef5fc39894c2c1b3599 | subsection | 37 | 153 | Floer homology from cycles of correspondences | Let {P}^{\mathrm {reg}} denote those products of t-dependent almost complex structures which induce regular almost complex structures for the pair (\mathbb {L},\Delta _l):{P}^{\mathrm {reg}}(M_1,\dots , M_l;\mathbb {L},\Delta _l ) = \mathrm {pl}^{-1}( {P}^{\mathrm {reg}}(\mathbb {M}; \mathbb {L},\Delta _l)).Lemma 3.5 {... | {
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ce693f838e0399c3c37fecb3701ae2f995d87bef | subsection | 38 | 153 | Floer homology from cycles of correspondences | Say (\beta _i,\gamma _i)\ne 0; then there exist
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80fc0f7959f41964265e168a11662b4e56e1c04a | subsection | 39 | 153 | Floer homology from cycles of correspondences | In particular, when l=2, we have\operatorname{CF}(L_{1,2}\times L_{2,1}, \Delta _2 ) \cong \operatorname{CF}(L_{1,2}, L_{2,1}).A generator for \operatorname{CF}(\mathbb {L},\Delta _l; \mathbb {J}) is an intersection point \left((x_1,x_2), (x_2,x_3),\dots , (x_l,x_1)\right) with (x_i,x_{i+1})\in L_{i,i+1}; we map this t... | {
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19f92dc40ce3084c4d8806fcd7c0fd08a4954196 | subsection | 40 | 153 | Floer homology from cycles of correspondences | From a cycle of normalised monotone Lagrangian correspondences L_{i,i+1}\subset (M_i)_- \times M_{i+1} between normalised monotone manifolds \lbrace M_i\rbrace _{i\in /l}, we form the product\mathbb {M} = \prod _{i\in /l} { (M_i)_- \times M_{i+1} } = \prod _{i\in / l} { M_i \times (M_i)_-}(also normalised monotone) and... | {
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76e9f09279fb6c0a3a5019c7c39039c725e5792a | subsection | 41 | 153 | Floer homology from cycles of correspondences | The last stage is the following: let u\colon [0,1]\times \rightarrow \mathbb {M} be a non-constant, finite-action (j,\mathbb {J})-holomorphic map satisfying the boundary conditions u(\lbrace 0 \rbrace \times )\subset \mathbb {L} and u(\lbrace 1 \rbrace \times )\subset \Delta _l. Let F\rightarrow [0,1]\times be the bund... | {
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