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Topic: SL(2,Z)

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	week125
		Associated to the hexagonal lattice is a special element of SL(2,Z) that corresponds to a 60 degree rotation.
		SL(2,Z) is a nonabelian group - this is how someone with a Ph.D. says that matrix multiplication doesn't commute - but suppose we abelianize it by imposing extra relations forcing commutativity.
		Now, the quotient space H/SL(2,Z) is not a smooth manifold, because while the upper halfplane H is a manifold and the group SL(2,Z) is discrete, the action of SL(2,Z) on H is not free: i.e., certain points in H don't move when you hit them with certain elements of SL(2,Z).
	math.ucr.edu /home/baez/week125.html (2712 words)

	NSDL Metadata Record -- $SL(2,Z)$ Self-duality of Super D3-brane Action on $AdS_5 \times S^5$ (via CobWeb/3.1 ... (Site not responding. Last check: 2007-10-08)
		To this end, we fix the $\kappa$-symmetry in a gauge which simplifies the classical action in order to perform an SO(2) rotation of the N=2 spinor index in a manifest way, though this may not be necessary.
		This situation is the same as the case of a super D-string on $AdS_5 \times S^5$ where it was shown that the super D-string action is transformed to a form of the IIB Green-Schwarz superstring action with the $SL(2,Z)$ covariant tension in the $AdS_5 \times S^5$ background through a duality transformation.
		These results strongly suggest that various duality relations originally found in the flat background may be independent of background geometry, in other words, the duality transformations in string and p-brane theories may exist even in general curved space-time.
	nsdl.org.cob-web.org:8888 /mr/134493 (205 words)

	Poincaré half-plane model - Wikipedia, the free encyclopedia
		First, it is a symmetry group of the square 2x2 lattice of points.
		Thus, functions that are periodic on a square grid, such as modular forms and elliptic functions, will thus inherit an SL(2,Z) symmetry from the grid.
		Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it.
	en.wikipedia.org /wiki/Poincar%C3%A9_half-plane_model (768 words)

	Luboš Motl's reference frame: S-duality and exceptional groups (Site not responding. Last check: 2007-10-08)
		This generates a group whose entries are combinations of rational numbers and rational multiples of sqrt(q), and this group - that carries Hecke's name - is not a subgroup of SL(2,Z) even though it is inside SL(2,R).
		There are subgroups of SL(2,Z) that act as transformations that do not change the gauge group.
		If you require the gauge group to be preserved, you obtain a smaller group - the intersection of SL(2,Z) with the Hecke group which is something like the Gamma(2) group.
	motls.blogspot.com /2006/01/s-duality-and-exceptional-groups.html (825 words)

	The Modular Group and Fractals
		Chapter 3: The Minkowski Question Mark and the Modular Group SL(2,Z) (PDF) (20 pages) shows that the distribution of Farey Fractions transforms under the dyadic representation of the Modular Group.
		Chapter 5: Rotations of the Binary Tree and the Modular Group SL(2,Z) (PDF) (5 pages) is a very rough pre-draft of the hyperbolic rotations of the binary tree.
		The goal here is to establish that these are once again given by the modular group, and that furthermore, these have utility in that they are automorphisms of the unit interval: they are both injective and surjective maps of the unit interval.
	linas.org /math/sl2z.html (1772 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		We then study the SL(2,Z) duality symmetry of type IIB theory for both open D3-brane (OD3) limit and open D5-brane (OD5) limit.
		But, under a special set of SL(2,Z) transformations for which {chi}{sub 0} is rational, OD3 theory goes over to a (5+1)-dimensional NCYM theory and these two theories in this case are related to each other by strong-weak duality symmetry.
		On the other hand, for OD5 theory, a generic SL(2,Z) duality gives another OD5 theory if {chi}{sub 0} is irrational, but when {chi}{sub 0} is rational it gives the little string theory limit indicating that OD5 theory is S dual to the type IIB little string theory.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20462437 (386 words)

	[No title]
		> SL(2,Z) is generated by x = [ 0 1 ] and y = [ 0 1 ] [-1 0 ] [-1 -1 ] with defining relations x^4 = y^3 = 1, x^2 y = y x^2.
		I can find you a reference if you like, but not right now because I am at home.) It follows that G/[G,G] is cyclic of order 12.
		Since its abelianization is cyclic of order 12, it has a unique homomorphism phi onto {1,-1}.
	www.math.niu.edu /~rusin/known-math/01_incoming/SL2Z (938 words)

	Fundamental domains and other pictures in the upper half plane, with Magma
		The second shows a tiling of the upper half plane using images of this domain, and random colouring.
		In the third picture, a tiling of the upper half plane is created by translating these 6 triangles by elements of Gamma(2).
		WARNING: Not all regions created by the method on the right give domains for SL(2,Z); you need to check that the edges of the polygon formed by this code do not intersect each other, apart from at their ends.
	www.math.lsu.edu /~verrill/fundomain/magmaFD.html (794 words)

	[No title]
		What can be said easily is that, according to the Shimura-Taniyama-Weil conjecture, the \emph{isogeny} classes of elliptic curves defined over $\Q$ are parameterized by certain modular forms.
		\section{Modular forms} The group $\Sl{2}{\Z}$ is an infinite group that is generated by the two elements \[ \mxtwo{0}{1}{-1}{0} \, \ \ \mxtwo{0}{-1}{1}{1} \, \] which have orders 4 and 6 respectively.
		The action of $\Sl{2}{\Z}$ on the upper-half plane $H$ by linear fractional transformations has kernel \[ \balbr{ \pm \mxtwo{1}{0}{0}{1} } \, \] and the quotient of $\Sl{2}{\Z}$ by this kernel is the group $\PSl{2}{\Z}$\@.
	www.albany.edu /~hammond/gellmu/examples/f356g.glm (5880 words)

	week233
		On other hand, the group SL(2,Z) consists of 2×2 integer matrices with determinant 1.
		And indeed, SL(2,Z) is a subgroup of SL(2,C), which is the double cover of the Lorentz group.
		Now, SL(2,Z) is famous for being the "mapping class group" of the torus - that is, the group of orientation-preserving diffeomorphisms, modulo diffeomorphisms connected to the identity.
	math.ucr.edu /home/baez/week233.html (2258 words)

	Wavelets and the Poincaré half-plane, by R. F. Streater and J. R. Klauder; coherent states, SL(2,Z), Blaschke ... (Site not responding. Last check: 2007-10-08)
		We show that this analytic function is determined by its scalar products with the discrete family of functions obtained by acting with SL(2,Z) on a cyclic vector, provided that the spin of the representation is less than 3.
		This says that the cyclic vector used in our work generates a frame over SL(2Z), a claim that was not made in our paper.
		Indeed, it has been shown by D. Kelly-Lyth that this total set of functions is not a frame.
	www.mth.kcl.ac.uk.cob-web.org:8888 /~streater/rfsjrk2.html (180 words)

	Continued Fractions and Modular Forms
		To this aim, let us recall a few facts about modular forms and Kloosterman sums, since these objects appeared to be the key to the asymptotic analysis.
		One now considers subgroups G of SL(2,Z) containing every matrices of SL(2,Z) congruent to the identity matrix in SL(2,Z/NZ).
		Sums of Kloosterman sums exhibit strong cancellations that can be estimated by making use of modular forms.
	algo.inria.fr /banderier/Seminar/vardi00.html (1322 words)

	UW-Madison Math Club
		Now, if anyone asks you what you study or would like to know a little more about math, you can regale them with these stories instead of waxing lyrical about the joys of group theory.
		Why We Like SL(2,Z) Alejandro Adem gave a rather in-depth talk about the group SL(2,Z) and its geometric interpretation.
		He also calculated the indices of SL(2,Z/pZ) in SL(2,Z) as well as the Euler characteristic for SL(2,Z).
	www.math.wisc.edu /~mathclub/2004.html (1239 words)

	Citebase - SL(2,Z) Action on Three-Dimensional CFTs and Holography
		Authors: Leigh, Robert G. Petkou, Anastasios C. We show that there is a natural action of SL(2,Z) on the two-point functions of the energy momentum tensor and of higher-spin conserved currents in three-dimensional CFTs.
		The dynamics behind the S-operation of SL(2,Z) is that of an irrelevant current-current deformation and we point out its similarity to the dynamics of a wide class of three-dimensional CFTs.
		The holographic interpretation of our results raises the possibility that many three-dimensional CFTs have duals on AdS4 with SL(2,Z) duality properties at the linearized level.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0309177 (151 words)

	NSDL Metadata Record -- $N=2$ Super Yang-Mills and Subgroups of $SL(2,Z)$ (via CobWeb/3.1 planetlab2.cs.umd.edu) (Site not responding. Last check: 2007-10-08)
		NSDL Metadata Record -- $N=2$ Super Yang-Mills and Subgroups of $SL(2,Z)$ (via CobWeb/3.1 planetlab2.cs.umd.edu)
		We discuss $SL(2,Z)$ subgroups appropriate for the study of $N=2$ Super Yang-Mills with $N_f=2n$ flavors.
		Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups.
	nsdl.org.cob-web.org:8888 /mr/226401 (104 words)

	Su Gao: papers (Site not responding. Last check: 2007-10-08)
		The Action of SL(2,Z) on the Subsets of Z
		We prove that the orbit equivalence relation of the canonical action of SL(2,Z) on the subsets of Z
		Preliminaries about the Action of SL(2,Z) on Z
	www.its.caltech.edu /~sugao/paper8.html (50 words)

	Math Forum Discussions
		And indeed, SL(2,Z) is a subgroup of SL(2,C), which is the double
		Now, SL(2,Z) is famous for being the "mapping class group" of the torus -
		In math jargon, we have this commutative diagram:
	mathforum.org /kb/thread.jspa?threadID=1385480&messageID=4727052 (1319 words)

	Dualities and the SL(2,Z) anomaly
		Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, U.S.A. Tata Institute of Fundamental Research, Homi Bhabha Rd, Bombay 400 005, India
		The SL(2,Z) anomaly recently derived for type IIB supergravity in 10 dimensions is shown to be a consequence of T-duality with the type IIA string, after compactification to 2 dimensions on an 8-fold.
		This explains the identity of the gravitational 8-forms appearing in different contexts in the effective actions of type IIA and IIB string theories.
	stacks.iop.org /1126-6708/1998/i=12/a=006 (310 words)

	Herman Verlinde, Princeton, SL(2,Z) Duality and Fractal Phase Diagram in SYM/NCOS Theory (Site not responding. Last check: 2007-10-08)
		Herman Verlinde, Princeton, SL(2,Z) Duality and Fractal Phase Diagram in SYM/NCOS Theory
		SL(2,Z) Duality and Fractal Phase Diagram in SYM/NCOS Theory
		Audio for this talk requires sound hardware, and RealPlayer or RealAudio by RealNetworks.
	online.itp.ucsb.edu /online/mtheory01/verlinde1 (54 words)

	GAP Forum: Re: 2x2 matrices mod n (Site not responding. Last check: 2007-10-08)
		Thus the most elegant way to compute SL(2,Z/nZ) as a matrix group in GAP 4.2
		is to use that fact that SL(2,Z/nZ) is an epimorphic image of SL(2,Z) under reduction of matrix entries modulo n (see, e.g., Theorem II.3.2 in
		Using the ``canonical generators'' T and J for SL(2,Z),
	www-groups.dcs.st-and.ac.uk /~gap/ForumArchive/Breuer.1/Thomas.1/Re__2x2_.1/1.html (414 words)

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