## New thread: "Topological String Theory"

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Dear Forum,\n\nI have been reading the reviews by Neitzke/Vafa and Marino on topological\nsigma models and have some (hopefully relatively minor) questions.\n\nSo we know from the superconformal algebra that we have a bunch of\nnilpotent charges, G^{\\pm}_{r}. As far as I can tell, the choice of a new\nstress-energy tensor can be motivated by wanting to be able to write\nT={Q,b} for some nilpotent charge Q and tensor b. I think then one wants\nto impose\n\ndZ/dg = 0 (*)\n\nto make this a topological theory\'. If T={Q,b} it is then necessary and\nsufficient that physical states are annihilated by Q in order to satisfy\n(*). Is that right, or is there some separate motivation for physical\nstates to be annihilated by Q?\n\nIn any case, in order to write T={G,b} for one of the nilpotent charges I\nthink it\'s necessary (but is it sufficient?) that {L_n,G}=0 for all n.\nAnd the way this is done is to invent new L_ns that satisfy\n\n{L_n,G^{+}_{-1/2}}=0 (**)\n\n(sorry for the double use of curly braces) by mixing together the modes of\nT and modes of the U(1) current J in an appropriate way. The commutation\nrelations of the J_n and G_r then tell you that (**) can be satisfied.\n\nHaving done this, imposed dZ/dg =0 and consequently imposed that physical\nstates are in the cohomology of this choice of Q, I think one finds that\nphysical states must be chiral primaries (effectively BPS highest weight\nstates, right?).\n\n(Making G conformally invariant in this sense also changes the conformal\nspin of the fermionic part of the stress energy tensor T_F from 3/2 to 1,\nand Neitzke/Vafa motivate this by saying only a spin 1 current will make\nsense\' in this context on an arbitrary Riemann surface. I\'m not sure what\nthis means or if it is equivalent to the above motivations?)\n\nIn the case of the bosonic string one already has a that T={Q,b} for some\nQ built out of c, b and L_n, and the motivation for physical states to be\nQ-closed is so that transition amplitudes are independent of gauge choice\n(of the world sheet metric). If I\'m not misunderstanding this point, does\nthis mean that bosonic string theory is a topological\' theory in the same\nsense as (*)? The statement doesn\'t sound right but certainly one wants\nthat bosonic string amplitudes are independent of choice of gauge of the\nworld sheet metric, right? Also, one can then write the partition function\nas something like the expectation value of the FP determinant, and I think\nthis is what motivates writing the topological string partition function\nin an analogous way. Is that roughly correct?\n\nAny comments on the above would be appreciated, particularly if I am\ngetting something completely wrong. One last question - why is this mixing\nof the U(1) current and stress-energy tensor modes called twisting\'?\n\n_______________________________\nDo you Yahoo!?\nDeclare Yourself - Register online to vote today!\nhttp://vote.yahoo.com\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Dear Forum,

I have been reading the reviews by Neitzke/Vafa and Marino on topological
$\sigma$ models and have some (hopefully relatively minor) questions.

So we know from the superconformal algebra that we have a bunch of
nilpotent charges, $G^{\pm}_{r}$. As far as I can tell, the choice of a new
stress-energy tensor can be motivated by wanting to be able to write
$T={Q,b}$ for some nilpotent charge Q and tensor b. I think then one wants
to impose

$$dZ/dg =[/itex] (*) to make this a topological theory'. If $T={Q,b} it$ is then necessary and sufficient that physical states are annihilated by Q in order to satisfy (*). Is that right, or is there some separate motivation for physical states to be annihilated by Q? In any case, in order to write $T={G,b}$ for one of the nilpotent charges I think it's necessary (but is it sufficient?) that ${L_n,G}=0$ for all n. And the way this is done is to invent new $L_{ns}$ that satisfy ${L_n,G^{+}_{-1/2}}=0 (**)$$ (sorry for the double use of curly braces) by mixing together the modes of T and modes of the U(1) current J in an appropriate way. The commutation relations of the [itex]J_n$ and $G_r$ then tell you that $(**)$ can be satisfied.

Having done this, imposed $dZ/dg =0$ and consequently imposed that physical
states are in the cohomology of this choice of Q, I think one finds that
physical states must be chiral primaries (effectively BPS highest weight
states, right?).

(Making G conformally invariant in this sense also changes the conformal
spin of the fermionic part of the stress energy tensor $T_F$ from 3/2 to 1,
and Neitzke/Vafa motivate this by saying only a spin 1 current will make
sense' in this context on an arbitrary Riemann surface. I'm not sure what
this means or if it is equivalent to the above motivations?)

In the case of the bosonic string one already has a that $T={Q,b}$ for some
Q built out of c, b and $L_n,$ and the motivation for physical states to be
Q-closed is so that transition amplitudes are independent of gauge choice
(of the world sheet metric). If I'm not misunderstanding this point, does
this mean that bosonic string theory is a topological' theory in the same
sense as (*)? The statement doesn't sound right but certainly one wants
that bosonic string amplitudes are independent of choice of gauge of the
world sheet metric, right? Also, one can then write the partition function
as something like the expectation value of the FP determinant, and I think
this is what motivates writing the topological string partition function
in an analogous way. Is that roughly correct?

Any comments on the above would be appreciated, particularly if I am
getting something completely wrong. One last question - why is this mixing
of the U(1) current and stress-energy tensor modes called twisting'?

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Red Bull wrote in message news:... > ...In any case, in order to write $T={G,b}$ for one of the nilpotent charges I > think it's necessary (but is it sufficient?) that ${L_n,G}=0$ for all n. > And the way this is done is to invent new $L_{ns}$ that satisfy > > ${L_n,G^{+}_{-1/2}}=0 (**)$ ... [Moderator's note: I believe that John Gonsowski's text below has absolutely nothing to do with topological string theory and Redbull's questions, but let's post it anyway to avoid controversy about rejected postings. LM] This from Tony Smith's website might be helpful to you: As discussed by Kaku in his books Strings, Conformal Field Theory, and Topology (Springer-Verlag 1991) and Introduction to Superstrings (Springer-Verlag 1988), in 26-dimensional String Theory:the action is the area of the world-sheet swept out by the string, and can be written as $S = - (1 / 4 \pi a) \INT d^{2x} \sqrt(g) g^{ab} (d_{aX_m}) (d_{bX_n}) N^{mn}$ where a is 1/2 for open strings and 1/4 for closed strings, $g^{ab}$ is the metric tensor on the world-sheet surface, $N^{mn}$ is the flat metric in 26-dimensional space with signature (25,1), x coordinates are world-sheet coordinates, and X coordinates are 26-dimensional space coordinates; the action can be invariant under 2-dimensional general coordinate transformations because the $\sqrt(g)$ factor cancels against the transformation of the 2-dimensional measure; the action is classically invariant under local scale transformations, and, after quantization, the conformal anomaly resulting from breakdown of the classical scale invariance disappears (for the Bosonic String) only in 26 dimensional space; after quantization, which can be done by Gupta-Bleuler, Light-Cone, or BRST methods, 26-dimensional Bosonic String Theory is seen to be Lorentz invariant, Conformal invariant, and Unitary with no non-physical states; Bosonic String interactions can be represented as an S matrix for which the Euler $\Beta$ Function is the lowest order term in a perturbation series that is a path integral summed over all conformally inequivalent Riemann surfaces; Light-Cone coordinates can be used, with twists, string lengths, and propagation times, to find specific moduli for high genus Riemann surfaces, thus solving the problem of triangulation of moduli space (whose dimension is $6g - 6 + 2N,$ where g is the genus and N is the space dimension so that here $N = 26);$ the Unoriented Closed Bosonic String spectrum contains Tachyons with imaginary mass, massless (at tree-level, but not necessarily after dynamical processes are considered) scalar Dilatons, and massless spin-2 MacroSpace Gravitons); Since Bosonic Unoriented Closed String Theory describes the physics of MacroSpace, there is no need to put in supersymmetry to make superstrings to get fermions - the World-Line Strings of MacroSpace are not fermionic; and Since the 26-dimensional subspace of MacroSpace is naturally 26-dimensional, there is no need to go to second quantization (string field theory) in order to reduce its dimensionality.



Red Bull wrote: > In the case of the bosonic string one already has a that $T={Q,b}$ for some > Q built out of c, b and $L_n,$ and the motivation for physical states to be > Q-closed is so that transition amplitudes are independent of gauge choice > (of the world sheet metric). If I'm not misunderstanding this point, does > this mean that bosonic string theory is a `topological' theory in the same > sense as (*)? The statement doesn't sound right but certainly one wants > that bosonic string amplitudes are independent of choice of gauge of the > world sheet metric, right? Also, one can then write the partition function > as something like the expectation value of the FP determinant, and I think > this is what motivates writing the topological string partition function > in an analogous way. Is that roughly correct? What you are suggesting above was pointed out and explained in detail in: Extended N=2 superconformal structure of gravity and W gravity coupled to matter. By M. Bershadsky, W. Lerche, D. Nemeschansky, N.P. Warner. Nucl.Phys.$B401:304-347,1993$. http://www.arxiv.org/abs/http://www....hep-th/9211040 There were some earlier papers with related observations such as in: Minimal models from W constrained hierarchies via the Kontsevich-Miwa transform By B. Gato-Rivera, A.M. Semikhatov (Lebedev Inst.) Phys.Lett.$B288:38-46,1992$. http://www.arxiv.org/abs/http://www....hep-th/9204085 This line of thinking has lead to the discovery that one can use the topological string theory to compute superpotential terms in the low energy effective action for superstring compactified on a Calabi-Yau manifold: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. By M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa. Commun.Math.Phys.$165:311-428,1994$. http://www.arxiv.org/abs/http://www....hep-th/9309140 See also: Topological amplitudes in string theory. By Ignatios Antoniadis, E. Gava, K.S. Narain, T.R. Taylor. Nucl.Phys.$B413:162-184,1994$. http://www.arxiv.org/abs/http://www....hep-th/9307158 This idea has been extended beyond the superpotential computation in: Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background. By Nathan Berkovits. Nucl.Phys.$B431:258-272,1994$. http://www.arxiv.org/abs/http://www....hep-th/9404162 With best regards, Hirosi Ooguri ---------------- [Moderator's note: Hirosi Ooguri is a co-founder and a leader of topological string theory. LM]

## New thread: "Topological String Theory"

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I did say "might be helpful" for a reason :-) Don\'t worry about\nrejecting my posts, I\'m an electrical engineer not a physicist. It\nwas my understanding that bosonic strings had problems with respect to\ntopological string theory, I was just pointing out that someone (Kaku)\nwas happy with a topological treatment for bosonic strings. Since I\nam actually more interested in bosonic strings than superstrings, I am\ngrateful for any comments on bosonic strings and topology.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>I did say "might be helpful" for a reason :-) Don't worry about
rejecting my posts, I'm an electrical engineer not a physicist. It
was my understanding that bosonic strings had problems with respect to
topological string theory, I was just pointing out that someone (Kaku)
was happy with a topological treatment for bosonic strings. Since I
am actually more interested in bosonic strings than superstrings, I am
grateful for any comments on bosonic strings and topology.



Dear Hirosi, thanks very much for the comments and references. I hope I was also on track with the rest of what I was saying. (Apologies the overuse of curly braces - I got carried away but I think it should be obvious from context where things should be altered). best wishes d70yxj __{_____________________________} Do you Yahoo!? Declare Yourself - Register online to vote today! http://vote.yahoo.com

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