“Nonperturbative aspects of 2D dilaton gravity”

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Final Report on J-2330-N08

“Nonperturbative aspects of 2D dilaton gravity”

Stipendiat: Daniel Grumiller

e-mail: grumiller@itp.uni-leipzig.de phone: +49-341-97-30-293

FAX: +49-341-97-32-548

Project start: 1.1.2004 Project end: 31.12.2005

See also: http://www.physik.uni-leipzig.de/∼grumiller/research.shtml

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1 Summary

1.1 Goals formulated in the project proposal

I recall that the main goal of this project has been an investigation of nonperturbative aspects of 2D dilaton gravity. In particular, according to the project proposal the specific goals had been

1. to collaborate with Dima Vassilevich on topics of relevance for the project 2. to study spherically reduced gravity in the framework of the Standard Model 3. to circumvent the no-go result on the exact string black hole by relaxing its

premises and to construct a target space action for it

4. to pursue loop calculations and to obtain nonperturbative results in the matter sector

5. to study deformations of 2D dilaton gravity

6. to start some follow-up project in Leipzig (originally envisaged with Michael Bordag on the Casimir effect)

7. to collaborate with members of the MPI (in particular with Eberhard Zeidler) 8. to keep contact to the Vienna group, to strengthen contact with the MIT group

and to get in touch with the Jena group

It is now my duty to clarify to what extend each of these aims could be realized.

Regarding 1., 2. and 8. the goals have been achieved as anticipated [1, 3–5, 8, 9, 13, 21, 22, 25, 26, 28]. Concerning 3. the goal also has been achieved [11], surpassing the hopes expressed in the project proposal (several of my talks were based upon this work [12,23,24,26–28,30]). Regarding 4. and 6. the goal has been achieved, although in a different manner than expected (we were not overly active regarding loop calculations, but we made considerable progress on other “nonperturbative fronts”, in particular supergravity with matter and Euclidean dilaton gravity [1, 3, 4, 8]; a follow-up project indeed has been granted by the DFG – however, it is not devoted to the Casimir effect but to the long time behavior of black holes [2]). Concerning 7., I have definitely benefitted from the MPI (in the form of visitors, talks and conferences), but Eberhard Zeidler retired in 2005. So although he was kind and interested we did not actually start a collaboration. Finally, the point 5. quickly turned out to be of less interest than originally hoped. Therefore, we abandoned this particular subject.

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1.2 Unexpected achievements

Unexpected developments necessarily lead to slight deviations from and additions to the aims formulated in a proposal. Quite often, these unexpected achievements are actually more interesting than the anticipated ones. Here I present a brief list of them:

• It was quite a surprise to us that our nonperturbative 2D solutions turned out to have applications in 3D, 4D and 11D SUGRA, as pointed out in our work with Luzi Bergamin, Alfredo Iorio and Carlos Nu˜nez [5].

• I have been invited to compile a brief review [6] on “Virtual Black Holes” for a Special Issue ofInternational Journal of Modern Physics D.

• I have started a fruitful collaboration with Dharam Ahluwalia-Khalilova, which led to the proposal of a new, ab initio, dark matter candidate [7, 10].

• The main project in the second year on nonperturbative dilaton gravity with boundaries (with Luzi Bergamin, Wolfgang Kummer and Dima Vassilevich) led to the unexpected result that on a black hole horizon physical degrees of freedom are converted into gauge degrees of freedom [13].

• After considering different possible advisors at the University of Leipizg and the MPI, the undergraduate student Rene Meyer decided to start a diploma thesis on nonperturbative quantization of 2D dilaton gravity with nonminimally coupled fermionic matter, so since June 2005 I am his advisor. I am very impressed by his depth and his diligence – until December 2005 he has been invited to three talks already (in Vienna, St. Petersburg and Leipzig) and submitted his first publication.

• I have been appointed as co-editor of a Special Issue of International Journal of Modern Physics Don the Thirring-Lense effect (see attachment for a copy of this issue). Also, I have been asked already to participate as co-editor for next year.

• Teaching activities [20, 31] and several invited talks [22–29, 32] were not anticipated, but very beneficial for making contact with students and other scientists.

• Last but not least I have received the Erich-Schmid award by the Austrian Academy of Sciences for my contributions to dilaton gravity.

Also, the number of publications (fourteen) and talks (eleven) definitely has exceeded my expectations. Finally, the fact that DFG has granted a project in Leipzig (on the long time behavior of black holes) may also be interpreted as an achievement of the current project.

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1.3 Financial issues

Travel money has been granted for the following conferences and scientific visits:

• GR17 in Dublin: 1000 Euro

• Visit at MIT: 450 Euro

• Conference in Sardinia: 737.65 Euro

• Visit to Prague: 0 Euro

Unfortunately, the last visit in the list above (about 65 Euro) has not been paid by FWF so I had to pay it privately.

Thus, a total of 2187.65 Euro has been spent on travel money, slightly less than the 2400 Euro that may be spent. In addition, some “Drittmittelgelder” (money from alternative sources) have been acquired to finance a conference in Prague (525 Euro), several scientific visits to Dresden, Jena, Prague and Vienna (about 1100 Euro), and also for the invitations of Herbert Balasin, Luzi Bergamin, Christian B¨ohmer and Alfredo Iorio (about 500 Euro). In total, about 50% of the travel money/money for invitations has been acquired from sources outside the Erwin-Schr¨odinger fellowship.

1.4 Development of collaborations

Vienna group As anticipated the contact to the group at the TU Vienna has re- mained a strong one. Several projects with Herbert Balasin, Luzi Bergamin, Christian B¨ohmer, Wolfgang Kummer and Daniel Mayerhofer have been pursued [1,3–5,8,9,13,15]

and numerous mutual visits have ocurred (I have been to Vienna seven times for scientific collaboration during the two years and I have invited Christian B¨ohmer once, Luzi Bergamin twice and Herbert Balasin also twice). In addition some first steps towards a collaboration with the University of Vienna have been made (with Peter Aichelburg and Michael P¨urrer) on numerical simulations involving the exact string black hole with scalar matter.

Leipzig group The interaction with the Leipzig group was primarily focussed on scientific collaboration with Dima Vassilevich. Besides joint publications [8, 13] the intensive discussions we had (simplified by the fact that we join the same room at the institute) also influenced several separate projects of both of us. I visited the MPI Leipzig several times for workshops or invited talks, and the contacts to Eberhard Zei- dler helped to invite Dharam Ahluwalia-Khalilova to Leipzig in 2005 (unfortunately, however, this visit had to be postponed due to health reasons). There were further noteworthy “interactions” with the Leipzig group: with Michael Bordag I co-organized a meeting in Spring 2004 to which Alfredo Iorio has been invited. Rene Meyer started

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his diploma thesis on a topic I suggested to him in June 2005 and he will continue to collaborate with me at least until Summer 2006. His supervision is performed in collaboration with Gerd Rudolph from the University of Leipzig. In addition the numerous seminars, invited talks and lectures (both actively and passively) were beneficial in several respects.

Other groups My visit to MIT in December 2004 helped to strengthen my ties to the MIT group [5, 21, 22], in particular to Alfredo Iorio, Roman Jackiw and Carlos Nuñez. Alfredo has moved to Prague in 2005 [32]. Roman Jackiw was available for discussion also at conferences (in Vienna and in Sardinia). At the time of writing this project summary I am on the top of the waiting list for a Marie-Curie Outgoing International Fellowship to MIT, which is also an off-spring of this connection. In addition, I have submitted a proposal for an APART Stipendium for MIT in October 2005. There has been a very active collaboration by e-mail with Dharam Ahluwalia- Khalilova fromZacatecas[7,10]. Due to personal reasons (family) I had to decline an offer for a position in Zacatecas, but I was delighted that Christian Böhmer, a former student of Wolfgang Kummer with whom I had collaborated [9], managed to obtain a grant for a post-doc position there. Also on a different front the interactions with the Zacatecas group were fruitful: not only have I been invited to compile a review article [6], but I was offered also co-editorship on a Special Issue about the Thirring- Lense effect (see attached volume). Further notable interactions at a quasi-local level includeJena(Thomas Strobl, Andreas Wipf, who also co-organize the “Mitteldeutsche Physik-Combo” in Halle, Jena and Leipzig) [25, 26],Berlin (Martin Bojowald, Harald Dorn) [23, 24] and Dresden (Ralf Schützhold, whom I invited also to Leipzig) [29].

These contacts will be very helpful for my next project granted by DFG.

2 Brief report on scientific work

2.1 Winter 2003/2004 – Spring 2005

For brief reports on the period Winter 2003/2004 – Spring 2005 see the first, second and third intermediate reports. A brief time table providing an overview of the whole project may be found in the subsequent section 3.

2.2 Summer 2005

The conference in Prague (see third intermediate report) in June led to valuable new contacts.¹ End of June I have been invited to the TU Vienna to give a talk on my solu-

1In fact, in November 2005 I have received an invitation to Brazil from one of the participants. It may be convenient to combine this trip with a visit to Sao Paolo (which is where one of my long-time

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tion for the exact string black hole and for scientific collaboration with Luzi Bergamin and Wolfgang Kummer, which helped a lot to focus on our common project on 2D dilaton gravity with boundaries. This topic has not been envisaged in the project proposal, but it turned out that boundary issues are quite important for a nonperturbative understanding of 2D dilaton gravity (in retrospect, as always, this is even obvious). The remaining part of the Summer (and also most of Autumn) we have been occupied mostly with calculations on this topic. We were able to finish this ambitious project in December and we discovered a remarkable property of black hole horizons:

they lead to an enhancement of the symmetries and thus to a conversion of physical degrees of freedom into gauge degrees of fredom. When we saw our results we were sceptical because they contradicted our expectations (we had the prejudice that there should be additional physical degrees of freedom on a horizon in order to explain the black hole entropy, and not fewer physical degrees of freedom). However after careful checks we had to accept the correctness of our results, and we were delighted to dis- cover that such a mechanism had been proposed by ’t Hooft in 2004 (he suggested that on or near a black hole horizon gauge degrees of freedom should be localized which perhaps represent “lost information”). Our paper has been submitted to the journal Classical and Quantum Gravity in December [13]. Besides this main research topic I had to prepare my proceedings contribution for the Prague conference in June [12].

In July I have been invited by Ralf Sch¨utzhold to a conference on “Quantum Simu- lations via Analogues”, which will be an important topic for my next research project.

I presented there an overview on black hole analogues in a twodimensional context.

Beginning of September I participated in the Fourth Meeting on Constrained Dy- namics and Quantum Gravity in Sardinia. This was not only a nice possibility to present my results on the exact string black hole, but especially an excellent oppor- tunity to discuss with Steve Carlip, who also made some progress on the contraint analysis in 2D dilaton gravity with boundaries (and, of course, also with other participants, e.g., with Roman Jackiw). The proceedings contribution [14] is on a slightly different topic, namely a definition of “volume” for black holes in 2D (this notion had not existed in a precise sense before).

The remaining spare-time in Summer has been occupied by editorial duties (I have been appointed as co-editor of a Special Issue on the Thirring-Lense effect for the International Journal of Modern Physics D, in collaboration with Dharam Ahluwalia- Khalilova, and by advisor duties (my diploma student Rene Meyer has made tremen- dous progress during the Summer, which culminated so far in three talks and one publication).

Additionally, all publications which were pending had been published in the mean- time, and the usual amount of referee reports had to be written.

collaborators, Dima Vassilevich, will be located as of 2006). But clearly this goes beyond the current project.

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2.3 Autumn 2005

Later in September I invited Herbert Balasin to Leipzig. We were able to start a new project on galactic rotation curves within general relativity and to make considerable progress. I visited him in Vienna in October and he visited me again in December, and we already have reached the stage where we think about making our results public [15].

The co-editorial duties peaked in that period, and we could close the Special Issue in November. One such Issue is attached in this final report.

Beginning of October there was the “International Symposium on Mathematical Sciences” in honor of Eberhard Zeidler with some very nice talks (for instance, by Robbert Dijkgraaf, one of the authors of the original “Exact String Black Hole” paper).

Later in October my lectures on “Black Holes” started (they are half-blocked, i.e., 4 academic hours per week, but on average only every second week in order to allow for visits). For beginning of October Eberhard Zeidler invited Dharam Ahluwalia- Khalilova to the MPI in Leipzig. Unfortunately, however, Dharam had to cancel the trip due to health problems. Also in October I have received the “Erich Schmid award”

from the Austrian Academy of Sciences for my contributions to dilaton gravity. Clearly, this award is also an off-spring of the FWF, since those contributions were achieved during this and also during a previous project granted by the FWF. The last two weeks in October I have been invited to TU Vienna for collaboration with Luzi Bergamin, Wolfgang Kummer and Herbert Balasin. This was a very active period, as I had also overlap with Dima Vassilevich in Vienna; therefore, after hours of discussions and calculations, we were able to essentially finish our project on boundaries. Additionally, Luzi Bergamin invited my diploma student Rene Meyer to TU Vienna, and Rene could present some first results on the constraint analysis of dilaton gravity with nonminimally coupled fermions (the topic of his diploma thesis) in a seminar talk. We expect to finish a joint work in Spring 2006 [16].

In November I have been invited to Charles University in Prague for a week to give a talk and to collaborate with Alfredo Iorio. This collaboration had rested because Alfredo moved from Boston to Italy and then to Prague, and also because of a “baby break” on his part. However, we were able to resurrect our project started a long time ago and, in fact, made some substantial progress; in many respects the break was quite healthy since some of the problems we had before could be solved within that week [17].

In order to keep the collaboration going I have invited already Alfredo to Leipzig (for February 2006). Apart from this visit, my lectures and discussions with Rene Meyer (who gave a talk on his results at an international conference in St. Petersburg in that period), I was mostly occupied with “polishing” our draft on boundaries.

In December we put the boundary paper on the Los Alamos preprint server [13].

Also, the second visit of Herbert Balasin took place, which gave our common project an important boost. Rene Meyer presented some of our results at the University of Leipizg (in two talks) and he finished his first publication.

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3 Time table of the project

Jan. 04: visit at TU Vienna; finishing [1]

Feb. 04: finishing [2]

Mar. 04: visit at TU Vienna; finishing [3]

Apr. 04: finishing [4]

May. 04: visit at the ESI Vienna; diploma thesis by Daniel Mayerhofer; scientific visit by Christian B¨ohmer

Jun. 04: co-organization of meeting with Michael Bordag; invitation of Alfredo Iorio and Luzi Bergamin; visit at TU Vienna

Jul. 04: conference GR17 in Dublin [18, 19]

Aug. 04: workshop at MPI Leipzig Sep. 04: finishing [5] and [6]

Oct. 04: start of my lectures “Black Holes”; finishing [7]; visit at the TU Vienna Nov. 04: conference in Vienna [21]

Dec. 04: visit at MIT [22]; intensive collaboration with Alfredo Iorio;

finishing all mayor projects of the first year [8–10]

Jan. 05: Marie Curie Outgoing International Fellowship proposal for MIT;

finishing construction of an action for the exact string black hole [11];

“Wolfgangfest” in Vienna on occasion of retirement of Wolfgang Kummer Feb. 05: visit at TU Vienna for scientific collaboration on boundary project

Mar. 05: visit at the MPI Golm [23]

Apr. 05: visit by Ralf Sch¨utzhold; visit by Luzi Bergamin; visit at Humboldt University, Berlin [24]

May. 05: visit at FSU Jena [25, 26]; preparation of DFG project proposal Jun. 05: conference in Prague [12, 27]; visit at TU Vienna [29];

start of collaboration with Rene Meyer Jul. 05: conference in Dresden [29]

Aug. 05: peak of editorial duties for the Special Issue on the Thirring-Lense effect Sep. 05: conference in Sardinia [14, 30]; visit by Herbert Balasin

Oct. 05: start of my lectures “Black Holes”; reception of Erich-Schmid award;

symposium in honor of Eberhard Zeidler in Leipzig; visit at TU Vienna to finalize [13]; invitation of Rene Meyer to TU Vienna [16];

APART Stipendium proposal for MIT

Nov. 05: visit at CU Prague [32]; collaboration with Alfredo Iorio [17];

closing the Special Issue on the Thirring-Lense effect (see attachment) Dec. 05: visit by Herbert Balasin for collaboration [15]; submission of [13];

DFG project in Leipzig is granted

See alsohttp://www.physik.uni-leipzig.de/∼grumiller/research.shtml

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References

Published in peer reviewed journals

[1] H. Balasin and D. Grumiller, “The ultrarelativistic limit of 2d dilaton gravity and its energy momentum tensor,” Class. Quant. Grav. 21 (2004) 2859–2872,

gr-qc/0312086. [17 pages]

[2] D. Grumiller, “Long time black hole evaporation with bounded Hawking flux,”

JCAP 05 (2004) 005,gr-qc/0307005. [32 pages]

[3] L. Bergamin, D. Grumiller, and W. Kummer, “Quantization of 2d dilaton supergravity with matter,” JHEP 05 (2004) 060, hep-th/0404004. [41 pages]

[4] D. Grumiller and D. Mayerhofer, “On static solutions in 2d dilaton gravity with scalar matter,” Class. Quant. Grav. 21 (2004) 5893–5914,gr-qc/0404013.

[28 pages]

[5] L. Bergamin, D. Grumiller, A. Iorio, and C. Nu˜nez, “Chemistry of Chern-Simons supergravity: Reduction to a BPS kink, oxidation to M-theory and

thermodynamical aspects,” JHEP 11 (2004) 021, hep-th/0409273. [38 pages]

[6] D. Grumiller, “Virtual Black Holes and the S-matrix,”Int. J. Mod. Phys. D13 (2004) 1973–2002, hep-th/0409231; invited review article. [27 pages]

[7] D. V. Ahluwalia-Khalilova and D. Grumiller, “Dark matter: A spin one half fermion field with mass dimension one?,” Phys. Rev. D72(2005) 067701, hep-th/0410192. [4 pages]

[8] L. Bergamin, D. Grumiller, W. Kummer, and D. V. Vassilevich, “Classical and quantum integrability of 2D dilaton gravities in Euclidean space,” Class. Quant.

Grav. 22 (2005) 1361–1382,hep-th/0412007. [27 pages]

[9] H. Balasin, C. G. Boehmer, and D. Grumiller, “The spherically symmetric standard model with gravity,” Gen. Rel. Grav. 37 (2005) 1435–1482, gr-qc/0412098. [58 pages]

[10] D. V. Ahluwalia-Khalilova and D. Grumiller, “Spin half fermions with mass dimension one: Theory, phenomenology, and dark matter,” JCAP 0507 (2005) 012, hep-th/0412080. [67 pages]

[11] D. Grumiller, “An action for the exact string black hole,” JHEP 05 (2005) 069, hep-th/0501208. [40 pages]

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Accepted for publication

[12] D. Grumiller, “Logarithmic corrections to the entropy of the exact string black hole,” hep-th/0506175; proceedings contribution for the conference “Path Integrals. From Quantum Information to Cosmology” in Prague, June 2005.

[20 pages]

[13] L. Bergamin, D. Grumiller, W. Kummer and D. V. Vassilevich,

“Physics-to-gauge conversion at black hole horizons,” hep-th/0512230; accepted for publication in Class. Quant. Grav. [31 pages]

Submitted for publication

[14] D. Grumiller, “The volume of 2D black holes,” gr-qc/0509077; proceedings contribution for the conference “Fourth Meeting on Constrained Dynamics and Quantum Gravity” in Sardinia, September 2005. [9 pages]

[15] H. Balasin and D. Grumiller, “Significant reduction of galactic dark matter by general relativity,” astro-ph/0602519.

In preparation

[16] D. Grumiller and R. Meyer, “Quantum dilaton gravity in 2D with non-minimally coupled fermionic matter,” in preparation (expected to be finished in Spring 2006).

[17] D. Grumiller and A. Iorio, “The Chern-Simons improved Kaluza-Klein Programme,” in preparation (expected to be finished in Summer 2006).

Three most important publications (see attachments)

First: [5] because it established a close link to MIT, Second: [10] because it established a close link to Zacatecas, Third (ex aequo): [11, 13] because they constitute unexpected achievements of direct relevance for the project

Posters

[18] D. Grumiller, “2d type 0A/0B as dilaton gravity in 2d,” poster presented at GR17 in Dublin, July 2004.

[19] D. Grumiller, “On the end point of black hole evaporation,” poster presented at

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Talks and lectures

[20] D. Grumiller, “Schwarze L¨ocher,” lectures given at the University of Leipzig (October 2004 – January 2005).

[21] D. Grumiller, “BPS-Kink and more solutions of the Chern-Simons

(Super)Gravity Term,” 2004. seminar talk presented at the 1^st Vienna Central European Seminar on Particle Physics and Quantum Field Theory “Advances in Quantum Field Theory” in Vienna, November 2004.

[22] D. Grumiller, “Fantastic Realism: The Vienna School of 2D dilaton gravity,”

invited seminar talk at the CTP group at MIT, December 2004.

[23] D. Grumiller, “An action for the exact string black hole,” invited seminar talk at the MPI Golm (Albert-Einstein-Institute), March 2005.

[24] D. Grumiller, “A (target space) action for the exact string black hole,” invited seminar talk at the Humboldt University, Berlin, April 2005.

[25] D. Grumiller, “2D dilaton gravity: An Overview,” invited lecture at the University of Jena, May 2005.

[26] D. Grumiller, “Mass and entropy of the exact string black hole,” invited seminar talk at the University of Jena, May 2005.

[27] D. Grumiller, “Path integral quantization of the exact string black hole,” invited seminar talk at the conference “Path Integrals. From Quantum Information to Cosmology” in Prague, June 2005.

[28] D. Grumiller, “An action for the exact string black hole,” invited seminar talk at the TU Vienna, June 2005.

[29] D. Grumiller, “Black Holes and Analogues in Two Dimensions,” invited seminar talk at the conference “Quantum Simulations via Analogues” in Dresden, July 2005.

[30] D. Grumiller, “Mass and entropy of the exact string black hole,” seminar talk at the conference “Fourth Meeting on Constrained Dynamics and Quantum Gravity”

in Sardinia, September 2005.

[31] D. Grumiller, “Schwarze L¨ocher,” lectures given at the University of Leipzig (October 2005 – January 2006).

[32] D. Grumiller, “(Super-)Gravity in lower dimensions,” invited lecture at the Charles University in Prague, November 2005.

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Non-scientific activities

Several non-scientific activities have been pursued which are related to the project in one way or another. Here is a brief list of them:

• About two dozen of referee reports (on average one per month) were compiled for the journals Classical and Quantum Gravity, Modern Physics Letters and International Journal of Modern Physics D.

• Popular talks on elementary particle physics and general relativity have been given at schools in Vienna (Radetzkyschule, contact teacher Barbara Hirss) and Leipzig (Leipzig International School, contact teacher Cheye Messner).

• After a request by the “Dramaturgie Schauspiel Leipzig” (contact person Birgit Rasch) I have performed advisory activities for a theatre production in Leipzig on particular aspects of the life of astrophysicists (“Drei mal Leben” by Yasmina Reza).

• Although I have transferred my administrational duties concerning the PR webpage by the “Fachausschuss Kern- und Teilchenphysik” of the Austrian Physical Society,

http://www.teilchen.at,

to an ex-student of mine, Wolfgang Waltenberger, I am still an active member of the Editorial Board and I have written and submitted several articles to the webpage in the past two years, in particular articles on the Nobel Prizes 2004 and 2005.

• In the beginning of the project I have prepared a new webpage, http://www.physik.uni-leipzig.de/∼grumiller,

which contains not only information relevant for the project, but also for teaching activities (and some private info). This webpage will be used as well for my next project.

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A Questionnaire: Collaboration with FWF

Guidelines for the proposal

Scale 2

Clarity 2

Comprehensibility 2

Pre-acceptance procedure

Guidance 2

Duration 1

Transparency X

Project monitoring

Availability 2 Elaborateness 2 Comprehensibility 1

Financial issues

Financial execution (Stipendium) 2 Financial execution (travel money) -1

Reasonable maintenance 0

Evaluation

Effort 0

Transparency X Usefulness for PR 0

Miscellaneous remarks

If I may suggest something to the FWF: in my humble opinion it would be much simpler if a total amount of “travel money” would granted automatically (e.g. the current 1200 Euro per year, although of course a slightly higher amount would be welcome). Whether or not it is used wisely depends of course on the project leader, but this can be checked in retrospect from the intermediate and final reports. This would reduce the bureaucratic overhead a bit and allow for a much better planning of visits. Also, in some cases this would perhaps allow to invite people with the travel money. Finally, this responsibility shift towards the project leader would be of some pedagogic value as well, I believe.

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B List of attachments

• Confirmation of the University of Leipzig about the duration of stay

• Copies of title pages/acknowledgments of all publications

• Print-outs of the publications [5, 10, 11, 13]

• One volume of the Special Issue of International Journal of Modern Physics D on the Thirring Lense efffect

• Copy of the DFG project granted in December 2005

• Copy of the Erich-Schmid award

• Short version of project summary for PR (in English)

• Short version of project summary for PR (in German)

“Nonperturbative aspects of 2D dilaton gravity”

Final Report on J-2330-N08