A Quantum Gauss-Bonnet Theorem

Transcription

1 A Quantum Gauss-Bonnet Theorem Tyler Fresen November 13, 2014

2 Curvature n the plane Let Γ be a smooth curve wth orentaton n R 2, parametrzed by arc length. The curvature k of Γ s ± Γ, where the sgn s postve f Γ s counterclockwse of Γ, and negatve f Γ s clockwse of Γ. Curvature measures the change n drecton per unt dstance along the curve. Γ'' Γ'

3 Hopf s Umlaufsatz A curve s smple f t has no self-ntersectons. Hopf s Umlaufsatz: If Γ s a smple smooth closed curve, S 1 k(s) ds = ±2π Umlaufsatz wth corners: If Γ s a smple pecewse-smooth closed curve, let C be the set of corners of Γ and for each c C, let ϕ c be the exteror angle at c. Then k(s) ds + ϕ c = ±2π S 1 c C

4 The Whtney-Grausten Theorem Two smooth curves are sad to be regular homotopc to each other f one can be contnuously deformed nto the other such that at any moment n tme, the ntermedate curve s a smooth curve. The rotaton number of a smooth closed curve s gven by the formula 1 k(s) ds 2π S 1 It s always an nteger, and ths formula can be vewed as the Umlaufsatz wth multplctes. Whtney-Grausten Theorem: Two smooth closed curves n the plane are regular homotopc f and only f they have the same rotaton number.

5 The J + nvarant A curve s generc f ts only self-ntersectons are transverse double ponts. Theorem: If two generc smooth closed curves n any surface are regular homotopc, then one can be deformed nto the other by dffeomorphsm of the surface and a fnte number of self-tangency moves and trple-pont moves. The J + nvarant assocates an nteger to each generc smooth curve n a gven orentable surface, such that ths nteger changes by 2 at drect self-tangency moves and s unchanged by opposte self-tangency moves and by trple-pont moves. Ths almost determnes J + unquely; we just need to specfy ts value on a representatve of each regular homotopy class. There s a standard such specfcaton for planar curves.

6 The J + nvarant (contnued) Fgure: Self-tangency and trple-pont moves Fgure: A drect self-tangency move and two opposte self-tangency moves Images due to Lanzat and Polyak

7 Wndng numbers Gven a curve Γ n the plane and a pont p not n Γ, we defne the wndng number or ndex nd Γ (p) to be the total number of (sgned) turns made by Γ around p. The wndng number changes by ±1 when p crosses over Γ, accordng to the orentaton of the secton of Γ t crosses. Gven Γ, the wndng number nd Γ thus gves a functon on R 2 \ Γ; we extend ths to all of R 2 by defnng nd Γ (p) for p Γ to be the average of the wndng numbers of the regons of R 2 \ Γ n a neghborhood of p.

8 Fgures /2 1 +1/2 +1 Ud d smooth Images due to Lanzat and Polyak

9 Lanzat and Polyak s Polynomal Invarant Let Γ be a generc smooth closed curve n the plane. Let X be ts set of double ponts, and for each d = Γ(t 1 ) = Γ(t 2 ) X, let θ d be the (non-orented) angle between Γ (t 1 ) and Γ (t 2 ). Then Lanzat and Polyak defne an assocated quantum nvarant I q (Γ) R[q 1/2, q 1/2 ] as follows: ( 1 2π k(t) S1 qndγ(γ(t)) dt d X θ d q ndγ(d) (q 1 2 q 1 2 )) They showed usng the Hopf Umlaufsatz wth corners that the expresson s nvarant under planar sotopy.

10 Its relaton to the rotaton number and J + Substtutng q = 1 nto Lanzat and Polyak s polynomal gves 1 k(s) ds, 2π S 1 the rotaton number. Hence we say t s a quantum deformaton of the rotaton number (or of the Umlaufsatz). Lanzat and Polyak showed that the frst dervatve of ther polynomal at q = 1, I 1 (Γ), changes by 1 under drect self-tangences and s nvarant under opposte self-tangences and trple-pont modfcatons, so 2I 1 (Γ) changes by 2 at drect self-tangences and s nvarant under opposte self-tangences and trple-pont moves. Thus J + (Γ) = 2I 1 (Γ) up to addton of some constant dependng only on the regular homotopy class of Γ.

11 Problems wth generalzng to curves n surfaces What takes the role of the wndng number n the defnton of the ntegral? What replaces the Umlaufsatz n the proof? Lanzat and Polyak s polynomal s a quantum deformaton of the formula for rotaton number; what should the generalzaton be a deformaton of?

12 Homologcally trval curves The most mportant facts about the wndng number are that t s locally constant on S \ Γ and t changes by the approprate amount when crossng over Γ. In some cases, there s no functon on S \ Γ satsfyng ths property: When there s, we say Γ s homologcally trval. Gven a homologcally trval curve Γ n a connected orented surface S and b S \ Γ, let nd Γ,b : S \ Γ Z be the unque locally constant functon whch sends b to 0 and changes by the approprate amount when crossng over Γ. Image copyrght Wkmeda Commons

13 Geodesc Curvature To generalze the Umlaufsatz to surfaces we need a concept of curvature. In order to talk about curvature we need to have a concept of lengths and angles. The followng concepts wll apply to any surface wth a Remannan metrc, but I wll descrbe them for the less general case of a surface embedded n R 3. As wth planar curvature, we parametrze Γ by arc length, but where planar curvature s the sgned magntude of Γ, geodesc curvature s the sgned magntude of the projecton of Γ onto the tangent plane T p S. Examples: The geodesc curvature of a curve n the plane s ts planar curvature. The geodesc curvature of a great crcle on a sphere s constantly zero.

14 The Gauss-Bonnet Theorem Gauss-Bonnet Theorem: If S s a closed subset of a surface and S s pecewse smooth wth fnte set C of corners, then ( χ(s) = 1 K da + k g ds + ) ϕ c 2π S S c C If S s the entre surface (wthout boundary) then ths reduces to the Gauss-Bonnet theorem presented earler by Dr. Farb. If S s a subset of the plane and S s homeomorphc to a dsk, then ths reduces to Hopf s Umlaufsatz wth corners. More generally, f S s homeomorphc to a dsk, the Gauss-Bonnet theorem s lke the Umlaufsatz wth a correctve term for Gaussan curvature.

15 Rotaton numbers of homologcally trval curves Gven an orented surface S, the rotaton number s the unque way of assgnng a value n Z/χ(S)Z (or Z f χ(s) = 0) to each homologcally trval curve n S such that 1. The rotaton number s nvarant under regular homotopes. 2. The rotaton number of a small counterclockwse curve s The rotaton number of the composton of two curves s the sum of ther rotaton numbers.

16 McIntyre and Carns s formula for the rotaton number Pck a base pont b n S \ Γ. For j 1 2 Z \ Z, let S j be the regon of S on whch nd Γ,b s greater than j. For j 1 2Z \ Z, let a j = { χ(sj ) χ(s) j < 0 χ(s j ) j > 0 Then the wndng number s gven by mod χ(s) j 1 2 Z\Z a j

17 The Gauss-Bonnet Theorem wth multplctes Let Γ be a homologcally trval generc smooth curve n a connected closed surface S wth Remannan metrc and orentaton. We can calculate the Euler characterstcs n McIntyre and Carns s formula usng the Gauss-Bonnet Theorem; ths gves the followng formula for the rotaton number: 1 2π ( k g (t) dt + S 1 S ) K nd Γ,b da mod χ(s) Ths can be vewed as the Gauss-Bonnet theorem wth multplctes. Note that f we change b so that nd Γ,b ncreases by 1 everywhere, the expresson before takng the modulus changes by χ(s).

18 The Quantum Gauss-Bonnet Theorem Instead of takng j 1 2 Z\Z a j, take j 1 2 Z\Z a jq j. Agan applyng the Gauss-Bonnet Theorem to calculate the a j s, we get ( 1 k g (t) q ndγ,b(γ(t)) dt 2π S 1 + (π θ d )q ndγ,b(d) (q 1 2 q 1 2 ) + d X S K qnd Γ,b 1 q 1 2 q 1 2 Ths s a topologcal nvarant and a quantum deformaton of the rotaton number, but t sn t qute a generalzaton of Lanzat and Polyak s formula. da )

19 The Quantum Gauss-Bonnet Theorem (contnued) The expresson 1 2 (q 1 2 q 1 2 )q nd Γ,b d X s a topologcal nvarant and s equal to 0 at q = 1, so subtractng t away from the expresson on the prevous page wll stll gve a topologcal nvarant and deformaton of the rotaton number. Here t s: d X I q (Γ, b) := 1 ( k g (t) q ndγ,b(γ(t)) dt 2π S 1 θ d q nd Γ,b(d) (q 1 2 q 1 2 ) + S K qnd Γ,b 1 q 1 2 q 1 2 da )

20 Its relaton to J + I q (Γ, b) changes the same way under self-tangency and trple-pont moves that Lanzat and Polyak s polynomal does, so we mght thnk that 2I 1 (Γ, b) gves us J+ (Γ) (up to a constant dependng on the regular homotopy class of Γ). However, I 1 (Γ, b) s not nvarant under a change of base pont b. I 1 (Γ, b) (the formula for rotaton number, before takng the modulus) can be used to produce a correctve term to gve a formula whch doesn t change under change of base pont. When χ(s) 0 J + (Γ) = I 1(Γ, b) 2 χ(s) 2I 1(Γ, b) up to a constant dependng on the regular homotopy class of Γ.

21 An explct formula for J + 1 π ( ) 1 2 4π 2 k g (t) dt + nd Γ,b da χ(s) S 1 S ( k g (t) nd Γ,b (Γ(t)) dt θ d + 1 ) K (nd Γ,b ) 2 da S 1 2 d X S